hypothesis testing calculator with data

World-Class

Data Science

Learn with instructors from:

t-test Calculator

Table of contents

Welcome to our t-test calculator! Here you can not only easily perform one-sample t-tests , but also two-sample t-tests , as well as paired t-tests .

Do you prefer to find the p-value from t-test, or would you rather find the t-test critical values? Well, this t-test calculator can do both! 😊

What does a t-test tell you? Take a look at the text below, where we explain what actually gets tested when various types of t-tests are performed. Also, we explain when to use t-tests (in particular, whether to use the z-test vs. t-test) and what assumptions your data should satisfy for the results of a t-test to be valid. If you've ever wanted to know how to do a t-test by hand, we provide the necessary t-test formula, as well as tell you how to determine the number of degrees of freedom in a t-test.

When to use a t-test?

A t-test is one of the most popular statistical tests for location , i.e., it deals with the population(s) mean value(s).

There are different types of t-tests that you can perform:

A one-sample t-test;
A two-sample t-test; and
A paired t-test.

In the next section , we explain when to use which. Remember that a t-test can only be used for one or two groups . If you need to compare three (or more) means, use the analysis of variance ( ANOVA ) method.

The t-test is a parametric test, meaning that your data has to fulfill some assumptions :

The data points are independent; AND
The data, at least approximately, follow a normal distribution .

If your sample doesn't fit these assumptions, you can resort to nonparametric alternatives. Visit our Mann–Whitney U test calculator or the Wilcoxon rank-sum test calculator to learn more. Other possibilities include the Wilcoxon signed-rank test or the sign test.

Which t-test?

Your choice of t-test depends on whether you are studying one group or two groups:

One sample t-test

Choose the one-sample t-test to check if the mean of a population is equal to some pre-set hypothesized value .

The average volume of a drink sold in 0.33 l cans — is it really equal to 330 ml?

The average weight of people from a specific city — is it different from the national average?

Two-sample t-test

Choose the two-sample t-test to check if the difference between the means of two populations is equal to some pre-determined value when the two samples have been chosen independently of each other.

In particular, you can use this test to check whether the two groups are different from one another .

The average difference in weight gain in two groups of people: one group was on a high-carb diet and the other on a high-fat diet.

The average difference in the results of a math test from students at two different universities.

This test is sometimes referred to as an independent samples t-test , or an unpaired samples t-test .

Paired t-test

A paired t-test is used to investigate the change in the mean of a population before and after some experimental intervention , based on a paired sample, i.e., when each subject has been measured twice: before and after treatment.

In particular, you can use this test to check whether, on average, the treatment has had any effect on the population .

The change in student test performance before and after taking a course.

The change in blood pressure in patients before and after administering some drug.

How to do a t-test?

So, you've decided which t-test to perform. These next steps will tell you how to calculate the p-value from t-test or its critical values, and then which decision to make about the null hypothesis.

Decide on the alternative hypothesis :

Use a two-tailed t-test if you only care whether the population's mean (or, in the case of two populations, the difference between the populations' means) agrees or disagrees with the pre-set value.

Use a one-tailed t-test if you want to test whether this mean (or difference in means) is greater/less than the pre-set value.

Compute your T-score value :

Formulas for the test statistic in t-tests include the sample size , as well as its mean and standard deviation . The exact formula depends on the t-test type — check the sections dedicated to each particular test for more details.

Determine the degrees of freedom for the t-test:

The degrees of freedom are the number of observations in a sample that are free to vary as we estimate statistical parameters. In the simplest case, the number of degrees of freedom equals your sample size minus the number of parameters you need to estimate . Again, the exact formula depends on the t-test you want to perform — check the sections below for details.

The degrees of freedom are essential, as they determine the distribution followed by your T-score (under the null hypothesis). If there are d degrees of freedom, then the distribution of the test statistics is the t-Student distribution with d degrees of freedom . This distribution has a shape similar to N(0,1) (bell-shaped and symmetric) but has heavier tails . If the number of degrees of freedom is large (>30), which generically happens for large samples, the t-Student distribution is practically indistinguishable from N(0,1).

💡 The t-Student distribution owes its name to William Sealy Gosset, who, in 1908, published his paper on the t-test under the pseudonym "Student". Gosset worked at the famous Guinness Brewery in Dublin, Ireland, and devised the t-test as an economical way to monitor the quality of beer. Cheers! 🍺🍺🍺

p-value from t-test

Recall that the p-value is the probability (calculated under the assumption that the null hypothesis is true) that the test statistic will produce values at least as extreme as the T-score produced for your sample . As probabilities correspond to areas under the density function, p-value from t-test can be nicely illustrated with the help of the following pictures:

The following formulae say how to calculate p-value from t-test. By cdf t,d we denote the cumulative distribution function of the t-Student distribution with d degrees of freedom:

p-value from left-tailed t-test:

p-value = cdf t,d (t score )

p-value from right-tailed t-test:

p-value = 1 − cdf t,d (t score )

p-value from two-tailed t-test:

p-value = 2 × cdf t,d (−|t score |)

or, equivalently: p-value = 2 − 2 × cdf t,d (|t score |)

However, the cdf of the t-distribution is given by a somewhat complicated formula. To find the p-value by hand, you would need to resort to statistical tables, where approximate cdf values are collected, or to specialized statistical software. Fortunately, our t-test calculator determines the p-value from t-test for you in the blink of an eye!

t-test critical values

Recall, that in the critical values approach to hypothesis testing, you need to set a significance level, α, before computing the critical values , which in turn give rise to critical regions (a.k.a. rejection regions).

Formulas for critical values employ the quantile function of t-distribution, i.e., the inverse of the cdf :

Critical value for left-tailed t-test: cdf t,d -1 (α)

critical region:

(-∞, cdf t,d -1 (α)]

Critical value for right-tailed t-test: cdf t,d -1 (1-α)

[cdf t,d -1 (1-α), ∞)

Critical values for two-tailed t-test: ±cdf t,d -1 (1-α/2)

(-∞, -cdf t,d -1 (1-α/2)] ∪ [cdf t,d -1 (1-α/2), ∞)

To decide the fate of the null hypothesis, just check if your T-score lies within the critical region:

If your T-score belongs to the critical region , reject the null hypothesis and accept the alternative hypothesis.

If your T-score is outside the critical region , then you don't have enough evidence to reject the null hypothesis.

How to use our t-test calculator

Choose the type of t-test you wish to perform:

A one-sample t-test (to test the mean of a single group against a hypothesized mean);

A two-sample t-test (to compare the means for two groups); or

A paired t-test (to check how the mean from the same group changes after some intervention).

Two-tailed;

Left-tailed; or

Right-tailed.

This t-test calculator allows you to use either the p-value approach or the critical regions approach to hypothesis testing!

Enter your T-score and the number of degrees of freedom . If you don't know them, provide some data about your sample(s): sample size, mean, and standard deviation, and our t-test calculator will compute the T-score and degrees of freedom for you .

Once all the parameters are present, the p-value, or critical region, will immediately appear underneath the t-test calculator, along with an interpretation!

One-sample t-test

The null hypothesis is that the population mean is equal to some value μ 0 \mu_0 μ 0 .

The alternative hypothesis is that the population mean is:

different from μ 0 \mu_0 μ 0 ;
smaller than μ 0 \mu_0 μ 0 ; or
greater than μ 0 \mu_0 μ 0 .

One-sample t-test formula :

μ 0 \mu_0 μ 0 — Mean postulated in the null hypothesis;
n n n — Sample size;
x ˉ \bar{x} x ˉ — Sample mean; and
s s s — Sample standard deviation.

Number of degrees of freedom in t-test (one-sample) = n − 1 n-1 n − 1 .

The null hypothesis is that the actual difference between these groups' means, μ 1 \mu_1 μ 1 , and μ 2 \mu_2 μ 2 , is equal to some pre-set value, Δ \Delta Δ .

The alternative hypothesis is that the difference μ 1 − μ 2 \mu_1 - \mu_2 μ 1 − μ 2 is:

Different from Δ \Delta Δ ;
Smaller than Δ \Delta Δ ; or
Greater than Δ \Delta Δ .

In particular, if this pre-determined difference is zero ( Δ = 0 \Delta = 0 Δ = 0 ):

The null hypothesis is that the population means are equal.

The alternate hypothesis is that the population means are:

μ 1 \mu_1 μ 1 and μ 2 \mu_2 μ 2 are different from one another;
μ 1 \mu_1 μ 1 is smaller than μ 2 \mu_2 μ 2 ; and
μ 1 \mu_1 μ 1 is greater than μ 2 \mu_2 μ 2 .

Formally, to perform a t-test, we should additionally assume that the variances of the two populations are equal (this assumption is called the homogeneity of variance ).

There is a version of a t-test that can be applied without the assumption of homogeneity of variance: it is called a Welch's t-test . For your convenience, we describe both versions.

Two-sample t-test if variances are equal

Use this test if you know that the two populations' variances are the same (or very similar).

Two-sample t-test formula (with equal variances) :

where s p s_p s p is the so-called pooled standard deviation , which we compute as:

Δ \Delta Δ — Mean difference postulated in the null hypothesis;
n 1 n_1 n 1 — First sample size;
x ˉ 1 \bar{x}_1 x ˉ 1 — Mean for the first sample;
s 1 s_1 s 1 — Standard deviation in the first sample;
n 2 n_2 n 2 — Second sample size;
x ˉ 2 \bar{x}_2 x ˉ 2 — Mean for the second sample; and
s 2 s_2 s 2 — Standard deviation in the second sample.

Number of degrees of freedom in t-test (two samples, equal variances) = n 1 + n 2 − 2 n_1 + n_2 - 2 n 1 + n 2 − 2 .

Two-sample t-test if variances are unequal (Welch's t-test)

Use this test if the variances of your populations are different.

Two-sample Welch's t-test formula if variances are unequal:

s 1 s_1 s 1 — Standard deviation in the first sample;
s 2 s_2 s 2 — Standard deviation in the second sample.

The number of degrees of freedom in a Welch's t-test (two-sample t-test with unequal variances) is very difficult to count. We can approximate it with the help of the following Satterthwaite formula :

Alternatively, you can take the smaller of n 1 − 1 n_1 - 1 n 1 − 1 and n 2 − 1 n_2 - 1 n 2 − 1 as a conservative estimate for the number of degrees of freedom.

🔎 The Satterthwaite formula for the degrees of freedom can be rewritten as a scaled weighted harmonic mean of the degrees of freedom of the respective samples: n 1 − 1 n_1 - 1 n 1 − 1 and n 2 − 1 n_2 - 1 n 2 − 1 , and the weights are proportional to the standard deviations of the corresponding samples.

As we commonly perform a paired t-test when we have data about the same subjects measured twice (before and after some treatment), let us adopt the convention of referring to the samples as the pre-group and post-group.

The null hypothesis is that the true difference between the means of pre- and post-populations is equal to some pre-set value, Δ \Delta Δ .

The alternative hypothesis is that the actual difference between these means is:

Typically, this pre-determined difference is zero. We can then reformulate the hypotheses as follows:

The null hypothesis is that the pre- and post-means are the same, i.e., the treatment has no impact on the population .

The alternative hypothesis:

The pre- and post-means are different from one another (treatment has some effect);
The pre-mean is smaller than the post-mean (treatment increases the result); or
The pre-mean is greater than the post-mean (treatment decreases the result).

Paired t-test formula

In fact, a paired t-test is technically the same as a one-sample t-test! Let us see why it is so. Let x 1 , . . . , x n x_1, ... , x_n x 1 , ... , x n be the pre observations and y 1 , . . . , y n y_1, ... , y_n y 1 , ... , y n the respective post observations. That is, x i , y i x_i, y_i x i , y i are the before and after measurements of the i -th subject.

For each subject, compute the difference, d i : = x i − y i d_i := x_i - y_i d i := x i − y i . All that happens next is just a one-sample t-test performed on the sample of differences d 1 , . . . , d n d_1, ... , d_n d 1 , ... , d n . Take a look at the formula for the T-score :

Δ \Delta Δ — Mean difference postulated in the null hypothesis;

n n n — Size of the sample of differences, i.e., the number of pairs;

x ˉ \bar{x} x ˉ — Mean of the sample of differences; and

s s s — Standard deviation of the sample of differences.

Number of degrees of freedom in t-test (paired): n − 1 n - 1 n − 1

t-test vs Z-test

We use a Z-test when we want to test the population mean of a normally distributed dataset, which has a known population variance . If the number of degrees of freedom is large, then the t-Student distribution is very close to N(0,1).

Hence, if there are many data points (at least 30), you may swap a t-test for a Z-test, and the results will be almost identical. However, for small samples with unknown variance, remember to use the t-test because, in such cases, the t-Student distribution differs significantly from the N(0,1)!

🙋 Have you concluded you need to perform the z-test? Head straight to our z-test calculator !

What is a t-test?

A t-test is a widely used statistical test that analyzes the means of one or two groups of data. For instance, a t-test is performed on medical data to determine whether a new drug really helps.

What are different types of t-tests?

Different types of t-tests are:

One-sample t-test;
Two-sample t-test; and
Paired t-test.

How to find the t value in a one sample t-test?

To find the t-value:

Subtract the null hypothesis mean from the sample mean value.
Divide the difference by the standard deviation of the sample.
Multiply the resultant with the square root of the sample size.

.css-m482sy.css-m482sy{color:#2B3148;background-color:transparent;font-family:var(--calculator-ui-font-family),Verdana,sans-serif;font-size:20px;line-height:24px;overflow:visible;padding-top:0px;position:relative;}.css-m482sy.css-m482sy:after{content:'';-webkit-transform:scale(0);-moz-transform:scale(0);-ms-transform:scale(0);transform:scale(0);position:absolute;border:2px solid #EA9430;border-radius:2px;inset:-8px;z-index:1;}.css-m482sy .js-external-link-button.link-like,.css-m482sy .js-external-link-anchor{color:inherit;border-radius:1px;-webkit-text-decoration:underline;text-decoration:underline;}.css-m482sy .js-external-link-button.link-like:hover,.css-m482sy .js-external-link-anchor:hover,.css-m482sy .js-external-link-button.link-like:active,.css-m482sy .js-external-link-anchor:active{text-decoration-thickness:2px;text-shadow:1px 0 0;}.css-m482sy .js-external-link-button.link-like:focus-visible,.css-m482sy .js-external-link-anchor:focus-visible{outline:transparent 2px dotted;box-shadow:0 0 0 2px #6314E6;}.css-m482sy p,.css-m482sy div{margin:0;display:block;}.css-m482sy pre{margin:0;display:block;}.css-m482sy pre code{display:block;width:-webkit-fit-content;width:-moz-fit-content;width:fit-content;}.css-m482sy pre:not(:first-child){padding-top:8px;}.css-m482sy ul,.css-m482sy ol{display:block margin:0;padding-left:20px;}.css-m482sy ul li,.css-m482sy ol li{padding-top:8px;}.css-m482sy ul ul,.css-m482sy ol ul,.css-m482sy ul ol,.css-m482sy ol ol{padding-top:0;}.css-m482sy ul:not(:first-child),.css-m482sy ol:not(:first-child){padding-top:4px;} .css-63uqft{margin:auto;background-color:white;overflow:auto;overflow-wrap:break-word;word-break:break-word;}.css-63uqft code,.css-63uqft kbd,.css-63uqft pre,.css-63uqft samp{font-family:monospace;}.css-63uqft code{padding:2px 4px;color:#444;background:#ddd;border-radius:4px;}.css-63uqft figcaption,.css-63uqft caption{text-align:center;}.css-63uqft figcaption{font-size:12px;font-style:italic;overflow:hidden;}.css-63uqft h3{font-size:1.75rem;}.css-63uqft h4{font-size:1.5rem;}.css-63uqft .mathBlock{font-size:24px;-webkit-padding-start:4px;padding-inline-start:4px;}.css-63uqft .mathBlock .katex{font-size:24px;text-align:left;}.css-63uqft .math-inline{background-color:#f0f0f0;display:inline-block;font-size:inherit;padding:0 3px;}.css-63uqft .videoBlock,.css-63uqft .imageBlock{margin-bottom:16px;}.css-63uqft .imageBlockimage-align--left,.css-63uqft .videoBlockvideo-align--left{float:left;}.css-63uqft .imageBlockimage-align--right,.css-63uqft .videoBlockvideo-align--right{float:right;}.css-63uqft .imageBlockimage-align--center,.css-63uqft .videoBlockvideo-align--center{display:block;margin-left:auto;margin-right:auto;clear:both;}.css-63uqft .imageBlockimage-align--none,.css-63uqft .videoBlockvideo-align--none{clear:both;margin-left:0;margin-right:0;}.css-63uqft .videoBlockvideo--wrapper{position:relative;padding-bottom:56.25%;height:0;}.css-63uqft .videoBlockvideo--wrapper iframe{position:absolute;top:0;left:0;width:100%;height:100%;}.css-63uqft .videoBlock__caption{text-align:left;}@font-face{font-family:'KaTeX_AMS';src:url(/katex-fonts/KaTeX_AMS-Regular.woff2) format('woff2'),url(/katex-fonts/KaTeX_AMS-Regular.woff) format('woff'),url(/katex-fonts/KaTeX_AMS-Regular.ttf) format('truetype');font-weight:normal;font-style:normal;}@font-face{font-family:'KaTeX_Caligraphic';src:url(/katex-fonts/KaTeX_Caligraphic-Bold.woff2) format('woff2'),url(/katex-fonts/KaTeX_Caligraphic-Bold.woff) format('woff'),url(/katex-fonts/KaTeX_Caligraphic-Bold.ttf) format('truetype');font-weight:bold;font-style:normal;}@font-face{font-family:'KaTeX_Caligraphic';src:url(/katex-fonts/KaTeX_Caligraphic-Regular.woff2) format('woff2'),url(/katex-fonts/KaTeX_Caligraphic-Regular.woff) format('woff'),url(/katex-fonts/KaTeX_Caligraphic-Regular.ttf) format('truetype');font-weight:normal;font-style:normal;}@font-face{font-family:'KaTeX_Fraktur';src:url(/katex-fonts/KaTeX_Fraktur-Bold.woff2) format('woff2'),url(/katex-fonts/KaTeX_Fraktur-Bold.woff) format('woff'),url(/katex-fonts/KaTeX_Fraktur-Bold.ttf) format('truetype');font-weight:bold;font-style:normal;}@font-face{font-family:'KaTeX_Fraktur';src:url(/katex-fonts/KaTeX_Fraktur-Regular.woff2) format('woff2'),url(/katex-fonts/KaTeX_Fraktur-Regular.woff) format('woff'),url(/katex-fonts/KaTeX_Fraktur-Regular.ttf) format('truetype');font-weight:normal;font-style:normal;}@font-face{font-family:'KaTeX_Main';src:url(/katex-fonts/KaTeX_Main-Bold.woff2) format('woff2'),url(/katex-fonts/KaTeX_Main-Bold.woff) format('woff'),url(/katex-fonts/KaTeX_Main-Bold.ttf) format('truetype');font-weight:bold;font-style:normal;}@font-face{font-family:'KaTeX_Main';src:url(/katex-fonts/KaTeX_Main-BoldItalic.woff2) format('woff2'),url(/katex-fonts/KaTeX_Main-BoldItalic.woff) format('woff'),url(/katex-fonts/KaTeX_Main-BoldItalic.ttf) format('truetype');font-weight:bold;font-style:italic;}@font-face{font-family:'KaTeX_Main';src:url(/katex-fonts/KaTeX_Main-Italic.woff2) format('woff2'),url(/katex-fonts/KaTeX_Main-Italic.woff) format('woff'),url(/katex-fonts/KaTeX_Main-Italic.ttf) format('truetype');font-weight:normal;font-style:italic;}@font-face{font-family:'KaTeX_Main';src:url(/katex-fonts/KaTeX_Main-Regular.woff2) format('woff2'),url(/katex-fonts/KaTeX_Main-Regular.woff) format('woff'),url(/katex-fonts/KaTeX_Main-Regular.ttf) format('truetype');font-weight:normal;font-style:normal;}@font-face{font-family:'KaTeX_Math';src:url(/katex-fonts/KaTeX_Math-BoldItalic.woff2) format('woff2'),url(/katex-fonts/KaTeX_Math-BoldItalic.woff) format('woff'),url(/katex-fonts/KaTeX_Math-BoldItalic.ttf) format('truetype');font-weight:bold;font-style:italic;}@font-face{font-family:'KaTeX_Math';src:url(/katex-fonts/KaTeX_Math-Italic.woff2) format('woff2'),url(/katex-fonts/KaTeX_Math-Italic.woff) format('woff'),url(/katex-fonts/KaTeX_Math-Italic.ttf) format('truetype');font-weight:normal;font-style:italic;}@font-face{font-family:'KaTeX_SansSerif';src:url(/katex-fonts/KaTeX_SansSerif-Bold.woff2) format('woff2'),url(/katex-fonts/KaTeX_SansSerif-Bold.woff) format('woff'),url(/katex-fonts/KaTeX_SansSerif-Bold.ttf) format('truetype');font-weight:bold;font-style:normal;}@font-face{font-family:'KaTeX_SansSerif';src:url(/katex-fonts/KaTeX_SansSerif-Italic.woff2) format('woff2'),url(/katex-fonts/KaTeX_SansSerif-Italic.woff) format('woff'),url(/katex-fonts/KaTeX_SansSerif-Italic.ttf) format('truetype');font-weight:normal;font-style:italic;}@font-face{font-family:'KaTeX_SansSerif';src:url(/katex-fonts/KaTeX_SansSerif-Regular.woff2) format('woff2'),url(/katex-fonts/KaTeX_SansSerif-Regular.woff) format('woff'),url(/katex-fonts/KaTeX_SansSerif-Regular.ttf) format('truetype');font-weight:normal;font-style:normal;}@font-face{font-family:'KaTeX_Script';src:url(/katex-fonts/KaTeX_Script-Regular.woff2) format('woff2'),url(/katex-fonts/KaTeX_Script-Regular.woff) format('woff'),url(/katex-fonts/KaTeX_Script-Regular.ttf) format('truetype');font-weight:normal;font-style:normal;}@font-face{font-family:'KaTeX_Size1';src:url(/katex-fonts/KaTeX_Size1-Regular.woff2) format('woff2'),url(/katex-fonts/KaTeX_Size1-Regular.woff) format('woff'),url(/katex-fonts/KaTeX_Size1-Regular.ttf) format('truetype');font-weight:normal;font-style:normal;}@font-face{font-family:'KaTeX_Size2';src:url(/katex-fonts/KaTeX_Size2-Regular.woff2) format('woff2'),url(/katex-fonts/KaTeX_Size2-Regular.woff) format('woff'),url(/katex-fonts/KaTeX_Size2-Regular.ttf) format('truetype');font-weight:normal;font-style:normal;}@font-face{font-family:'KaTeX_Size3';src:url(/katex-fonts/KaTeX_Size3-Regular.woff2) format('woff2'),url(/katex-fonts/KaTeX_Size3-Regular.woff) format('woff'),url(/katex-fonts/KaTeX_Size3-Regular.ttf) format('truetype');font-weight:normal;font-style:normal;}@font-face{font-family:'KaTeX_Size4';src:url(/katex-fonts/KaTeX_Size4-Regular.woff2) format('woff2'),url(/katex-fonts/KaTeX_Size4-Regular.woff) format('woff'),url(/katex-fonts/KaTeX_Size4-Regular.ttf) format('truetype');font-weight:normal;font-style:normal;}@font-face{font-family:'KaTeX_Typewriter';src:url(/katex-fonts/KaTeX_Typewriter-Regular.woff2) format('woff2'),url(/katex-fonts/KaTeX_Typewriter-Regular.woff) format('woff'),url(/katex-fonts/KaTeX_Typewriter-Regular.ttf) format('truetype');font-weight:normal;font-style:normal;}.css-63uqft .katex{font:normal 1.21em KaTeX_Main,Times New Roman,serif;line-height:1.2;text-indent:0;text-rendering:auto;}.css-63uqft .katex *{-ms-high-contrast-adjust:none!important;border-color:currentColor;}.css-63uqft .katex .katex-version::after{content:'0.13.13';}.css-63uqft .katex .katex-mathml{position:absolute;clip:rect(1px,1px,1px,1px);padding:0;border:0;height:1px;width:1px;overflow:hidden;}.css-63uqft .katex .katex-html>.newline{display:block;}.css-63uqft .katex .base{position:relative;display:inline-block;white-space:nowrap;width:-webkit-min-content;width:-moz-min-content;width:-webkit-min-content;width:-moz-min-content;width:min-content;}.css-63uqft .katex .strut{display:inline-block;}.css-63uqft .katex .textbf{font-weight:bold;}.css-63uqft .katex .textit{font-style:italic;}.css-63uqft .katex .textrm{font-family:KaTeX_Main;}.css-63uqft .katex .textsf{font-family:KaTeX_SansSerif;}.css-63uqft .katex .texttt{font-family:KaTeX_Typewriter;}.css-63uqft .katex .mathnormal{font-family:KaTeX_Math;font-style:italic;}.css-63uqft .katex .mathit{font-family:KaTeX_Main;font-style:italic;}.css-63uqft .katex .mathrm{font-style:normal;}.css-63uqft .katex .mathbf{font-family:KaTeX_Main;font-weight:bold;}.css-63uqft .katex .boldsymbol{font-family:KaTeX_Math;font-weight:bold;font-style:italic;}.css-63uqft .katex .amsrm{font-family:KaTeX_AMS;}.css-63uqft .katex .mathbb,.css-63uqft .katex .textbb{font-family:KaTeX_AMS;}.css-63uqft .katex .mathcal{font-family:KaTeX_Caligraphic;}.css-63uqft .katex .mathfrak,.css-63uqft .katex .textfrak{font-family:KaTeX_Fraktur;}.css-63uqft .katex .mathtt{font-family:KaTeX_Typewriter;}.css-63uqft .katex .mathscr,.css-63uqft .katex .textscr{font-family:KaTeX_Script;}.css-63uqft .katex .mathsf,.css-63uqft .katex .textsf{font-family:KaTeX_SansSerif;}.css-63uqft .katex .mathboldsf,.css-63uqft .katex .textboldsf{font-family:KaTeX_SansSerif;font-weight:bold;}.css-63uqft .katex .mathitsf,.css-63uqft .katex .textitsf{font-family:KaTeX_SansSerif;font-style:italic;}.css-63uqft .katex .mainrm{font-family:KaTeX_Main;font-style:normal;}.css-63uqft .katex .vlist-t{display:inline-table;table-layout:fixed;border-collapse:collapse;}.css-63uqft .katex .vlist-r{display:table-row;}.css-63uqft .katex .vlist{display:table-cell;vertical-align:bottom;position:relative;}.css-63uqft .katex .vlist>span{display:block;height:0;position:relative;}.css-63uqft .katex .vlist>span>span{display:inline-block;}.css-63uqft .katex .vlist>span>.pstrut{overflow:hidden;width:0;}.css-63uqft .katex .vlist-t2{margin-right:-2px;}.css-63uqft .katex .vlist-s{display:table-cell;vertical-align:bottom;font-size:1px;width:2px;min-width:2px;}.css-63uqft .katex .vbox{display:-webkit-inline-box;display:-webkit-inline-flex;display:-ms-inline-flexbox;display:inline-flex;-webkit-flex-direction:column;-ms-flex-direction:column;flex-direction:column;-webkit-align-items:baseline;-webkit-box-align:baseline;-ms-flex-align:baseline;align-items:baseline;}.css-63uqft .katex .hbox{display:-webkit-inline-box;display:-webkit-inline-flex;display:-ms-inline-flexbox;display:inline-flex;-webkit-flex-direction:row;-ms-flex-direction:row;flex-direction:row;width:100%;}.css-63uqft .katex .thinbox{display:-webkit-inline-box;display:-webkit-inline-flex;display:-ms-inline-flexbox;display:inline-flex;-webkit-flex-direction:row;-ms-flex-direction:row;flex-direction:row;width:0;max-width:0;}.css-63uqft .katex .msupsub{text-align:left;}.css-63uqft .katex .mfrac>span>span{text-align:center;}.css-63uqft .katex .mfrac .frac-line{display:inline-block;width:100%;border-bottom-style:solid;}.css-63uqft .katex .mfrac .frac-line,.css-63uqft .katex .overline .overline-line,.css-63uqft .katex .underline .underline-line,.css-63uqft .katex .hline,.css-63uqft .katex .hdashline,.css-63uqft .katex .rule{min-height:1px;}.css-63uqft .katex .mspace{display:inline-block;}.css-63uqft .katex .llap,.css-63uqft .katex .rlap,.css-63uqft .katex .clap{width:0;position:relative;}.css-63uqft .katex .llap>.inner,.css-63uqft .katex .rlap>.inner,.css-63uqft .katex .clap>.inner{position:absolute;}.css-63uqft .katex .llap>.fix,.css-63uqft .katex .rlap>.fix,.css-63uqft .katex .clap>.fix{display:inline-block;}.css-63uqft .katex .llap>.inner{right:0;}.css-63uqft .katex .rlap>.inner,.css-63uqft .katex .clap>.inner{left:0;}.css-63uqft .katex .clap>.inner>span{margin-left:-50%;margin-right:50%;}.css-63uqft .katex .rule{display:inline-block;border:solid 0;position:relative;}.css-63uqft .katex .overline .overline-line,.css-63uqft .katex .underline .underline-line,.css-63uqft .katex .hline{display:inline-block;width:100%;border-bottom-style:solid;}.css-63uqft .katex .hdashline{display:inline-block;width:100%;border-bottom-style:dashed;}.css-63uqft .katex .sqrt>.root{margin-left:0.27777778em;margin-right:-0.55555556em;}.css-63uqft .katex .sizing.reset-size1.size1,.css-63uqft .katex .fontsize-ensurer.reset-size1.size1{font-size:1em;}.css-63uqft .katex .sizing.reset-size1.size2,.css-63uqft .katex .fontsize-ensurer.reset-size1.size2{font-size:1.2em;}.css-63uqft .katex .sizing.reset-size1.size3,.css-63uqft .katex .fontsize-ensurer.reset-size1.size3{font-size:1.4em;}.css-63uqft .katex .sizing.reset-size1.size4,.css-63uqft .katex .fontsize-ensurer.reset-size1.size4{font-size:1.6em;}.css-63uqft .katex .sizing.reset-size1.size5,.css-63uqft .katex .fontsize-ensurer.reset-size1.size5{font-size:1.8em;}.css-63uqft .katex .sizing.reset-size1.size6,.css-63uqft .katex .fontsize-ensurer.reset-size1.size6{font-size:2em;}.css-63uqft .katex .sizing.reset-size1.size7,.css-63uqft .katex .fontsize-ensurer.reset-size1.size7{font-size:2.4em;}.css-63uqft .katex .sizing.reset-size1.size8,.css-63uqft .katex .fontsize-ensurer.reset-size1.size8{font-size:2.88em;}.css-63uqft .katex .sizing.reset-size1.size9,.css-63uqft .katex .fontsize-ensurer.reset-size1.size9{font-size:3.456em;}.css-63uqft .katex .sizing.reset-size1.size10,.css-63uqft .katex .fontsize-ensurer.reset-size1.size10{font-size:4.148em;}.css-63uqft .katex .sizing.reset-size1.size11,.css-63uqft .katex .fontsize-ensurer.reset-size1.size11{font-size:4.976em;}.css-63uqft .katex .sizing.reset-size2.size1,.css-63uqft .katex .fontsize-ensurer.reset-size2.size1{font-size:0.83333333em;}.css-63uqft .katex .sizing.reset-size2.size2,.css-63uqft .katex .fontsize-ensurer.reset-size2.size2{font-size:1em;}.css-63uqft .katex .sizing.reset-size2.size3,.css-63uqft .katex .fontsize-ensurer.reset-size2.size3{font-size:1.16666667em;}.css-63uqft .katex .sizing.reset-size2.size4,.css-63uqft .katex .fontsize-ensurer.reset-size2.size4{font-size:1.33333333em;}.css-63uqft .katex .sizing.reset-size2.size5,.css-63uqft .katex .fontsize-ensurer.reset-size2.size5{font-size:1.5em;}.css-63uqft .katex .sizing.reset-size2.size6,.css-63uqft .katex .fontsize-ensurer.reset-size2.size6{font-size:1.66666667em;}.css-63uqft .katex .sizing.reset-size2.size7,.css-63uqft .katex .fontsize-ensurer.reset-size2.size7{font-size:2em;}.css-63uqft .katex .sizing.reset-size2.size8,.css-63uqft .katex .fontsize-ensurer.reset-size2.size8{font-size:2.4em;}.css-63uqft .katex .sizing.reset-size2.size9,.css-63uqft .katex .fontsize-ensurer.reset-size2.size9{font-size:2.88em;}.css-63uqft .katex .sizing.reset-size2.size10,.css-63uqft .katex .fontsize-ensurer.reset-size2.size10{font-size:3.45666667em;}.css-63uqft .katex .sizing.reset-size2.size11,.css-63uqft .katex .fontsize-ensurer.reset-size2.size11{font-size:4.14666667em;}.css-63uqft .katex .sizing.reset-size3.size1,.css-63uqft .katex .fontsize-ensurer.reset-size3.size1{font-size:0.71428571em;}.css-63uqft .katex .sizing.reset-size3.size2,.css-63uqft .katex .fontsize-ensurer.reset-size3.size2{font-size:0.85714286em;}.css-63uqft .katex .sizing.reset-size3.size3,.css-63uqft .katex .fontsize-ensurer.reset-size3.size3{font-size:1em;}.css-63uqft .katex .sizing.reset-size3.size4,.css-63uqft .katex .fontsize-ensurer.reset-size3.size4{font-size:1.14285714em;}.css-63uqft .katex .sizing.reset-size3.size5,.css-63uqft .katex .fontsize-ensurer.reset-size3.size5{font-size:1.28571429em;}.css-63uqft .katex .sizing.reset-size3.size6,.css-63uqft .katex .fontsize-ensurer.reset-size3.size6{font-size:1.42857143em;}.css-63uqft .katex .sizing.reset-size3.size7,.css-63uqft .katex .fontsize-ensurer.reset-size3.size7{font-size:1.71428571em;}.css-63uqft .katex .sizing.reset-size3.size8,.css-63uqft .katex .fontsize-ensurer.reset-size3.size8{font-size:2.05714286em;}.css-63uqft .katex .sizing.reset-size3.size9,.css-63uqft .katex .fontsize-ensurer.reset-size3.size9{font-size:2.46857143em;}.css-63uqft .katex .sizing.reset-size3.size10,.css-63uqft .katex .fontsize-ensurer.reset-size3.size10{font-size:2.96285714em;}.css-63uqft .katex .sizing.reset-size3.size11,.css-63uqft .katex .fontsize-ensurer.reset-size3.size11{font-size:3.55428571em;}.css-63uqft .katex .sizing.reset-size4.size1,.css-63uqft .katex .fontsize-ensurer.reset-size4.size1{font-size:0.625em;}.css-63uqft .katex .sizing.reset-size4.size2,.css-63uqft .katex .fontsize-ensurer.reset-size4.size2{font-size:0.75em;}.css-63uqft .katex .sizing.reset-size4.size3,.css-63uqft .katex .fontsize-ensurer.reset-size4.size3{font-size:0.875em;}.css-63uqft .katex .sizing.reset-size4.size4,.css-63uqft .katex .fontsize-ensurer.reset-size4.size4{font-size:1em;}.css-63uqft .katex .sizing.reset-size4.size5,.css-63uqft .katex .fontsize-ensurer.reset-size4.size5{font-size:1.125em;}.css-63uqft .katex .sizing.reset-size4.size6,.css-63uqft .katex .fontsize-ensurer.reset-size4.size6{font-size:1.25em;}.css-63uqft .katex .sizing.reset-size4.size7,.css-63uqft .katex .fontsize-ensurer.reset-size4.size7{font-size:1.5em;}.css-63uqft .katex .sizing.reset-size4.size8,.css-63uqft .katex .fontsize-ensurer.reset-size4.size8{font-size:1.8em;}.css-63uqft .katex .sizing.reset-size4.size9,.css-63uqft .katex .fontsize-ensurer.reset-size4.size9{font-size:2.16em;}.css-63uqft .katex .sizing.reset-size4.size10,.css-63uqft .katex .fontsize-ensurer.reset-size4.size10{font-size:2.5925em;}.css-63uqft .katex .sizing.reset-size4.size11,.css-63uqft .katex .fontsize-ensurer.reset-size4.size11{font-size:3.11em;}.css-63uqft .katex .sizing.reset-size5.size1,.css-63uqft .katex .fontsize-ensurer.reset-size5.size1{font-size:0.55555556em;}.css-63uqft .katex .sizing.reset-size5.size2,.css-63uqft .katex .fontsize-ensurer.reset-size5.size2{font-size:0.66666667em;}.css-63uqft .katex .sizing.reset-size5.size3,.css-63uqft .katex .fontsize-ensurer.reset-size5.size3{font-size:0.77777778em;}.css-63uqft .katex .sizing.reset-size5.size4,.css-63uqft .katex .fontsize-ensurer.reset-size5.size4{font-size:0.88888889em;}.css-63uqft .katex .sizing.reset-size5.size5,.css-63uqft .katex .fontsize-ensurer.reset-size5.size5{font-size:1em;}.css-63uqft .katex .sizing.reset-size5.size6,.css-63uqft .katex .fontsize-ensurer.reset-size5.size6{font-size:1.11111111em;}.css-63uqft .katex .sizing.reset-size5.size7,.css-63uqft .katex .fontsize-ensurer.reset-size5.size7{font-size:1.33333333em;}.css-63uqft .katex .sizing.reset-size5.size8,.css-63uqft .katex .fontsize-ensurer.reset-size5.size8{font-size:1.6em;}.css-63uqft .katex .sizing.reset-size5.size9,.css-63uqft .katex .fontsize-ensurer.reset-size5.size9{font-size:1.92em;}.css-63uqft .katex .sizing.reset-size5.size10,.css-63uqft .katex .fontsize-ensurer.reset-size5.size10{font-size:2.30444444em;}.css-63uqft .katex .sizing.reset-size5.size11,.css-63uqft .katex .fontsize-ensurer.reset-size5.size11{font-size:2.76444444em;}.css-63uqft .katex .sizing.reset-size6.size1,.css-63uqft .katex .fontsize-ensurer.reset-size6.size1{font-size:0.5em;}.css-63uqft .katex .sizing.reset-size6.size2,.css-63uqft .katex .fontsize-ensurer.reset-size6.size2{font-size:0.6em;}.css-63uqft .katex .sizing.reset-size6.size3,.css-63uqft .katex .fontsize-ensurer.reset-size6.size3{font-size:0.7em;}.css-63uqft .katex .sizing.reset-size6.size4,.css-63uqft .katex .fontsize-ensurer.reset-size6.size4{font-size:0.8em;}.css-63uqft .katex .sizing.reset-size6.size5,.css-63uqft .katex .fontsize-ensurer.reset-size6.size5{font-size:0.9em;}.css-63uqft .katex .sizing.reset-size6.size6,.css-63uqft .katex .fontsize-ensurer.reset-size6.size6{font-size:1em;}.css-63uqft .katex .sizing.reset-size6.size7,.css-63uqft .katex .fontsize-ensurer.reset-size6.size7{font-size:1.2em;}.css-63uqft .katex .sizing.reset-size6.size8,.css-63uqft .katex .fontsize-ensurer.reset-size6.size8{font-size:1.44em;}.css-63uqft .katex .sizing.reset-size6.size9,.css-63uqft .katex .fontsize-ensurer.reset-size6.size9{font-size:1.728em;}.css-63uqft .katex .sizing.reset-size6.size10,.css-63uqft .katex .fontsize-ensurer.reset-size6.size10{font-size:2.074em;}.css-63uqft .katex .sizing.reset-size6.size11,.css-63uqft .katex .fontsize-ensurer.reset-size6.size11{font-size:2.488em;}.css-63uqft .katex .sizing.reset-size7.size1,.css-63uqft .katex .fontsize-ensurer.reset-size7.size1{font-size:0.41666667em;}.css-63uqft .katex .sizing.reset-size7.size2,.css-63uqft .katex .fontsize-ensurer.reset-size7.size2{font-size:0.5em;}.css-63uqft .katex .sizing.reset-size7.size3,.css-63uqft .katex .fontsize-ensurer.reset-size7.size3{font-size:0.58333333em;}.css-63uqft .katex .sizing.reset-size7.size4,.css-63uqft .katex .fontsize-ensurer.reset-size7.size4{font-size:0.66666667em;}.css-63uqft .katex .sizing.reset-size7.size5,.css-63uqft .katex .fontsize-ensurer.reset-size7.size5{font-size:0.75em;}.css-63uqft .katex .sizing.reset-size7.size6,.css-63uqft .katex .fontsize-ensurer.reset-size7.size6{font-size:0.83333333em;}.css-63uqft .katex .sizing.reset-size7.size7,.css-63uqft .katex .fontsize-ensurer.reset-size7.size7{font-size:1em;}.css-63uqft .katex .sizing.reset-size7.size8,.css-63uqft .katex .fontsize-ensurer.reset-size7.size8{font-size:1.2em;}.css-63uqft .katex .sizing.reset-size7.size9,.css-63uqft .katex .fontsize-ensurer.reset-size7.size9{font-size:1.44em;}.css-63uqft .katex .sizing.reset-size7.size10,.css-63uqft .katex .fontsize-ensurer.reset-size7.size10{font-size:1.72833333em;}.css-63uqft .katex .sizing.reset-size7.size11,.css-63uqft .katex .fontsize-ensurer.reset-size7.size11{font-size:2.07333333em;}.css-63uqft .katex .sizing.reset-size8.size1,.css-63uqft .katex .fontsize-ensurer.reset-size8.size1{font-size:0.34722222em;}.css-63uqft .katex .sizing.reset-size8.size2,.css-63uqft .katex .fontsize-ensurer.reset-size8.size2{font-size:0.41666667em;}.css-63uqft .katex .sizing.reset-size8.size3,.css-63uqft .katex .fontsize-ensurer.reset-size8.size3{font-size:0.48611111em;}.css-63uqft .katex .sizing.reset-size8.size4,.css-63uqft .katex .fontsize-ensurer.reset-size8.size4{font-size:0.55555556em;}.css-63uqft .katex .sizing.reset-size8.size5,.css-63uqft .katex .fontsize-ensurer.reset-size8.size5{font-size:0.625em;}.css-63uqft .katex .sizing.reset-size8.size6,.css-63uqft .katex .fontsize-ensurer.reset-size8.size6{font-size:0.69444444em;}.css-63uqft .katex .sizing.reset-size8.size7,.css-63uqft .katex .fontsize-ensurer.reset-size8.size7{font-size:0.83333333em;}.css-63uqft .katex .sizing.reset-size8.size8,.css-63uqft .katex .fontsize-ensurer.reset-size8.size8{font-size:1em;}.css-63uqft .katex .sizing.reset-size8.size9,.css-63uqft .katex .fontsize-ensurer.reset-size8.size9{font-size:1.2em;}.css-63uqft .katex .sizing.reset-size8.size10,.css-63uqft .katex .fontsize-ensurer.reset-size8.size10{font-size:1.44027778em;}.css-63uqft .katex .sizing.reset-size8.size11,.css-63uqft .katex .fontsize-ensurer.reset-size8.size11{font-size:1.72777778em;}.css-63uqft .katex .sizing.reset-size9.size1,.css-63uqft .katex .fontsize-ensurer.reset-size9.size1{font-size:0.28935185em;}.css-63uqft .katex .sizing.reset-size9.size2,.css-63uqft .katex .fontsize-ensurer.reset-size9.size2{font-size:0.34722222em;}.css-63uqft .katex .sizing.reset-size9.size3,.css-63uqft .katex .fontsize-ensurer.reset-size9.size3{font-size:0.40509259em;}.css-63uqft .katex .sizing.reset-size9.size4,.css-63uqft .katex .fontsize-ensurer.reset-size9.size4{font-size:0.46296296em;}.css-63uqft .katex .sizing.reset-size9.size5,.css-63uqft .katex .fontsize-ensurer.reset-size9.size5{font-size:0.52083333em;}.css-63uqft .katex .sizing.reset-size9.size6,.css-63uqft .katex .fontsize-ensurer.reset-size9.size6{font-size:0.5787037em;}.css-63uqft .katex .sizing.reset-size9.size7,.css-63uqft .katex .fontsize-ensurer.reset-size9.size7{font-size:0.69444444em;}.css-63uqft .katex .sizing.reset-size9.size8,.css-63uqft .katex .fontsize-ensurer.reset-size9.size8{font-size:0.83333333em;}.css-63uqft .katex .sizing.reset-size9.size9,.css-63uqft .katex .fontsize-ensurer.reset-size9.size9{font-size:1em;}.css-63uqft .katex .sizing.reset-size9.size10,.css-63uqft .katex .fontsize-ensurer.reset-size9.size10{font-size:1.20023148em;}.css-63uqft .katex .sizing.reset-size9.size11,.css-63uqft .katex .fontsize-ensurer.reset-size9.size11{font-size:1.43981481em;}.css-63uqft .katex .sizing.reset-size10.size1,.css-63uqft .katex .fontsize-ensurer.reset-size10.size1{font-size:0.24108004em;}.css-63uqft .katex .sizing.reset-size10.size2,.css-63uqft .katex .fontsize-ensurer.reset-size10.size2{font-size:0.28929605em;}.css-63uqft .katex .sizing.reset-size10.size3,.css-63uqft .katex .fontsize-ensurer.reset-size10.size3{font-size:0.33751205em;}.css-63uqft .katex .sizing.reset-size10.size4,.css-63uqft .katex .fontsize-ensurer.reset-size10.size4{font-size:0.38572806em;}.css-63uqft .katex .sizing.reset-size10.size5,.css-63uqft .katex .fontsize-ensurer.reset-size10.size5{font-size:0.43394407em;}.css-63uqft .katex .sizing.reset-size10.size6,.css-63uqft .katex .fontsize-ensurer.reset-size10.size6{font-size:0.48216008em;}.css-63uqft .katex .sizing.reset-size10.size7,.css-63uqft .katex .fontsize-ensurer.reset-size10.size7{font-size:0.57859209em;}.css-63uqft .katex .sizing.reset-size10.size8,.css-63uqft .katex .fontsize-ensurer.reset-size10.size8{font-size:0.69431051em;}.css-63uqft .katex .sizing.reset-size10.size9,.css-63uqft .katex .fontsize-ensurer.reset-size10.size9{font-size:0.83317261em;}.css-63uqft .katex .sizing.reset-size10.size10,.css-63uqft .katex .fontsize-ensurer.reset-size10.size10{font-size:1em;}.css-63uqft .katex .sizing.reset-size10.size11,.css-63uqft .katex .fontsize-ensurer.reset-size10.size11{font-size:1.19961427em;}.css-63uqft .katex .sizing.reset-size11.size1,.css-63uqft .katex .fontsize-ensurer.reset-size11.size1{font-size:0.20096463em;}.css-63uqft .katex .sizing.reset-size11.size2,.css-63uqft .katex .fontsize-ensurer.reset-size11.size2{font-size:0.24115756em;}.css-63uqft .katex .sizing.reset-size11.size3,.css-63uqft .katex .fontsize-ensurer.reset-size11.size3{font-size:0.28135048em;}.css-63uqft .katex .sizing.reset-size11.size4,.css-63uqft .katex .fontsize-ensurer.reset-size11.size4{font-size:0.32154341em;}.css-63uqft .katex .sizing.reset-size11.size5,.css-63uqft .katex .fontsize-ensurer.reset-size11.size5{font-size:0.36173633em;}.css-63uqft .katex .sizing.reset-size11.size6,.css-63uqft .katex .fontsize-ensurer.reset-size11.size6{font-size:0.40192926em;}.css-63uqft .katex .sizing.reset-size11.size7,.css-63uqft .katex .fontsize-ensurer.reset-size11.size7{font-size:0.48231511em;}.css-63uqft .katex .sizing.reset-size11.size8,.css-63uqft .katex .fontsize-ensurer.reset-size11.size8{font-size:0.57877814em;}.css-63uqft .katex .sizing.reset-size11.size9,.css-63uqft .katex .fontsize-ensurer.reset-size11.size9{font-size:0.69453376em;}.css-63uqft .katex .sizing.reset-size11.size10,.css-63uqft .katex .fontsize-ensurer.reset-size11.size10{font-size:0.83360129em;}.css-63uqft .katex .sizing.reset-size11.size11,.css-63uqft .katex .fontsize-ensurer.reset-size11.size11{font-size:1em;}.css-63uqft .katex .delimsizing.size1{font-family:KaTeX_Size1;}.css-63uqft .katex .delimsizing.size2{font-family:KaTeX_Size2;}.css-63uqft .katex .delimsizing.size3{font-family:KaTeX_Size3;}.css-63uqft .katex .delimsizing.size4{font-family:KaTeX_Size4;}.css-63uqft .katex .delimsizing.mult .delim-size1>span{font-family:KaTeX_Size1;}.css-63uqft .katex .delimsizing.mult .delim-size4>span{font-family:KaTeX_Size4;}.css-63uqft .katex .nulldelimiter{display:inline-block;width:0.12em;}.css-63uqft .katex .delimcenter{position:relative;}.css-63uqft .katex .op-symbol{position:relative;}.css-63uqft .katex .op-symbol.small-op{font-family:KaTeX_Size1;}.css-63uqft .katex .op-symbol.large-op{font-family:KaTeX_Size2;}.css-63uqft .katex .op-limits>.vlist-t{text-align:center;}.css-63uqft .katex .accent>.vlist-t{text-align:center;}.css-63uqft .katex .accent .accent-body{position:relative;}.css-63uqft .katex .accent .accent-body:not(.accent-full){width:0;}.css-63uqft .katex .overlay{display:block;}.css-63uqft .katex .mtable .vertical-separator{display:inline-block;min-width:1px;}.css-63uqft .katex .mtable .arraycolsep{display:inline-block;}.css-63uqft .katex .mtable .col-align-c>.vlist-t{text-align:center;}.css-63uqft .katex .mtable .col-align-l>.vlist-t{text-align:left;}.css-63uqft .katex .mtable .col-align-r>.vlist-t{text-align:right;}.css-63uqft .katex .svg-align{text-align:left;}.css-63uqft .katex svg{display:block;position:absolute;width:100%;height:inherit;fill:currentColor;stroke:currentColor;fill-rule:nonzero;fill-opacity:1;stroke-width:1;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0;stroke-opacity:1;}.css-63uqft .katex svg path{stroke:none;}.css-63uqft .katex img{border-style:none;min-width:0;min-height:0;max-width:none;max-height:none;}.css-63uqft .katex .stretchy{width:100%;display:block;position:relative;overflow:hidden;}.css-63uqft .katex .stretchy::before,.css-63uqft .katex .stretchy::after{content:'';}.css-63uqft .katex .hide-tail{width:100%;position:relative;overflow:hidden;}.css-63uqft .katex .halfarrow-left{position:absolute;left:0;width:50.2%;overflow:hidden;}.css-63uqft .katex .halfarrow-right{position:absolute;right:0;width:50.2%;overflow:hidden;}.css-63uqft .katex .brace-left{position:absolute;left:0;width:25.1%;overflow:hidden;}.css-63uqft .katex .brace-center{position:absolute;left:25%;width:50%;overflow:hidden;}.css-63uqft .katex .brace-right{position:absolute;right:0;width:25.1%;overflow:hidden;}.css-63uqft .katex .x-arrow-pad{padding:0 0.5em;}.css-63uqft .katex .cd-arrow-pad{padding:0 0.55556em 0 0.27778em;}.css-63uqft .katex .x-arrow,.css-63uqft .katex .mover,.css-63uqft .katex .munder{text-align:center;}.css-63uqft .katex .boxpad{padding:0 0.3em 0 0.3em;}.css-63uqft .katex .fbox,.css-63uqft .katex .fcolorbox{box-sizing:border-box;border:0.04em solid;}.css-63uqft .katex .cancel-pad{padding:0 0.2em 0 0.2em;}.css-63uqft .katex .cancel-lap{margin-left:-0.2em;margin-right:-0.2em;}.css-63uqft .katex .sout{border-bottom-style:solid;border-bottom-width:0.08em;}.css-63uqft .katex .angl{box-sizing:border-box;border-top:0.049em solid;border-right:0.049em solid;margin-right:0.03889em;}.css-63uqft .katex .anglpad{padding:0 0.03889em 0 0.03889em;}.css-63uqft .katex .eqn-num::before{counter-increment:katexEqnNo;content:'(' counter(katexEqnNo) ')';}.css-63uqft .katex .mml-eqn-num::before{counter-increment:mmlEqnNo;content:'(' counter(mmlEqnNo) ')';}.css-63uqft .katex .mtr-glue{width:50%;}.css-63uqft .katex .cd-vert-arrow{display:inline-block;position:relative;}.css-63uqft .katex .cd-label-left{display:inline-block;position:absolute;right:calc(50% + 0.3em);text-align:left;}.css-63uqft .katex .cd-label-right{display:inline-block;position:absolute;left:calc(50% + 0.3em);text-align:right;}.css-63uqft .katex-display{display:block;margin:1em 0;text-align:center;}.css-63uqft .katex-display>.katex{display:block;white-space:nowrap;}.css-63uqft .katex-display>.katex>.katex-html{display:block;position:relative;}.css-63uqft .katex-display>.katex>.katex-html>.tag{position:absolute;right:0;}.css-63uqft .katex-display.leqno>.katex>.katex-html>.tag{left:0;right:auto;}.css-63uqft .katex-display.fleqn>.katex{text-align:left;padding-left:2em;}.css-63uqft body{counter-reset:katexEqnNo mmlEqnNo;}.css-63uqft table{width:-webkit-max-content;width:-moz-max-content;width:max-content;}.css-63uqft .tableBlock{max-width:100%;margin-bottom:1rem;overflow-y:scroll;}.css-63uqft .tableBlock thead,.css-63uqft .tableBlock thead th{border-bottom:1px solid #333!important;}.css-63uqft .tableBlock th,.css-63uqft .tableBlock td{padding:10px;text-align:left;}.css-63uqft .tableBlock th{font-weight:bold!important;}.css-63uqft .tableBlock caption{caption-side:bottom;color:#555;font-size:12px;font-style:italic;text-align:center;}.css-63uqft .tableBlock caption>p{margin:0;}.css-63uqft .tableBlock th>p,.css-63uqft .tableBlock td>p{margin:0;}.css-63uqft .tableBlock [data-background-color='aliceblue']{background-color:#f0f8ff;color:#000;}.css-63uqft .tableBlock [data-background-color='black']{background-color:#000;color:#fff;}.css-63uqft .tableBlock [data-background-color='chocolate']{background-color:#d2691e;color:#fff;}.css-63uqft .tableBlock [data-background-color='cornflowerblue']{background-color:#6495ed;color:#fff;}.css-63uqft .tableBlock [data-background-color='crimson']{background-color:#dc143c;color:#fff;}.css-63uqft .tableBlock [data-background-color='darkblue']{background-color:#00008b;color:#fff;}.css-63uqft .tableBlock [data-background-color='darkseagreen']{background-color:#8fbc8f;color:#000;}.css-63uqft .tableBlock [data-background-color='deepskyblue']{background-color:#00bfff;color:#000;}.css-63uqft .tableBlock [data-background-color='gainsboro']{background-color:#dcdcdc;color:#000;}.css-63uqft .tableBlock [data-background-color='grey']{background-color:#808080;color:#fff;}.css-63uqft .tableBlock [data-background-color='lemonchiffon']{background-color:#fffacd;color:#000;}.css-63uqft .tableBlock [data-background-color='lightpink']{background-color:#ffb6c1;color:#000;}.css-63uqft .tableBlock [data-background-color='lightsalmon']{background-color:#ffa07a;color:#000;}.css-63uqft .tableBlock [data-background-color='lightskyblue']{background-color:#87cefa;color:#000;}.css-63uqft .tableBlock [data-background-color='mediumblue']{background-color:#0000cd;color:#fff;}.css-63uqft .tableBlock [data-background-color='omnigrey']{background-color:#f0f0f0;color:#000;}.css-63uqft .tableBlock [data-background-color='white']{background-color:#fff;color:#000;}.css-63uqft .tableBlock [data-text-align='center']{text-align:center;}.css-63uqft .tableBlock [data-text-align='left']{text-align:left;}.css-63uqft .tableBlock [data-text-align='right']{text-align:right;}.css-63uqft .tableBlock [data-vertical-align='bottom']{vertical-align:bottom;}.css-63uqft .tableBlock [data-vertical-align='middle']{vertical-align:middle;}.css-63uqft .tableBlock [data-vertical-align='top']{vertical-align:top;}.css-63uqft .tableBlockfont-size--xxsmall{font-size:10px;}.css-63uqft .tableBlockfont-size--xsmall{font-size:12px;}.css-63uqft .tableBlockfont-size--small{font-size:14px;}.css-63uqft .tableBlockfont-size--large{font-size:18px;}.css-63uqft .tableBlockborder--some tbody tr:not(:last-child){border-bottom:1px solid #e2e5e7;}.css-63uqft .tableBlockborder--bordered td,.css-63uqft .tableBlockborder--bordered th{border:1px solid #e2e5e7;}.css-63uqft .tableBlockborder--borderless tbody+tbody,.css-63uqft .tableBlockborder--borderless td,.css-63uqft .tableBlockborder--borderless th,.css-63uqft .tableBlockborder--borderless tr,.css-63uqft .tableBlockborder--borderless thead,.css-63uqft .tableBlockborder--borderless thead th{border:0!important;}.css-63uqft .tableBlock:not(.tableBlocktable-striped) tbody tr{background-color:unset!important;}.css-63uqft .tableBlocktable-striped tbody tr:nth-of-type(odd){background-color:#f9fafc!important;}.css-63uqft .tableBlocktable-compactl th,.css-63uqft .tableBlocktable-compact td{padding:3px!important;}.css-63uqft .tableBlockfull-size{width:100%;}.css-63uqft .textBlock{margin-bottom:16px;}.css-63uqft .textBlocktext-formatting--finePrint{font-size:12px;}.css-63uqft .textBlocktext-infoBox{padding:0.75rem 1.25rem;margin-bottom:1rem;border:1px solid transparent;border-radius:0.25rem;}.css-63uqft .textBlocktext-infoBox p{margin:0;}.css-63uqft .textBlocktext-infoBox--primary{background-color:#cce5ff;border-color:#b8daff;color:#004085;}.css-63uqft .textBlocktext-infoBox--secondary{background-color:#e2e3e5;border-color:#d6d8db;color:#383d41;}.css-63uqft .textBlocktext-infoBox--success{background-color:#d4edda;border-color:#c3e6cb;color:#155724;}.css-63uqft .textBlocktext-infoBox--danger{background-color:#f8d7da;border-color:#f5c6cb;color:#721c24;}.css-63uqft .textBlocktext-infoBox--warning{background-color:#fff3cd;border-color:#ffeeba;color:#856404;}.css-63uqft .textBlocktext-infoBox--info{background-color:#d1ecf1;border-color:#bee5eb;color:#0c5460;}.css-63uqft .textBlocktext-infoBox--dark{background-color:#d6d8d9;border-color:#c6c8ca;color:#1b1e21;}.css-63uqft .text-overline{-webkit-text-decoration:overline;text-decoration:overline;}.css-63uqft.css-63uqft{color:#2B3148;background-color:transparent;font-family:var(--calculator-ui-font-family),Verdana,sans-serif;font-size:20px;line-height:24px;overflow:visible;padding-top:0px;position:relative;}.css-63uqft.css-63uqft:after{content:'';-webkit-transform:scale(0);-moz-transform:scale(0);-ms-transform:scale(0);transform:scale(0);position:absolute;border:2px solid #EA9430;border-radius:2px;inset:-8px;z-index:1;}.css-63uqft .js-external-link-button.link-like,.css-63uqft .js-external-link-anchor{color:inherit;border-radius:1px;-webkit-text-decoration:underline;text-decoration:underline;}.css-63uqft .js-external-link-button.link-like:hover,.css-63uqft .js-external-link-anchor:hover,.css-63uqft .js-external-link-button.link-like:active,.css-63uqft .js-external-link-anchor:active{text-decoration-thickness:2px;text-shadow:1px 0 0;}.css-63uqft .js-external-link-button.link-like:focus-visible,.css-63uqft .js-external-link-anchor:focus-visible{outline:transparent 2px dotted;box-shadow:0 0 0 2px #6314E6;}.css-63uqft p,.css-63uqft div{margin:0;display:block;}.css-63uqft pre{margin:0;display:block;}.css-63uqft pre code{display:block;width:-webkit-fit-content;width:-moz-fit-content;width:fit-content;}.css-63uqft pre:not(:first-child){padding-top:8px;}.css-63uqft ul,.css-63uqft ol{display:block margin:0;padding-left:20px;}.css-63uqft ul li,.css-63uqft ol li{padding-top:8px;}.css-63uqft ul ul,.css-63uqft ol ul,.css-63uqft ul ol,.css-63uqft ol ol{padding-top:0;}.css-63uqft ul:not(:first-child),.css-63uqft ol:not(:first-child){padding-top:4px;} Test setup

Choose test type

t-test for the population mean, μ, based on one independent sample . Null hypothesis H 0 : μ = μ 0

Alternative hypothesis H 1

Test details

Significance level α

The probability that we reject a true H 0 (type I error).

Degrees of freedom

Calculated as sample size minus one.

Test results

Hypothesis Testing Calculator Online

Hypothesis testing is a foundational method used in statistics to infer the validity of a hypothesis about a population parameter. The Hypothesis Testing Calculator facilitates this process by automating the computations necessary for the t-test , a method used to compare sample means against a hypothesized mean or against each other. Let’s delve into the formulas this calculator uses to execute one-sample and two-sample t-tests.

One-Sample t-Test

This test is used to determine if the mean (x̄) of your sample is statistically different from a hypothesized population mean (μ₀).

Two-Sample t-Test

Equal variances:, unequal variances (welch’s t-test):.

t = (x̄₁ – x̄₂) / (√((s₁² / n₁) + (s₂² / n₂)))

Table of Critical t-Values

Confidence Level (%)	df=10	df=30	df=50	df=100
90	1.812	1.697	1.676	1.660
95	2.228	2.042	2.009	1.984
99	3.169	2.750	2.678	2.626

These values are crucial in hypothesis testing as they help define the threshold for significance, assisting users of the calculator in interpreting their results accurately.

Using the one-sample t-test formula:

t = (74 – 70) / (8 / √36) = (4 / 1.333) = 3.00

Most Common FAQs

The p-value represents the probability of obtaining test results at least as extreme as the results observed, under the assumption that the null hypothesis is correct. A low p-value (typically below 0.05) indicates strong evidence against the null hypothesis, hence it is usually rejected.

Yes, while the t-test is specifically design for means, the principles of hypothesis testing apply to other parameters such as proportions and variances. Which can also be tested using appropriate versions of hypothesis tests such as the z-test and F-test.

Related Calculators

Metric Variables:

Ordinal variables:, nominal variables:, hypothesis test calculator.

Do you want to calculate a hypothesis test such as a t-test , Chi Square test or an ANOVA ? You can do that easily here in the browser.

If you want to use your own data just clear the upper table

Clear the table in the Hypothesis test calculator.
Copy your data into the table.
Select the variables.

In the hypothesis test calculator you can calculate e.g. a t-test, a chi-square test, a binomial test or an analysis of variance. If you need a more detailed explanation, you can find more information in the tutorials.

In order to use the hypothesis test calculator, you must first formulate your hypothesis and collect your data. DATAtab will then suggest the hypothesis test you need based on the data entered into the statistics calculator.

p value calculator

With the p value calculator you can calculate the p value for different tests. There is a wide range of methods for this. Just click on the variables you want to evaluate above and DATAtab will give you the tests you can use.

For example, if you select a metric and a categorical variable, the Independent t-Test calculator is automatically selected. If your data is not normally distributed, simply use the Mann-Whitney U-test calculator.

H0 and H1 calculator

With the h0 and h1 calculator for the different hypothesis test you can calculate the p-value which gives you an indication if you can reject the H0 or not.

Cite DATAtab: DATAtab Team (2024). DATAtab: Online Statistics Calculator. DATAtab e.U. Graz, Austria. URL https://datatab.net

Calculators

Hypothesis Testing Calculator

Enhanced hypothesis test calculator, hypothesis test calculator: your essential tool for statistical analysis.

The Hypothesis Test Calculator is an intuitive, web-based tool crafted to simplify the process of conducting statistical hypothesis tests. Whether you’re a researcher, student, or educator, this tool streamlines the process, providing accurate results and clear visualizations with minimal effort.

How to Use the Hypothesis Test Calculator

Step 1: Start by entering the required data into the respective input fields. You’ll need to provide the sample mean, sample size, population mean, population standard deviation, and the significance level (alpha). If you have raw data, you can also upload a CSV file directly.
Step 2: Choose whether you’re conducting a one-tailed or two-tailed test. If you’re interested in knowing the confidence interval, select the option to calculate it.
Step 3: Click the “Perform Test” button. The tool will quickly process the information, performing the hypothesis test and generating a detailed report.
Step 4: Review the results. The calculator provides the calculated t-value, critical value, and the test’s conclusion. You can also visualize the sample distribution with a dynamic chart that compares it to the population mean.

Understanding Hypothesis Testing

Hypothesis testing is a core statistical method used to determine whether there is enough evidence in a sample to infer that a certain condition is true for the entire population. The test compares the sample data against the null hypothesis, which typically represents the status quo or a baseline condition.

In this context, the Hypothesis Test Calculator is particularly useful for t-tests, which assess whether the sample mean significantly differs from a known population mean. By inputting your data, the tool computes the necessary values, helping you determine whether to reject the null hypothesis.

Hypothesis Test Formula:

$$ t = \frac{\text{Sample Mean} – \text{Population Mean}}{\text{Standard Error}} $$

Features and Advantages

User-Friendly Design: The calculator is built with simplicity in mind, allowing users to quickly enter data and obtain results without any hassle.
Rapid Calculations: With just a few clicks, the tool performs complex statistical calculations, delivering results almost instantaneously.
Comprehensive Results: Not only does the calculator provide the final conclusion, but it also breaks down the steps involved in the calculation, aiding in understanding the statistical process.
Visual Representation: The tool includes a dynamic chart that illustrates the sample distribution against the population mean, helping users visualize the data and results.
Versatile Input Options: You can input data manually or upload a CSV file, making it easy to work with real datasets.
Educational Utility: Perfect for both students and educators, the Hypothesis Test Calculator serves as an excellent learning tool, offering insights into the mechanics of hypothesis testing.

Enhancing Your Statistical Toolkit

The Hypothesis Test Calculator is an invaluable resource for anyone involved in data analysis. It simplifies the hypothesis testing process, delivering clear, accurate results and visual aids that enhance your understanding of statistical concepts.

Try the Hypothesis Test Calculator today and take your statistical analysis to the next level!

How useful was this Tool?

Click on a star to rate it!

Average rating / 5. Vote count:

No votes so far! Be the first to rate this post.

As you found this tool useful...

Share it on social media!

We are sorry that this tool was not useful for you!

Let us improve this tool!

Tell us how we can improve this tool?

Related Tools:

Website Responsive Preview Testing Tool
ANOVA Calculator
Time Duration Calculator
Protein Calculator
Biweekly Payment Calculator
Meal planning and recipe calculator

Username or Email Address

Remember Me

Hypothesis Testing Calculator

Understanding Hypothesis Testing: A Guide to the Hypothesis Testing Calculator

Hypothesis testing is a crucial statistical method used to make informed decisions about data and draw conclusions. Whether you’re a student, researcher, or professional, a Hypothesis Testing Calculator can be an invaluable tool in your statistical toolkit. Let’s explore what hypothesis testing is and how this calculator can assist you:

Hypothesis Testing Basics:

Null Hypothesis (H0): This is the default assumption or claim that there is no significant difference or effect. It’s often denoted as H0.
Alternative Hypothesis (Ha): This is the statement that contradicts the null hypothesis. It suggests that there is a significant difference or effect. It’s denoted as Ha.
Significance Level (α): This is the predetermined threshold (e.g., 0.05 or 5%) used to determine statistical significance. If the calculated p-value is less than α, you reject the null hypothesis.
p-value: This is the probability of observing the results (or more extreme results) if the null hypothesis is true. A small p-value suggests that the results are unlikely under the null hypothesis.

Key Features of the Hypothesis Testing Calculator:

Input Parameters: The calculator typically requires you to input sample data, choose the type of test (e.g., t-test, chi-square test), specify the null and alternative hypotheses, and set the significance level.
Calculations: Once you input the data and parameters, the calculator performs the necessary statistical tests and calculations. It generates results such as the test statistic, degrees of freedom, and the p-value.
Interpretation: Based on the results, the calculator helps you determine whether to reject or fail to reject the null hypothesis. It provides an interpretation of the findings, which is crucial for drawing conclusions.
Visual Representation: Some calculators may offer visual aids like graphs or charts to help you better understand the data distribution and test results.

Significance of the Hypothesis Testing Calculator:

Scientific Research: Researchers across various fields use hypothesis testing to validate their hypotheses and draw meaningful conclusions from data.
Quality Control: Industries use hypothesis testing to ensure the quality and consistency of products and processes.
Medical Studies: In medical research, hypothesis testing helps assess the effectiveness of treatments or interventions.
Academics: Students and educators use hypothesis testing to teach and learn statistical concepts and conduct experiments.
Data-Driven Decisions: Businesses use hypothesis testing to make data-driven decisions, such as whether to launch a new product based on market research.

Conclusion:

The Hypothesis Testing Calculator is a powerful tool that simplifies complex statistical analysis and enables data-driven decision-making. Whether you’re conducting experiments, analyzing survey data, or performing quality control, understanding hypothesis testing and using this calculator can help you make informed choices and contribute to evidence-based research and decision-making.

One Sample T Test Calculator

Enter sample data

Reporting results in APA style

One sample t-test, what is a one sample t-test, how to use the one sample t test calculator, calculators.

Two sample t test

Kruskal wallis

Pearson product moment

Two sample f test

One way anova

Bartlett test

Chi square goodness of fit

Fligner killeen

Hypothesis Test Calculator

Upload your data set below to get started

Or input your data as csv

Sharing helps us build more free tools

Calculators
Descriptive Statistics
Merchandise
Which Statistics Test?

Single Sample T-Test Calculator

A single sample t-test (or one sample t-test) is used to compare the mean of a single sample of scores to a known or hypothetical population mean. So, for example, it could be used to determine whether the mean diastolic blood pressure of a particular group differs from 85, a value determined by a previous study.

Requirements

The data is normally distributed
Scale of measurement should be interval or ratio
A randomized sample from a defined population

Null Hypothesis

H 0 : M - μ = 0, where M is the sample mean and μ is the population or hypothesized mean.

As above, the null hypothesis is that there is no difference between the sample mean and the known or hypothesized population mean.

t-Test Calculator

T-test - work with steps.

Input Data : Data set x = 3, 11, 17, 28, 34 Data set y = 5, 8, 13, 19, 28 Total number of elements = 5 Objective : Find the t-score by using mean and standard deviation. Solution : Mean 1 = (3 + 11 + 17 + 28 + 34)/5 = 93/5 Mean 1 = 18.6 Mean 2 = (5 + 8 + 13 + 19 + 28)/5 = 73/5 Mean 2 = 14.6 SD1 = √(1/5 - 1) x ((3 - 18.6) 2 + ( 11 - 18.6) 2 + ( 17 - 18.6) 2 + ( 28 - 18.6) 2 + ( 34 - 18.6) 2 ) = √(1/4) x ((-15.6) 2 + (-7.6) 2 + (-1.6) 2 + (9.4) 2 + (15.4) 2 ) = √(0.25) x ((243.36) + (57.76) + (2.56) + (88.36) + (237.16)) = √(0.25) x 629.2 = √157.3 SD1 = 12.5419 SD2 = √(1/5 - 1) x ((5 - 14.6) 2 + ( 8 - 14.6) 2 + ( 13 - 14.6) 2 + ( 19 - 14.6) 2 + ( 28 - 14.6) 2 ) = √(1/4) x ((-9.6) 2 + (-6.6) 2 + (-1.6) 2 + (4.4) 2 + (13.4) 2 ) = √(0.25) x ((92.16) + (43.56) + (2.56) + (19.36) + (179.56)) = √(0.25) x 337.2 = √84.3 SD2 = 9.1815 t-score = x 1 - x 2 √(SD1 2 /n1 + SD2 2 /n2) = 18.6 - 14.6 √((12.5419) 2 /5 + (9.1815) 2 /5) = 4 √((157.3)/5 + (84.3)/5) = 4 √(31.46 + 16.86) = 4 √(48.32) = 4 6.9513 t-score = 0.5754

What is t-Test?

A hypothesis test consists of two hypotheses, the null hypothesis and the alternative hypothesis or research hypothesis. The symbol $H_0$ represents the null hypothesis. The null hypothesis reflects that there will be no observed effect on the experiment. The null hypothesis consists of an equal sign. The alternative hypothesis reflects that there is an observed effect on the experiment. The symbol $H_a$ represents the alternative hypothesis. The first step in testing is to determine the null hypothesis and the alternative hypothesis. Regarding the testing hypothesis, there are some important terms. Rejection region is the set of values leads to rejection of the null hypothesis. Non-rejection region is the set of values that leads to nonrejection of the null hypothesis. Critical values are the value that separates the rejection and non-rejection regions. The t-Test is used in comparing the means of two populations. There are two approaches:

How to Find t-Critical Value

T-test with mean and standard deviation.

A t-Test is one of the most frequently used tests in statistics. A t-Test is useful to conclude if the results are correct and applicable to the entire population. If we want to analyze simple experiments or when making simple comparisons between levels of independent variable we use the t-Test. It's used in comparison between two separate groups of individuals, for example: male vs female, experimental vs control group, etc. Practice Problem 1: There are two company A and B. We want to test average age of employees at these companies so we use a random sample of employee ages from each company.

	Company A	Company B
Mean	43.2	36.7
Standard Deviation	7	8.3
Number of Employess	50	66

	Serbia	United States
Mean	43.2	5.2
Standard Deviation	1.2	8.3
Number of Employess	67	166

Math Lessons
Math Formulas
Calculators

Math Calculators, Lessons and Formulas

It is time to solve your math problem

HW Help (paid service)
Statistics and probability
T-test calculator

T-Test calculator

The Student's t-test is used to determine if means of two data sets differ significantly. This calculator will generate a step by step explanation on how to apply t – test.

Groups Have Equal Variance (default)

Groups Have Unequal Variance (Welch t-test)

Two Tailed Test (default)

One Tailed Test

0.05 (default)

0.01

0.001

Unpaired T Test (default)

Paired (Dependent) T Test

Long Division
Evaluate Expressions
Fraction Calculator
Greatest Common Divisor GCD
Least Common Multiple LCM
Prime Factorization
Scientific Notation
Percentage Calculator
Dec / Bin / Hex
Factoring Polynomials
Polynomial Roots
Synthetic Division
Polynomial Operations
Graphing Polynomials
Simplify Polynomials
Generate From Roots
Simplify Expression
Multiplication / Division
Addition / Subtraction
Rationalize Denominator
Simplifying
Quadratic Equations Solver
Polynomial Equations
Solving Equations - With Steps
Solving (with steps)
Quadratic Plotter
Factoring Trinomials
Equilateral Triangle
Right Triangle
Oblique Triangle
Square Calculator
Rectangle Calculator
Circle Calculator
Hexagon Calculator
Rhombus Calculator
Trapezoid Calculator
Triangular Prism
Distance calculator
Midpoint Calculator
Triangle Calculator
Graphing Lines
Lines Intersection
Two Point Form
Line-Point Distance
Parallel/Perpendicular
Circle Equation
Circle From 3 Points
Circle-line Intersection
Modulus, inverse, polar form
Vectors (2D & 3D)
Add, Subtract, Multiply
Determinant Calculator
Matrix Inverse
Characteristic Polynomial
Eigenvalues
Eigenvectors
Matrix Decomposition
Limit Calculator
Derivative Calculator
Integral Calculator
Arithmetic Sequences
Geometric Sequences
Find n th Term
Degrees to Radians
Trig. Equations
Probability Calculator
Probability Distributions
Descriptive Statistics
Standard Deviation
Z - score Calculator
Normal Distribution
T-Test Calculator
Correlation & Regression
Simple Interest
Compound Interest
Amortization Calculator
Annuity Calculator
Work Problems

Hire MATHPORTAL experts to do math homework for you.

Prices start at $3 per problem.

Twelve younger adults and twelve older adults conducted a life satisfaction test. The data are presented in the table below. Compute the appropriate t-test.

Are the means between two data sets are significantly different at level $\alpha < 0.05$.

Welcome to MathPortal. This website's owner is mathematician Miloš Petrović. I designed this website and wrote all the calculators, lessons, and formulas .

If you want to contact me, probably have some questions, write me using the contact form or email me on [email protected]

Email (optional)

Informative

Conversion Calculator

Test Statistic Calculator

Choose the method, enter the values into the test statistic calculator, and click on the “Calculate” button to calculate the statistical value for hypothesis evaluation.

Add this calculator to your site

This test statistic calculator helps to find the static value for hypothesis testing. The calculated test value shows if there’s enough evidence to reject a null hypothesis. Also, this calculator performs calculations of either for one population mean, comparing two means, single population proportion, and two population proportions.

Our tool is highly useful in various fields like research, experimentation, quality control, and data analysis.

What is Test Statistics?

A test statistic is a numerical value obtained from the sample data set. It summarizes the differences between what you observe within your sample and what would be expected if a hypothesis were true.

The t-test statistic also shows how closely your data matches the predicted distribution among the sample tests you perform.

How to Calculate Test Statistics Value?

Collect the data from the populations
Use the data to find the standard deviation of the population
Calculate the mean (μ) of the population using this data
Determine the z-value or sample size
Use the suitable test statistic formula and get the results

Test Statistic For One Population Mean:

Test statistics for a single population mean is calculated when a variable is numeric and involves one population or a group.

x̄ - µ 0 σ / √n

x̄ = Mean of your sample data
µ 0 = Hypothesized population mean that you are comparing to your sample mean
σ = Population standard deviation
n = number of observations (sample size) in your data set

Suppose we want to test if the average height of adult males in a city is 70 inches. We take a sample of 25 adult males and find the sample mean height to be 71 inches with a sample standard deviation of 3 inches. We use a significance level of 0.05.

t = 70 - 71 3√25

Test Statistic Comparing Two Population Means:

This test is applied when the numeric value is compared across the various populations or groups. To compute the resulting t statistic, two distinct random samples must be chosen, one from each population.

$\frac{√x̄ - √ȳ}{\sqrt{\frac{\sigma_{1}^{2}}{n_{1}}+\frac{\sigma_{2}^{2}}{n_{2}}}}$

ȳ = means of hypothesized population

Suppose we want to test if there is a difference in average test scores between two schools. We take a sample of 30 students from school A with an average score of 85 and a standard deviation of 5, and a sample of 35 students from school B with an average score of 82 and a standard deviation of 6.

t = 85 - 82 √5 2 / 30 + 6 2 / 35

t = 3 √ 25/30 + 36/35

t = 3 √0.833 + 1.029

t = 3 √1.862

Test Statistic For a Single Population Proportion:

This test is used to determine if a single population's proportion differs from a specified standard. The t statistic calculator works for a population proportion when dealing with data by having a limit of P₀ because proportions represent parts of a whole and cannot logically exceed the total or be negative.

$\frac{\hat{p}-p_{0}}{\sqrt{\frac{p_{0}(1-p_{0})}{n}}}$

P̂ = Sample proportion
P 0 = Population proportion

Suppose we want to test if the proportion of left-handed people in a population is 10%. We take a sample of 100 people and find that 8 are left-handed. We use a significance level of 0.05.

= P̂ - P₀ √0.10 (1 - 0.10)/100

= 0.08 - 0.10 √0.10 (1 - 0.10)/100

= -0.02 √0.10 (0.9)/100

= -0.02 √0.009

= -0.02 0.03

= −0.67

Test Statistic For Two Population Proportion:

This test identifies the difference in proportions between two independent groups to assess their significance.

$\frac{\hat{p}_{1}-\hat{p}_{2}}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_{1}}+\frac{1}{n_{2}})}}$

P̂ 1 and P̂ 2 = Sample proportions for two groups

Suppose we want to test if the proportion of smokers is different between two cities. We take a sample of 150 people from City A and find that 30 are smokers, and a sample of 200 people from City B and find that 50 are smokers.

P̂ 1 = 30 / 150 = 0.20
P̂ 2 = 50 / 200 = 0.25
P̂ = 30 + 50 / 150 + 200 = 0.229

Calculation:

= 0.20 - 0.25 √0.229 (1 - 0.229) (1 / 150 + 1/200)

= -0.05 √0.229 (0.771) (1 / 150 + 1 / 200)

= -0.05 √0.176 (1/150 + 1/200)

= -0.05 √0.176 (0.0113)

= -0.05 √0.002

= -0.05 0.045

= −1.11

Sample Distribution Calculator

Sample Size Calculator

Effect Size Calculator

Chi-Square Calculator

Critical Value Calculator

Probability Calculator

Pie Chart Calculator

Add this calculator to your site.

Just copy a given code & paste it right now into your website HTML (source) for suitable page.

Give Us Your Feedback

Share Result

Get the ease of calculating anything from the source of calculator online

Email us at

Easy to use statistics calculator

Our website provides various statistical calculators to perform various statistical calculations quickly and accurately. These calculators are useful for students, researchers, and professionals who need to analyze data without complex software.

Most useful calculator

The p-value and critical values are most useful calculator as they are used in Hypothesis Testing, Confidence Intervals and Decision Making.

Hypothesis Testing

Hypothesis testing is a method used in statistics to make a decision about a population based on sample data. By using this calculator, we can make a decision about whether the null and alternative hypotheses can be rejected or not.

Regression Analysis

The Regression analysis is a statistical technique used in statistics to explore the relationship between a dependent variable and independent variables. It helps in predicting outcomes and understanding relationships.

Descriptive Statistics

The descriptive statistics summarize and describe the main features of a dataset. They provide simple summaries about the sample and the measures also provides rovides insights into data trends and patterns.

Distribution

In statistics, a distribution describes how values of a dataset are spread out. It shows the possible values and their frequencies.

Explore additional statistical tools to enhance your data analysis:

Key Features

Types of Tests: Supports various tests such as t-tests, z-tests, chi-square tests, and ANOVA.
Input Data: Allows you to enter sample data or summary statistics.
Results: Provides test statistics, p-values, and confidence intervals.
Interpretation: Helps you understand the results and make decisions about your hypothesis.
Research: Validates research hypotheses with statistical evidence.
Quality Control: Assesses manufacturing processes and product quality.
Business: Analyzes market research data and business performance.
Efficiency: Saves time by automating complex calculations.
Accuracy: Reduces human error with precise computations.
Accessibility: Easily accessible online for quick analysis.

These calculators are valuable tools for students, researchers, and professionals needing to make data-driven decisions.

Two Population Calculator

Confidence interval.

$n_1$

x̄₁p̄₁

$n_2$

x̄₂

σ₂

Hypothesis Testing

$H_o$:	$\mu_d$
$H_a$:	μ₁ - μ₂$\mu_d$	≠	$D_o$

$n_1$

$\bar{x}_1$$\bar{p}_1$

$n_2$

$\bar{x}_2$

σ₂

$\text{Level of Significance:}$

$\alpha$ =

When computing confidence intervals for two population means, we are interested in the difference between the population means ($ \mu_1 - \mu_2 $). A confidence interval is made up of two parts, the point estimate and the margin of error. The point estimate of the difference between two population means is simply the difference between two sample means ($ \bar{x}_1 - \bar{x}_2 $). The standard error of $ \bar{x}_1 - \bar{x}_2 $, which is used in computing the margin of error, is given by the formula below.

Point Estimate	Standard Error
$ \bar{x}_1 - \bar{x}_2 $	$ \sqrt{\dfrac{\sigma_1^2}{n_1}+\dfrac{\sigma_2^2}{n_2}} $

The formula for the margin of error depends on whether the population standard deviations ($\sigma_1$ and $\sigma_2$) are known or unknown. If the population standard deviations are known, then they are used in the formula. If they are unknown, then the sample standard deviations ($s_1$ and $s_2$)are used in their place. To change from $\sigma$ known to $\sigma$ unknown, click on $\boxed{σ}$ and select $\boxed{s}$ in the Two Population Calculator.

	$\sigma$ Known	$\sigma$ Unknown
Margin of Error	$ z_{\alpha/2} \sqrt{\dfrac{\sigma_1^2}{n_1}+\dfrac{\sigma_2^2}{{\color{Black}n_2}}} $	$ t_{\alpha/2} \sqrt{\dfrac{s_1^2}{n_1}+\dfrac{s_2^2}{n_2}} $

While the formulas for the margin of error in the two population case are similar to those in the one population case, the formula for the degrees of freedom is quite a bit more complicated. Although this formula does seem intimidating at first sight, there is a shortcut to get the answer faster. Notice that the terms $\frac{s_1^2}{n_1}$ and $\frac{s_2^2}{n_2}$ each repeat twice. The terms are actually computed previously when finding the margin of error so they don't need to be calculated again.

Degrees of Freedom

$ df = \frac{\left(\dfrac{s_1^2}{n_1}+\dfrac{s_2^2}{n_2}\right)^2}{\dfrac{1}{n_1-1}\left(\dfrac{s_1^2}{n_1}\right)^2 + \dfrac{1}{n_2-1}\left(\dfrac{s_2^2}{n_2}\right)^2} $

If the two population variances are assumed to be equal, an alternative formula for computing the degrees of freedom is used. It's simply df = n1 + n2 - 2. This is a simple extension of the formula for the one population case. In the one population case the degrees of freedom is given by df = n - 1. If we add up the degrees of freedom for the two samples we would get df = (n1 - 1) + (n2 - 1) = n1 + n2 - 2. This formula gives a pretty good approximation of the more complicated formula above.

Just like in hypothesis tests about a single population mean, there are lower-tail, upper-tail and two tailed tests. However, the null and alternative are slightly different. First of all, instead of having mu on the left side of the equality, we have $\mu_1 - \mu_2$. On the right side of the equality, we don't have $\mu_0$, the hypothesized value of the population mean. Instead we have $D_0$, the hypothesized difference between the population means. To switch from a lower tail test to an upper tail or two-tailed test, click on $\boxed{\geq}$ and select $\boxed{\leq}$ or $\boxed{=}$, respectively.

Lower Tail Test	Upper Tail Test
$H_0 \colon \mu_1 - \mu_2 \geq D_0$	$H_0 \colon \mu_1 - \mu_2 \leq D_0$
$H_a \colon \mu_1 - \mu_2

Again, hypothesis testing for a single population mean is very similar to hypothesis testing for two population means. For a single population mean, the test statistics is the difference between mu and mu0 dividied by the standard error. For two population means, the test statistic is the difference between $\bar{x}_1 - \bar{x}_2$ and $D_0$ divided by the standard error. The procedure after computing the test statistic is identical to the one population case. That is, you proceed with the p-value approach or critical value approach in the same exact way.

$\sigma$ Known	$\sigma$ Unknown
$ z = \dfrac{(\bar{x}_1 - \bar{x}_2)-D_0}{\sqrt{\dfrac{\sigma_1^2}{n_1}+\dfrac{\sigma_2^2}{n_2}}} $	$ t = \dfrac{(\bar{x}_1 - \bar{x}_2)-D_0}{\sqrt{\dfrac{s_1^2}{n_1}+\dfrac{s_2^2}{n_2}}} $

The calculator above computes confidence intervals and hypothesis tests for the difference between two population means. The simpler version of this is confidence intervals and hypothesis tests for a single population mean. For confidence intervals about a single population mean, visit the Confidence Interval Calculator . For hypothesis tests about a single population mean, visit the Hypothesis Testing Calculator .

Step-by-step guide to hypothesis testing in statistics

Hypothesis testing in statistics helps us use data to make informed decisions. It starts with an assumption or guess about a group or population—something we believe might be true. We then collect sample data to check if there is enough evidence to support or reject that guess. This method is useful in many fields, like science, business, and healthcare, where decisions need to be based on facts.

Learning how to do hypothesis testing in statistics step-by-step can help you better understand data and make smarter choices, even when things are uncertain. This guide will take you through each step, from creating your hypothesis to making sense of the results, so you can see how it works in practical situations.

What is Hypothesis Testing?

Table of Contents

Hypothesis testing is a method for determining whether data supports a certain idea or assumption about a larger group. It starts by making a guess, like an average or a proportion, and then uses a small sample of data to see if that guess seems true or not.

For example, if a company wants to know if its new product is more popular than its old one, it can use hypothesis testing. They start with a statement like “The new product is not more popular than the old one” (this is the null hypothesis) and compare it with “The new product is more popular” (this is the alternative hypothesis). Then, they look at customer feedback to see if there’s enough evidence to reject the first statement and support the second one.

Simply put, hypothesis testing is a way to use data to help make decisions and understand what the data is really telling us, even when we don’t have all the answers.

Importance Of Hypothesis Testing In Decision-Making And Data Analysis

Hypothesis testing is important because it helps us make smart choices and understand data better. Here’s why it’s useful:

Reduces Guesswork : It helps us see if our guesses or ideas are likely correct, even when we don’t have all the details.
Uses Real Data : Instead of just guessing, it checks if our ideas match up with real data, which makes our decisions more reliable.
Avoids Errors : It helps us avoid mistakes by carefully checking if our ideas are right so we don’t make costly errors.
Shows What to Do Next : It tells us if our ideas work or not, helping us decide whether to keep, change, or drop something. For example, a company might test a new ad and decide what to do based on the results.
Confirms Research Findings : It makes sure that research results are accurate and not just random chance so that we can trust the findings.

Here’s a simple guide to understanding hypothesis testing, with an example:

1. Set Up Your Hypotheses

Explanation: Start by defining two statements:

Null Hypothesis (H0): This is the idea that there is no change or effect. It’s what you assume is true.
Alternative Hypothesis (H1): This is what you want to test. It suggests there is a change or effect.

Example: Suppose a company says their new batteries last an average of 500 hours. To check this:

Null Hypothesis (H0): The average battery life is 500 hours.
Alternative Hypothesis (H1): The average battery life is not 500 hours.

2. Choose the Test

Explanation: Pick a statistical test that fits your data and your hypotheses. Different tests are used for various kinds of data.

Example: Since you’re comparing the average battery life, you use a one-sample t-test .

3. Set the Significance Level

Explanation: Decide how much risk you’re willing to take if you make a wrong decision. This is called the significance level, often set at 0.05 or 5%.

Example: You choose a significance level of 0.05, meaning you’re okay with a 5% chance of being wrong.

4. Gather and Analyze Data

Explanation: Collect your data and perform the test. Calculate the test statistic to see how far your sample result is from what you assumed.

Example: You test 30 batteries and find they last an average of 485 hours. You then calculate how this average compares to the claimed 500 hours using the t-test.

5. Find the p-Value

Explanation: The p-value tells you the probability of getting a result as extreme as yours if the null hypothesis is true.

Example: You find a p-value of 0.0001. This means there’s a very small chance (0.01%) of getting an average battery life of 485 hours or less if the true average is 500 hours.

6. Make Your Decision

Explanation: Compare the p-value to your significance level. If the p-value is smaller, you reject the null hypothesis. If it’s larger, you do not reject it.

Example: Since 0.0001 is much less than 0.05, you reject the null hypothesis. This means the data suggests the average battery life is different from 500 hours.

7. Report Your Findings

Explanation: Summarize what the results mean. State whether you rejected the null hypothesis and what that implies.

Example: You conclude that the average battery life is likely different from 500 hours. This suggests the company’s claim might not be accurate.

Hypothesis testing is a way to use data to check if your guesses or assumptions are likely true. By following these steps—setting up your hypotheses, choosing the right test, deciding on a significance level, analyzing your data, finding the p-value, making a decision, and reporting results—you can determine if your data supports or challenges your initial idea.

Understanding Hypothesis Testing: A Simple Explanation

Hypothesis testing is a way to use data to make decisions. Here’s a straightforward guide:

1. What is the Null and Alternative Hypotheses?

Null Hypothesis (H0): This is your starting assumption. It says that nothing has changed or that there is no effect. It’s what you assume to be true until your data shows otherwise. Example: If a company says their batteries last 500 hours, the null hypothesis is: “The average battery life is 500 hours.” This means you think the claim is correct unless you find evidence to prove otherwise.
Alternative Hypothesis (H1): This is what you want to find out. It suggests that there is an effect or a difference. It’s what you are testing to see if it might be true. Example: To test the company’s claim, you might say: “The average battery life is not 500 hours.” This means you think the average battery life might be different from what the company says.

2. One-Tailed vs. Two-Tailed Tests

One-Tailed Test: This test checks for an effect in only one direction. You use it when you’re only interested in finding out if something is either more or less than a specific value. Example: If you think the battery lasts longer than 500 hours, you would use a one-tailed test to see if the battery life is significantly more than 500 hours.
Two-Tailed Test: This test checks for an effect in both directions. Use this when you want to see if something is different from a specific value, whether it’s more or less. Example: If you want to see if the battery life is different from 500 hours, whether it’s more or less, you would use a two-tailed test. This checks for any significant difference, regardless of the direction.

3. Common Misunderstandings

Clarification: Hypothesis testing doesn’t prove that the null hypothesis is true. It just helps you decide if you should reject it. If there isn’t enough evidence against it, you don’t reject it, but that doesn’t mean it’s definitely true.
Clarification: A small p-value shows that your data is unlikely if the null hypothesis is true. It suggests that the alternative hypothesis might be right, but it doesn’t prove the null hypothesis is false.
Clarification: The significance level (alpha) is a set threshold, like 0.05, that helps you decide how much risk you’re willing to take for making a wrong decision. It should be chosen carefully, not randomly.
Clarification: Hypothesis testing helps you make decisions based on data, but it doesn’t guarantee your results are correct. The quality of your data and the right choice of test affect how reliable your results are.

Benefits and Limitations of Hypothesis Testing

Clear Decisions: Hypothesis testing helps you make clear decisions based on data. It shows whether the evidence supports or goes against your initial idea.
Objective Analysis: It relies on data rather than personal opinions, so your decisions are based on facts rather than feelings.
Concrete Numbers: You get specific numbers, like p-values, to understand how strong the evidence is against your idea.
Control Risk: You can set a risk level (alpha level) to manage the chance of making an error, which helps avoid incorrect conclusions.
Widely Used: It can be used in many areas, from science and business to social studies and engineering, making it a versatile tool.

Limitations

Sample Size Matters: The results can be affected by the size of the sample. Small samples might give unreliable results, while large samples might find differences that aren’t meaningful in real life.
Risk of Misinterpretation: A small p-value means the results are unlikely if the null hypothesis is true, but it doesn’t show how important the effect is.
Needs Assumptions: Hypothesis testing requires certain conditions, like data being normally distributed . If these aren’t met, the results might not be accurate.
Simple Decisions: It often results in a basic yes or no decision without giving detailed information about the size or impact of the effect.
Can Be Misused: Sometimes, people misuse hypothesis testing, tweaking data to get a desired result or focusing only on whether the result is statistically significant.
No Absolute Proof: Hypothesis testing doesn’t prove that your hypothesis is true. It only helps you decide if there’s enough evidence to reject the null hypothesis, so the conclusions are based on likelihood, not certainty.

Final Thoughts

Hypothesis testing helps you make decisions based on data. It involves setting up your initial idea, picking a significance level, doing the test, and looking at the results. By following these steps, you can make sure your conclusions are based on solid information, not just guesses.

This approach lets you see if the evidence supports or contradicts your initial idea, helping you make better decisions. But remember that hypothesis testing isn’t perfect. Things like sample size and assumptions can affect the results, so it’s important to be aware of these limitations.

In simple terms, using a step-by-step guide for hypothesis testing is a great way to better understand your data. Follow the steps carefully and keep in mind the method’s limits.

What is the difference between one-tailed and two-tailed tests?

A one-tailed test assesses the probability of the observed data in one direction (either greater than or less than a certain value). In contrast, a two-tailed test looks at both directions (greater than and less than) to detect any significant deviation from the null hypothesis.

How do you choose the appropriate test for hypothesis testing?

The choice of test depends on the type of data you have and the hypotheses you are testing. Common tests include t-tests, chi-square tests, and ANOVA. You get more details about ANOVA, you may read Complete Details on What is ANOVA in Statistics ? It’s important to match the test to the data characteristics and the research question.

What is the role of sample size in hypothesis testing?

Sample size affects the reliability of hypothesis testing. Larger samples provide more reliable estimates and can detect smaller effects, while smaller samples may lead to less accurate results and reduced power.

Can hypothesis testing prove that a hypothesis is true?

Hypothesis testing cannot prove that a hypothesis is true. It can only provide evidence to support or reject the null hypothesis. A result can indicate whether the data is consistent with the null hypothesis or not, but it does not prove the alternative hypothesis with certainty.

How to Find the Best Online Statistics Homework Help

why-spss-homework-help-is-an-important-aspects-for-students

Why SPSS Homework Help Is An Important aspect for Students?

Two Sample T-Test Calculator

Two sample t-test is used to check whether the means of two groups are significantly different from each other. For example, if you want to see if mean weight of males and females have statistically significant difference between them.

Independent T-test assumes that the two samples have equal variances. Welch's t-test is used if you have unequal variances.

Either enter raw data or summary information to calculate two sample t-test. You can directly paste data from MS Excel.

Enter Raw Data

Enter Summary Data

Scores are normally distributed within each of the two groups
Each score is sampled independently and randomly.
Data must be continuous

A Sampling-Based Framework for Hypothesis Testing on Large Attributed Graphs

New Citation Alert added!

This alert has been successfully added and will be sent to:

You will be notified whenever a record that you have chosen has been cited.

To manage your alert preferences, click on the button below.

New Citation Alert!

Please log in to your account

Information & Contributors

Bibliometrics & citations, index terms.

Computing methodologies

Machine learning

Machine learning approaches

Learning in probabilistic graphical models

Information systems

Data management systems

Database design and models

Graph-based database models

Information retrieval

Retrieval models and ranking

Top-k retrieval in databases

Information systems applications

Data mining

Mathematics of computing

Probability and statistics

Probabilistic inference problems

Statistical paradigms

Recommendations

Sampling from large graphs.

Given a huge real graph, how can we derive a representative sample? There are many known algorithms to compute interesting measures (shortest paths, centrality, betweenness, etc.), but several of them become impractical for large graphs. Thus graph ...

Sampling from Large Graphs with a Reservoir

Sampling is a process of choosing a suitable representative subset from a population and uniformity is a basic requirement of representative ness. A sampling process produces a uniform random sample when all possible samples of the same size have the ...

Large Induced Forests in Graphs

In this article, we prove three theorems. The first is that every connected graph of order n and size m has an induced forest of order at least 8n-2m-2/9 with equality if and only if such a graph is obtained from a tree by expanding every vertex to a ...

Information

Published in.

cover image Proceedings of the VLDB Endowment

Beijing Institute of Technology

University of Southern California

VLDB Endowment

Publication History

Check for updates.

Research-article

Contributors

Other metrics, bibliometrics, article metrics.

0 Total Citations
3 Total Downloads
Downloads (Last 12 months) 3
Downloads (Last 6 weeks) 3

View Options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

View options.

View or Download as a PDF file.

View online with eReader .

Share this Publication link

Copying failed.

Share on social media

Affiliations, export citations.

Please download or close your previous search result export first before starting a new bulk export. Preview is not available. By clicking download, a status dialog will open to start the export process. The process may take a few minutes but once it finishes a file will be downloadable from your browser. You may continue to browse the DL while the export process is in progress. Download
Download citation
Copy citation

We are preparing your search results for download ...

We will inform you here when the file is ready.

Your file of search results citations is now ready.

Your search export query has expired. Please try again.

Search term Advanced Search Citation Search
Individual login
Institutional login

How does plant chemodiversity evolve? Testing five hypotheses in one population genetic model

Corresponding Author

Meike J. Wittmann

[email protected]
orcid.org/0000-0002-7209-9172

Faculty of Biology, Theoretical Biology, Bielefeld University, Universitätsstraße 25, 33615 Bielefeld, Germany

Joint Institute for Individualisation in a Changing Environment (JICE), University of Münster and Bielefeld University, 33615 Bielefeld, Germany

Author for correspondence:

Email: [email protected]

Andrea Bräutigam

orcid.org/0000-0002-5309-0527

Faculty of Biology, Computational Biology, Bielefeld University, Universitätsstraße 25, 33615 Bielefeld, Germany

Center for Biotechnology, Bielefeld University, Universitätsstraße 25, 33615 Bielefeld, Germany

Plant chemodiversity, the diversity of plant-specialized metabolites, is an important dimension of biodiversity. However, there are so far few mathematical models to test verbal hypotheses on how chemodiversity evolved. Here, we develop such a model to test predictions of five hypotheses: the ‘fluctuating selection hypothesis’, the ‘dominance reversal hypothesis’, the interaction diversity hypothesis, the synergy hypothesis, and the screening hypothesis.
We build a population genetic model of a plant population attacked by herbivore species whose occurrence fluctuates over time. We study the model using mathematical analysis and individual-based simulations.
As predicted by the ‘dominance reversal hypothesis’, chemodiversity can be maintained if alleles conferring a defense metabolite are dominant with respect to the benefits, but recessive with respect to costs. However, even smaller changes in dominance can maintain polymorphism. Moreover, our results underpin and elaborate predictions of the synergy and interaction diversity hypotheses, and, to the extent that our model can address it, the screening hypotheses. By contrast, we found only partial support for the ‘fluctuating selection hypothesis’.
In summary, we have developed a flexible model and tested various verbal models for the evolution of chemodiversity. Next, more mechanistic models are needed that explicitly consider the organization of metabolic pathways.

Introduction

Plants harbor an amazing diversity of so-called specialized (or secondary) metabolites. These are not involved in basic functions like photosynthesis that are shared by (almost) all plants. Rather, different plant lineages have different sets of specialized metabolites to defend themselves against herbivores, attract pollinators, or fulfill other yet unknown functions (Huang & Dudareva, 2023 ). This ‘chemodiversity’ raises several evolutionary questions: Why are there so many metabolites? And why do individuals differ qualitatively and quantitatively in the metabolites that they produce? That is, what maintains chemical polymorphism in populations?

Potential answers to such questions are offered by a number of hypotheses and verbal models (for reviews, see Moore et al ., 2014 ; Wetzel & Whitehead, 2020 ; Thon et al ., 2024 ). To test whether these verbal models ‘work’, that is whether the claimed predictions follow from the assumptions, and to find out what exact patterns of chemodiversity they predict, mathematical models, and computer simulations are needed (Servedio et al ., 2014 ). Such models exist for genetic variation and for species diversity, but so far, there are few mathematical and simulation models for chemodiversity (reviewed in Thon et al ., 2024 ). These include optimality models that predict investment in defenses based on resource constraints (e.g. Coley et al ., 1985 ; Yamamura & Tsuji, 1995 ; Orrock et al ., 2015 ), evolutionary game theory, and frequency-dependent selection models that explain coexistence between defended and undefended plants based on direct and indirect interactions between plants in neighborhoods (e.g. Augner et al ., 1991 ; Lankau, 2009 ; Sato et al ., 2017 ), simulation models for plant–herbivore coevolution (Speed et al ., 2015 ; Zu et al ., 2020 ; see also Bass & Kessler, 2022 ; Zu et al ., 2021 ), as well as differential-equation models for eco-evolutionary dynamics between plants, herbivores, and pollinators (McPeek et al ., 2022 ). While some models specifically address variation in chemical traits, others more generally address absence or presence of defense traits, but could be also be applied to absence or presence of a single defense metabolite.

However, several important verbal hypotheses still lack underpinning by mathematical models. First, the interaction diversity hypothesis (Iason et al ., 2011 ; Whitehead et al ., 2021 ) posits that plants produce many different metabolites because they engage in many ecological interactions mediated by different metabolites. Second, the synergy hypothesis suggests that plants produce many metabolites because the antiherbivore effects of mixtures are often larger than expected based on adding the effects of individual metabolites (Richards et al ., 2016 ). Third, the screening hypothesis (Jones & Firn, 1991 ; Firn & Jones, 2003 ) suggests that each new metabolite has only a small probability of having biological activity in a relevant ecological interaction, for example against a herbivore, forcing plants to produce a large number of metabolites to ‘find’ the few useful ones. Since the machinery for producing metabolites is costly, the screening hypothesis further predicts that plants should evolve grid-like pathways with promiscuous enzymes that can efficiently produce a large diversity of metabolites.

Furthermore, although much of chemodiversity has a genetic basis, there is a surprising lack of population genetic models for chemodiversity. One powerful mechanism for the maintenance of genetic variation is temporally fluctuating selection where heterozygotes have the highest fitness in the geometric mean over time (called marginal overdominance), although they may not have the highest fitness at any particular time (e.g. see Haldane & Jayakar, 1963 ; Hedrick, 1976 ; Wittmann et al ., 2017 ; Johnson et al ., 2023 ). Since many insect herbivores fluctuate in abundance over time (Root, 1996 ; Stange et al ., 2011 ; De-la-Cruz & Núñez-Farfán, 2023 ), for example due to temperature variation between seasons or across years, fluctuating selection could maintain chemodiversity between individuals in a population. For example, if a metabolite repels generalist herbivores, but attracts specialist herbivores, a genetic polymorphism at an underlying locus could potentially be maintained by fluctuations in the presence of specialist and generalist herbivores (as suggested for the glucosinolate sinigrin by Lankau, 2007 ). However, this ‘fluctuating selection hypothesis’ has not been explored in a mathematical model.

Finally, we know from population genetic theory that the maintenance of polymorphism can strongly depend on patterns of genetic dominance. Dominance quantifies the phenotype or fitness of heterozygotes relative to homozygotes. Consider a locus with two alleles, 1 (presence) and 0 (absence), where ‘11’ homozygotes produce a metabolite, but ‘00’ homozygotes do not. The 1 allele would be called (partially) dominant if ‘10’ heterozygotes produce more than half of what 11 homozygotes produce and recessive if heterozygotes produce less than half. If a locus influences two or more phenotypes (pleiotropy), dominance coefficients can differ between phenotypes (Grieshop et al ., 2024 ). Antagonistic pleiotropy with reversals of dominance across contexts can be a powerful mechanism for the maintenance of genetic variation (Rose, 1982 ; Hoekstra et al ., 1985 ; Curtsinger et al ., 1994 ; Wittmann et al ., 2017 ; Connallon & Chenoweth, 2019 ; Grieshop et al ., 2024 ). For example, in the context of chemodiversity, if the 1 allele confers herbivore resistance and is also dominant for herbivore resistance, but has costs which reduce fecundity and is recessive for fecundity, similar to the case of fluctuating selection, heterozygote advantage (overdominance) could emerge at the level of overall fitness such that polymorphism is maintained. So far, there are no models exploring this ‘dominance reversal hypothesis’ for chemodiversity. From crossing experiments, we know that specialized metabolites and plant defense traits often exhibit dominance (Kondra & Stefansson, 1970 ; Orians, 2000 ; van Dam & Baldwin, 2003 ; Olson-Manning et al ., 2013 ), although to our knowledge, nothing is known on dominance reversals.

To help fill these gaps in chemodiversity modeling, here we develop a population genetic modeling approach focused on the evolution of chemodiversity via presence–absence polymorphisms, that is polymorphisms where some individuals in the population produce a metabolite while others do not. We use this model to explore qualitatively how patterns of chemodiversity are affected by herbivore fluctuations, dominance patterns, herbivore numbers, and synergistic effects of metabolites. For each scenario, we will analytically (if possible) determine the potential for genetic polymorphism and use individual-based simulations to quantify the expected total number of metabolites produced in the population (γ diversity), the average number of metabolites produced per individual (α diversity), and the average number of different metabolites between two individuals (β diversity). Thereby, we perform proof-of-concept tests for the dominance reversal hypothesis, the fluctuating selection hypothesis, the interaction diversity hypothesis and the synergy hypothesis. Additionally, we test whether plant populations with higher chemodiversity are better able to cope with a new herbivore, as suggested by the screening hypothesis.

Model description

We assume a diploid, randomly mating plant population of constant size N (Table 1 for a parameter overview). Each time step corresponds to one plant generation time. Adult plants die with probability θ $$ \theta $$ per time step and there is no seed bank. We focus on the case of nonoverlapping generations θ = 1 , $$ \left(\theta =1\right), $$ but also consider overlapping generations θ < 1 . $$ \left(\theta <1\right). $$ The plant population can be attacked by n h $$ {n}_{\mathrm{h}} $$ herbivores. Time is divided into phases of g $$ g $$ generations. We assume that herbivore i is present and active with probability p i $$ {p}_i $$ independently for each phase and independently from other herbivores. If herbivore fluctuations are at the same time scale as plant generations, for example annually for an annual plant, g = 1 $$ g=1 $$ . If g > 1 $$ g>1 $$ (our default is g = 5 $$ g=5 $$ ), herbivore fluctuations are slower than plant generations, as in annual herbivore fluctuations for a weed with multiple generations per year or changes in the herbivore community every few years for an annual plant.

Parameter	Explanation	Default value
	Population size	500
	Adult death probability per time step	1
	Number of herbivores	1
	Number of generations per phase	5
	Probability that herbivore is present and active during a particular phase	0.2
	Number of loci responsible for metabolite production = maximum possible number of different metabolites	10
	Protective effect of metabolite against herbivore	1
	Dominance for effect on herbivores	Always varied
	Dominance for costs	Always varied
	Baseline probability to escape herbivore	0
	Half-saturation parameter of the benefit function	0
	Sensitivity parameter of the benefit function	1
	Cost parameter	0.02
	Mutation probability per allele copy per generation	0.0001

The plant genome has L loci responsible for the production of metabolites that affect herbivores. They could be either the genes coding for the biosynthetic enzymes or regulatory loci affecting such genes. The L × n h $$ L\times {n}_{\mathrm{h}} $$ matrix M = m li $$ \mathbf{M}=\left({m}_{li}\right) $$ specifies antiherbivore activity of a certain locus l against a certain herbivore species i . If m li > 0 $$ {m}_{li}>0 $$ , the metabolite acts as a repellent for herbivore i , if m li = 0 $$ {m}_{li}=0 $$ , it has no effect, and if m li < 0 $$ {m}_{li}<0 $$ , the metabolite attracts the herbivore. At each locus, there can be two alleles: 0 (absence, that is a deletion or a nonfunctional gene-variant) and 1 (presence). That is, individuals can have three possible genotypes at each locus: 00 or 11 homozygote or 01 heterozygote/hemizygote.

Details are in the caption following the image

Based on the death probability θ $$ \theta $$ , the number of sites F that free up at the end of the time step is drawn from a binomial distribution with parameters N and θ $$ \theta $$ . Individuals then reproduce in proportion to their fitness. That is, for each of the free sites, we draw one mother and one father individual (allowing incidental selfing) with weights proportional to their fitness. Such selection is called ‘soft’ because an individual's absolute fitness, that is number of offspring in the next time step, depends not only on its own trait but via competition for a limited number of places in the population also on the traits of its conspecifics (Bell et al ., 2021 ). Mutations happen with probability u per generation per allele copy and then change an allele into the respective other allele. Finally, F adults are randomly killed to complete the time step.

Mathematical analysis

Next, we investigate whether and when there is scope for genetic polymorphism and thus also for variation in the chemical composition between individuals. For this, we use an invasion analysis similar to that in Wittmann et al . ( 2017 ). Such invasion arguments are also used in game theoretical or frequency-dependent selection models (Augner et al ., 1991 ; Sato et al ., 2017 ), but usually under the explicit or implicit assumption of haploidy and asexual reproduction, whereas here, we consider sexually reproducing diploids and thus have to consider heterozygotes and dominance.

Individual-based simulations

The analytical results only tell us under which conditions the population would be expected to be fully monomorphic or somewhat polymorphic. But it cannot tell us at how many loci polymorphism will arise and how patterns of allele frequencies will look like in the face of random genetic drift. Also, the analytical approach does not work with different effect sizes of the different loci against the different herbivores. To investigate these aspects, we built a stochastic individual-based simulation in C++. For each parameter combination, we ran five replicate simulations for 5000 generations with a population size N = 500 $$ N=500 $$ , g = 5 $$ g=5 $$ generations per phase, a mutation rate of 0.0001, and 10 unlinked loci. Like many population genetic simulations, we work with a relatively small population size and a relatively large mutation rate to keep simulation run times down. Since evolutionary processes often only depend on the product of population size and mutation rates, these settings can approximate larger natural populations with a lower mutation rate (Johnson et al ., 2024 ).

For each parameter combination, we then quantified or estimated (1) the total number of ‘metabolites’, that is how many of the 10 loci have some 1 alleles in the population (γ diversity), (2) the average number of ‘metabolites’ per individual (also between 0 and 10), estimated as ∑ l = 1 L 1 − 1 − π l 2 $$ {\sum}_{l=1}^L1-{\left(1-{\pi}_l\right)}^2 $$ , where π l $$ {\pi}_l $$ is the frequency of the 1 allele at locus l (α diversity), and (3) the average number of ‘metabolites’ that are not shared in a randomly drawn pair of individuals (i.e. one individual has it, the other one does not), estimated as ∑ l = 1 L 2 1 − π l 2 ⋅ 1 − 1 − π l 2 $$ {\sum}_{l=1}^L2{\left(1-{\pi}_l\right)}^2\cdot \left(1-{\left(1-{\pi}_l\right)}^2\right) $$ (β diversity). We evaluated each measure at the end of the simulation run and averaged across the five replicates. To generate null expectations for the diversity measures, we ran 500 replicates of a neutral model with all protective effects m li $$ {m}_{li} $$ as well as the cost parameter c set to zero. We also ran simulations where dominance coefficients were not the same across loci but independently drawn for each locus and independently for activity and costs from a uniform distribution on the interval between 0 and 1.

Testing the fluctuating selection and dominance reversal hypotheses

We first focus on the simplest case of a single herbivore n h = 1 $$ \left({n}_{\mathrm{h}}=1\right) $$ with every locus having the same antiherbivore effect m l , 1 = 1 $$ {m}_{l,1}=1 $$ and address the two population genetic hypotheses, that is the fluctuating selection hypothesis and the dominance reversal hypothesis (Fig. 2 ). To test for the effect of fluctuations, we compared three scenarios. In the fluctuating herbivory scenario (left column of Fig. 2 ), the herbivore was present in 20% of phases and the baseline probability of escaping the herbivore when it is present was low ( b 0 = 0 $$ {b}_0=0 $$ , blue curve in Fig. 1a ). In the constant high herbivory scenario (middle column), we used the same benefit function, but the herbivore was present in every phase, that is there were no fluctuations. In the constant low herbivory scenario (right column), the herbivore was also present in all phases, but the baseline probability to escape the herbivore was higher (green curve in Fig. 1a ) such that the overall herbivory pressure was similar to the fluctuating herbivory scenario. As might be expected, in the fluctuating herbivory scenario and in the constant low herbivory scenario, the optimum number of metabolites based on geometric mean fitness was similar and lower than in the constant high herbivory scenario (upper row of Fig. 2 ).

To test the dominance reversal hypothesis, we varied the dominance coefficients for benefits and costs independently (second row of Fig. 2 ). According to the analytic approach, polymorphism requires the dominance coefficient for activity d a $$ {d}_{\mathrm{a}} $$ to be roughly as high or higher than the dominance coefficient for costs d c $$ {d}_{\mathrm{c}} $$ (black region). Based on the dominance reversal hypothesis, one would have expected that polymorphism is only possible when dominance for activity is above 0.5 and dominance for costs is below 0.5 (red squares). However, the actual coexistence region was larger, while also excluding some parameter combinations with d a > 0.5 $$ {d}_{\mathrm{a}}>0.5 $$ and d c < 0.5 $$ {d}_{\mathrm{c}}<0.5 $$ . Against the expectation from the fluctuating selection hypothesis, the region where polymorphism is possible was only slightly higher in the fluctuating herbivory scenario compared with the two constant herbivory scenarios. A sufficient difference between dominance for activity and dominance for costs could maintain polymorphism even in a constant environment.

The corresponding individual-based simulation results (bottom three rows of Fig. 2 ) are qualitatively consistent with the analytical results. Again, there was little difference between the fluctuating herbivory and constant low herbivory scenario. This result also held up with overlapping generations and under a generalist-specialist trade-off (Supporting Information Notes S1 with Fig. S1 and Table S1 and Notes S2 with Figs S2 , S3 ). The total number of metabolites in the population (third row) and the average number of metabolites per individual (fourth row) were highest in the constant high herbivory scenario. The average number of metabolites not shared between individuals (bottom row) was close to zero in regions where the analytical results predict that polymorphism is not possible, but reached values of four or more in regions where polymorphism is predicted. Roughly in this region, the average number of metabolites in the population and the average number of differences between individuals were also higher than expected under neutrality. The average number of metabolites per individual exceeded neutral expectations only in the constant high herbivory scenario. In regions not conducive to polymorphism, all diversity levels were generally below neutral expectations.

Testing the interaction diversity hypothesis

Next, to test the interaction diversity hypothesis, we ran simulations with one, two, or five herbivores. In addition to the three fluctuation (or not) scenarios, we considered two scenarios for the effect size distributions: In the ‘only repellent’ scenario, each locus had protective effects 0.2, 0.4, 0.6, 0.8, or 1 against each herbivore with probability 0.1 each, and was neutral with probability 0.5. In the ‘repellent + attractive’ scenario, the probabilities for the different protective effect sizes were the same, but each metabolite also had attractive effects of magnitude 0.2, 0.4, 0.6, 0.8, and 1 with probability 0.02 each, and the probability to be neutral was 0.4 (see Fig. S4 for a visualization of the two distributions). The values were independently drawn from these distributions for each combination of locus, herbivore, and replicate. That is, a single metabolite could have different effects on the different herbivores.

The results were qualitatively similar to those in Fig. 2 here and in other scenarios below, with the difference between dominance for activity and dominance for costs as main driver for polymorphism. To better understand the effect of the number of herbivores, we summarized all the results by averaging the three quantities of interest (total metabolites, average number per individual, and average number of differences between individuals) across all dominance parameter combinations. We also obtained SE of the mean based on average chemodiversity measures across dominance combinations for each of the five replicates. Because of the many independent simulations going into each replicate, SE are very small. In addition, we obtained the scope of polymorphism analytically as the % of dominance parameter space for which the analytical calculations predict polymorphism ( c . numbers in Fig. 2d–f ).

As predicted under the interaction diversity hypothesis, the average number of metabolites per individual increased with the number of herbivores, albeit only very weakly under constant low herbivory (Fig. 3b ). The interaction diversity hypothesis does not make a clear prediction for differences between individuals, but our model indicates that the average number of differences between individuals increased with the number of herbivores under all scenarios. The effect of the number of herbivores on the total number of metabolites in the population differed among scenarios: More herbivores led to more metabolites under constant high herbivory, but had little effect under fluctuating herbivory. Unexpectedly, an increasing number of herbivores slightly decreased metabolite numbers under constant low herbivory in the repel + attract scenario. With increasing herbivore numbers, the probability for a metabolite to attract at least one herbivore increases (0.1 for 1 herbivore, 0.19 for 2 herbivores, and 0.41 for 5 herbivores), while the probability that such loci are lost stays roughly constant (0.49, 0.48, 0.52). Thus, with more herbivores, there are fewer metabolites that are unconditionally beneficial. This can lead to a higher loss probability of presence alleles and therefore lower total numbers of metabolites in the population.

Diversity levels were again in most cases lower than under neutrality. Since these diversity levels are averages over the entire dominance parameter space, it seems that the diversity increases in the region conducive to polymorphism are often outweighed by the diversity decreases in the rest of parameter space (Fig. 2 ).

For the case of a single herbivore, we also explored the relationship between antiherbivore effect of a metabolite and final allele frequency of the corresponding presence allele (Fig. S5 ). While metabolites with an attractive or small protective effect were restricted to low allele frequencies, metabolites with increasing protective effect tended to have higher final allele frequencies.

The results were similar when dominance coefficients were not constant across loci but drawn independently from a uniform distribution (Fig. S6 ), presumably because still roughly the same overall proportion of loci had parameters conducive to polymorphism. In the fluctuating herbivory scenario with only repellent effects (the other scenarios yield very similar results), for most loci that are polymorphic (which we define here as having allele frequencies between 0.1 and 0.9), dominance for activity was larger than dominance for costs (Fig. S7 ).

Testing the synergy hypothesis

To explore patterns of chemodiversity under the synergy hypothesis, we focused on a single herbivore and compared different shapes of the benefit function (small plots below Fig. 4 ). Scenarios with a low half-saturation constant a half $$ {a}_{\mathrm{half}} $$ had diminishing returns of antiherbivore protection with increasing number of metabolites. With high half-saturation constant, the benefit of having multiple metabolites was larger than the benefit expected from single metabolite effects over most of the range (synergistic effects). Had we chosen b 0 = 0 $$ {b}_0=0 $$ , like in most other simulations, the scenarios would have differed not only in the shape but more strongly also in the overall protection level against herbivores. Thus, we chose b 0 $$ {b}_0 $$ for each scenario such that individuals with activity 0 always had probability 0.01 to escape the herbivore when it is present.

Both under constant or fluctuating herbivory, the total number of metabolites in the population and the average number of metabolites per individual increased with the half-saturation parameter of the benefit function, that is as the benefit function becomes more synergistic, and also exceeded neutral levels for high half-saturation parameters. By contrast, the average number of differences between individuals was above the neutral level for low half-saturation constants but decreased as the benefit function becomes more synergistic. Fluctuating herbivory and constant low herbivory produced similar chemodiversity patterns, but for strongly synergistic benefit functions, the total number and average number per individual were higher under fluctuating herbivory. Constant high herbivory led to the highest number of metabolites, both in total and per individual, but the fewest differences between individuals. The analytic approximation (Fig. 4d ) did not capture the between-individual chemodiversity well and predicted that the scope for polymorphism is lowest in the constant low herbivory scenario. For the synergistic benefit function, the geometric mean fitness function can be nonmonotonic (see inset), suggesting that the extreme genotype without any metabolites might be noninvasible by all one-step mutants, but could be invasible by mutants differing in multiple positions. In the simulations, the mutation rate seems to be large enough that multiple mutants differing in several positions segregate in the population. Apparently, other genotypes invaded the no-metabolite genotype in many cases where the analytical results suggest it is not possible. Again, the results were similar to those with dominance coefficients drawn from a uniform distribution (Fig. S8 ) and most of the loci with allele frequencies between 0.1 and 0.9 had a larger dominance coefficient for activity than for costs (Fig. S9 ).

Testing the screening hypothesis

A more complex model with metabolic pathways is needed to test most predictions of the screening hypothesis. However, we can test one prediction: Based on the screening hypothesis, plants with more metabolites should have a higher probability of having a metabolite that is active against a newly colonizing herbivore and therefore a higher fitness after such an invasion (Jones & Firn, 1991 ). To test this, we ran simulations under the fluctuating herbivory scenario and with varying dominance parameters, where 1, 2, or 5 herbivores were present until time 5000 and an additional herbivore was introduced at time 5000 and present until the end of the simulation at time 5500. For these simulations, we used the distributions of effect sizes from Fig. S4 and the effect of each metabolite on each herbivore was drawn independently, also for the additional herbivore.

As predicted, plant populations with a higher chemodiversity (independently of whether it was measured as total metabolites produced, average number of metabolites produced per individual, or average number of differences between individuals) maintained higher average fitness in the time period after the invasion of an additional herbivore (Fig. 5 ). This was the case both when metabolites had only repellent effects (top row of Fig. 5 ) and when metabolites had both repellent and attractive effects (bottom row of Fig. 5 ).

Support for and new predictions of the five hypotheses

We have presented a stochastic population genetics modeling approach to qualitatively test various verbal models and hypotheses on the evolution of chemodiversity.

Our analytical and simulation results agree with the dominance reversal hypothesis in that differences in dominance between different traits, here the costs and benefits of defense metabolites, are a key factor for the maintenance of polymorphism. However, the exact conditions did not match with the hypothesis. We had expected polymorphism when dominance for costs is below 0.5 and dominance for activity above 0.5. But polymorphism actually emerged when the dominance for antiherbivore activity was roughly as large or larger than the genetic dominance for the costs of producing metabolites, that is heterozygotes experienced relatively large benefits and relatively low costs of their one copy of the metabolite-producing allele. This led to a high total number of metabolites produced in the population and a high average number of differences between individuals. Beneficial reversal of dominance was not necessary, but also not completely sufficient for the maintenance of polymorphism.

Antagonistic pleiotropy with changes in dominance between traits is known to be a powerful mechanism for the maintenance of genetic polymorphism (Rose, 1982 ; Curtsinger et al ., 1994 ). Although the focus has been mostly on reversals of dominance, a closer examination of the model by Rose ( 1982 ) shows that also in a simpler model with antagonistic pleiotropy at a single locus in a constant environment, smaller quantitative differences in dominance between traits can be sufficient to maintain polymorphism (Notes S3 with Table S2 and Fig. S10 ). Similar observations were made by Van Dooren ( 2006 ) and Brud ( 2023 ). Thus, one should talk less about beneficial dominance reversals and more about beneficial dominance shifts as a factor promoting genetic variation. Until recently, the maintenance of polymorphism by antagonistic pleiotropy with dominance reversals has been seen more as a strange exception rather than as an important mechanism contributing to diversity levels in natural populations (Curtsinger et al ., 1994 ; Hedrick, 1999 ). However, there are more and more empirical examples of dominance reversal across contexts and theoretical models showing how it can easily arise (see recent reviews by Connallon & Chenoweth, 2019 ; Grieshop et al ., 2024 ).

Differences in dominance that according to our model are often sufficient for maintenance of polymorphism, can arise even more easily than a reversal of dominance. Such differences in dominance have been observed for example for loci influencing mandibular morphology in mice (Ehrich et al ., 2003 ) and can arise in branched enzyme pathways with nonlinearities and feedbacks (Keightley & Kacser, 1987 ). In fact, since many traits and fitness components emerge from highly nonlinear processes, there is no particular reason why dominance should be equal for all traits affected by a pleiotropic locus. For metabolic pathways in particular, heterozygotes might be expected to be intermediate with respect to the costs of genes and enzymes (corresponding to a dominance for costs of 0.5 in our model). Flux through the respective pathway is often a concave (diminishing returns) function of enzyme concentration (Kacser & Burns, 1981 ). Thus heterozygotes will be closer in flux to the homozygote with the larger flux (in our model the presence-presence homozygotes) and dominance for activity is expected to be larger than 0.5 (Kacser & Burns, 1981 ). This is the case as long as the enzyme is not saturated with substrate (Wright, 1934 ; Kacser & Burns, 1981 ; Cornish-Bowden, 1987 ). In specialized metabolism, it is indeed expected that most enzymes operate far from saturation. First, flux through a specialized metabolite pathway is frequently controlled at gateway entry points (Olson-Manning et al ., 2013 ; Dixon & Dickinson, 2024 ). Second, compared with enzymes in primary metabolism, enzymes in specialized metabolism have on average less optimized kinetic parameters, suggesting that selection pressure on efficiency in specialized metabolism is relatively low, presumably because of the lower flow through these pathways (Bar-Even et al ., 2011 ; Bar-Even & Salah Tawfik, 2013 ). Thus, it appears plausible that enzymes have not evolved Michaelis constants K M $$ {K}_M $$ low enough to match the physiological substrate concentrations and thus do not operate at saturation, again making dominance for activity likely.

Unfortunately, there are to our knowledge only few studies so far that quantify the phenotypes or fitness of heterozygotes relative to homozygotes at loci underlying plant secondary metabolite variation. Kondra & Stefansson ( 1970 ) crossed two cultivars of Brassica napus , one with very low and one with relatively high concentrations of three glucosinolates. Offspring glucosinolate content was more than the average between the two parents and the results were consistent with an inheritance model with recessive absence alleles. Similarly, heterozygotes for loss-of-function mutations in the aliphatic glucosinolate pathway of Arabidopsis thaliana produced substantially more than half of wild-type levels for most metabolites although gene expression was often close to half or even less, which would be consistent with dominance for costs at or below 0.5 and dominance for antiherbivore activity above 0.5 (Olson-Manning et al ., 2013 ). More such studies are urgently needed to make progress in understanding the evolution of chemodiversity, but so far both theoretical enzyme kinetic considerations and the available empirical data suggest that changes in dominance conducive to high chemodiversity and genetic polymorphism frequently occur in nature. To our knowledge, this is the first time that dominance changes are highlighted as a potential key factor in the evolution of plant chemodiversity.

Contrary to the expectation from the fluctuating selection hypothesis, temporal fluctuations in herbivore presence had surprisingly small effects for chemodiversity in our model. The reason seems to be that selection pressures under fluctuating herbivore occurrence are in the long run similar to scenarios with continuous presence of herbivores but lower herbivory pressure (Figs 2 , 3 ). Had previous studies on the interplay of temporally fluctuating selection and dominance changes (Hedrick, 1976 ; Wittmann et al ., 2017 ) also included such a constant control treatment, they might have found similar maintenance of polymorphism even without fluctuations. However with multiple herbivores or strong synergy, fluctuating herbivore occurrence slightly promoted chemodiversity compared with constant low herbivory both within individuals and within populations, but not between individuals (Figs 3 , 4 ). While we have focused on a scenario where herbivore presence fluctuates on a longer time scale than plant generations ( g = 5 $$ g=5 $$ plant generation before a random shift in herbivore absence or presence), the fact that diversity levels under fluctuating herbivore were very similar to constant low herbivory and that our analytic condition for polymorphism was entirely independent of g suggests that long-term chemodiversity patterns are robust to the details of the fluctuation regime. In a model with antagonistic pleiotropy driven by a fecundity–viability trade-off of flowering time, Brown & Kelly ( 2018 ) similarly found only a small effect of environmental fluctuations on the maintenance of polymorphism. Since many studies on plant chemodiversity speculate about an important contribution of fluctuating selection to maintaining chemodiversity, it is important to recognize that the conditions for this to happen might be rather limited.

As expected from the interaction diversity hypothesis (Whitehead et al ., 2021 ), with increasing number of herbivores, the average number of metabolites per individual increased (Fig. 3 ). Surprisingly, this did not always lead to a higher total number of metabolites in the population, apparently because some of the scenarios with more herbivores also caused a larger probability of loss of rare alleles. Maybe the most interesting result of the simulations with multiple herbivores is that increasing herbivore numbers do not just promote an increase in the number of metabolites per individual, a straightforward prediction, but also lead to more differences in metabolite composition between individuals, a nontrivial prediction that would not be possible without mathematical models such as ours.

Consistent with the synergy hypothesis, our model produced generally larger numbers of metabolites in the population and per individual when metabolites had synergistic effects on protection against herbivores (Fig. 4 ). However, the average number of differences between individuals decreased with increasing synergy. This is also expected since when most individuals have most metabolites, they cannot differ in many metabolites.

Lastly, our simulation results support one of the predictions of the screening hypothesis, namely that more chemodiverse plant populations cope better with the invasion of an additional herbivore with randomly drawn traits. However, this benefit of chemodiversity is not the reason for the evolution of chemodiversity in our model. Chemodiversity evolved in response to the ‘native’ herbivores and if, because of the parameter settings, a higher diversity evolved, this coincidentally also helped to cope with a new herbivore, which is rather self-evident. To tackle the core of the screening hypothesis, more sophisticated models also involving the structure of metabolic pathways are needed.

Limitations and outlook

Here, we have taken a simplified approach where the genotype directly determines whether or not a metabolite is produced. In reality, the link between genotype and chemotype may not be so direct. For example, if metabolites are produced by multi-step enzymatic pathways, a metabolite can only be produced if the enzyme catalyzing its synthesis from its direct precursor is expressed, but also all precursors need to be present and the enzymes necessary to produce them need to be expressed. Thus, metabolites produced in complex pathways do not evolve independently. A logical next step is to develop more mechanistic models that do not take the shortcut from genotype to metabolite profile, but model the underlying proteome and metabolic pathways. Such models could then be used to test other predictions of the screening hypothesis and to model quantitative variation in addition to qualitative presence–absence variation. Such models should also allow for the evolution of gene regulation to test whether plant populations evolve to defend themselves with few highly expressed defenses or many weakly expressed ones. Our analysis of the relationship between metabolite effect strength and final allele frequency (Fig. S5 ) suggests that small-effect defenses tend rare in the population, thereby contributing to the average number of metabolites in the population, but not so much to the other chemodiversity measures. Metabolites with intermediate effect tend to have also intermediate allele frequencies and thus contribute substantially to differences between individuals, while large effect metabolites tend to have the highest frequencies and thus contribute especially to the average number of metabolites per individual.

Second, we have not considered that plants may be affected by the chemical profile and repellent or attractive effects of their neighbors, either through associational resistance or associational susceptibility (Underwood et al ., 2014 ). These phenomena can under some conditions maintain plant defense polymorphisms via negative frequency-dependent selection (Sato, 2018 ). So far, models for associational effects (e.g. Sato et al ., 2017 ) have assumed asexual reproduction where dominance does not have to be considered. Since both frequency dependence from ecological interactions and genetic dominance patterns can be important for the maintenance of chemodiversity, an interesting future direction is to build diploid sexual reproduction models with associational effects that allow us to explore their interaction.

Third, we have assumed that plant population sizes are constant and herbivore pressures are independent of evolution in the plant population. Since eco-evolutionary feedbacks for example with specialist herbivores can be important for the evolution of plant chemodiversity (Agrawal et al ., 2013 ), it would be interesting to include population dynamics of plants and herbivores as well as plant–herbivore coevolution in future models.

Finally, our model allowed us to generate qualitative insights into the effects of ecological and genetic parameters on chemodiversity patterns. Now, empirical quantification of key model parameters such as dominance coefficients and the shape of the cost and benefit functions is needed in order to make quantitative predictions for chemodiversity patterns in natural plant populations.

Acknowledgements

This work was facilitated by the DFG research unit FOR 3000 Ecology and Evolution of Intraspecific Chemodiversity of Plants, project no.: 415496540, WI 4544/2 (to MJW) and BR 4627/2-2 (to AB). We thank Anke Steppuhn and other FOR 3000 members for helpful discussions and Frans Thon and three anonymous reviewers for very constructive comments on the manuscript. Open Access funding enabled and organized by Projekt DEAL.

Competing interests

None declared.

Author contributions

MJW and AB designed the research. MJW developed, programmed, and analyzed the model, and wrote the manuscript, with input from AB.

Open Research

Data availability.

Simulation code and analysis scripts are provided in a zip folder in Dataset S1 .

Supporting Information

Filename

Description

Zip archive, 965.2 KB

Zip folder containing simulation code and analysis scripts.

PDF document, 7.9 MB

Dependence of chemodiversity patterns on dominance coefficients and fluctuations in the presence of one herbivore and with overlapping generations with .

Patterns of chemodiversity in a generalist-specialist scenario and different fluctuation scenarios.

Allele-frequency spectra for generalist-specialist scenarios and different fluctuation scenarios.

Effect size distributions used for simulations with multiple herbivores.

Relationship between the protective effect of a metabolite and its final allele frequency.

Effect of the number of herbivores and the distribution of effect sizes on chemodiversity patterns when the two dominance coefficients are independently drawn from a uniform distribution on [0,1].

Combinations of dominance for activity and dominance for costs for loci that were polymorphic and those that were not for different numbers of herbivores.

Effect of the shape of the benefit function on patterns of chemodiversity in the case of a single herbivore when the two dominance coefficients are independently drawn from a uniform distribution on [0,1].

Combinations of dominance for activity and dominance for costs for loci that were polymorphic and those that were not for different half-saturation constants.

Changes in dominance can maintain polymorphism at a single locus with antagonistic pleiotropy, even without reversal of dominance.

Results for the case with overlapping generations.

Simulations with a generalist-specialist trade-off.

Conditions for maintenance of polymorphism at a single locus with antagonistic pleiotropy in a constant environment.

Average chemodiversity levels across all dominance parameter combinations for different herbivory scenarios and with different adult death probabilities .

Fitness effects in a simple single-locus antagonistic pleiotropy model with two fitness components ( and ) that act either additively or multiplicatively.

Please note: Wiley is not responsible for the content or functionality of any Supporting Information supplied by the authors. Any queries (other than missing material) should be directed to the Central Office.

Please note: The publisher is not responsible for the content or functionality of any supporting information supplied by the authors. Any queries (other than missing content) should be directed to the corresponding author for the article.

Agrawal AA , Johnson MTJ , Hastings AP , Maron JL . 2013 . A field experiment demonstrating plant life-history evolution and its eco-evolutionary feedback to seed predator populations . The American Naturalist 181 : S35 – S45 . 10.1086/666727 PubMed Web of Science® Google Scholar
Augner M , Fagerström T , Tuomi J . 1991 . Competition, defense and games between plants . Behavioral Ecology and Sociobiology 29 : 231 – 234 . 10.1007/BF00163979 Web of Science® Google Scholar
Bar-Even A , Noor E , Savir Y , Liebermeister W , Davidi D , Tawfik DS , Milo R . 2011 . The moderately efficient enzyme: evolutionary and physicochemical trends shaping enzyme parameters . Biochemistry 50 : 4402 – 4410 . 10.1021/bi2002289 CAS PubMed Web of Science® Google Scholar
Bar-Even A , Tawfik DS . 2013 . Engineering specialized metabolic pathways—is there a room for enzyme improvements? Current Opinion in Biotechnology 24 : 310 – 319 . 10.1016/j.copbio.2012.10.006 CAS PubMed Web of Science® Google Scholar
Bass E , Kessler A . 2022 . Comment on “Information arms race explains plant–herbivore chemical communication in ecological communities” . Peer Community Journal 2 : e19. 10.24072/pcjournal.102 Google Scholar
Bell DA , Kovach RP , Robinson ZL , Whiteley AR , Reed TE . 2021 . The ecological causes and consequences of hard and soft selection . Ecology Letters 24 : 1505 – 1521 . 10.1111/ele.13754 PubMed Web of Science® Google Scholar
Brown KE , Kelly JK . 2018 . Antagonistic pleiotropy can maintain fitness variation in annual plants . Journal of Evolutionary Biology 31 : 46 – 56 . 10.1111/jeb.13192 CAS PubMed Web of Science® Google Scholar
Brud E . 2023 . Dominance and the potential for seasonally balanced polymorphism . bioRxiv . doi: 10.1101/2023.11.20.567918 . Google Scholar
Coley PD , Bryant JP , Chapin FS . 1985 . Resource availability and plant antiherbivore defense . Science 230 : 895 – 899 . 10.1126/science.230.4728.895 CAS PubMed Web of Science® Google Scholar
Connallon T , Chenoweth SF . 2019 . Dominance reversals and the maintenance of genetic variation for fitness . PLoS Biology 17 : e3000118. 10.1371/journal.pbio.3000118 PubMed Web of Science® Google Scholar
Cornish-Bowden A . 1987 . Dominance is not inevitable . Journal of Theoretical Biology 125 : 333 – 338 . 10.1016/S0022-5193(87)80065-6 CAS PubMed Web of Science® Google Scholar
Curtsinger JW , Service PM , Prout T . 1994 . Antagonistic pleiotropy, reversal of dominance, and genetic polymorphism . The American Naturalist 144 : 210 – 228 . 10.1086/285671 Web of Science® Google Scholar
van Dam NM , Baldwin IT . 2003 . Heritability of a quantitative and qualitative protease inhibitor polymorphism in Nicotiana attenuata . Plant Biology 5 : 179 – 185 . 10.1055/s-2003-40719 Web of Science® Google Scholar
De-la-Cruz IM , Núñez-Farfán J . 2023 . Inter-annual variation in the abundance of specialist herbivores determines plant resistance in Datura stramonium . Ecology and Evolution 13 : e10794. 10.1002/ece3.10794 PubMed Web of Science® Google Scholar
Dixon RA , Dickinson AJ . 2024 . A century of studying plant secondary metabolism—from “what?” to “where, how, and why?” . Plant Physiology 195 : 48 – 66 . 10.1093/plphys/kiad596 CAS PubMed Google Scholar
Ehrich TH , Vaughn TT , Koreishi SF , Linsey RB , Pletscher LS , Cheverud JM . 2003 . Pleiotropic effects on mandibular morphology I. Developmental morphological integration and differential dominance . Journal of Experimental Zoology 296B : 58 – 79 . 10.1002/jez.b.9 Web of Science® Google Scholar
Firn RD , Jones CG . 2003 . Natural products – a simple model to explain chemical diversity . Natural Product Reports 20 : 382 – 391 . 10.1039/b208815k CAS PubMed Web of Science® Google Scholar
Gillespie JH . 1973 . Polymorphism in random environments . Theoretical Population Biology 4 : 193 – 195 . 10.1016/0040-5809(73)90028-2 Web of Science® Google Scholar
Grieshop K , Ho EK , Kasimatis KR . 2024 . Dominance reversals: the resolution of genetic conflict and maintenance of genetic variation . Proceedings of the Royal Society B: Biological Sciences 291 : 20232816. 10.1098/rspb.2023.2816 CAS PubMed Google Scholar
Haldane JBS , Jayakar SD . 1963 . Polymorphism due to selection of varying direction . Journal of Genetics 58 : 237 – 242 . 10.1007/BF02986143 Web of Science® Google Scholar
Hedrick PW . 1976 . Genetic variation in a heterogeneous environment. II. Temporal heterogeneity and directional selection . Genetics 84 : 145 – 157 . 10.1093/genetics/84.1.145 CAS PubMed Web of Science® Google Scholar
Hedrick PW . 1999 . Antagonistic pleiotropy and genetic polymorphism: a perspective . Heredity 82 : 126 – 133 . 10.1046/j.1365-2540.1999.00440.x Web of Science® Google Scholar
Hoekstra RF , Bijlsma R , Dolman AJ . 1985 . Polymorphism from environmental heterogeneity – models are only robust if the heterozygote is close in fitness to the favored homozygote in each environment . Genetical Research 45 : 299 – 314 . 10.1017/S001667230002228X CAS PubMed Web of Science® Google Scholar
Huang X-Q , Dudareva N . 2023 . Plant specialized metabolism . Current Biology 33 : R473 – R478 . 10.1016/j.cub.2023.01.057 PubMed Google Scholar
Iason GR , O'Reilly-Wapstra JM , Brewer MJ , Summers RW , Moore B . 2011 . Do multiple herbivores maintain chemical diversity of scots pine monoterpenes? Philosophical Transactions of the Royal Society B: Biological Sciences 366 : 1337 – 1345 . 10.1098/rstb.2010.0236 PubMed Web of Science® Google Scholar
Johnson OL , Tobler R , Schmidt JM , Huber CD . 2023 . Fluctuating selection and the determinants of genetic variation . Trends in Genetics 39 : 491 – 504 . 10.1016/j.tig.2023.02.004 CAS PubMed Google Scholar
Johnson OL , Tobler R , Schmidt JM , Huber CD . 2024 . Population genetic simulation: benchmarking frameworks for non-standard models of natural selection . Molecular Ecology Resources 24 : e13930. 10.1111/1755-0998.13930 PubMed Google Scholar
Jones CG , Firn RD . 1991 . On the evolution of plant secondary chemical diversity . Philosophical Transactions of the Royal Society of London. Series B: Biological Sciences 333 : 273 – 280 . 10.1098/rstb.1991.0077 Web of Science® Google Scholar
Kacser H , Burns JA . 1981 . The molecular basis of dominance . Genetics 97 : 639 – 666 . 10.1093/genetics/97.3-4.639 CAS PubMed Web of Science® Google Scholar
Keightley PD , Kacser H . 1987 . Dominance, pleiotropy and metabolic structure . Genetics 117 : 319 – 329 . 10.1093/genetics/117.2.319 CAS PubMed Web of Science® Google Scholar
Kondra Z , Stefansson B . 1970 . Inheritance of the major glucosinolates of rapeseed ( Brassica napus ) meal . Canadian Journal of Plant Science 50 : 643 – 647 . 10.4141/cjps70-122 CAS Web of Science® Google Scholar
Lankau RA . 2007 . Specialist and generalist herbivores exert opposing selection on a chemical defense . New Phytologist 175 : 176 – 184 . 10.1111/j.1469-8137.2007.02090.x PubMed Web of Science® Google Scholar
Lankau RA . 2009 . Genetic variation promotes long-term coexistence of Brassica nigra and its competitors . The American Naturalist 174 : E40 – E53 . 10.1086/600083 PubMed Web of Science® Google Scholar
McPeek SJ , Bronstein JL , McPeek MA . 2022 . Eco-evolutionary feedbacks among pollinators, herbivores, and their plant resources . Evolution 76 : 1287 – 1300 . 10.1111/evo.14492 PubMed Web of Science® Google Scholar
Moore BD , Andrew RL , Kulheim C , Foley WJ . 2014 . Explaining intraspecific diversity in plant secondary metabolites in an ecological context . New Phytologist 201 : 733 – 750 . 10.1111/nph.12526 PubMed Web of Science® Google Scholar
Olson-Manning CF , Lee C-R , Rausher MD , Mitchell-Olds T . 2013 . Evolution of flux control in the glucosinolate pathway in Arabidopsis thaliana . Molecular Biology and Evolution 30 : 14 – 23 . 10.1093/molbev/mss204 CAS PubMed Web of Science® Google Scholar
Orians C . 2000 . The effects of hybridization in plants on secondary chemistry: implications for the ecology and evolution of plant-herbivore interactions . American Journal of Botany 87 : 1749 – 1756 . 10.2307/2656824 CAS PubMed Web of Science® Google Scholar
Orrock JL , Sih A , Ferrari MC , Karban R , Preisser EL , Sheriff MJ , Thaler JS . 2015 . Error management in plant allocation to herbivore defense . Trends in Ecology & Evolution 30 : 441 – 445 . 10.1016/j.tree.2015.06.005 PubMed Web of Science® Google Scholar
Pearse IS , Paul R , Ode PJ . 2018 . Variation in plant defense suppresses herbivore performance . Current Biology 28 : 1981 – 1986 . 10.1016/j.cub.2018.04.070 CAS PubMed Web of Science® Google Scholar
Richards LA , Glassmire AE , Ochsenrider KM , Smilanich AM , Dodson CD , Jeffrey CS , Dyer LA . 2016 . Phytochemical diversity and synergistic effects on herbivores . Phytochemistry Reviews 15 : 1153 – 1166 . 10.1007/s11101-016-9479-8 CAS Web of Science® Google Scholar
Root R . 1996 . Herbivore pressure on goldenrods ( Solidago altissima ): its variation and cumulative effects . Ecology 77 : 1074 – 1087 . 10.2307/2265577 Web of Science® Google Scholar
Rose MR . 1982 . Antagonistic pleiotropy, dominance, and genetic variation . Heredity 48 : 63 – 78 . 10.1038/hdy.1982.7 Web of Science® Google Scholar
Sato Y . 2018 . Associational effects and the maintenance of polymorphism in plant defense against herbivores: review and evidence . Plant Species Biology 33 : 91 – 108 . 10.1111/1442-1984.12201 Web of Science® Google Scholar
Sato Y , Ito K , Kudoh H . 2017 . Optimal foraging by herbivores maintains polymorphism in defence in a natural plant population . Functional Ecology 31 : 2233 – 2243 . 10.1111/1365-2435.12937 Web of Science® Google Scholar
Servedio MR , Brandvain Y , Dhole S , Fitzpatrick CL , Goldberg EE , Stern CA , Van Cleve J , Yeh DJ . 2014 . Not just a theory—the utility of mathematical models in evolutionary biology . PLoS Biology 12 : e1002017. 10.1371/journal.pbio.1002017 PubMed Web of Science® Google Scholar
Speed MP , Fenton A , Jones MG , Ruxton GD , Brockhurst MA . 2015 . Coevolution can explain defensive secondary metabolite diversity in plants . New Phytologist 208 : 1251 – 1263 . 10.1111/nph.13560 PubMed Web of Science® Google Scholar
Stange EE , Ayres MP , Bess JA . 2011 . Concordant population dynamics of Lepidoptera herbivores in a forest ecosystem . Ecography 34 : 772 – 779 . 10.1111/j.1600-0587.2010.06940.x Web of Science® Google Scholar
Strauss SY , Rudgers JA , Lau JA , Irwin RE . 2002 . Direct and ecological costs of resistance to herbivory . Trends in Ecology & Evolution 17 : 278 – 285 . 10.1046/j.0028-646X.2001.00319.x CAS Web of Science® Google Scholar
Thon FM , Müller C , Wittmann MJ . 2024 . The evolution of chemodiversity in plants—from verbal to quantitative models . Ecology Letters 27 : e14365. 10.1111/ele.14365 PubMed Web of Science® Google Scholar
Underwood N , Inouye BD , Hamback PA . 2014 . A conceptual framework for associational effects: when do neighbors matter and how would we know? Quarterly Review of Biology 89 : 1 – 19 . 10.1086/674991 PubMed Web of Science® Google Scholar
Van Dooren TJM . 2006 . Protected polymorphism and evolutionary stability in pleiotropic models with trait-specific dominance . Evolution 60 : 1991 – 2003 . 10.1554/05-259.1 CAS PubMed Web of Science® Google Scholar
Wetzel WC , Kharouba HM , Robinson M , Holyoak M , Karban R . 2016 . Variability in plant nutrients reduces insect herbivore performance . Nature 539 : 425 – 427 . 10.1038/nature20140 CAS PubMed Web of Science® Google Scholar
Wetzel WC , Whitehead SR . 2020 . The many dimensions of phytochemical diversity: linking theory to practice . Ecology Letters 23 : 16 – 32 . 10.1111/ele.13422 PubMed Web of Science® Google Scholar
Whitehead SR , Bass E , Corrigan A , Kessler A , Poveda K . 2021 . Interaction diversity explains the maintenance of phytochemical diversity . Ecology Letters 24 : 1205 – 1214 . 10.1111/ele.13736 PubMed Web of Science® Google Scholar
Wittmann MJ , Bergland AO , Feldman MW , Schmidt PS , Petrov DA . 2017 . Seasonally fluctuating selection can maintain polymorphism at many loci via segregation lift . Proceedings of the National Academy of Sciences, USA 114 : E9932 – E9941 . 10.1073/pnas.1702994114 CAS PubMed Web of Science® Google Scholar
Wright S . 1934 . Physiological and evolutionary theories of dominance . The American Naturalist 68 : 24 – 53 . 10.1086/280521 Web of Science® Google Scholar
Yamamura N , Tsuji N . 1995 . Optimal strategy of plant antiherbivore defense: implications for apparency and resource-availability theories . Ecological Research 10 : 19 – 30 . 10.1007/BF02347652 Web of Science® Google Scholar
Zu P , Boege K , del Val E , Schuman M , Stevenson P , Zaldivar-Riverón A , Saavedra S . 2021 . Response to comment on “Information arms race explains plant-herbivore chemical communication in ecological communities” . EcoEvoRxiv . doi: 10.32942/osf.io/4watk . Google Scholar
Zu P , Boege K , del Val E , Schuman MC , Stevenson PC , Zaldivar-Riverón A , Saavedra S . 2020 . Information arms race explains plant-herbivore chemical communication in ecological communities . Science 368 : 1377 – 1381 . 10.1126/science.aba2965 CAS PubMed Web of Science® Google Scholar

Online Version of Record before inclusion in an issue

Information

Log in to Wiley Online Library

Change password, your password must have 10 characters or more:.

a lower case character,
an upper case character,
a special character

Password Changed Successfully

Your password has been changed

Create a new account

Forgot your password.

Enter your email address below.

Please check your email for instructions on resetting your password. If you do not receive an email within 10 minutes, your email address may not be registered, and you may need to create a new Wiley Online Library account.

Request Username

Can't sign in? Forgot your username?

Enter your email address below and we will send you your username

If the address matches an existing account you will receive an email with instructions to retrieve your username

Open access
Published: 04 September 2024

Hypothesis paper: GDF15 demonstrated promising potential in Cancer diagnosis and correlated with cardiac biomarkers

Xiaohe Hao 1 na1 ,
Zhenyu Zhang 1 na1 ,
Jing Kong 2 na1 ,
Rufei Ma 3 ,
Cuiping Mao 1 ,
Xun Peng 1 ,
Lisheng Liu 1 ,
Chuanxi Zhao 1 ,
Xinkai Mo 1 ,
Meijuan Cai 4 ,
Xiangguo Yu 1 &
Qinghai Lin 1

Cardio-Oncology volume 10 , Article number: 56 ( 2024 ) Cite this article

Metrics details

Cardiovascular toxicity represents a significant adverse consequence of cancer therapies, yet there remains a paucity of effective biomarkers for its timely monitoring and diagnosis. To give a first evidence able to elucidate the role of Growth Differentiation Factor 15 (GDF15) in the context of cancer diagnosis and its specific association with cardiac indicators in cancer patients, thereby testing its potential in predicting the risk of CTRCD (cancer therapy related cardiac dysfunction).

Analysis of differentially expressed genes (DEGs), including GDF15, was performed by utilizing data from the public repositories of the Cancer Genome Atlas (TCGA) and the Gene Expression Omnibus (GEO). Cardiomyopathy is the most common heart disease and its main clinical manifestations, such as heart failure and arrhythmia, are similar to those of CTRCD. Examination of GDF15 expression was conducted in various normal and cancerous tissues or sera, using available database and serum samples. The study further explored the correlation between GDF15 expression and the combined detection of cardiac troponin-T (c-TnT) and N-terminal prohormone of brain natriuretic peptide (NT-proBNP), assessing the combined diagnostic utility of these markers in predicting risk of CTRCD through longitudinal electrocardiograms (ECG).

GDF15 emerged as a significant DEG in both cancer and cardiomyopathy disease models, demonstrating good diagnostic efficacy across multiple cancer types compared to healthy controls. GDF15 levels in cancer patients correlated with the established cardiac biomarkers c-TnT and NT-proBNP. Moreover, higher GDF15 levels correlated with an increased risk of ECG changes in the cancer cohort.

GDF15 demonstrated promising diagnostic potential in cancer identification; higher GDF15, combined with elevated cardiac markers, may play a role in the monitoring and prediction of CTRCD risk.

Introduction

Cancer and cardiovascular disease (CVD), share common risk factors such as obesity, smoking, and diabetes, and exhibit overlap in the signaling pathways that govern both normal cardiovascular physiology and tumor growth [ 1 , 2 ]. The incidence of cardiovascular toxicity during or after cancer treatment has been on the rise, with heart failure (HF) being the most prevalent and severe cardiovascular complication associated with cancer therapy [ 3 ]. This trend may be attributed to improved survival rates among cancer patients, which has led to an increased prevalence of cardiomyopathy associated with aging and changes in immune function. Additionally, the cardiotoxic effects of specific cancer treatments (including chemotherapy, targeted therapy, biological agents, and irradiation) have become more pronounced [ 4 ]. Consequently, there is a pressing need to enhance the prevention, surveillance, and early management of cardiovascular diseases in patients who are at high risk of cardiac dysfunction related to cancer therapeutics throughout their treatment journey [ 5 ]. While cardiac biomarkers such as cardiac troponin-T (c-TnT) and N-terminal prohormone of brain natriuretic peptide (NT-proBNP) have been somewhat effective in guiding the initiation and monitoring of heart-protective therapy in cancer patients, there is a high demand for more sensitive and specific markers [ 6 ].

Growth Differentiation Factor 15 (GDF-15), a member of the transforming growth factor-beta (TGF-β) superfamily, is also known as macrophage inhibitory cytokine-1 (MIC-1) due to its role in inhibiting macrophage secretion of pro-inflammatory factors [ 7 ]. GDF15 is associated with a wide range of biological functions in both physiological and pathological processes, as evidenced by its alternative names [ 8 ]. Under normal conditions, GDF15 expression remains low in various tissues and serum but markedly increases in response to inflammation, tissue damage, and various disease states, including malignant tumors, CVD, diabetes, and obesity, thus acting as a stress response molecule [ 9 , 10 , 11 , 12 , 13 ]. As a diagnostic and prognostic marker for tumors, the expression of GDF15 correlates with the degree of cachexia [ 14 ]. Similarly, as a cardiovascular disease marker, it is closely related to heart failure and myocardial infarction severity [ 15 , 16 ]. However, the potential of GDF15 as a predictive biomarker for CTRCD (cancer therapy related cardiac dysfunction) remains unclear, and its efficacy in assessing and monitoring cardiovascular toxicity during cancer treatment necessitates further experimental validation [ 17 , 18 ].

In the current study, we identified cardiac biomarkers (including GDF15) in cancer patients using TCGA and GEO databases, and confirmed the efficiency of GDF15 in cancer identification by using serum samples, revealing the potential of GDF15 in the monitoring and predicting risk of CTRCD.

Materials and methods

Overall method framework.

Recent cancer statistics indicate that approximately a quarter of all estimated cancer deaths can be attributed to digestive system tumors (DSTs) [ 19 ]. To identify cardiac biomarkers in cancer patients, we analyzed differentially expressed genes (DEGs) common to various DSTs using the TCGA database, supplemented by GEO database data related to cardiomyopathy (Figure S1 ). Among the five extracellular differential molecules identified across the two disease model datasets, GDF15 emerged as the most significant. Subsequent analysis revealed widespread expression of GDF15 across various normal and tumor tissues. Serum samples from 30 healthy donors and 507 cancer patients indicated that GDF15 is highly expressed in nearly all tumors, signifying significant diagnostic efficacy. Moreover, GDF15 serum levels showed a significant correlation with the cardiac markers NT-proBNP and c-TnT, particularly in cases involving acute heart failure and myocardial injury (Figure S2 ). An evaluation of electrocardiogram (ECG) results for cancer patients with varying GDF15 expression levels revealed a higher incidence of arrhythmic (e.g., sinus bradycardia, sinus tachycardia) and ischemic (e.g., ST changes, T-wave alterations) conditions among patients with elevated GDF15 levels, whereas patients with lower expression levels frequently exhibited normal ECG results.

Patients and healthy donors

This study enrolled a cohort comprising 30 healthy donors and 507 cancer patients treated at the Shandong Cancer Hospital and Institute from January 2022 to June 2023. The healthy donors consisted of individuals undergoing routine physical examinations, all of whom underwent serum GDF15 testing. Of these, five cases were also assessed for both c-TnT and NT-proBNP. The cancer patient group encompassed various malignancies, with 450 patients completing the full spectrum of tests for GDF15, cTnT, and NT-proBNP, and 57 liver cancer patients undergoing testing for GDF15 expression only. Comprehensive clinical data for the 450 patients across various tumor types (Table S1 ) were extracted from electronic medical records and the Ruimei Laboratory Information System version 6.0 (rmlis, Huangpu District, Shanghai, China), as summarized in Table 1 . Informed consent was obtained from all participants prior to the study, with ethical approval granted by the Ethics Committee of the Shandong Cancer Hospital and Institute, aligning with the Declaration of Helsinki.

Data collection

Data for this research were sourced from electronic medical records and the Ruimei Laboratory Information System. The c-TnT and NT-proBNP levels in cancer patients were determined using Electrochemiluminescence on the Cobas e801 analyzer (Roche Diagnostics, GmbH, Mannheim, Germany). The normal range for NT-proBNP was considered to be 0-125 pg/ml, with < 125 pg/ml exclusionary of chronic heart failure and < 300 pg/ml exclusionary of acute heart failure. For the diagnosis of acute heart failure using NT-proBNP levels, the criteria vary by age: for individuals younger than 50 years, a level greater than 450 pg/ml is indicative; for those aged between 50 and 75 years, the threshold is above 900 pg/ml; and for those over 75 years, a level exceeding 1800 pg/ml is suggestive of acute heart failure. Additionally, for patients with a glomerular filtration rate (GFR) below 60 ml/min, a NT-proBNP level greater than 1200 pg/ml is indicative. Regarding cardiac troponin-T (c-TnT), normal levels range from 0 to 14 pg/ml. Levels between 15 and 52 pg/ml suggest myocardial injury, while levels above 52 pg/ml are indicative of acute myocardial injury. The delineation of test range and criteria, which are diagnostic criteria based on many previous studies and the long-term data accumulation of Roch company in hospital detection operation [ 20 , 21 ].

Enzyme‑linked immunosorbent assay (ELISA)

Plasma samples were collected, centrifuged again at 3500 ×g for 10 min to eliminate hemocytes, including red blood cells, white blood cells, and platelets, and the supernatant was retained. The plasma levels of GDF15 from cancer patients and healthy donors were quantified using Human GDF-15 ELISA Kits (ab155432, Abcam, Cambridge, UK). Samples underwent a 10-fold dilution prior to testing, with subsequent procedures conducted as per the kit’s protocol.

Database and processing

Analysis of differentially expressed genes (DEGs) in colon adenocarcinoma (COAD), esophageal carcinoma (ESCA), liver hepatocellular carcinoma (LIHC), stomach adenocarcinoma (STAD), lung adenocarcinoma (LUAD) tissues, and adjacent tissues was conducted using the GEPIA2 online database ( http://gepia2.cancer-pku.cn/#dataset ). Detailed clinical data of cancer patients were obtained from The Cancer Genome Atlas (TCGA) database.

Additionally, public gene expression profiles (GSE116250) covering 14 non-failing donors (NF), 37 dilated cardiomyopathy (DCM), and 13 ischemic cardiomyopathy (ICM) were examined. DEGs identification and visualization between NF, DCM and ICM were executed through volcano plots and heatmaps. Extracellular gene analysis for protein subcellular localization utilized Hum-mPLoc 3.0.

ECG analysis

ECG data, collected on the day of or within one day before or after blood sampling, were recorded with 12-lead ECG devices and interpreted by a minimum of two cardiologists. The analysis included normal ECG readings and identification of arrhythmic (e.g., sinus bradycardia, sinus tachycardia, incomplete right bundle branch block, intraventricular block), ischemic (e.g., ST changes, T-wave alterations), and non-specific (e.g., low-voltage QRS, QT interval variations) findings.

Statistical analysis

Statistical analysis was performed using GraphPad Prism version 9.0 (GraphPad Software, CA, USA). Data were analyzed using the unpaired t-test for two groups with normal distribution, the Mann-Whitney test for non-normally distributed data, the Kruskal-Wallis test for multiple groups with non-normally distributed data, and one-way ANOVA for normally distributed datasets. Results are presented as mean ± standard deviation (SD), with a p-value of < 0.05 considered statistically significant.

Screening of DEGs in TCGA database

Given the high incidence and mortality associated with digestive system tumors (DSTs), we sourced data for colon adenocarcinoma (COAD), esophageal carcinoma (ESCA), liver hepatocellular carcinoma (LIHC), and stomach adenocarcinoma (STAD) from the TCGA database, comprising a total of 1234 cancerous and 1006 non-cancerous tissue samples. The data included colon adenocarcinoma (COAD; 275 cancer vs. 349 control tissues), esophageal carcinoma (ESCA; 182 cancer vs. 286 control tissues), liver hepatocellular carcinoma (LIHC; 369 cancer vs. 160 control tissues), and stomach adenocarcinoma (STAD; 408 cancer vs. 211 control tissues), as detailed in Table S2 . DEGs were determined based on a fold change > 2.0 and a P-value < 0.05, and their distribution was illustrated in volcano plots (Fig. 1 A-D, Tables S3 - 6 ). An intersection of DEGs across these DSTs revealed 381 common genes (Fig. 1 E, Tables S7 - 10 ), with LIHC-specific DEGs showcased in a heatmap (Fig. 1 F). Given the need for biomarkers detectable in serum, we conducted a sub-localization analysis using Hum-mPLoc 3.0, identifying 48 extracellularly localized DEGs (Tables S11 - 14 ), among which GDF15 was highlighted as a significant finding (Fig. 1 G).

DEGs identification in the TCGA database. Volcano plots comparing the expression fold-change of DEGs in COAD tissues (A) , ESCA tissues (B) , LIHC tissues (C) , STAD tissues (D) compared with adjacent normal tissues. (E) Venn plot showing the shared genes among the DEGs in 4 kinds of DST tissues vs. healthy control tissues, and displayed in a heatmap ( F , up-regulated marked in red or down-regulated marked in blue). (G) Shared DEGs that extracellular localized were screened and presented by heatmap (LIHC vs. normal tissues), and GDF15 was among them

Identification of DEGs clusters attributed to cardiomyopathy and DSTs patients

Integrating cardiomyopathy data from GSE116250, including 14 non-failing donors (NF), 37 dilated cardiomyopathy (DCM), and 13 ischemic cardiomyopathy (ICM) cases, we screened for DEGs with fold changes > 2.0 and P < 0.05, as depicted in volcano maps (Fig. 2 A-B, Tables S15 - 16 ). This analysis identified 629 DEGs common between the two cardiomyopathy types (Fig. 2 C). A further screen of serum biomarkers using Hum-mPLoc 3.0 revealed 105 extracellular DEGs, as displayed in heatmaps (Fig. 2 D-E, Tables S17 - 18 ). A Venn diagram pinpointed 19 DEGs shared between 4 types of DST and 2 types cardiomyopathy patients, with 5 extracellular molecules identified for potential clinical blood detection, most notably GDF15 (Fig. 3 B-C, Tables S19 - 22 ). The differential expression of GDF15 in lung adenocarcinoma (LUAD)/adjacent tissues was not as pronounced as in DSTs (Fig. 3 D).

Screening of DEG clusters attributed to cardiomyopathy. Volcano plots comparing the expression fold-change of DEGs in dilated cardiomyopathy (DCM) vs. non-failing donors (NF) (A) , and ischemic cardiomyopathy (ICM) vs. NF (B) . (C) Venn plot showing the shared genes among the DEGs in DCM and ICM vs. NF, and the shared extracellular localized DEGs, presented in heatmaps ( D , E ); these include GDF15

Identification of DEG clusters attributed to cardiomyopathy and DST patients. Venn plot showing DEGs shared among DST patients (A) , from which five differential extracellular molecules were identified (B) and are depicted in a heatmap (C) . Additionally, panel (D) displays GDF15 expression levels in various tumor tissues compared to control tissues, as recorded in the TCGA database. Statistical data are expressed as mean ± standard deviation (SD), with significance determined by an unpaired two-tailed t-test: * indicates p < 0.05

GDF15 as a diagnostic biomarker for cancer patients

In pursuit of clinical insights on GDF15, we utilized integrated databases such as the Protein Atlas ( https://www.proteinatlas.org/ENSG00000130513-GDF15 ) to examine its expression across 44 diverse tissues. This analysis, underpinned by knowledge-based annotations, utilized color-coding to demarcate tissue groups sharing functional similarities. Notably, GDF15 exhibited significant spatial expression specificity, prominently within most digestive system tissues, albeit with notable exceptions in the liver and esophagus. Immunohistochemical assays revealed distinct cytoplasmic staining patterns in colorectal and prostate cancers, among others, demonstrating the presence of GDF15. In contrast, lung cancer and certain other tumors displayed minimal to no GDF15 staining, a finding that aligns with previous analyses from the TCGA database (Fig. 4 A-B). We utilized the proximity extension assay (PEA) to measure plasma concentrations of GDF15 across various cancer types. Notably, the serum levels of GDF15 in different cancer types did not always align with the patterns observed in our tissue staining (Fig. 4 C). Serum samples from 30 healthy donors and 507 cancer patients representing a diverse array of cancers were analyzed to ascertain GDF15 concentrations (Table 1 ). In the healthy cohort ( N = 30), the mean GDF15 concentration was 760.5 ± 60.30 pg/mL. In cancer patients ( N = 507), however, GDF15 levels were markedly elevated, with mean concentrations ranging from 1710 ± 656.2 to 4162 ± 214.8 pg/mL (Fig. 4 D). Earlier observations had indicated that normal liver tissues exhibited minimal or no GDF15 staining, whereas liver cancer tissues and serum samples displayed significantly higher GDF15 levels, the highest among all examined cancer types. Similarly, GDF15 staining was either weak or absent in both healthy and cancerous lung tissues, yet serum levels of GDF15 were considerably increased in lung cancer patients.

Expression profile of GDF15 in human normal / tumor tissues and blood. The expression of GDF15 provided by available analysis platforms in normal human tissues (A) against tumor tissues (B) . Panel (C) shows the serum GDF15 levels in patients with various tumors, as reported in an online database, while panel (D) illustrates the concentrations of GDF15 that we determined in the serum of cancer patients in comparison to healthy controls. The data are expressed as mean ± standard deviation (SD), with statistical analysis conducted via an unpaired two-tailed t-test, where * signifies p < 0.05, *** signifies p < 0.001, and **** signifies p < 0.0001

To evaluate the diagnostic utility of GDF15 for cancer, we compared serum GDF15 levels between representative cancer patient groups and healthy controls, employing receiver operating characteristic (ROC) curves. GDF15 demonstrated significant discriminative ability for pan-cancer detection, achieving an area under the curve (AUC) of 0.91 (95% confidence interval [CI] 0.8719–0.9396), with a sensitivity of 84.02% and a specificity of 86.67% compared to healthy controls (Fig. 5 A). The diagnostic performance of GDF15 varied across the various cancers, exhibiting particularly strong diagnostic efficacy in liver cancer, with an AUC of 0.99 (95% confidence interval [CI] 0.9731–1.001), 92.86% sensitivity, and 96.67% specificity (Fig. 5 B-F). However, in breast cancer, the diagnostic value of GDF15 was notably lower, evidenced by an AUC of 0.51 ( p = 0.9578), with 30.77% sensitivity and 96.67% specificity (Fig. 5 E), possibly due to the limited number of breast cancer cases ( N = 13) included in the study.

Diagnostic utility of GDF15 in various cancers. (A) ROC curves revealed the AUC of pan-cancer to be 0.9057, P < 0.0001. ROC curves for GDF15 in the diagnosis of lung cancer (B) , liver cancer (C) , esophageal cancer (D) , breast cancer (E) , and lymphoma (F)

Serum GDF15 levels in cancer patients correlated with cardiac indicators

Cancer therapy related cardiovascular toxicity includes myocardial injury and heart failure, immune myocarditis, hypertension, arrhythmia, coronary heart disease, venous thromboembolism, dyslipidemia. The correlation of these markers with cardiac dysfunction, as detailed in the Methods and Materials section, varies across the spectrum of potential values. Despite most c-TnT and NT-proBNP values falling within or near the normal range, with outliers presenting scattered results, it was challenging to deduce a straightforward relationship between GDF15 levels and these cardiac markers using a simple correlation analysis approach. Thus, we observed GDF15 expression across various degrees of cardiac disease in a pan-cancer context. We discovered that as c-TnT and NT-proBNP levels rose, indicating worsening cardiac disease, GDF15 expression also significantly increased (Fig. 6 A-B). The mean concentration of GDF15 in individuals with NT-proBNP within the normal range (< 125 pg/ml, N = 287) was approximately 2507 ± 110.4 pg/mL. This included individuals with NT-proBNP levels below 10 pg/ml ( N = 35), where GDF15 levels averaged about 1523 ± 184.4 pg/mL. When NT-proBNP levels suggested chronic heart failure (> 125 & <300 pg/ml, N = 88), GDF15 concentrations averaged around 3044 ± 226.0 pg/mL, P = 0.0230. For NT-proBNP levels indicating potential acute heart failure (> 450 pg/ml, N = 55), GDF15 levels rose to an average of 4850 ± 306.7 pg/mL, P < 0.0001. Notably, in cases with NT-proBNP exceeding 1500 pg/ml ( N = 17), GDF15 levels reached a particularly high average of 5933 ± 395.4 pg/mL, P < 0.0001 (Fig. 6 A).

Association Between Serum GDF15 Levels and Cardiac Biomarkers in Cancer Patients. This figure displays the correlation of serum GDF15 levels with cardiac biomarkers across all cancer types, with panels (A) and (B) illustrating GDF15 levels across varying NT-proBNP and c-TnT ranges, respectively. Panels (C) and (D) detail GDF15 levels in lung cancer patients across different NT-proBNP ranges, while panels (E) and (F) depict these levels in liver cancer patients across varying c-TnT ranges. Data represent mean ± standard error of the mean (SEM), with normal ranges for c-TnT and NT-proBNP serving as controls. N: represented the number of cases. Statistical significance was assessed using an unpaired two-tailed t-test, where * indicates p < 0.05, ** indicates p < 0.01, and **** indicates P < 0.0001

When evaluating c-TnT levels, individuals within the normal range (< 14 pg/ml, N = 366) had a mean GDF15 concentration of 2615 ± 101.1 pg/mL. This included a subset ( N = 13) with c-TnT levels below 3 pg/ml, where GDF15 averaged approximately 1283 ± 277.5 pg/mL. For patients with c-TnT levels indicative of myocardial injury (15–52 pg/ml, N = 82), GDF15 levels averaged around 4303 ± 242.2 pg/mL, P < 0.0001. In cases of acute myocardial injury (c-TnT > 52 pg/ml, N = 4), GDF15 concentrations escalated to an average of 5789 ± 1151 pg/mL, P = 0.0012 (Fig. 6 B).

Further analysis was conducted on lung and liver cancer patients, representing the largest subsets in this study, to more closely explore the correlation between GDF15 expression and cardiac biomarkers. Given that most c-TnT or NT-proBNP values were within or near normal ranges, with a limited number of cases showing abnormal expression, the correlation findings for these specific cancer types were not as pronounced as those observed in the broader cancer cohort. GDF15 levels were only significantly correlated with acute heart failure (NT-proBNP > 450 pg/ml) in lung cancer patients (Fig. 6 C-D). In instances of myocardial injury (c-TnT > 14 pg/ml), GDF15 expression markedly increased in both lung and liver cancer (Fig. 6 E-F).

The levels of serum GDF15 associated with ECG changes

Electrocardiography (ECG) has historically been used for screening and monitoring of cardiovascular toxicity caused by cancer treatments [ 22 ], however it could not be completed on all patients at the times of blood collection due to various circumstances. For instance, the patient with the highest serum level of GDF15 (7493.66 pg/ml) died the day following blood collection, leaving no ECG data. Consequently, we selected a subset of 26 patients, characterized by either relatively high or low serum GDF15 levels, for analysis and comparison of ECG results, as summarized in Table 2 . The ECG findings of six representative cases are illustrated in Fig. 7 . This analysis revealed that patients with lower serum levels of GDF15 generally had low levels of c-TnT or NT-proBNP, correlating with predominantly normal ECG outcomes (Fig. 7 A-B). Conversely, when serum GDF15 exceeded 600 pg/ml, ECGs exhibited sinus rhythm and ST changes, despite c-TnT and NT-proBNP values remaining within normal limits (Table 2 ). Moreover, patients presenting with higher GDF15 expression were found to have elevated levels of c-TnT or NT-proBNP, alongside arrhythmic alterations (such as sinus bradycardia and tachycardia) and significant ischemic changes (including ST changes and T-wave alterations), suggesting a notable correlation among these biomarkers (Fig. 7 C-D; Table 2 ). One particular case involved a patient with a high GDF15 expression (6637.38 pg/ml) whose serum levels of c-TnT (6.84 pg/ml) and NT-proBNP (16.57 pg/ml) were within normal ranges, yet the ECG revealed ST depression in leads III and avF, along with T wave inversion among other abnormal alterations (Fig. 7 F; Table 2 ). Another significant observation was made in a patient with an exceptionally high serum GDF15 level (7493.66 pg/ml); while the c-TnT (23.03 pg/ml) and NT-proBNP (277.70 pg/ml) levels were not markedly elevated, the ECG indicated serious myocardial ischemia, as evidenced by ST segment elevation in leads II, III, avF, V5-V6 (Fig. 7 E).

ECG Outcomes Related to Serum GDF15 Concentrations in Cancer Patients. This figure shows ECG findings in cancer patients, categorized by serum GDF15 levels. Panels (A) and (B) demonstrate patients with low GDF15 levels and corresponding low c-TnT and NT-proBNP levels, resulting in normal ECG outcomes. In contrast, panels ( C - F ) present cases with elevated GDF15 levels alongside more complex c-TnT and NT-proBNP readings, showing a heightened incidence of ECG abnormalities. All ECG recordings were conducted at a standard speed of 25 mm/s

Cardiovascular disease and cancer are the two leading causes of death worldwide [ 23 ]. Accumulating clinical evidence demonstrates the increased risk of developing cardiac disease during cancer treatment. Cancer incidence and mortality were significantly increased in patients with heart failure [ 24 , 25 ]. Addressing the cardiotoxic effects associated with anti-cancer therapies represents a formidable challenge currently confronting cardiologists and oncologists [ 26 ]. This underscores the need for a reliable serum biomarker for monitoring cardiovascular toxicity during cancer treatment. Research indicates that elevated GDF15 levels are linked to a spectrum of cardiovascular conditions, including myocardial hypertrophy, heart failure, atherosclerosis, and endothelial dysfunction. Moreover, GDF15 has been shown to precipitate cachexia and provide protection against obesity and insulin resistance in murine models [ 9 , 27 ]. Notably, cardiovascular diseases, heart failure, and organ failure emerge as the prevalent clinical manifestations of cachexia induced by malignant tumors [ 28 ].

In this study, we elucidated the role of DEGs across various DSTs utilizing the TCGA database, highlighting 48 secreted proteins as potential serum biomarkers. Subsequent analyses, leveraging the GEO database, allowed us to delve into DEGs pertinent to cardiomyopathy. This dual-disease model approach culminated in the identification of five extracellular molecules, with GDF15 emerging as a notably significant biomarker for clinical detection.

Despite the variability observed in GDF15 immunohistochemical staining across normal and cancerous tissues, serum levels of GDF15 in patients with tumors significantly exceeded those in healthy controls. This disparity might be attributed to the limited sample size of healthy individuals ( n = 30) and the complexity surrounding the treatment histories of cancer patients, potentially influencing GDF15 expression. Nonetheless, GDF15 demonstrated robust diagnostic utility across a spectrum of cancers, particularly standing out in DSTs where its levels were markedly elevated, aligning with findings from prior research [ 29 ].

Our results suggested an aberrant expression of GDF15 in both cancerous conditions and cardiomyopathy, posing the question of its utility as a marker for cardiac dysfunction induced by cancer treatments. While existing studies suggest GDF15’s potential as a predictive marker for CTRCD [ 18 , 30 ], our analysis extends this narrative by showing a correlation between increased GDF15 levels and enhanced cardiac markers across a pan-cancer cohort. Notably, patients exhibiting the highest GDF15 serum levels also displayed elevated cardiac dysfunction markers but succumbed shortly after testing, underscoring the marker’s prognostic significance. Specifically, the patient with the highest serum GDF15 concentration (7937.295 pg/ml) presented with significantly high levels of c-TnT (342.40 pg/ml) and NT-proBNP (4348.00 pg/ml), yet died merely two days following testing. Similarly, another patient, registering extremely high GDF15 levels (7804.815 pg/ml) alongside a history of premature cardiac beats and serum levels of c-TnT (15.24 pg/ml) and NT-proBNP (1265.00 pg/ml), succumbed within a month following discharge. Through medical records review, it was found that these two patients with extremely high level of GDF15 were terminal-stage (stage IV) patients with multiple organ failure caused by bone metastasis and cachexia, which may be an important cause of their death. In contrast, when analyzing lung and liver cancer patients independently, the link between GDF15 levels and cardiac biomarkers appeared to be less pronounced. This reduced correlation suggests that the majority of these cancer patients may not exhibit pronounced cardiomyopathy post-treatment, with most cardiac indicators values falling into the normal range, except for few samples showing aberrant expressions, thereby influencing the overall statistical outcomes.

ECG has historically been critical for the diagnosis and management of cardiac injury, cardiomyopathy, and cardiovascular toxicity [ 22 , 31 , 32 ]. In this study, we observed the ECG patterns in patients exhibiting varying levels of GDF15 expression. Generally, we noted a correlation where elevated GDF15 levels were associated with increased c-TnT and NT-proBNP levels, alongside more pronounced arrhythmic alterations (such as sinus bradycardia) and ischemic changes (including ST segment and T-wave variations). However, there were instances where GDF15 levels were markedly high, while c-TnT and NT-proBNP levels remained within normal limits or did not show significant elevation, yet the ECG demonstrated substantial changes.

It is important to acknowledge certain limitations: not every patient had their ECG recorded in close proximity to the blood collection time, leading to disparate data without a temporally-proximal correlation. Echocardiography is common monitoring methods of CTRCD according to the Guidelines of ESC [ 33 ]. Artificial intelligence electrocardiogram served as a screening tool to detect a newly abnormal LVEF (left ventricular ejection fraction) after anthracycline-based cancer therapy [ 34 ] and lack of LVEF evaluation is indeed a limitation of this study. GDF15 has been proved to be a keen pro-inflammatory factor in many studies, and both cardiac disease and cancer have links to inflammation. The results of GDF15 and cardiac biomarkers may be the case, but whether the results can be applied accurately and specifically is certainly worth discussing. In addition, cancer subtypes and comorbidities may affect the expression of GDF15.

In conclusion, this hypothesis generating study attempts to reinforce the association between GDF15 expression and both cancer and cardiac damage after chemotherapy, underscoring its diagnostic efficacy in cancer patients and its potential in monitoring cardiovascular toxicity. While GDF15 levels generally correlate with traditional cardiac markers, the study reveals instances of discordance, suggesting a complementary role for GDF15 in the complex landscape of cancer treatment-related cardiac care.

Data availability

No datasets were generated or analysed during the current study.

Bertero E, Canepa M, Maack C, Ameri P. Linking heart failure to Cancer: background evidence and research perspectives. Circulation. 2018;138:735–42.

Article PubMed Google Scholar

de Wit S, Glen C, de Boer RA, Lang NN. Mechanisms shared between cancer, heart failure, and targeted anti-cancer therapies. Cardiovasc Res. 2023;118:3451–66.

Otto CM. Heartbeat: heart failure induced by cancer therapy. Heart. 2019;105:1–3.

Zamorano JL, Lancellotti P, Rodriguez MD, Aboyans V, Asteggiano R, Galderisi M, et al. 2016 ESC position paper on cancer treatments and cardiovascular toxicity developed under the auspices of the ESC Committee for Practice Guidelines: the Task Force for cancer treatments and cardiovascular toxicity of the European Society of Cardiology (ESC). Eur J Heart Fail. 2017;19:9–42.

Finet JE. Management of Heart failure in Cancer patients and Cancer survivors. Heart Fail Clin. 2017;13:253–88.

Pudil R, Mueller C, Celutkiene J, Henriksen PA, Lenihan D, Dent S, et al. Role of serum biomarkers in cancer patients receiving cardiotoxic cancer therapies: a position statement from the Cardio-Oncology Study Group of the Heart Failure Association and the Cardio-Oncology Council of the European Society of Cardiology. Eur J Heart Fail. 2020;22:1966–83.

Article CAS PubMed Google Scholar

Bootcov MR, Bauskin AR, Valenzuela SM, Moore AG, Bansal M, He XY, et al. MIC-1, a novel macrophage inhibitory cytokine, is a divergent member of the TGF-beta superfamily. Proc Natl Acad Sci U S A. 1997;94:11514–9.

Article CAS PubMed PubMed Central Google Scholar

Assadi A, Zahabi A, Hart RA. .GDF15, an update of the physiological and pathological roles it plays: a review. Pflugers Arch. 2020;472:1535–46.

Wang D, Day EA, Townsend LK, Djordjevic D, Jorgensen SB, Steinberg GR. 2021.GDF15: emerging biology and therapeutic applications for obesity and cardiometabolic disease. Nat Rev Endocrinol 17: 592–607.

Asrih M, Wei S, Nguyen TT, Yi HS, Ryu D, Gariani K. Overview of growth differentiation factor 15 in metabolic syndrome. J Cell Mol Med. 2023;27:1157–67.

Breit SN, Brown DA, Tsai V. .GDF15 analogs as obesity therapeutics. Cell Metab. 2023;35:227–8.

Sjoberg KA, Sigvardsen CM, Alvarado-Diaz A, Andersen NR, Larance M, Seeley RJ et al. 2023.GDF15 increases insulin action in the liver and adipose tissue via a beta-adrenergic receptor-mediated mechanism. Cell Metab 35: 1327–e13405.

Wang D, Townsend LK, DesOrmeaux GJ, Frangos SM, Batchuluun B, Dumont L et al. 2023.GDF15 promotes weight loss by enhancing energy expenditure in muscle. Nature 619: 143–50.

Ling T, Zhang J, Ding F, Ma L. Role of growth differentiation factor 15 in cancer cachexia (review). Oncol Lett. 2023;26:462.

Wollert KC, Kempf T, Wallentin L. 2017.Growth differentiation factor 15 as a Biomarker in Cardiovascular Disease. Clin Chem 63: 140–51.

Sawalha K, Norgard NB, Drees BM, Lopez-Candales A. Growth differentiation factor 15 (GDF-15), a new biomarker in heart failure management. Curr Heart Fail Rep. 2023;20:287–99.

Yu LR, Cao Z, Makhoul I, Daniels JR, Klimberg S, Wei JY, et al. Immune response proteins as predictive biomarkers of doxorubicin-induced cardiotoxicity in breast cancer patients. Exp Biol Med (Maywood). 2018;243:248–55.

Ananthan K, Lyon AR. The role of biomarkers in Cardio-Oncology. J Cardiovasc Transl Res. 2020;13:431–50.

Article PubMed PubMed Central Google Scholar

Siegel RL, Miller KD, Wagle NS, Jemal A. 2023.Cancer statistics, 2023. CA Cancer J Clin 73: 17–48.

Januzzi JJ, Chen-Tournoux AA, Moe G. Amino-terminal pro-B-type natriuretic peptide testing for the diagnosis or exclusion of heart failure in patients with acute symptoms. Am J Cardiol. 2008;101:29–38.

Hochholzer W, Morrow DA, Giugliano RP. Novel biomarkers in cardiovascular disease: update 2010. Am Heart J. 2010;160:583–94.

2018.Global, regional, and national age-sex-specific mortality for 282 causes of death in 195 countries and territories, 1980–2017: a systematic analysis for the global burden of Disease Study 2017. Lancet 392: 1736–88.

Camilli M, Chiabrando JG, Lombardi M, Del BM, Montone RA, Lombardo A, et al. Cancer incidence and mortality in patients diagnosed with heart failure: results from an updated systematic review and meta-analysis. Cardiooncology. 2023;9:8.

PubMed PubMed Central Google Scholar

Sayour NV, Paal AM, Ameri P, Meijers WC, Minotti G, Andreadou I, et al. Heart failure pharmacotherapy and cancer: pathways and pre-clinical/clinical evidence. Eur Heart J. 2024;45:1224–40.

Fogarassy G, Vathy-Fogarassy A, Kenessey I, Veress G, Polgar C, Forster T. Preventing cancer therapy-related heart failure: the need for novel studies. J Cardiovasc Med (Hagerstown). 2021;22:459–68.

Hsu JY, Crawley S, Chen M, Ayupova DA, Lindhout DA, Higbee J, et al. Non-homeostatic body weight regulation through a brainstem-restricted receptor for GDF15. Nature. 2017;550:255–9.

Laird B, Jatoi A. Cancer cachexia: learn from yesterday, live for today and hope for tomorrow. Curr Opin Support Palliat Care. 2023;17:161.

Wang Y, Jiang T, Jiang M, Gu S. Appraising growth differentiation factor 15 as a promising biomarker in digestive system tumors: a meta-analysis. BMC Cancer. 2019;19:177.

Cartas-Espinel I, Telechea-Fernandez M, Manterola DC, Avila BA, Saavedra CN, Riffo-Campos AL. Novel molecular biomarkers of cancer therapy-induced cardiotoxicity in adult population: a scoping review. ESC Heart Fail. 2022;9:1651–65.

Valentini F, Anselmi F, Metra M, Cavigli L, Giacomin E, Focardi M, et al. Diagnostic and prognostic value of low QRS voltages in cardiomyopathies: old but gold. Eur J Prev Cardiol. 2022;29:1177–87.

Chaudhari GR, Mayfield JJ, Barrios JP, Abreau S, Avram R, Olgin JE, et al. Deep learning augmented ECG analysis to identify biomarker-defined myocardial injury. Sci Rep. 2023;13:3364.

Lyon AR, Lopez-Fernandez T, Couch LS, Asteggiano R, Aznar MC, Bergler-Klein J et al. 2022.2022 ESC guidelines on cardio-oncology developed in collaboration with the European Hematology Association (EHA), the European Society for Therapeutic Radiology and Oncology (ESTRO) and the International Cardio-Oncology Society (IC-OS). Eur Heart J 43: 4229–361.

Jacobs J, Greason G, Mangold KE, Wildiers H, Willems R, Janssens S, et al. Artificial intelligence electrocardiogram as a novel screening tool to detect a newly abnormal left ventricular ejection fraction after anthracycline-based cancer therapy. Eur J Prev Cardiol. 2024;31:560–6.

Download references

Acknowledgements

We are grateful to all the patients participated in the study and China Scholarship Council.

This work was supported by the National Natural Science Foundation of China (No. 82202836), Natural Science Foundation of Shandong Province (No. ZR202103020544), Shandong Province Medical and Health Technology Development Plan Project (No. 202002070690), Qingdao Science and Technology Demonstration and Guidance Project (22-3-7-smjk-10-nsh).

Author information

Xiaohe Hao, Zhenyu Zhang and Jing Kong contributed equally to this work.

Authors and Affiliations

Department of Clinical Laboratory, Shandong Cancer Hospital and Institute, Shandong First Medical University, Shandong Academy of Medical Sciences, 440 Ji-Yan Road, Jinan, Shandong Province, 250117, PR China

Xiaohe Hao, Zhenyu Zhang, Cuiping Mao, Xun Peng, Kun Ru, Lisheng Liu, Chuanxi Zhao, Xinkai Mo, Xiangguo Yu & Qinghai Lin

Department of Cardiology, Qilu Hospital of Shandong University, Jinan, Shandong, 250012, China

Electrocardiogram Room, Shandong Cancer Hospital and Institute, Shandong First Medical University, Shandong Academy of Medical Sciences, Jinan, Shandong, 250117, China

Department of Clinical Laboratory, Qilu Hospital of Shandong University, Jinan, Shandong, 250012, China

Meijuan Cai

You can also search for this author in PubMed Google Scholar

Contributions

Qinghai Lin designed experiments; Xiaohe Hao, Jing Kong, Xun Peng, Zhenyu Zhang and Cuiping Mao collected the laboratory results; Kun Ru, Chuanxi Zhao, Xinkai Mo and Rufei Ma provided the information of cancer patients; Qinghai Lin and Xiaohe Haoprepared Figs. 1 , 2 , 3 , 4 and 5 ， Jing Kong and Rufei Ma prepared Figs. 6 and 7 ; Table 1 , and 2 . Qinghai Lin, Xiangguo Yu, Meijuan Cai and Lisheng Liu wrote the manuscript and analyzed the data; Xiaohe Hao polished the manuscript. All authors reviewed the manuscript.

Corresponding authors

Correspondence to Xiangguo Yu or Qinghai Lin .

Ethics declarations

Ethics approval and consent to participate.

All procedures performed in studies involving human samples were reviewed and approved by the Ethics Committee of Shandong Cancer Hospital and Institute, in accordance with the Declaration of Helsinki. The participants provided their written informed consent to participate in this study.

Consent for publication

Not applicable.

Competing interests

The authors declare no competing interests.

Additional information

Publisher’s note.

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary Material 1

Supplementary material 2, supplementary material 3, supplementary material 4, supplementary material 5, supplementary material 6, supplementary material 7, supplementary material 8, supplementary material 9, supplementary material 10, supplementary material 11, supplementary material 12, supplementary material 13, supplementary material 14, supplementary material 15, supplementary material 16, supplementary material 17, supplementary material 18, supplementary material 19, supplementary material 20, supplementary material 21, supplementary material 22, supplementary material 23.

Supplementary Material 24

Supplementary Material 25

Supplementary material 26, supplementary material 27, rights and permissions.

Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/ .

Reprints and permissions

About this article

Cite this article.

Hao, X., Zhang, Z., Kong, J. et al. Hypothesis paper: GDF15 demonstrated promising potential in Cancer diagnosis and correlated with cardiac biomarkers. Cardio-Oncology 10 , 56 (2024). https://doi.org/10.1186/s40959-024-00263-9

Download citation

Received : 07 June 2024

Accepted : 30 August 2024

Published : 04 September 2024

DOI : https://doi.org/10.1186/s40959-024-00263-9

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Cardiovascular toxicity

Cardio-Oncology

ISSN: 2057-3804

General enquiries: [email protected]

IMAGES

Hypothesis Testing Population Mean
Two tailed hypothesis test calculator using p value
One Tailed Binomial Hypothesis Testing on Casio fx-CG50 Calculator
Hypothesis Testing Formula
hypothesis test formula statistics
Your Guide to Master Hypothesis Testing in Statistics

VIDEO

Hypothesis Testing
Perform and Interpret Results of a Hypothesis Test Using a Calculator
Perform and Interpret Results of a Hypothesis Test Using a Calculator
Practical Research 2 Module 5 Activity 4 Am I Accepted or Rejected Answer Key
2 Independent Means Hypothesis Test With Excel
Hypothesis Testing 🔥 Explained in 60 Seconds

COMMENTS

Hypothesis Testing Calculator with Steps
Hypothesis Testing Calculator. The first step in hypothesis testing is to calculate the test statistic. The formula for the test statistic depends on whether the population standard deviation (σ) is known or unknown. If σ is known, our hypothesis test is known as a z test and we use the z distribution. If σ is unknown, our hypothesis test is ...
Hypothesis Test Calculator
Calculation Example: There are six steps you would follow in hypothesis testing: Formulate the null and alternative hypotheses in three different ways: H 0: θ = θ 0 v e r s u s H 1: θ ≠ θ 0. H 0: θ ≤ θ 0 v e r s u s H 1: θ> θ 0. H 0: θ ≥ θ 0 v e r s u s H 1: θ <θ 0.
Hypothesis testing calculator
How to Use Our Hypothesis Testing Calculator. Using our hypothesis testing calculator is straightforward. Simply select the type of test you need from the list above, input your data, and follow the prompts. Our calculators will guide you through each step, ensuring you understand the process and obtain accurate results.
t-test Calculator
t-test Calculator | Formula | p-value
One sample t test calculator
Select Data Type: Choose whether to input summary statistics directly or provide a data set. Input Your Data: Enter the required values such as population mean, sample size, sample mean, and sample standard deviation. Set Hypotheses: Specify the null and alternative hypotheses. Calculate: Click the "Calculate" button to see the test statistic ...
Online Statistics Calculator: Hypothesis testing, t-test, chi-square
Online Statistics Calculator: Hypothesis testing, t-test, chi ...
Hypothesis Testing Calculator Online
Hypothesis testing is a foundational method used in statistics to infer the validity of a hypothesis about a population parameter. The Hypothesis Testing Calculator facilitates this process by automating the computations necessary for the t-test, a method used to compare sample means against a hypothesized mean or against each other.Let's delve into the formulas this calculator uses to ...
Hypothesis Test Calculator: t-test, chi-square, analysis of variance
Copy your data into the table. Select the variables. In the hypothesis test calculator you can calculate e.g. a t-test, a chi-square test, a binomial test or an analysis of variance. If you need a more detailed explanation, you can find more information in the tutorials. In order to use the hypothesis test calculator, you must first formulate ...
Hypothesis Testing Calculator
The Hypothesis Test Calculator is an invaluable resource for anyone involved in data analysis. It simplifies the hypothesis testing process, delivering clear, accurate results and visual aids that enhance your understanding of statistical concepts. Try the Hypothesis Test Calculator today and take your statistical analysis to the next level!
Hypothesis Testing Calculator
Calculations: Once you input the data and parameters, the calculator performs the necessary statistical tests and calculations. It generates results such as the test statistic, degrees of freedom, and the p-value. Interpretation: Based on the results, the calculator helps you determine whether to reject or fail to reject the null hypothesis.
28: Hypothesis Test for a Population Mean With Data Calculator
Hypothesis Test for a Population Mean With Data Calculator. Type in the values from the data set separated by commas, for example, 2,4,5,8,11,2. Then type in the population standard deviation σ σ if it is known or leave it blank if it is not known. Then choose a left, right or two tailed test, and the hypothesized mean.
One Sample T Test Calculator
One Sample T Test Calculator
Hypothesis Test Calculator
Hypothesis Test Calculator. Upload your data set below to get started. Upload File. Or input your data as csv. column_one,column_two,column_three 1,2,3 4,5,6 7,8,9. Submit CSV. Sharing helps us build more free tools.
T-Test Calculator for 2 Independent Means
T-Test Calculator for 2 Independent Means
Single Sample T-Test Calculator
Single Sample T-Test Calculator. A single sample t-test (or one sample t-test) is used to compare the mean of a single sample of scores to a known or hypothetical population mean. So, for example, it could be used to determine whether the mean diastolic blood pressure of a particular group differs from 85, a value determined by a previous study.
t-Test Calculator
t-test calculator is an online statistics tool to estimate the significance of observed differences between the means of two samples when there is a null hypothesis that is no significant difference between the means by using standard deviation. It is necessary to follow the next steps: Enter two samples (observed values) in the box. These values must be real numbers or variables and may be ...
T-Test calculator
Two sample and one sample t-test calculator with step by step explanation. Site map; Math Tests; Math Lessons; Math Formulas; Calculators; Math Calculators, Lessons and Formulas ... Twelve younger adults and twelve older adults conducted a life satisfaction test. The data are presented in the table below. Compute the appropriate t-test ...
One Sample t-test Calculator
A one sample t-test is used to test whether or not the mean of a population is equal to some value. To perform a one sample t-test, simply fill in the information below and then click the "Calculate" button. Enter raw data Enter summary data. 301, 298, 295, 297, 304, 305, 309, 298, 291, 299, 293, 304. μ0 (hypothesized population mean) t ...
Test Statistic Calculator
Test Statistic Calculator. Choose the method, enter the values into the test statistic calculator, and click on the "Calculate" button to calculate the statistical value for hypothesis evaluation. Select Format. Enter the data in row. Enter mean, SEM and n. Enter mean, SD and n. Group One.
Two Sample t-test Calculator
Two Sample t-test Calculator
Statistics calculator
These calculators are valuable tools for students, researchers, and professionals needing to make data-driven decisions. Our site provides various statistics calculators to help you with your statistical analysis needs, including critical value, p-value, hypothesis testing, regression analysis, and descriptive statistics calculators.
Two Population Calculator with Steps
If the two population variances are assumed to be equal, an alternative formula for computing the degrees of freedom is used. It's simply df = n1 + n2 - 2. This is a simple extension of the formula for the one population case. In the one population case the degrees of freedom is given by df = n - 1. If we add up the degrees of freedom for the ...
Step-by-step guide to hypothesis testing in statistics
Hypothesis testing is a way to use data to check if your guesses or assumptions are likely true. By following these steps—setting up your hypotheses, choosing the right test, deciding on a significance level, analyzing your data, finding the p-value, making a decision, and reporting results—you can determine if your data supports or ...
Two Sample T-Test Calculator
Two Sample T-Test Calculator
8.3: Sampling distribution and hypothesis testing
Introduction. Understanding the relationship between sampling distributions, probability distributions, and hypothesis testing is the crucial concept in the NHST — Null Hypothesis Significance Testing — approach to inferential statistics. is crucial, and many introductory text books are excellent here. I will add some here to their discussion, perhaps with a different approach, but the ...
A Sampling-Based Framework for Hypothesis Testing on Large Attributed
Hypothesis testing is a statistical method used to draw conclusions about populations from sample data, typically represented in tables. With the prevalence of graph representations in real-life applications, hypothesis testing on graphs is gaining importance. In this work, we formalize node, edge, and path hypotheses on attributed graphs.
How does plant chemodiversity evolve? Testing five hypotheses in one
A more complex model with metabolic pathways is needed to test most predictions of the screening hypothesis. However, we can test one prediction: Based on the screening hypothesis, plants with more metabolites should have a higher probability of having a metabolite that is active against a newly colonizing herbivore and therefore a higher ...
Hypothesis paper: GDF15 demonstrated promising potential in Cancer
Background Cardiovascular toxicity represents a significant adverse consequence of cancer therapies, yet there remains a paucity of effective biomarkers for its timely monitoring and diagnosis. To give a first evidence able to elucidate the role of Growth Differentiation Factor 15 (GDF15) in the context of cancer diagnosis and its specific association with cardiac indicators in cancer patients ...