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## Understanding Analysis of Variance (ANOVA) and the F-test

Topics: ANOVA , Hypothesis Testing , Data Analysis

Analysis of variance (ANOVA) can determine whether the means of three or more groups are different. ANOVA uses F-tests to statistically test the equality of means. In this post, I’ll show you how ANOVA and F-tests work using a one-way ANOVA example.

But wait a minute...have you ever stopped to wonder why you’d use an analysis of variance to determine whether means are different? I'll also show how variances provide information about means.

As in my posts about understanding t-tests , I’ll focus on concepts and graphs rather than equations to explain ANOVA F-tests.

## What are F-statistics and the F-test?

F-tests are named after its test statistic, F, which was named in honor of Sir Ronald Fisher. The F-statistic is simply a ratio of two variances. Variances are a measure of dispersion, or how far the data are scattered from the mean. Larger values represent greater dispersion.

Variance is the square of the standard deviation. For us humans, standard deviations are easier to understand than variances because they’re in the same units as the data rather than squared units. However, many analyses actually use variances in the calculations.

F-statistics are based on the ratio of mean squares. The term “ mean squares ” may sound confusing but it is simply an estimate of population variance that accounts for the degrees of freedom (DF) used to calculate that estimate.

Despite being a ratio of variances, you can use F-tests in a wide variety of situations. Unsurprisingly, the F-test can assess the equality of variances. However, by changing the variances that are included in the ratio, the F-test becomes a very flexible test. For example, you can use F-statistics and F-tests to test the overall significance for a regression model , to compare the fits of different models, to test specific regression terms, and to test the equality of means.

## Using the F-test in One-Way ANOVA

To use the F-test to determine whether group means are equal, it’s just a matter of including the correct variances in the ratio. In one-way ANOVA, the F-statistic is this ratio:

F = variation between sample means / variation within the samples

The best way to understand this ratio is to walk through a one-way ANOVA example.

We’ll analyze four samples of plastic to determine whether they have different mean strengths. You can download the sample data if you want to follow along. (If you don't have Minitab, you can download a free 30-day trial .) I'll refer back to the one-way ANOVA output as I explain the concepts.

In Minitab, choose Stat > ANOVA > One-Way ANOVA... In the dialog box, choose "Strength" as the response, and "Sample" as the factor. Press OK, and Minitab's Session Window displays the following output:

## Numerator: Variation Between Sample Means

One-way ANOVA has calculated a mean for each of the four samples of plastic. The group means are: 11.203, 8.938, 10.683, and 8.838. These group means are distributed around the overall mean for all 40 observations, which is 9.915. If the group means are clustered close to the overall mean, their variance is low. However, if the group means are spread out further from the overall mean, their variance is higher.

Clearly, if we want to show that the group means are different, it helps if the means are further apart from each other. In other words, we want higher variability among the means.

Imagine that we perform two different one-way ANOVAs where each analysis has four groups. The graph below shows the spread of the means. Each dot represents the mean of an entire group. The further the dots are spread out, the higher the value of the variability in the numerator of the F-statistic.

What value do we use to measure the variance between sample means for the plastic strength example? In the one-way ANOVA output, we’ll use the adjusted mean square (Adj MS) for Factor, which is 14.540. Don’t try to interpret this number because it won’t make sense. It’s the sum of the squared deviations divided by the factor DF. Just keep in mind that the further apart the group means are, the larger this number becomes.

## Denominator: Variation Within the Samples

We also need an estimate of the variability within each sample. To calculate this variance, we need to calculate how far each observation is from its group mean for all 40 observations. Technically, it is the sum of the squared deviations of each observation from its group mean divided by the error DF.

If the observations for each group are close to the group mean, the variance within the samples is low. However, if the observations for each group are further from the group mean, the variance within the samples is higher.

In the graph, the panel on the left shows low variation in the samples while the panel on the right shows high variation. The more spread out the observations are from their group mean, the higher the value in the denominator of the F-statistic.

If we’re hoping to show that the means are different, it's good when the within-group variance is low. You can think of the within-group variance as the background noise that can obscure a difference between means.

For this one-way ANOVA example, the value that we’ll use for the variance within samples is the Adj MS for Error, which is 4.402. It is considered “error” because it is the variability that is not explained by the factor.

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## The F-Statistic: Variation Between Sample Means / Variation Within the Samples

The F-statistic is the test statistic for F-tests. In general, an F-statistic is a ratio of two quantities that are expected to be roughly equal under the null hypothesis, which produces an F-statistic of approximately 1.

The F-statistic incorporates both measures of variability discussed above. Let's take a look at how these measures can work together to produce low and high F-values. Look at the graphs below and compare the width of the spread of the group means to the width of the spread within each group.

The low F-value graph shows a case where the group means are close together (low variability) relative to the variability within each group. The high F-value graph shows a case where the variability of group means is large relative to the within group variability. In order to reject the null hypothesis that the group means are equal, we need a high F-value.

For our plastic strength example, we'll use the Factor Adj MS for the numerator (14.540) and the Error Adj MS for the denominator (4.402), which gives us an F-value of 3.30.

Is our F-value high enough? A single F-value is hard to interpret on its own. We need to place our F-value into a larger context before we can interpret it. To do that, we’ll use the F-distribution to calculate probabilities.

## F-distributions and Hypothesis Testing

For one-way ANOVA, the ratio of the between-group variability to the within-group variability follows an F-distribution when the null hypothesis is true.

When you perform a one-way ANOVA for a single study, you obtain a single F-value. However, if we drew multiple random samples of the same size from the same population and performed the same one-way ANOVA, we would obtain many F-values and we could plot a distribution of all of them. This type of distribution is known as a sampling distribution .

Because the F-distribution assumes that the null hypothesis is true, we can place the F-value from our study in the F-distribution to determine how consistent our results are with the null hypothesis and to calculate probabilities.

The probability that we want to calculate is the probability of observing an F-statistic that is at least as high as the value that our study obtained. That probability allows us to determine how common or rare our F-value is under the assumption that the null hypothesis is true. If the probability is low enough, we can conclude that our data is inconsistent with the null hypothesis. The evidence in the sample data is strong enough to reject the null hypothesis for the entire population.

This probability that we’re calculating is also known as the p-value!

To plot the F-distribution for our plastic strength example, I’ll use Minitab’s probability distribution plots . In order to graph the F-distribution that is appropriate for our specific design and sample size, we'll need to specify the correct number of DF. Looking at our one-way ANOVA output, we can see that we have 3 DF for the numerator and 36 DF for the denominator.

The graph displays the distribution of F-values that we'd obtain if the null hypothesis is true and we repeat our study many times. The shaded area represents the probability of observing an F-value that is at least as large as the F-value our study obtained. F-values fall within this shaded region about 3.1% of the time when the null hypothesis is true. This probability is low enough to reject the null hypothesis using the common significance level of 0.05. We can conclude that not all the group means are equal.

Learn how to correctly interpret the p-value.

## Assessing Means by Analyzing Variation

ANOVA uses the F-test to determine whether the variability between group means is larger than the variability of the observations within the groups. If that ratio is sufficiently large, you can conclude that not all the means are equal.

This brings us back to why we analyze variation to make judgments about means. Think about the question: "Are the group means different?" You are implicitly asking about the variability of the means. After all, if the group means don't vary, or don't vary by more than random chance allows, then you can't say the means are different. And that's why you use analysis of variance to test the means.

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## Chapter 6. F-Test and One-Way ANOVA

F-distribution.

Years ago, statisticians discovered that when pairs of samples are taken from a normal population, the ratios of the variances of the samples in each pair will always follow the same distribution. Not surprisingly, over the intervening years, statisticians have found that the ratio of sample variances collected in a number of different ways follow this same distribution, the F-distribution. Because we know that sampling distributions of the ratio of variances follow a known distribution, we can conduct hypothesis tests using the ratio of variances.

The F-statistic is simply:

[latex]F = s^2_1 / s^2_2[/latex]

where s 1 2 is the variance of sample 1. Remember that the sample variance is:

[latex]s^2 = \sum(x - \overline{x})^2 / (n-1)[/latex]

Think about the shape that the F-distribution will have. If s 1 2 and s 2 2 come from samples from the same population, then if many pairs of samples were taken and F-scores computed, most of those F-scores would be close to one. All of the F-scores will be positive since variances are always positive — the numerator in the formula is the sum of squares, so it will be positive, the denominator is the sample size minus one, which will also be positive. Thinking about ratios requires some care. If s 1 2 is a lot larger than s 2 2 , F can be quite large. It is equally possible for s 2 2 to be a lot larger than s 1 2 , and then F would be very close to zero. Since F goes from zero to very large, with most of the values around one, it is obviously not symmetric; there is a long tail to the right, and a steep descent to zero on the left.

There are two uses of the F-distribution that will be discussed in this chapter. The first is a very simple test to see if two samples come from populations with the same variance. The second is one-way analysis of variance (ANOVA), which uses the F-distribution to test to see if three or more samples come from populations with the same mean.

## A simple test: Do these two samples come from populations with the same variance?

Because the F-distribution is generated by drawing two samples from the same normal population, it can be used to test the hypothesis that two samples come from populations with the same variance. You would have two samples (one of size n 1 and one of size n 2 ) and the sample variance from each. Obviously, if the two variances are very close to being equal the two samples could easily have come from populations with equal variances. Because the F-statistic is the ratio of two sample variances, when the two sample variances are close to equal, the F-score is close to one. If you compute the F-score, and it is close to one, you accept your hypothesis that the samples come from populations with the same variance.

This is the basic method of the F-test. Hypothesize that the samples come from populations with the same variance. Compute the F-score by finding the ratio of the sample variances. If the F-score is close to one, conclude that your hypothesis is correct and that the samples do come from populations with equal variances. If the F-score is far from one, then conclude that the populations probably have different variances.

The basic method must be fleshed out with some details if you are going to use this test at work. There are two sets of details: first, formally writing hypotheses, and second, using the F-distribution tables so that you can tell if your F-score is close to one or not. Formally, two hypotheses are needed for completeness. The first is the null hypothesis that there is no difference (hence null ). It is usually denoted as H o . The second is that there is a difference, and it is called the alternative, and is denoted H 1 or H a .

Using the F-tables to decide how close to one is close enough to accept the null hypothesis (truly formal statisticians would say “fail to reject the null”) is fairly tricky because the F-distribution tables are fairly tricky. Before using the tables, the researcher must decide how much chance he or she is willing to take that the null will be rejected when it is really true. The usual choice is 5 per cent, or as statisticians say, “ α – .05″. If more or less chance is wanted, α can be varied. Choose your α and go to the F-tables. First notice that there are a number of F-tables, one for each of several different levels of α (or at least a table for each two α ’s with the F-values for one α in bold type and the values for the other in regular type). There are rows and columns on each F-table, and both are for degrees of freedom. Because two separate samples are taken to compute an F-score and the samples do not have to be the same size, there are two separate degrees of freedom — one for each sample. For each sample, the number of degrees of freedom is n -1, one less than the sample size. Going to the table, how do you decide which sample’s degrees of freedom (df) are for the row and which are for the column? While you could put either one in either place, you can save yourself a step if you put the sample with the larger variance (not necessarily the larger sample) in the numerator , and then that sample’s df determines the column and the other sample’s df determines the row. The reason that this saves you a step is that the tables only show the values of F that leave α in the right tail where F > 1, the picture at the top of most F-tables shows that. Finding the critical F-value for left tails requires another step, which is outlined in the interactive Excel template in Figure 6.1. Simply change the numerator and the denominator degrees of freedom, and the α in the right tail of the F-distribution in the yellow cells.

F-tables are virtually always printed as one-tail tables, showing the critical F-value that separates the right tail from the rest of the distribution. In most statistical applications of the F-distribution, only the right tail is of interest, because most applications are testing to see if the variance from a certain source is greater than the variance from another source, so the researcher is interested in finding if the F-score is greater than one. In the test of equal variances, the researcher is interested in finding out if the F-score is close to one, so that either a large F-score or a small F-score would lead the researcher to conclude that the variances are not equal. Because the critical F-value that separates the left tail from the rest of the distribution is not printed, and not simply the negative of the printed value, researchers often simply divide the larger sample variance by the smaller sample variance, and use the printed tables to see if the quotient is “larger than one”, effectively rigging the test into a one-tail format. For purists, and occasional instances, the left-tail critical value can be computed fairly easily.

The left-tail critical value for x , y degrees of freedom (df) is simply the inverse of the right-tail (table) critical value for y , x df. Looking at an F-table, you would see that the F-value that leaves α – .05 in the right tail when there are 10 , 20 df is F=2.35. To find the F-value that leaves α – .05 in the left tail with 10, 20 df, look up F=2.77 for α – .05, 20 , 10 df. Divide one by 2.77, finding .36. That means that 5 per cent of the F-distribution for 10 , 20 df is below the critical value of .36, and 5 per cent is above the critical value of 2.35.

Putting all of this together, here is how to conduct the test to see if two samples come from populations with the same variance. First, collect two samples and compute the sample variance of each, s 1 2 and s 2 2 . Second, write your hypotheses and choose α . Third find the F-score from your samples, dividing the larger s 2 by the smaller so that F>1. Fourth, go to the tables, find the table for α /2, and find the critical (table) F-score for the proper degrees of freedom ( n -1 and n -1). Compare it to the samples’ F-score. If the samples’ F is larger than the critical F, the samples’ F is not “close to one”, and H a the population variances are not equal, is the best hypothesis. If the samples’ F is less than the critical F, H o , that the population variances are equal, should be accepted.

Lin Xiang, a young banker, has moved from Saskatoon, Saskatchewan, to Winnipeg, Manitoba, where she has recently been promoted and made the manager of City Bank, a newly established bank in Winnipeg with branches across the Prairies. After a few weeks, she has discovered that maintaining the correct number of tellers seems to be more difficult than it was when she was a branch assistant manager in Saskatoon. Some days, the lines are very long, but on other days, the tellers seem to have little to do. She wonders if the number of customers at her new branch is simply more variable than the number of customers at the branch where she used to work. Because tellers work for a whole day or half a day (morning or afternoon), she collects the following data on the number of transactions in a half day from her branch and the branch where she used to work:

Winnipeg branch: 156, 278, 134, 202, 236, 198, 187, 199, 143, 165, 223

Saskatoon branch: 345, 332, 309, 367, 388, 312, 355, 363, 381

She hypothesizes:

[latex]H_o: \sigma^2_W = \sigma^2_S[/latex]

[latex]H_a: \sigma^2_W \neq \sigma^2_S[/latex]

She decides to use α – .05. She computes the sample variances and finds:

[latex]s^2_W =1828.56[/latex]

[latex]s^2_S =795.19[/latex]

Following the rule to put the larger variance in the numerator, so that she saves a step, she finds:

[latex]F = s^2_W/s^2_S = 1828.56/795.19 = 2.30[/latex]

Using the interactive Excel template in Figure 6.2 (and remembering to use the α – .025 table because the table is one-tail and the test is two-tail), she finds that the critical F for 10,8 df is 4.30. Because her F-calculated score from Figure 6.2 is less than the critical score, she concludes that her F-score is “close to one”, and that the variance of customers in her office is the same as it was in the old office. She will need to look further to solve her staffing problem.

## Analysis of variance (ANOVA)

The importance of anova.

A more important use of the F-distribution is in analyzing variance to see if three or more samples come from populations with equal means. This is an important statistical test, not so much because it is frequently used, but because it is a bridge between univariate statistics and multivariate statistics and because the strategy it uses is one that is used in many multivariate tests and procedures.

## One-way ANOVA: Do these three (or more) samples all come from populations with the same mean?

This seems wrong — we will test a hypothesis about means by analyzing variance . It is not wrong, but rather a really clever insight that some statistician had years ago. This idea — looking at variance to find out about differences in means — is the basis for much of the multivariate statistics used by researchers today. The ideas behind ANOVA are used when we look for relationships between two or more variables, the big reason we use multivariate statistics.

Testing to see if three or more samples come from populations with the same mean can often be a sort of multivariate exercise. If the three samples came from three different factories or were subject to different treatments, we are effectively seeing if there is a difference in the results because of different factories or treatments — is there a relationship between factory (or treatment) and the outcome?

Think about three samples. A group of x ’s have been collected, and for some good reason (other than their x value) they can be divided into three groups. You have some x ’s from group (sample) 1, some from group (sample) 2, and some from group (sample) 3. If the samples were combined, you could compute a grand mean and a total variance around that grand mean. You could also find the mean and (sample) variance within each of the groups. Finally, you could take the three sample means, and find the variance between them. ANOVA is based on analyzing where the total variance comes from. If you picked one x , the source of its variance, its distance from the grand mean, would have two parts: (1) how far it is from the mean of its sample, and (2) how far its sample’s mean is from the grand mean. If the three samples really do come from populations with different means, then for most of the x ’s, the distance between the sample mean and the grand mean will probably be greater than the distance between the x and its group mean. When these distances are gathered together and turned into variances, you can see that if the population means are different, the variance between the sample means is likely to be greater than the variance within the samples.

By this point in the book, it should not surprise you to learn that statisticians have found that if three or more samples are taken from a normal population, and the variance between the samples is divided by the variance within the samples, a sampling distribution formed by doing that over and over will have a known shape. In this case, it will be distributed like F with m -1, n – m df, where m is the number of samples and n is the size of the m samples altogether. Variance between is found by:

where x j is the mean of sample j , and x is the grand mean.

The numerator of the variance between is the sum of the squares of the distance between each x ’s sample mean and the grand mean. It is simply a summing of one of those sources of variance across all of the observations.

The variance within is found by:

Double sums need to be handled with care. First (operating on the inside or second sum sign) find the mean of each sample and the sum of the squares of the distances of each x in the sample from its mean. Second (operating on the outside sum sign), add together the results from each of the samples.

The strategy for conducting a one-way analysis of variance is simple. Gather m samples. Compute the variance between the samples, the variance within the samples, and the ratio of between to within, yielding the F-score. If the F-score is less than one, or not much greater than one, the variance between the samples is no greater than the variance within the samples and the samples probably come from populations with the same mean. If the F-score is much greater than one, the variance between is probably the source of most of the variance in the total sample, and the samples probably come from populations with different means.

The details of conducting a one-way ANOVA fall into three categories: (1) writing hypotheses, (2) keeping the calculations organized, and (3) using the F-tables. The null hypothesis is that all of the population means are equal, and the alternative is that not all of the means are equal. Quite often, though two hypotheses are really needed for completeness, only H o is written:

[latex]H_o: m_1=m_2=\ldots=m_m[/latex]

Keeping the calculations organized is important when you are finding the variance within. Remember that the variance within is found by squaring, and then summing, the distance between each observation and the mean of its sample . Though different people do the calculations differently, I find the best way to keep it all straight is to find the sample means, find the squared distances in each of the samples, and then add those together. It is also important to keep the calculations organized in the final computing of the F-score. If you remember that the goal is to see if the variance between is large, then its easy to remember to divide variance between by variance within.

Using the F-tables is the third detail. Remember that F-tables are one-tail tables and that ANOVA is a one-tail test. Though the null hypothesis is that all of the means are equal, you are testing that hypothesis by seeing if the variance between is less than or equal to the variance within. The number of degrees of freedom is m -1, n – m , where m is the number of samples and n is the total size of all the samples together.

The young bank manager in Example 1 is still struggling with finding the best way to staff her branch. She knows that she needs to have more tellers on Fridays than on other days, but she is trying to find if the need for tellers is constant across the rest of the week. She collects data for the number of transactions each day for two months. Here are her data:

Mondays: 276, 323, 298, 256, 277, 309, 312, 265, 311

Tuesdays: 243, 279, 301, 285, 274, 243, 228, 298, 255

Wednesdays: 288, 292, 310, 267, 243, 293, 255, 273

Thursdays: 254, 279, 241, 227, 278, 276, 256, 262

She tests the null hypothesis:

[latex]H_o: m_m=m_{tu}=m_w=m_{th}[/latex]

and decides to use α – .05. She finds:

and the grand mean = 274.3

She computes variance within:

[(276-291.8)2+(323-291.8)2+…+(243-267.6)2+…+(288-277.6)2+…+(254-259.1)2]/[34-4]=15887.6/30=529.6

Then she computes variance between:

[9(291.8-274.3)2+9(267.3-274.3)2+8(277.6-274.3)2+8(259.1-274.3)2]/[4-1]

= 5151.8/3 = 1717.3

She computes her F-score:

You can enter the number of transactions each day in the yellow cells in Figure 6.3, and select the α . As you can then see in Figure 6.3, the calculated F-value is 3.24, while the F-table (F-Critical) for α – .05 and 3, 30 df, is 2.92. Because her F-score is larger than the critical F-value, or alternatively since the p-value (0.036) is less than α – .05, she concludes that the mean number of transactions is not equal on different days of the week, or at least there is one day that is different from others. She will want to adjust her staffing so that she has more tellers on some days than on others.

The F-distribution is the sampling distribution of the ratio of the variances of two samples drawn from a normal population. It is used directly to test to see if two samples come from populations with the same variance. Though you will occasionally see it used to test equality of variances, the more important use is in analysis of variance (ANOVA). ANOVA, at least in its simplest form as presented in this chapter, is used to test to see if three or more samples come from populations with the same mean. By testing to see if the variance of the observations comes more from the variation of each observation from the mean of its sample or from the variation of the means of the samples from the grand mean, ANOVA tests to see if the samples come from populations with equal means or not.

ANOVA has more elegant forms that appear in later chapters. It forms the basis for regression analysis, a statistical technique that has many business applications; it is covered in later chapters. The F-tables are also used in testing hypotheses about regression results.

This is also the beginning of multivariate statistics. Notice that in the one-way ANOVA, each observation is for two variables: the x variable and the group of which the observation is a part. In later chapters, observations will have two, three, or more variables.

The F-test for equality of variances is sometimes used before using the t-test for equality of means because the t-test, at least in the form presented in this text, requires that the samples come from populations with equal variances. You will see it used along with t-tests when the stakes are high or the researcher is a little compulsive.

Introductory Business Statistics with Interactive Spreadsheets - 1st Canadian Edition by Mohammad Mahbobi and Thomas K. Tiemann is licensed under a Creative Commons Attribution 4.0 International License , except where otherwise noted.

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## Hypothesis Testing - Analysis of Variance (ANOVA)

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## The ANOVA Approach

Test statistic for anova.

All Modules

Table of F-Statistic Values

Consider an example with four independent groups and a continuous outcome measure. The independent groups might be defined by a particular characteristic of the participants such as BMI (e.g., underweight, normal weight, overweight, obese) or by the investigator (e.g., randomizing participants to one of four competing treatments, call them A, B, C and D). Suppose that the outcome is systolic blood pressure, and we wish to test whether there is a statistically significant difference in mean systolic blood pressures among the four groups. The sample data are organized as follows:

The hypotheses of interest in an ANOVA are as follows:

- H 0 : μ 1 = μ 2 = μ 3 ... = μ k
- H 1 : Means are not all equal.

where k = the number of independent comparison groups.

In this example, the hypotheses are:

- H 0 : μ 1 = μ 2 = μ 3 = μ 4
- H 1 : The means are not all equal.

The null hypothesis in ANOVA is always that there is no difference in means. The research or alternative hypothesis is always that the means are not all equal and is usually written in words rather than in mathematical symbols. The research hypothesis captures any difference in means and includes, for example, the situation where all four means are unequal, where one is different from the other three, where two are different, and so on. The alternative hypothesis, as shown above, capture all possible situations other than equality of all means specified in the null hypothesis.

The test statistic for testing H 0 : μ 1 = μ 2 = ... = μ k is:

and the critical value is found in a table of probability values for the F distribution with (degrees of freedom) df 1 = k-1, df 2 =N-k. The table can be found in "Other Resources" on the left side of the pages.

NOTE: The test statistic F assumes equal variability in the k populations (i.e., the population variances are equal, or s 1 2 = s 2 2 = ... = s k 2 ). This means that the outcome is equally variable in each of the comparison populations. This assumption is the same as that assumed for appropriate use of the test statistic to test equality of two independent means. It is possible to assess the likelihood that the assumption of equal variances is true and the test can be conducted in most statistical computing packages. If the variability in the k comparison groups is not similar, then alternative techniques must be used.

The F statistic is computed by taking the ratio of what is called the "between treatment" variability to the "residual or error" variability. This is where the name of the procedure originates. In analysis of variance we are testing for a difference in means (H 0 : means are all equal versus H 1 : means are not all equal) by evaluating variability in the data. The numerator captures between treatment variability (i.e., differences among the sample means) and the denominator contains an estimate of the variability in the outcome. The test statistic is a measure that allows us to assess whether the differences among the sample means (numerator) are more than would be expected by chance if the null hypothesis is true. Recall in the two independent sample test, the test statistic was computed by taking the ratio of the difference in sample means (numerator) to the variability in the outcome (estimated by Sp).

The decision rule for the F test in ANOVA is set up in a similar way to decision rules we established for t tests. The decision rule again depends on the level of significance and the degrees of freedom. The F statistic has two degrees of freedom. These are denoted df 1 and df 2 , and called the numerator and denominator degrees of freedom, respectively. The degrees of freedom are defined as follows:

df 1 = k-1 and df 2 =N-k,

where k is the number of comparison groups and N is the total number of observations in the analysis. If the null hypothesis is true, the between treatment variation (numerator) will not exceed the residual or error variation (denominator) and the F statistic will small. If the null hypothesis is false, then the F statistic will be large. The rejection region for the F test is always in the upper (right-hand) tail of the distribution as shown below.

Rejection Region for F Test with a =0.05, df 1 =3 and df 2 =36 (k=4, N=40)

For the scenario depicted here, the decision rule is: Reject H 0 if F > 2.87.

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Content ©2019. All Rights Reserved. Date last modified: January 23, 2019. Wayne W. LaMorte, MD, PhD, MPH

F test is a statistical test that is used in hypothesis testing to check whether the variances of two populations or two samples are equal or not. In an f test, the data follows an f distribution. This test uses the f statistic to compare two variances by dividing them. An f test can either be one-tailed or two-tailed depending upon the parameters of the problem.

The f value obtained after conducting an f test is used to perform the one-way ANOVA (analysis of variance) test. In this article, we will learn more about an f test, the f statistic, its critical value, formula and how to conduct an f test for hypothesis testing.

## What is F Test in Statistics?

F test is statistics is a test that is performed on an f distribution. A two-tailed f test is used to check whether the variances of the two given samples (or populations) are equal or not. However, if an f test checks whether one population variance is either greater than or lesser than the other, it becomes a one-tailed hypothesis f test.

## F Test Definition

F test can be defined as a test that uses the f test statistic to check whether the variances of two samples (or populations) are equal to the same value. To conduct an f test, the population should follow an f distribution and the samples must be independent events. On conducting the hypothesis test, if the results of the f test are statistically significant then the null hypothesis can be rejected otherwise it cannot be rejected.

## F Test Formula

The f test is used to check the equality of variances using hypothesis testing . The f test formula for different hypothesis tests is given as follows:

Left Tailed Test:

- Null Hypothesis: \(H_{0}\) : \(\sigma_{1}^{2} = \sigma_{2}^{2}\)

Alternate Hypothesis: \(H_{1}\) : \(\sigma_{1}^{2} < \sigma_{2}^{2}\)

Decision Criteria: If the f statistic < f critical value then reject the null hypothesis

Right Tailed test:

- Alternate Hypothesis: \(H_{1}\) : \(\sigma_{1}^{2} > \sigma_{2}^{2}\)

Decision Criteria: If the f test statistic > f test critical value then reject the null hypothesis

Two Tailed test:

Alternate Hypothesis: \(H_{1}\) : \(\sigma_{1}^{2} ≠ \sigma_{2}^{2}\)

Decision Criteria: If the f test statistic > f test critical value then the null hypothesis is rejected

## F Statistic

The f test statistic or simply the f statistic is a value that is compared with the critical value to check if the null hypothesis should be rejected or not. The f test statistic formula is given below:

F statistic for large samples: F = \(\frac{\sigma_{1}^{2}}{\sigma_{2}^{2}}\), where \(\sigma_{1}^{2}\) is the variance of the first population and \(\sigma_{2}^{2}\) is the variance of the second population.

F statistic for small samples: F = \(\frac{s_{1}^{2}}{s_{2}^{2}}\), where \(s_{1}^{2}\) is the variance of the first sample and \(s_{2}^{2}\) is the variance of the second sample.

The selection criteria for the \(\sigma_{1}^{2}\) and \(\sigma_{2}^{2}\) for an f statistic is given below:

- For a right-tailed and a two-tailed f test, the variance with the greater value will be in the numerator. Thus, the sample corresponding to \(\sigma_{1}^{2}\) will become the first sample. The smaller value variance will be the denominator and belongs to the second sample.
- For a left-tailed test, the smallest variance becomes the numerator (sample 1) and the highest variance goes in the denominator (sample 2).

## F Test Critical Value

A critical value is a point that a test statistic is compared to in order to decide whether to reject or not to reject the null hypothesis. Graphically, the critical value divides a distribution into the acceptance and rejection regions. If the test statistic falls in the rejection region then the null hypothesis can be rejected otherwise it cannot be rejected. The steps to find the f test critical value at a specific alpha level (or significance level), \(\alpha\), are as follows:

- Find the degrees of freedom of the first sample. This is done by subtracting 1 from the first sample size. Thus, x = \(n_{1} - 1\).
- Determine the degrees of freedom of the second sample by subtracting 1 from the sample size. This given y = \(n_{2} - 1\).
- If it is a right-tailed test then \(\alpha\) is the significance level. For a left-tailed test 1 - \(\alpha\) is the alpha level. However, if it is a two-tailed test then the significance level is given by \(\alpha\) / 2.
- The F table is used to find the critical value at the required alpha level.
- The intersection of the x column and the y row in the f table will give the f test critical value.

## ANOVA F Test

The one-way ANOVA is an example of an f test. ANOVA stands for analysis of variance. It is used to check the variability of group means and the associated variability in observations within that group. The F test statistic is used to conduct the ANOVA test. The hypothesis is given as follows:

\(H_{0}\): The means of all groups are equal.

\(H_{1}\): The means of all groups are not equal.

Test Statistic: F = explained variance / unexplained variance

Decision rule: If F > F critical value then reject the null hypothesis.

To determine the critical value of an ANOVA f test the degrees of freedom are given by \(df_{1}\) = K - 1 and \(df_{1}\) = N - K, where N is the overall sample size and K is the number of groups.

## F Test vs T-Test

F test and t-test are different types of statistical tests used for hypothesis testing depending on the distribution followed by the population data. The table given below outlines the differences between the F test and the t-test.

Related Articles:

- Probability and Statistics
- Data Handling
- Summary Statistics

Important Notes on F Test

- The f test is a statistical test that is conducted on an F distribution in order to check the equality of variances of two populations.
- The f test formula for the test statistic is given by F = \(\frac{\sigma_{1}^{2}}{\sigma_{2}^{2}}\).
- The f critical value is a cut-off value that is used to check whether the null hypothesis can be rejected or not.
- A one-way ANOVA is an example of an f test that is used to check the variability of group means and the associated variability in the group observations.

## Examples on F Test

- Example 2: Pizza delivery times of two cities are given below City 1: Number of delivery times observed = 28, Variance = 38 City 2: Number of delivery times observed = 25, Variance = 83 Check if the delivery times of city 1 are lesser than city 2 at a 0.05 alpha level. Solution: This is an example of a left-tailed F test. Thus, the alpha level is 1 - 0.05 = 0.95 \(H_{0}\) : \(s_{1}^{2} = s_{2}^{2}\) \(H_{1}\) : \(s_{1}^{2} < s_{2}^{2}\) As 38 < 83 thus, city 1 will be sample 1 and city 2 is sample 2. \(n_{1}\) = 28, \(n_{2}\) = 25 \(df_{1}\) = 28 - 1 = 27 \(df_{2}\) = 25 - 1 = 24 \(s_{1}^{2}\) = 38, \(s_{2}^{2}\) = 83 F = \(\frac{s_{1}^{2}}{s_{2}^{2}}\) = 38 / 83 F = 0.4578 As an F table for 0.95 alpha level is not available, the critical value is determined as follows: F(0.95, 27, 24) = 1 / F(0.05, 24, 27) F(0.05, 24, 27) = 1.93 F(0.95, 27, 24) = 1 / 1.93 = 0.5181 As 0.4578 < 0.5181, thus, the null hypothesis can be rejected and it can be concluded that there is enough evidence to support the claim that the delivery times in city 1 are less than in city 2. Answer: Reject the null hypothesis
- Example 3: A toy manufacturer wants to get batteries for toys. A team collected 41 samples from supplier A and the variance was 110 hours. The team also collected 21 samples from supplier B with a variance of 65 hours. At a 0.05 alpha level determine if there is a difference in the variances. Solution: This is an example of a two-tailed F test. Thus, the alpha level is 0.05 / 2 = 0.025 \(H_{0}\) : \(s_{1}^{2} = s_{2}^{2}\) \(H_{1}\) : \(s_{1}^{2} \neq s_{2}^{2}\) \(n_{1}\) = 41, \(n_{2}\) = 21 \(df_{1}\) = 41 - 1 = 40 \(df_{2}\) = 21 - 1 = 20 \(s_{1}^{2}\) = 110, \(s_{2}^{2}\) = 65 F = \(\frac{s_{1}^{2}}{s_{2}^{2}}\) = 110 / 65 F = 1.69 Using the F table F(0.025, 40, 20) = 2.287 As 1.69 < 2.287 thus, the null hypothesis cannot be rejected, Answer: Fail to reject the null hypothesis.

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## FAQs on F Test

What is the f test.

The f test in statistics is used to find whether the variances of two populations are equal or not by using a one-tailed or two-tailed hypothesis test.

## What is the F Test Formula?

The f test formula can be used to find the f statistic. The f test formula is given as follows:

- F statistic for large samples: F = \(\frac{\sigma_{1}^{2}}{\sigma_{2}^{2}}\)
- F statistic for small samples: F = \(\frac{s_{1}^{2}}{s_{2}^{2}}\)

## What is the Decision Criterion for a Right Tailed F Test?

The algorithm to set up an right tailed f test hypothesis along with the decision criteria are given as follows:

- Decision Criteria: Reject \(H_{0}\) if the f test statistic > f test critical value.

## What is the Critical Value for an F Test?

The F critical value for an f test can be defined as the cut-off value that is compared with the test statistic to decide if the null hypothesis should be rejected or not.

## Why is an F Test Used in ANOVA?

A one-way ANOVA test uses the f test to compare if there is a difference between the variability of group means and the associated variability of observations of those groups.

## Can the F statistic in an F Test be Negative?

As the f test statistic is the ratio of variances thus, it cannot be negative. This is because the square of a number will always be positive.

## What is the Difference Between F Test and T-Test?

An F test is conducted on an f distribution to determine the equality of variances of two samples. The t-test is performed on a student t distribution when the number of samples is less and the population standard deviation is not known. It is used to compare means.

Statistics Made Easy

## How to Interpret the F-Value and P-Value in ANOVA

An ANOVA (“analysis of variance”) is used to determine whether or not the means of three or more independent groups are equal.

An ANOVA uses the following null and alternative hypotheses:

- H 0 : All group means are equal.
- H A : At least one group mean is different from the rest.

Whenever you perform an ANOVA, you will end up with a summary table that looks like the following:

Two values that we immediately analyze in the table are the F-statistic and the corresponding p-value .

## Understanding the F-Statistic in ANOVA

The F-statistic is the ratio of the mean squares treatment to the mean squares error:

- F-statistic: Mean Squares Treatment / Mean Squares Error

Another way to write this is:

- F-statistic: Variation between sample means / Variation within samples

The larger the F-statistic, the greater the variation between sample means relative to the variation within the samples.

Thus, the larger the F-statistic, the greater the evidence that there is a difference between the group means.

## Understanding the P-Value in ANOVA

To determine if the difference between group means is statistically significant, we can look at the p-value that corresponds to the F-statistic.

To find the p-value that corresponds to this F-value, we can use an F Distribution Calculator with numerator degrees of freedom = df Treatment and denominator degrees of freedom = df Error.

For example, the p-value that corresponds to an F-value of 2.358, numerator df = 2, and denominator df = 27 is 0.1138 .

If this p-value is less than α = .05, we reject the null hypothesis of the ANOVA and conclude that there is a statistically significant difference between the means of the three groups.

Otherwise, if the p-value is not less than α = .05 then we fail to reject the null hypothesis and conclude that we do not have sufficient evidence to say that there is a statistically significant difference between the means of the three groups.

In this particular example, the p-value is 0.1138 so we would fail to reject the null hypothesis. This means we don’t have sufficient evidence to say that there is a statistically significant difference between the group means.

## On Using Post-Hoc Tests with an ANOVA

If the p-value of an ANOVA is less than .05, then we reject the null hypothesis that each group mean is equal.

In this scenario, we can then perform post-hoc tests to determine exactly which groups differ from each other.

There are several potential post-hoc tests we can use following an ANOVA, but the most popular ones include:

- Bonferroni Test
- Scheffe Test

Refer to this guide to understand which post-hoc test you should use depending on your particular situation.

## Additional Resources

The following resources offer additional information about ANOVA tests:

An Introduction to the One-Way ANOVA An Introduction to the Two-Way ANOVA The Complete Guide: How to Report ANOVA Results ANOVA vs. Regression: What’s the Difference?

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## Keyboard Shortcuts

6.2 - the general linear f-test, the general linear f-test section .

The " general linear F-test " involves three basic steps, namely:

- Define a larger full model . (By "larger," we mean one with more parameters.)
- Define a smaller reduced model . (By "smaller," we mean one with fewer parameters.)
- Use an F- statistic to decide whether or not to reject the smaller reduced model in favor of the larger full model.

As you can see by the wording of the third step, the null hypothesis always pertains to the reduced model, while the alternative hypothesis always pertains to the full model.

The easiest way to learn about the general linear test is to first go back to what we know, namely the simple linear regression model. Once we understand the general linear test for the simple case, we then see that it can be easily extended to the multiple-case model. We take that approach here.

## The Full Model Section

The " full model ", which is also sometimes referred to as the " unrestricted model ," is the model thought to be most appropriate for the data. For simple linear regression, the full model is:

\(y_i=(\beta_0+\beta_1x_{i1})+\epsilon_i\)

Here's a plot of a hypothesized full model for a set of data that we worked with previously in this course (student heights and grade point averages):

And, here's another plot of a hypothesized full model that we previously encountered (state latitudes and skin cancer mortalities):

In each plot, the solid line represents what the hypothesized population regression line might look like for the full model. The question we have to answer in each case is "does the full model describe the data well?" Here, we might think that the full model does well in summarizing the trend in the second plot but not the first.

## The Reduced Model Section

The " reduced model ," which is sometimes also referred to as the " restricted model ," is the model described by the null hypothesis \(H_{0}\). For simple linear regression, a common null hypothesis is \(H_{0} : \beta_{1} = 0\). In this case, the reduced model is obtained by "zeroing out" the slope \(\beta_{1}\) that appears in the full model. That is, the reduced model is:

\(y_i=\beta_0+\epsilon_i\)

This reduced model suggests that each response \(y_{i}\) is a function only of some overall mean, \(\beta_{0}\), and some error \(\epsilon_{i}\).

Let's take another look at the plot of student grade point average against height, but this time with a line representing what the hypothesized population regression line might look like for the reduced model:

Not bad — there (fortunately?!) doesn't appear to be a relationship between height and grade point average. And, it appears as if the reduced model might be appropriate in describing the lack of a relationship between heights and grade point averages. What does the reduced model do for the skin cancer mortality example?

It doesn't appear as if the reduced model would do a very good job of summarizing the trend in the population.

## F-Statistic Test Section

How do we decide if the reduced model or the full model does a better job of describing the trend in the data when it can't be determined by simply looking at a plot? What we need to do is to quantify how much error remains after fitting each of the two models to our data. That is, we take the general linear test approach:

- Obtain the least squares estimates of \(\beta_{0}\) and \(\beta_{1}\).
- Determine the error sum of squares, which we denote as " SSE ( F )."
- Obtain the least squares estimate of \(\beta_{0}\).
- Determine the error sum of squares, which we denote as " SSE ( R )."

Recall that, in general, the error sum of squares is obtained by summing the squared distances between the observed and fitted (estimated) responses:

\(\sum(\text{observed } - \text{ fitted})^2\)

Therefore, since \(y_i\) is the observed response and \(\hat{y}_i\) is the fitted response for the full model :

\(SSE(F)=\sum(y_i-\hat{y}_i)^2\)

And, since \(y_i\) is the observed response and \(\bar{y}\) is the fitted response for the reduced model :

\(SSE(R)=\sum(y_i-\bar{y})^2\)

Let's get a better feel for the general linear F-test approach by applying it to two different datasets. First, let's look at the Height and GPA data . The following plot of grade point averages against heights contains two estimated regression lines — the solid line is the estimated line for the full model, and the dashed line is the estimated line for the reduced model:

As you can see, the estimated lines are almost identical. Calculating the error sum of squares for each model, we obtain:

\(SSE(F)=\sum(y_i-\hat{y}_i)^2=9.7055\)

\(SSE(R)=\sum(y_i-\bar{y})^2=9.7331\)

The two quantities are almost identical. Adding height to the reduced model to obtain the full model reduces the amount of error by only 0.0276 (from 9.7331 to 9.7055). That is, adding height to the model does very little in reducing the variability in grade point averages. In this case, there appears to be no advantage in using the larger full model over the simpler reduced model.

Look what happens when we fit the full and reduced models to the skin cancer mortality and latitude dataset :

Here, there is quite a big difference between the estimated equation for the full model (solid line) and the estimated equation for the reduced model (dashed line). The error sums of squares quantify the substantial difference in the two estimated equations:

\(SSE(F)=\sum(y_i-\hat{y}_i)^2=17173\)

\(SSE(R)=\sum(y_i-\bar{y})^2=53637\)

Adding latitude to the reduced model to obtain the full model reduces the amount of error by 36464 (from 53637 to 17173). That is, adding latitude to the model substantially reduces the variability in skin cancer mortality. In this case, there appears to be a big advantage in using the larger full model over the simpler reduced model.

Where are we going with this general linear test approach? In short:

- The general linear test involves a comparison between SSE ( R ) and SSE ( F ).
- If SSE ( F ) is close to SSE ( R ), then the variation around the estimated full model regression function is almost as large as the variation around the estimated reduced model regression function. If that's the case, it makes sense to use the simpler reduced model.
- On the other hand, if SSE ( F ) and SSE ( R ) differ greatly, then the additional parameter(s) in the full model substantially reduce the variation around the estimated regression function. In this case, it makes sense to go with the larger full model.

How different does SSE ( R ) have to be from SSE ( F ) in order to justify using the larger full model? The general linear F -statistic:

\(F^*=\left( \dfrac{SSE(R)-SSE(F)}{df_R-df_F}\right)\div\left( \dfrac{SSE(F)}{df_F}\right)\)

helps answer this question. The F -statistic intuitively makes sense — it is a function of SSE ( R )- SSE ( F ), the difference in the error between the two models. The degrees of freedom — denoted \(df_{R}\) and \(df_{F}\) — are those associated with the reduced and full model error sum of squares, respectively.

We use the general linear F -statistic to decide whether or not:

- to reject the null hypothesis \(H_{0}\colon\) The reduced model
- in favor of the alternative hypothesis \(H_{A}\colon\) The full model

In general, we reject \(H_{0}\) if F * is large — or equivalently if its associated P -value is small.

## The test applied to the simple linear regression model Section

For simple linear regression, it turns out that the general linear F -test is just the same ANOVA F -test that we learned before. As noted earlier for the simple linear regression case, the full model is:

and the reduced model is:

Therefore, the appropriate null and alternative hypotheses are specified either as:

- \(H_{0} \colon y_i = \beta_{0} + \epsilon_{i}\)
- \(H_{A} \colon y_i = \beta_{0} + \beta_{1} x_{i} + \epsilon_{i}\)
- \(H_{0} \colon \beta_{1} = 0 \)
- \(H_{A} \colon \beta_{1} ≠ 0 \)

The degrees of freedom associated with the error sum of squares for the reduced model is n -1, and:

\(SSE(R)=\sum(y_i-\bar{y})^2=SSTO\)

The degrees of freedom associated with the error sum of squares for the full model is n -2, and:

\(SSE(F)=\sum(y_i-\hat{y}_i)^2=SSE\)

Now, we can see how the general linear F -statistic just reduces algebraically to the ANOVA F -test that we know:

Can be rewritten by substituting...

\(\begin{aligned} &&df_{R} = n - 1\\ &&df_{F} = n - 2\\ &&SSE(R)=SSTO\\&&SSE(F)=SSE\end{aligned}\)

\(F^*=\left( \dfrac{SSTO-SSE}{(n-1)-(n-2)}\right)\div\left( \dfrac{SSE}{(n-2)}\right)=\frac{MSR}{MSE}\)

That is, the general linear F -statistic reduces to the ANOVA F -statistic:

\(F^*=\dfrac{MSR}{MSE}\)

For the student height and grade point average example:

\( F^*=\dfrac{MSR}{MSE}=\dfrac{0.0276/1}{9.7055/33}=\dfrac{0.0276}{0.2941}=0.094\)

For the skin cancer mortality example:

\( F^*=\dfrac{MSR}{MSE}=\dfrac{36464/1}{17173/47}=\dfrac{36464}{365.4}=99.8\)

The P -value is calculated as usual. The P -value answers the question: "what is the probability that we’d get an F* statistic as large as we did if the null hypothesis were true?" The P -value is determined by comparing F * to an F distribution with 1 numerator degree of freedom and n -2 denominator degrees of freedom. For the student height and grade point average example, the P -value is 0.761 (so we fail to reject \(H_{0}\) and we favor the reduced model), while for the skin cancer mortality example, the P -value is 0.000 (so we reject \(H_{0}\) and we favor the full model).

## Example 6-2: Alcohol and muscle Strength Section

Does alcoholism have an effect on muscle strength? Some researchers (Urbano-Marquez, et al , 1989) who were interested in answering this question collected the following data ( Alcohol Arm data ) on a sample of 50 alcoholic men:

- x = the total lifetime dose of alcohol ( kg per kg of body weight) consumed
- y = the strength of the deltoid muscle in the man's non-dominant arm

The full model is the model that would summarize a linear relationship between alcohol consumption and arm strength. The reduced model, on the other hand, is the model that claims there is no relationship between alcohol consumption and arm strength.

- \(H_0 \colon y_i = \beta_0 + \epsilon_i \)
- \(H_A \colon y_i = \beta_0 + \beta_{1}x_i + \epsilon_i\)
- \(H_0 \colon \beta_1 = 0\)
- \(H_A \colon \beta_1 ≠ 0\)

Upon fitting the reduced model to the data, we obtain:

\(SSE(R)=\sum(y_i-\bar{y})^2=1224.32\)

Note that the reduced model does not appear to summarize the trend in the data very well.

Upon fitting the full model to the data, we obtain:

\(SSE(F)=\sum(y_i-\hat{y}_i)^2=720.27\)

The full model appears to describe the trend in the data better than the reduced model.

The good news is that in the simple linear regression case, we don't have to bother with calculating the general linear F -statistic. Minitab does it for us in the ANOVA table.

Click on the light bulb to see the error in the full and reduced models.

## Analysis of Variance

As you can see, Minitab calculates and reports both SSE ( F ) — the amount of error associated with the full model — and SSE ( R ) — the amount of error associated with the reduced model. The F -statistic is:

\( F^*=\dfrac{MSR}{MSE}=\dfrac{504.04/1}{720.27/48}=\dfrac{504.04}{15.006}=33.59\)

and its associated P -value is < 0.001 (so we reject \(H_{0}\) and favor the full model). We can conclude that there is a statistically significant linear association between lifetime alcohol consumption and arm strength.

This concludes our discussion of our first aside from the general linear F-test. Now, we move on to our second aside from sequential sums of squares.

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One-way ANOVA compares three or more unmatched groups, based on the assumption that the populations are Gaussian.

The P value tests the null hypothesis that data from all groups are drawn from populations with identical means. Therefore, the P value answers this question:

If all the populations really have the same mean (the treatments are ineffective), what is the chance that random sampling would result in means as far apart (or more so) as observed in this experiment?

If the overall P value is large, the data do not give you any reason to conclude that the means differ. Even if the population means were equal, you would not be surprised to find sample means this far apart just by chance. This is not the same as saying that the true means are the same. You just don't have compelling evidence that they differ.

If the overall P value is small, then it is unlikely that the differences you observed are due to random sampling. You can reject the idea that all the populations have identical means. This doesn't mean that every mean differs from every other mean, only that at least one differs from the rest. Look at the results of post tests to identify where the differences are.

## F ratio and ANOVA table

The P value is computed from the F ratio which is computed from the ANOVA table.

ANOVA partitions the variability among all the values into one component that is due to variability among group means (due to the treatment) and another component that is due to variability within the groups (also called residual variation). Variability within groups (within the columns) is quantified as the sum of squares of the differences between each value and its group mean. This is the residual sum-of-squares. Variation among groups (due to treatment) is quantified as the sum of the squares of the differences between the group means and the grand mean (the mean of all values in all groups). Adjusted for the size of each group, this becomes the treatment sum-of-squares.

Each sum-of-squares is associated with a certain number of degrees of freedom (df, computed from number of subjects and number of groups), and the mean square (MS) is computed by dividing the sum-of-squares by the appropriate number of degrees of freedom. These can be thought of as variances. The square root of the mean square residual can be thought of as the pooled standard deviation.

The F ratio is the ratio of two mean square values. If the null hypothesis is true, you expect F to have a value close to 1.0 most of the time. A large F ratio means that the variation among group means is more than you'd expect to see by chance. You'll see a large F ratio both when the null hypothesis is wrong (the data are not sampled from populations with the same mean) and when random sampling happened to end up with large values in some groups and small values in others.

The P value is determined from the F ratio and the two values for degrees of freedom shown in the ANOVA table.

## Tests for equal variances

ANOVA is based on the assumption that the data are sampled from populations that all have the same standard deviations. Prism tests this assumption with two tests. It computes the Brown-Forsythe test and also (if every group has at least five values) computes Bartlett's test. There are no options for whether to run these tests. Prism automatically does so and always reports the results.

Both these tests compute a P value designed to answer this question:

If the populations really have the same standard deviations, what is the chance that you'd randomly select samples whose standard deviations are as different from one another (or more different) as they are in your experiment?

## Bartlett's test

Prism reports the results of the "corrected" Barlett's test as explained in section 10.6 of Zar(1). Bartlett's test works great if the data really are sampled from Gaussian distributions. But if the distributions deviate even slightly from the Gaussian ideal, Bartett's test may report a small P value even when the differences among standard deviations is trivial. For this reason, many do not recommend that test. That's why we added the test of Brown and Forsythe. It has the same goal as the Bartlett's test, but is less sensitive to minor deviations from normality. We suggest that you pay attention to the Brown-Forsythe result, and ignore Bartlett's test (which we left in to be consistent with prior versions of Prism).

## Brown-Forsythe test

The Brown-Forsythe test is conceptually simple. Each value in the data table is transformed by subtracting from it the median of that column, and then taking the absolute value of that difference. One-way ANOVA is run on these values, and the P value from that ANOVA is reported as the result of the Brown-Forsythe test.

How does it work. By subtracting the medians, any differences between medians have been subtracted away, so the only distinction between groups is their variability.

Why subtract the median and not the mean of each group? If you subtract the column mean instead of the column median, the test is called the Levene test for equal variances . Which is better? If the distributions are not quite Gaussian, it depends on what the distributions are. Simulations from several groups of statisticians show that using the median works well with many types of nongaussian data. Prism only uses the median (Brown-Forsythe) and not the mean (Levene).

## Interpreting the results

If the P value is small, you must decide whether you will conclude that the standard deviations of the populations are different. Obviously the tests of equal variances are based only on the values in this one experiment. Think about data from other similar experiments before making a conclusion.

If you conclude that the populations have different variances, you have four choices:

• Conclude that the populations are different. In many experimental contexts, the finding of different standard deviations is as important as the finding of different means. If the standard deviations are truly different, then the populations are different regardless of what ANOVA concludes about differences among the means. This may be the most important conclusion from the experiment.

• Transform the data to equalize the standard deviations, and then rerun the ANOVA. Often you'll find that converting values to their reciprocals or logarithms will equalize the standard deviations and also make the distributions more Gaussian.

• Use the Welch or Brown-Forsythe versions of one-way ANOVA that do not assume that all standard deviations are equal.

• Switch to the nonparametric Kruskal-Wallis test. The problem with this is that if your groups have very different standard deviations, it is difficult to interpret the results of the Kruskal-Wallis test. If the standard deviations are very different, then the shapes of the distributions are very different, and the kruskal-Wallis results cannot be interpreted as comparing medians.

R 2 is the fraction of the overall variance (of all the data, pooling all the groups) attributable to differences among the group means. It compares the variability among group means with the variability within the groups. A large value means that a large fraction of the variation is due to the treatment that defines the groups. The R 2 value is calculated from the ANOVA table and equals the between group sum-of-squares divided by the total sum-of-squares. Some programs (and books) don't bother reporting this value. Others refer to it as η 2 (eta squared) rather than R 2 . It is a descriptive statistic that quantifies the strength of the relationship between group membership and the variable you measured.

J.H. Zar, Biostatistical Analysis , Fifth edition 2010, ISBN: 0131008463.

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Home » ANOVA (Analysis of variance) – Formulas, Types, and Examples

## ANOVA (Analysis of variance) – Formulas, Types, and Examples

Table of Contents

## Analysis of Variance (ANOVA)

Analysis of Variance (ANOVA) is a statistical method used to test differences between two or more means. It is similar to the t-test, but the t-test is generally used for comparing two means, while ANOVA is used when you have more than two means to compare.

ANOVA is based on comparing the variance (or variation) between the data samples to the variation within each particular sample. If the between-group variance is high and the within-group variance is low, this provides evidence that the means of the groups are significantly different.

## ANOVA Terminology

When discussing ANOVA, there are several key terms to understand:

- Factor : This is another term for the independent variable in your analysis. In a one-way ANOVA, there is one factor, while in a two-way ANOVA, there are two factors.
- Levels : These are the different groups or categories within a factor. For example, if the factor is ‘diet’ the levels might be ‘low fat’, ‘medium fat’, and ‘high fat’.
- Response Variable : This is the dependent variable or the outcome that you are measuring.
- Within-group Variance : This is the variance or spread of scores within each level of your factor.
- Between-group Variance : This is the variance or spread of scores between the different levels of your factor.
- Grand Mean : This is the overall mean when you consider all the data together, regardless of the factor level.
- Treatment Sums of Squares (SS) : This represents the between-group variability. It is the sum of the squared differences between the group means and the grand mean.
- Error Sums of Squares (SS) : This represents the within-group variability. It’s the sum of the squared differences between each observation and its group mean.
- Total Sums of Squares (SS) : This is the sum of the Treatment SS and the Error SS. It represents the total variability in the data.
- Degrees of Freedom (df) : The degrees of freedom are the number of values that have the freedom to vary when computing a statistic. For example, if you have ‘n’ observations in one group, then the degrees of freedom for that group is ‘n-1’.
- Mean Square (MS) : Mean Square is the average squared deviation and is calculated by dividing the sum of squares by the corresponding degrees of freedom.
- F-Ratio : This is the test statistic for ANOVAs, and it’s the ratio of the between-group variance to the within-group variance. If the between-group variance is significantly larger than the within-group variance, the F-ratio will be large and likely significant.
- Null Hypothesis (H0) : This is the hypothesis that there is no difference between the group means.
- Alternative Hypothesis (H1) : This is the hypothesis that there is a difference between at least two of the group means.
- p-value : This is the probability of obtaining a test statistic as extreme as the one that was actually observed, assuming that the null hypothesis is true. If the p-value is less than the significance level (usually 0.05), then the null hypothesis is rejected in favor of the alternative hypothesis.
- Post-hoc tests : These are follow-up tests conducted after an ANOVA when the null hypothesis is rejected, to determine which specific groups’ means (levels) are different from each other. Examples include Tukey’s HSD, Scheffe, Bonferroni, among others.

## Types of ANOVA

Types of ANOVA are as follows:

## One-way (or one-factor) ANOVA

This is the simplest type of ANOVA, which involves one independent variable . For example, comparing the effect of different types of diet (vegetarian, pescatarian, omnivore) on cholesterol level.

## Two-way (or two-factor) ANOVA

This involves two independent variables. This allows for testing the effect of each independent variable on the dependent variable , as well as testing if there’s an interaction effect between the independent variables on the dependent variable.

## Repeated Measures ANOVA

This is used when the same subjects are measured multiple times under different conditions, or at different points in time. This type of ANOVA is often used in longitudinal studies.

## Mixed Design ANOVA

This combines features of both between-subjects (independent groups) and within-subjects (repeated measures) designs. In this model, one factor is a between-subjects variable and the other is a within-subjects variable.

## Multivariate Analysis of Variance (MANOVA)

This is used when there are two or more dependent variables. It tests whether changes in the independent variable(s) correspond to changes in the dependent variables.

## Analysis of Covariance (ANCOVA)

This combines ANOVA and regression. ANCOVA tests whether certain factors have an effect on the outcome variable after removing the variance for which quantitative covariates (interval variables) account. This allows the comparison of one variable outcome between groups, while statistically controlling for the effect of other continuous variables that are not of primary interest.

## Nested ANOVA

This model is used when the groups can be clustered into categories. For example, if you were comparing students’ performance from different classrooms and different schools, “classroom” could be nested within “school.”

## ANOVA Formulas

ANOVA Formulas are as follows:

Sum of Squares Total (SST)

This represents the total variability in the data. It is the sum of the squared differences between each observation and the overall mean.

- yi represents each individual data point
- y_mean represents the grand mean (mean of all observations)

Sum of Squares Within (SSW)

This represents the variability within each group or factor level. It is the sum of the squared differences between each observation and its group mean.

- yij represents each individual data point within a group
- y_meani represents the mean of the ith group

Sum of Squares Between (SSB)

This represents the variability between the groups. It is the sum of the squared differences between the group means and the grand mean, multiplied by the number of observations in each group.

- ni represents the number of observations in each group
- y_mean represents the grand mean

Degrees of Freedom

The degrees of freedom are the number of values that have the freedom to vary when calculating a statistic.

For within groups (dfW):

For between groups (dfB):

For total (dfT):

- N represents the total number of observations
- k represents the number of groups

Mean Squares

Mean squares are the sum of squares divided by the respective degrees of freedom.

Mean Squares Between (MSB):

Mean Squares Within (MSW):

F-Statistic

The F-statistic is used to test whether the variability between the groups is significantly greater than the variability within the groups.

If the F-statistic is significantly higher than what would be expected by chance, we reject the null hypothesis that all group means are equal.

## Examples of ANOVA

Examples 1:

Suppose a psychologist wants to test the effect of three different types of exercise (yoga, aerobic exercise, and weight training) on stress reduction. The dependent variable is the stress level, which can be measured using a stress rating scale.

Here are hypothetical stress ratings for a group of participants after they followed each of the exercise regimes for a period:

- Yoga: [3, 2, 2, 1, 2, 2, 3, 2, 1, 2]
- Aerobic Exercise: [2, 3, 3, 2, 3, 2, 3, 3, 2, 2]
- Weight Training: [4, 4, 5, 5, 4, 5, 4, 5, 4, 5]

The psychologist wants to determine if there is a statistically significant difference in stress levels between these different types of exercise.

To conduct the ANOVA:

1. State the hypotheses:

- Null Hypothesis (H0): There is no difference in mean stress levels between the three types of exercise.
- Alternative Hypothesis (H1): There is a difference in mean stress levels between at least two of the types of exercise.

2. Calculate the ANOVA statistics:

- Compute the Sum of Squares Between (SSB), Sum of Squares Within (SSW), and Sum of Squares Total (SST).
- Calculate the Degrees of Freedom (dfB, dfW, dfT).
- Calculate the Mean Squares Between (MSB) and Mean Squares Within (MSW).
- Compute the F-statistic (F = MSB / MSW).

3. Check the p-value associated with the calculated F-statistic.

- If the p-value is less than the chosen significance level (often 0.05), then we reject the null hypothesis in favor of the alternative hypothesis. This suggests there is a statistically significant difference in mean stress levels between the three exercise types.

4. Post-hoc tests

- If we reject the null hypothesis, we conduct a post-hoc test to determine which specific groups’ means (exercise types) are different from each other.

Examples 2:

Suppose an agricultural scientist wants to compare the yield of three varieties of wheat. The scientist randomly selects four fields for each variety and plants them. After harvest, the yield from each field is measured in bushels. Here are the hypothetical yields:

The scientist wants to know if the differences in yields are due to the different varieties or just random variation.

Here’s how to apply the one-way ANOVA to this situation:

- Null Hypothesis (H0): The means of the three populations are equal.
- Alternative Hypothesis (H1): At least one population mean is different.
- Calculate the Degrees of Freedom (dfB for between groups, dfW for within groups, dfT for total).
- If the p-value is less than the chosen significance level (often 0.05), then we reject the null hypothesis in favor of the alternative hypothesis. This would suggest there is a statistically significant difference in mean yields among the three varieties.
- If we reject the null hypothesis, we conduct a post-hoc test to determine which specific groups’ means (wheat varieties) are different from each other.

## How to Conduct ANOVA

Conducting an Analysis of Variance (ANOVA) involves several steps. Here’s a general guideline on how to perform it:

- Null Hypothesis (H0): The means of all groups are equal.
- Alternative Hypothesis (H1): At least one group mean is different from the others.
- The significance level (often denoted as α) is usually set at 0.05. This implies that you are willing to accept a 5% chance that you are wrong in rejecting the null hypothesis.
- Data should be collected for each group under study. Make sure that the data meet the assumptions of an ANOVA: normality, independence, and homogeneity of variances.
- Calculate the Degrees of Freedom (df) for each sum of squares (dfB, dfW, dfT).
- Compute the Mean Squares Between (MSB) and Mean Squares Within (MSW) by dividing the sum of squares by the corresponding degrees of freedom.
- Compute the F-statistic as the ratio of MSB to MSW.
- Determine the critical F-value from the F-distribution table using dfB and dfW.
- If the calculated F-statistic is greater than the critical F-value, reject the null hypothesis.
- If the p-value associated with the calculated F-statistic is smaller than the significance level (0.05 typically), you reject the null hypothesis.
- If you rejected the null hypothesis, you can conduct post-hoc tests (like Tukey’s HSD) to determine which specific groups’ means (if you have more than two groups) are different from each other.
- Regardless of the result, report your findings in a clear, understandable manner. This typically includes reporting the test statistic, p-value, and whether the null hypothesis was rejected.

## When to use ANOVA

ANOVA (Analysis of Variance) is used when you have three or more groups and you want to compare their means to see if they are significantly different from each other. It is a statistical method that is used in a variety of research scenarios. Here are some examples of when you might use ANOVA:

- Comparing Groups : If you want to compare the performance of more than two groups, for example, testing the effectiveness of different teaching methods on student performance.
- Evaluating Interactions : In a two-way or factorial ANOVA, you can test for an interaction effect. This means you are not only interested in the effect of each individual factor, but also whether the effect of one factor depends on the level of another factor.
- Repeated Measures : If you have measured the same subjects under different conditions or at different time points, you can use repeated measures ANOVA to compare the means of these repeated measures while accounting for the correlation between measures from the same subject.
- Experimental Designs : ANOVA is often used in experimental research designs when subjects are randomly assigned to different conditions and the goal is to compare the means of the conditions.

Here are the assumptions that must be met to use ANOVA:

- Normality : The data should be approximately normally distributed.
- Homogeneity of Variances : The variances of the groups you are comparing should be roughly equal. This assumption can be tested using Levene’s test or Bartlett’s test.
- Independence : The observations should be independent of each other. This assumption is met if the data is collected appropriately with no related groups (e.g., twins, matched pairs, repeated measures).

## Applications of ANOVA

The Analysis of Variance (ANOVA) is a powerful statistical technique that is used widely across various fields and industries. Here are some of its key applications:

Agriculture

ANOVA is commonly used in agricultural research to compare the effectiveness of different types of fertilizers, crop varieties, or farming methods. For example, an agricultural researcher could use ANOVA to determine if there are significant differences in the yields of several varieties of wheat under the same conditions.

Manufacturing and Quality Control

ANOVA is used to determine if different manufacturing processes or machines produce different levels of product quality. For instance, an engineer might use it to test whether there are differences in the strength of a product based on the machine that produced it.

Marketing Research

Marketers often use ANOVA to test the effectiveness of different advertising strategies. For example, a marketer could use ANOVA to determine whether different marketing messages have a significant impact on consumer purchase intentions.

Healthcare and Medicine

In medical research, ANOVA can be used to compare the effectiveness of different treatments or drugs. For example, a medical researcher could use ANOVA to test whether there are significant differences in recovery times for patients who receive different types of therapy.

ANOVA is used in educational research to compare the effectiveness of different teaching methods or educational interventions. For example, an educator could use it to test whether students perform significantly differently when taught with different teaching methods.

Psychology and Social Sciences

Psychologists and social scientists use ANOVA to compare group means on various psychological and social variables. For example, a psychologist could use it to determine if there are significant differences in stress levels among individuals in different occupations.

Biology and Environmental Sciences

Biologists and environmental scientists use ANOVA to compare different biological and environmental conditions. For example, an environmental scientist could use it to determine if there are significant differences in the levels of a pollutant in different bodies of water.

## Advantages of ANOVA

Here are some advantages of using ANOVA:

Comparing Multiple Groups: One of the key advantages of ANOVA is the ability to compare the means of three or more groups. This makes it more powerful and flexible than the t-test, which is limited to comparing only two groups.

Control of Type I Error: When comparing multiple groups, the chances of making a Type I error (false positive) increases. One of the strengths of ANOVA is that it controls the Type I error rate across all comparisons. This is in contrast to performing multiple pairwise t-tests which can inflate the Type I error rate.

Testing Interactions: In factorial ANOVA, you can test not only the main effect of each factor, but also the interaction effect between factors. This can provide valuable insights into how different factors or variables interact with each other.

Handling Continuous and Categorical Variables: ANOVA can handle both continuous and categorical variables . The dependent variable is continuous and the independent variables are categorical.

Robustness: ANOVA is considered robust to violations of normality assumption when group sizes are equal. This means that even if your data do not perfectly meet the normality assumption, you might still get valid results.

Provides Detailed Analysis: ANOVA provides a detailed breakdown of variances and interactions between variables which can be useful in understanding the underlying factors affecting the outcome.

Capability to Handle Complex Experimental Designs: Advanced types of ANOVA (like repeated measures ANOVA, MANOVA, etc.) can handle more complex experimental designs, including those where measurements are taken on the same subjects over time, or when you want to analyze multiple dependent variables at once.

## Disadvantages of ANOVA

Some limitations or disadvantages that are important to consider:

Assumptions: ANOVA relies on several assumptions including normality (the data follows a normal distribution), independence (the observations are independent of each other), and homogeneity of variances (the variances of the groups are roughly equal). If these assumptions are violated, the results of the ANOVA may not be valid.

Sensitivity to Outliers: ANOVA can be sensitive to outliers. A single extreme value in one group can affect the sum of squares and consequently influence the F-statistic and the overall result of the test.

Dichotomous Variables: ANOVA is not suitable for dichotomous variables (variables that can take only two values, like yes/no or male/female). It is used to compare the means of groups for a continuous dependent variable.

Lack of Specificity: Although ANOVA can tell you that there is a significant difference between groups, it doesn’t tell you which specific groups are significantly different from each other. You need to carry out further post-hoc tests (like Tukey’s HSD or Bonferroni) for these pairwise comparisons.

Complexity with Multiple Factors: When dealing with multiple factors and interactions in factorial ANOVA, interpretation can become complex. The presence of interaction effects can make main effects difficult to interpret.

Requires Larger Sample Sizes: To detect an effect of a certain size, ANOVA generally requires larger sample sizes than a t-test.

Equal Group Sizes: While not always a strict requirement, ANOVA is most powerful and its assumptions are most likely to be met when groups are of equal or similar sizes.

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## One-way ANOVA | When and How to Use It (With Examples)

Published on March 6, 2020 by Rebecca Bevans . Revised on June 22, 2023.

ANOVA , which stands for Analysis of Variance, is a statistical test used to analyze the difference between the means of more than two groups.

A one-way ANOVA uses one independent variable , while a two-way ANOVA uses two independent variables.

## Table of contents

When to use a one-way anova, how does an anova test work, assumptions of anova, performing a one-way anova, interpreting the results, post-hoc testing, reporting the results of anova, other interesting articles, frequently asked questions about one-way anova.

Use a one-way ANOVA when you have collected data about one categorical independent variable and one quantitative dependent variable . The independent variable should have at least three levels (i.e. at least three different groups or categories).

ANOVA tells you if the dependent variable changes according to the level of the independent variable. For example:

- Your independent variable is social media use , and you assign groups to low , medium , and high levels of social media use to find out if there is a difference in hours of sleep per night .
- Your independent variable is brand of soda , and you collect data on Coke , Pepsi , Sprite , and Fanta to find out if there is a difference in the price per 100ml .
- You independent variable is type of fertilizer , and you treat crop fields with mixtures 1 , 2 and 3 to find out if there is a difference in crop yield .

The null hypothesis ( H 0 ) of ANOVA is that there is no difference among group means. The alternative hypothesis ( H a ) is that at least one group differs significantly from the overall mean of the dependent variable.

If you only want to compare two groups, use a t test instead.

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ANOVA determines whether the groups created by the levels of the independent variable are statistically different by calculating whether the means of the treatment levels are different from the overall mean of the dependent variable.

If any of the group means is significantly different from the overall mean, then the null hypothesis is rejected.

ANOVA uses the F test for statistical significance . This allows for comparison of multiple means at once, because the error is calculated for the whole set of comparisons rather than for each individual two-way comparison (which would happen with a t test).

The F test compares the variance in each group mean from the overall group variance. If the variance within groups is smaller than the variance between groups , the F test will find a higher F value, and therefore a higher likelihood that the difference observed is real and not due to chance.

The assumptions of the ANOVA test are the same as the general assumptions for any parametric test:

- Independence of observations : the data were collected using statistically valid sampling methods , and there are no hidden relationships among observations. If your data fail to meet this assumption because you have a confounding variable that you need to control for statistically, use an ANOVA with blocking variables.
- Normally-distributed response variable : The values of the dependent variable follow a normal distribution .
- Homogeneity of variance : The variation within each group being compared is similar for every group. If the variances are different among the groups, then ANOVA probably isn’t the right fit for the data.

While you can perform an ANOVA by hand , it is difficult to do so with more than a few observations. We will perform our analysis in the R statistical program because it is free, powerful, and widely available. For a full walkthrough of this ANOVA example, see our guide to performing ANOVA in R .

The sample dataset from our imaginary crop yield experiment contains data about:

- fertilizer type (type 1, 2, or 3)
- planting density (1 = low density, 2 = high density)
- planting location in the field (blocks 1, 2, 3, or 4)
- final crop yield (in bushels per acre).

This gives us enough information to run various different ANOVA tests and see which model is the best fit for the data.

For the one-way ANOVA, we will only analyze the effect of fertilizer type on crop yield.

Sample dataset for ANOVA

After loading the dataset into our R environment, we can use the command aov() to run an ANOVA. In this example we will model the differences in the mean of the response variable , crop yield, as a function of type of fertilizer.

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To view the summary of a statistical model in R, use the summary() function.

The summary of an ANOVA test (in R) looks like this:

The ANOVA output provides an estimate of how much variation in the dependent variable that can be explained by the independent variable.

- The first column lists the independent variable along with the model residuals (aka the model error).
- The Df column displays the degrees of freedom for the independent variable (calculated by taking the number of levels within the variable and subtracting 1), and the degrees of freedom for the residuals (calculated by taking the total number of observations minus 1, then subtracting the number of levels in each of the independent variables).
- The Sum Sq column displays the sum of squares (a.k.a. the total variation) between the group means and the overall mean explained by that variable. The sum of squares for the fertilizer variable is 6.07, while the sum of squares of the residuals is 35.89.
- The Mean Sq column is the mean of the sum of squares, which is calculated by dividing the sum of squares by the degrees of freedom.
- The F value column is the test statistic from the F test: the mean square of each independent variable divided by the mean square of the residuals. The larger the F value, the more likely it is that the variation associated with the independent variable is real and not due to chance.
- The Pr(>F) column is the p value of the F statistic. This shows how likely it is that the F value calculated from the test would have occurred if the null hypothesis of no difference among group means were true.

Because the p value of the independent variable, fertilizer, is statistically significant ( p < 0.05), it is likely that fertilizer type does have a significant effect on average crop yield.

ANOVA will tell you if there are differences among the levels of the independent variable, but not which differences are significant. To find how the treatment levels differ from one another, perform a TukeyHSD (Tukey’s Honestly-Significant Difference) post-hoc test.

The Tukey test runs pairwise comparisons among each of the groups, and uses a conservative error estimate to find the groups which are statistically different from one another.

The output of the TukeyHSD looks like this:

First, the table reports the model being tested (‘Fit’). Next it lists the pairwise differences among groups for the independent variable.

Under the ‘$fertilizer’ section, we see the mean difference between each fertilizer treatment (‘diff’), the lower and upper bounds of the 95% confidence interval (‘lwr’ and ‘upr’), and the p value , adjusted for multiple pairwise comparisons.

The pairwise comparisons show that fertilizer type 3 has a significantly higher mean yield than both fertilizer 2 and fertilizer 1, but the difference between the mean yields of fertilizers 2 and 1 is not statistically significant.

When reporting the results of an ANOVA, include a brief description of the variables you tested, the F value, degrees of freedom, and p values for each independent variable, and explain what the results mean.

If you want to provide more detailed information about the differences found in your test, you can also include a graph of the ANOVA results , with grouping letters above each level of the independent variable to show which groups are statistically different from one another:

If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples.

- Chi square test of independence
- Statistical power
- Descriptive statistics
- Degrees of freedom
- Pearson correlation
- Null hypothesis

Methodology

- Double-blind study
- Case-control study
- Research ethics
- Data collection
- Hypothesis testing
- Structured interviews

Research bias

- Hawthorne effect
- Unconscious bias
- Recall bias
- Halo effect
- Self-serving bias
- Information bias

The only difference between one-way and two-way ANOVA is the number of independent variables . A one-way ANOVA has one independent variable, while a two-way ANOVA has two.

- One-way ANOVA : Testing the relationship between shoe brand (Nike, Adidas, Saucony, Hoka) and race finish times in a marathon.
- Two-way ANOVA : Testing the relationship between shoe brand (Nike, Adidas, Saucony, Hoka), runner age group (junior, senior, master’s), and race finishing times in a marathon.

All ANOVAs are designed to test for differences among three or more groups. If you are only testing for a difference between two groups, use a t-test instead.

A factorial ANOVA is any ANOVA that uses more than one categorical independent variable . A two-way ANOVA is a type of factorial ANOVA.

Some examples of factorial ANOVAs include:

- Testing the combined effects of vaccination (vaccinated or not vaccinated) and health status (healthy or pre-existing condition) on the rate of flu infection in a population.
- Testing the effects of marital status (married, single, divorced, widowed), job status (employed, self-employed, unemployed, retired), and family history (no family history, some family history) on the incidence of depression in a population.
- Testing the effects of feed type (type A, B, or C) and barn crowding (not crowded, somewhat crowded, very crowded) on the final weight of chickens in a commercial farming operation.

In ANOVA, the null hypothesis is that there is no difference among group means. If any group differs significantly from the overall group mean, then the ANOVA will report a statistically significant result.

Significant differences among group means are calculated using the F statistic, which is the ratio of the mean sum of squares (the variance explained by the independent variable) to the mean square error (the variance left over).

If the F statistic is higher than the critical value (the value of F that corresponds with your alpha value, usually 0.05), then the difference among groups is deemed statistically significant.

Quantitative variables are any variables where the data represent amounts (e.g. height, weight, or age).

Categorical variables are any variables where the data represent groups. This includes rankings (e.g. finishing places in a race), classifications (e.g. brands of cereal), and binary outcomes (e.g. coin flips).

You need to know what type of variables you are working with to choose the right statistical test for your data and interpret your results .

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## 11.3: Hypotheses in ANOVA

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- Michelle Oja
- Taft College

So far we have seen what ANOVA is used for, why we use it, and how we use it. Now we can turn to the formal hypotheses we will be testing. As with before, we have a null and a research hypothesis to lay out.

## Research Hypotheses

Our research hypothesis for ANOVA is more complex with more than two groups. Let’s take a look at it and then dive deeper into what it means.

What the ANOVA tests is whether there is a difference between any one set of means, but usually we still have expected directions of what means we think will be bigger than what other means. Let's work out an example. Let's say that my IV is mindset, and the three groups (levels) are:

- Growth Mindset
- Mixed Mindset (some Growth ideas and some Fixed ideas)
- Fixed Mindset

If we are measuring passing rates in math, we could write this all out in one sentence and one line of symbols:

- Research Hypothesis: Students with Growth Mindset with have higher average passing rates in math than students with either a mixed mindset or Fixed Mindset, but Fixed Mindset will have similar average passing rates to students with mixed mindset.
- Symbols: \( \overline{X}_{G} > \overline{X}_{M} = \overline{X}_{F} \)

But it ends up being easier to write out each pair of means:

- Research Hypothesis: Students with Growth Mindset with have higher average passing rates in math than students with a mixed mindset. Students with Growth Mindset with have higher average passing rates in math than students with a Fixed Mindset. Students with a Fixed Mindset will have similar average passing rates to students with mixed mindset.
- \( \overline{X}_{G} > \overline{X}_{M} \)
- \( \overline{X}_{G} > \overline{X}_{F} \)
- \( \overline{X}_{M} = \overline{X}_{F} \)

What you might notice is that one of these looks like a null hypothesis (no difference between the means)! And that is okay, as long as the research hypothesis predicts that at least one mean will differ from at least one other mean. It doesn't matter what order you list these means in; it helps to match the research hypothesis, but it's really to help you conceptualize the relationships that you are predicting so put it in the order that makes the most sense to you!

Why is it better to list out each pair of means? Well, look at this research hypothesis:

- Research Hypothesis: Students with Growth Mindset with have a similar average passing rate in math as students with a mixed mindset. Students with Growth Mindset with have higher average passing rates in math than students with a Fixed Mindset. Students with a Fixed Mindset will have similar average passing rates to students with mixed mindset.
- \( \overline{X}_{G} = \overline{X}_{M} \)

If you try to write that out in one line of symbols, it'll get confusing because you won't be able to easily show all three predictions. And if you have more than three groups, many research hypotheses won't be able to be represented in one line.

Another reason that this makes more sense is that each mean will be statistically compared with each other mean if the ANOVA results end up rejecting the null hypothesis. If you set up your research hypotheses this way in the first place (in pairs of means), then these pairwise comparisons make more sense later.

## Null Hypotheses

Our null hypothesis is still the idea of “no difference” in our data. Because we have multiple group means, we simply list them out as equal to each other:

- Null Hypothesis: Students with Growth Mindset, mixed mindset, and Fixed Mindset will have similar average passing rates in math .
- Symbols: \( \overline{X}_{G} = \overline{X}_{M} = \overline{X}_{F} \)

You can list them all out, as well, but it's less necessary with a null hypothesis:

- Research Hypothesis: Students with Growth Mindset with have a similar average passing rate in math as students with a mixed mindset. Students with Growth Mindset with have a similar average passing rates in math than students with a Fixed Mindset. Students with a Fixed Mindset will have similar average passing rates to students with mixed mindset.
- \( \overline{X}_{G} = \overline{X}_{F} \)

## Null Hypothesis Significance Testing

In our studies so far, when we've calculated an inferential test statistics, like a t-score, what do we do next? Compare it to a critical value in a table! And that's the same thing that we do with our calculated F-value. We compare our calculated value to our critical value to determine if we retain or reject the null hypothesis that all of the means are similar.

(Critical \(<\) Calculated) \(=\) Reject null \(=\) At least one mean is different from at least one other mean. \(= p<.05\)

(Critical \(>\) Calculated) \(=\) Retain null \(=\) All of the means are similar. \(= p>.05\)

## What does Rejecting the Null Hypothesis Mean for a Research Hypothesis with Three or More Groups?

Remember when we rejected the null hypothesis when comparing two means with a t-test that we didn't have to do any additional comparisons; rejecting the null hypothesis with a t-test tells us that the two means are statistically significantly different, which means that the bigger mean was statistically significantly bigger. All we had to do was make sure that the means were in the direction that the research hypothesis predicted.

Unfortunately, with three or more group means, we do have to do additional statistical comparisons to test which means are statistically significantly different from which other means. The ANOVA only tells us that at least one mean is different from one other mean. So, rejecting the null hypothesis doesn't really tell us whether our research hypothesis is (fully) supported, partially supported, or not supported. When the null hypothesis is rejected, we will know that a difference exists somewhere, but we will not know where that difference is. Is Growth Mindset different from mixed mindset and Fixed Mindset, but mixed and Fixed are the same? Is Growth Mindset different from both mixed and Fixed Mindset? Are all three of them different from each other? And even if the means are different, are they different in the hypothesized direction? Does Growth Mindset always have a higher mean? We will come back to this issue later and see how to find out specific differences. For now, just remember that an ANOVA tests for any difference in group means, and it does not matter where that difference occurs. We must follow-up with any significant ANOVA to see which means are different from each other, and whether those mean differences (fully) support, partially support, or do not support the research hypothesis.

- 9.1 Null and Alternative Hypotheses
- Introduction
- 1.1 Definitions of Statistics, Probability, and Key Terms
- 1.2 Data, Sampling, and Variation in Data and Sampling
- 1.3 Frequency, Frequency Tables, and Levels of Measurement
- 1.4 Experimental Design and Ethics
- 1.5 Data Collection Experiment
- 1.6 Sampling Experiment
- Chapter Review
- Bringing It Together: Homework
- 2.1 Stem-and-Leaf Graphs (Stemplots), Line Graphs, and Bar Graphs
- 2.2 Histograms, Frequency Polygons, and Time Series Graphs
- 2.3 Measures of the Location of the Data
- 2.4 Box Plots
- 2.5 Measures of the Center of the Data
- 2.6 Skewness and the Mean, Median, and Mode
- 2.7 Measures of the Spread of the Data
- 2.8 Descriptive Statistics
- Formula Review
- 3.1 Terminology
- 3.2 Independent and Mutually Exclusive Events
- 3.3 Two Basic Rules of Probability
- 3.4 Contingency Tables
- 3.5 Tree and Venn Diagrams
- 3.6 Probability Topics
- Bringing It Together: Practice
- 4.1 Probability Distribution Function (PDF) for a Discrete Random Variable
- 4.2 Mean or Expected Value and Standard Deviation
- 4.3 Binomial Distribution (Optional)
- 4.4 Geometric Distribution (Optional)
- 4.5 Hypergeometric Distribution (Optional)
- 4.6 Poisson Distribution (Optional)
- 4.7 Discrete Distribution (Playing Card Experiment)
- 4.8 Discrete Distribution (Lucky Dice Experiment)
- 5.1 Continuous Probability Functions
- 5.2 The Uniform Distribution
- 5.3 The Exponential Distribution (Optional)
- 5.4 Continuous Distribution
- 6.1 The Standard Normal Distribution
- 6.2 Using the Normal Distribution
- 6.3 Normal Distribution—Lap Times
- 6.4 Normal Distribution—Pinkie Length
- 7.1 The Central Limit Theorem for Sample Means (Averages)
- 7.2 The Central Limit Theorem for Sums (Optional)
- 7.3 Using the Central Limit Theorem
- 7.4 Central Limit Theorem (Pocket Change)
- 7.5 Central Limit Theorem (Cookie Recipes)
- 8.1 A Single Population Mean Using the Normal Distribution
- 8.2 A Single Population Mean Using the Student's t-Distribution
- 8.3 A Population Proportion
- 8.4 Confidence Interval (Home Costs)
- 8.5 Confidence Interval (Place of Birth)
- 8.6 Confidence Interval (Women's Heights)
- 9.2 Outcomes and the Type I and Type II Errors
- 9.3 Distribution Needed for Hypothesis Testing
- 9.4 Rare Events, the Sample, and the Decision and Conclusion
- 9.5 Additional Information and Full Hypothesis Test Examples
- 9.6 Hypothesis Testing of a Single Mean and Single Proportion
- 10.1 Two Population Means with Unknown Standard Deviations
- 10.2 Two Population Means with Known Standard Deviations
- 10.3 Comparing Two Independent Population Proportions
- 10.4 Matched or Paired Samples (Optional)
- 10.5 Hypothesis Testing for Two Means and Two Proportions
- 11.1 Facts About the Chi-Square Distribution
- 11.2 Goodness-of-Fit Test
- 11.3 Test of Independence
- 11.4 Test for Homogeneity
- 11.5 Comparison of the Chi-Square Tests
- 11.6 Test of a Single Variance
- 11.7 Lab 1: Chi-Square Goodness-of-Fit
- 11.8 Lab 2: Chi-Square Test of Independence
- 12.1 Linear Equations
- 12.2 The Regression Equation
- 12.3 Testing the Significance of the Correlation Coefficient (Optional)
- 12.4 Prediction (Optional)
- 12.5 Outliers
- 12.6 Regression (Distance from School) (Optional)
- 12.7 Regression (Textbook Cost) (Optional)
- 12.8 Regression (Fuel Efficiency) (Optional)
- 13.1 One-Way ANOVA
- 13.2 The F Distribution and the F Ratio
- 13.3 Facts About the F Distribution
- 13.4 Test of Two Variances
- 13.5 Lab: One-Way ANOVA
- A | Appendix A Review Exercises (Ch 3–13)
- B | Appendix B Practice Tests (1–4) and Final Exams
- C | Data Sets
- D | Group and Partner Projects
- E | Solution Sheets
- F | Mathematical Phrases, Symbols, and Formulas
- G | Notes for the TI-83, 83+, 84, 84+ Calculators

The actual test begins by considering two hypotheses . They are called the null hypothesis and the alternative hypothesis . These hypotheses contain opposing viewpoints.

H 0 , the — null hypothesis: a statement of no difference between sample means or proportions or no difference between a sample mean or proportion and a population mean or proportion. In other words, the difference equals 0.

H a —, the alternative hypothesis: a claim about the population that is contradictory to H 0 and what we conclude when we reject H 0 .

Since the null and alternative hypotheses are contradictory, you must examine evidence to decide if you have enough evidence to reject the null hypothesis or not. The evidence is in the form of sample data.

After you have determined which hypothesis the sample supports, you make a decision. There are two options for a decision. They are reject H 0 if the sample information favors the alternative hypothesis or do not reject H 0 or decline to reject H 0 if the sample information is insufficient to reject the null hypothesis.

Mathematical Symbols Used in H 0 and H a :

H 0 always has a symbol with an equal in it. H a never has a symbol with an equal in it. The choice of symbol depends on the wording of the hypothesis test. However, be aware that many researchers use = in the null hypothesis, even with > or < as the symbol in the alternative hypothesis. This practice is acceptable because we only make the decision to reject or not reject the null hypothesis.

## Example 9.1

H 0 : No more than 30 percent of the registered voters in Santa Clara County voted in the primary election. p ≤ 30 H a : More than 30 percent of the registered voters in Santa Clara County voted in the primary election. p > 30

A medical trial is conducted to test whether or not a new medicine reduces cholesterol by 25 percent. State the null and alternative hypotheses.

## Example 9.2

We want to test whether the mean GPA of students in American colleges is different from 2.0 (out of 4.0). The null and alternative hypotheses are the following: H 0 : μ = 2.0 H a : μ ≠ 2.0

We want to test whether the mean height of eighth graders is 66 inches. State the null and alternative hypotheses. Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

- H 0 : μ __ 66
- H a : μ __ 66

## Example 9.3

We want to test if college students take fewer than five years to graduate from college, on the average. The null and alternative hypotheses are the following: H 0 : μ ≥ 5 H a : μ < 5

We want to test if it takes fewer than 45 minutes to teach a lesson plan. State the null and alternative hypotheses. Fill in the correct symbol ( =, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

- H 0 : μ __ 45
- H a : μ __ 45

## Example 9.4

An article on school standards stated that about half of all students in France, Germany, and Israel take advanced placement exams and a third of the students pass. The same article stated that 6.6 percent of U.S. students take advanced placement exams and 4.4 percent pass. Test if the percentage of U.S. students who take advanced placement exams is more than 6.6 percent. State the null and alternative hypotheses. H 0 : p ≤ 0.066 H a : p > 0.066

On a state driver’s test, about 40 percent pass the test on the first try. We want to test if more than 40 percent pass on the first try. Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

- H 0 : p __ 0.40
- H a : p __ 0.40

## Collaborative Exercise

Bring to class a newspaper, some news magazines, and some internet articles. In groups, find articles from which your group can write null and alternative hypotheses. Discuss your hypotheses with the rest of the class.

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## ANOVA for Hypothesis Testing

- January 30, 2024
- Machine Learning

ANOVA, which stands for Analysis of Variance, is a statistical method used to analyze differences among group means and their associated procedures. It’s an essential tool in Data Science, particularly when comparing three or more groups for statistical significance. So, if you want to learn about ANOVA and how to implement it for Hypothesis Testing , this article is for you. In this article, I’ll take you through a guide to ANOVA for Hypothesis Testing with implementation using Python.

## Introduction to ANOVA

ANOVA is a Hypothesis Testing technique used when the objective is to compare the means of three or more different groups to determine if they are statistically different from each other. It’s a technique that helps understand whether the differences observed in data samples are meaningful or simply due to random variation.

The primary purpose of ANOVA is to test if there is any statistically significant difference between the means of different groups. It is done under the hypothesis that any observed differences in sample means are due to random variation. ANOVA helps in determining whether the evidence is strong enough to reject this hypothesis, thus concluding that the differences in means are statistically significant.

There are two main types of ANOVA:

- One-way ANOVA : Used when there is only one independent variable. It assesses the impact of this independent variable on a single dependent variable. For instance, it can be used to compare the test scores of students from different classrooms (the independent variable being the classroom and the dependent variable being the test scores). The “one-way” refers to the fact that there is only one factor being considered.
- Two-way ANOVA : When the analysis involves two independent variables, we use two-way ANOVA. It not only assesses the individual effects of each independent variable on the dependent variable, but also looks at the interaction effect between the two independent variables. For example, in a study looking at the effect of diet and exercise on weight loss, a two-way ANOVA could determine not only the individual effects of diet and exercise but also whether there is a combined effect of diet and exercise on weight loss.

## How does ANOVA work?

Let’s understand how ANOVA works step by step:

- Step 1: Assume that all group means are equal (there’s no effect or difference).
- Step 2: Determine the variance within each group and the variance between groups.
- Step 3: Compute the F-ratio, a ratio of the variance between the groups to the variance within the groups. A higher F-ratio indicates that the group means are not all the same.
- Step 4: Obtain the P-value associated with the calculated F-ratio. The P-value indicates the probability of observing the data if the null hypothesis is true.
- Step 5: If the P-value is below a predetermined threshold (commonly 0.05), reject the null hypothesis, suggesting that at least one group mean is different from the others.

## An Example Use Case of ANOVA

Let’s consider a scenario where a data science professional might use ANOVA.

Situation : A nutritionist wants to test the effectiveness of three different diets (Diet A, Diet B, and Diet C) on weight loss. She selects a sample of 60 people, dividing them equally into three groups, with each group following one of the diets for three months.

Objective : To determine if there is a significant difference in the mean weight loss among the three diet plans.

The three diets represent the independent variable (with three levels: A, B, and C), and the weight loss in pounds is the dependent variable. The null hypothesis will be that there is no difference in the mean weight loss among the three diets. We can perform a one-way ANOVA to compare the mean weight loss across the three diets. If the P-value from the ANOVA is less than 0.05, the nutritionist can conclude that there is a statistically significant difference in weight loss among at least two of the diet groups.

## Implementation using Python

Let’s create sample data based on the example use case of ANOVA I mentioned above. We will create a sample dataset based on the example scenario where a nutritionist compares the effectiveness of three different diets on weight loss. We’ll then perform a one-way ANOVA using Python.

We will simulate a dataset for 60 people, with 20 people in each of the three diet groups (Diet A, Diet B, Diet C). For simplicity, let’s assume that the weight loss data follows a normal distribution, with different means and standard deviations for each group as shown below:

- Diet A : Mean weight loss = 5 lbs, Standard deviation = 1.5 lbs
- Diet B : Mean weight loss = 6 lbs, Standard deviation = 1.2 lbs
- Diet C : Mean weight loss = 4.5 lbs, Standard deviation = 1.8 lbs

Here’s how to create sample data based on this scenario:

We’ll now use Python’s scipy library to perform the one-way ANOVA. The key function we’ll use is scipy.stats.f_oneway :

The results of the one-way ANOVA on our sample data are as follows:

- F-Value : Approximately 14.665
- P-Value : Approximately 7.28e-06

Since the P-value is significantly less than 0.05, we can reject the null hypothesis. It suggests that there is a statistically significant difference in weight loss among at least two of the diet groups (Diet A, Diet B, and Diet C). It indicates that the effectiveness of these diets in terms of weight loss is not the same.

So, The primary purpose of ANOVA is to test if there is any statistically significant difference between the means of different groups. It is done under the hypothesis that any observed differences in sample means are due to random variation. ANOVA helps in determining whether the evidence is strong enough to reject this hypothesis, thus concluding that the differences in means are statistically significant.

I hope you liked this article on AVOVA for Hypothesis Testing with implementation using Python. Feel free to ask valuable questions in the comments section below. You can follow me on Instagram for many more resources.

## Aman Kharwal

I'm a writer and data scientist on a mission to educate others about the incredible power of data📈.

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An F-value is the ratio of two variances, or technically, two mean squares. Mean squares are simply variances that account for the degrees of freedom (DF) used to estimate the variance. F-values are the test statistic for F-tests. Learn more about Test Statistics. Think of it this way. Variances are the sum of the squared deviations from the mean.

Understanding the Null Hypothesis for ANOVA Models A one-way ANOVA is used to determine if there is a statistically significant difference between the mean of three or more independent groups. A one-way ANOVA uses the following null and alternative hypotheses: H0: μ1 = μ2 = μ3 = … = μk (all of the group means are equal)

F test of analysis of variance (ANOVA) follows three assumptions Normality (statistics) Homogeneity of variance Independence of errors and random sampling The hypothesis that a proposed regression model fits the data well. See Lack-of-fit sum of squares.

The hypotheses of interest in an ANOVA are as follows: H 0: μ 1 = μ 2 = μ 3 ... = μ k H 1: Means are not all equal. where k = the number of independent comparison groups. In this example, the hypotheses are: H 0: μ 1 = μ 2 = μ 3 = μ 4 H 1: The means are not all equal. The null hypothesis in ANOVA is always that there is no difference in means.

F = variation between sample means / variation within the samples The best way to understand this ratio is to walk through a one-way ANOVA example. We'll analyze four samples of plastic to determine whether they have different mean strengths. You can download the sample data if you want to follow along.

The one-way ANOVA F-test is a statistical test for testing the equality of \(k\) population means from 3 or more groups within one variable or factor. ... (ANOVA) are set up with all the means equal to one another in the null hypothesis and at least one mean is different in the alternative hypothesis. \(H_{0}: \mu_{1} = \mu_{2} = \mu_{3 ...

The thing that you really need to understand is that the F-test, as it is used in both ANOVA and regression, is really a comparison of two statistical models. One of these models is the full model (alternative hypothesis), and the other model is a simpler model that is missing one or more of the terms that the full model includes (null hypothesis).

Because the F-distribution is generated by drawing two samples from the same normal population, it can be used to test the hypothesis that two samples come from populations with the same variance. You would have two samples (one of size n1 and one of size n2) and the sample variance from each.

Marius 11 years ago Hey, can someone explain to me why t^2=F for a ANOVA analysis? I can use it to solve problems, but I don't know why the p-values of both density curves predict the same p-values. How can I transform the t^2 distribution to the corresponding F distribution?

The null hypothesis in ANOVA is always that there is no difference in means. The research or alternative hypothesis is always that the means are not all equal and is usually written in words rather than in mathematical symbols. ... The decision rule for the F test in ANOVA is set up in a similar way to decision rules we established for t tests ...

On conducting the hypothesis test, if the results of the f test are statistically significant then the null hypothesis can be rejected otherwise it cannot be rejected. F Test Formula The f test is used to check the equality of variances using hypothesis testing. The f test formula for different hypothesis tests is given as follows:

In this section, we will learn a new method called analysis of variance (ANOVA) and a new test statistic called F. ANOVA uses a single hypothesis test to check whether the means across many groups are equal: H 0: The mean outcome is the same across all groups. In statistical notation, \(\mu _1 = \mu _2 = \dots = \mu _k\) where \(\mu _i ...

An ANOVA ("analysis of variance") is used to determine whether or not the means of three or more independent groups are equal. An ANOVA uses the following null and alternative hypotheses: H0: All group means are equal. HA: At least one group mean is different from the rest.

Back to Top F Test to Compare Two Variances A Statistical F Test uses an F Statistic to compare two variances, s 1 and s 2, by dividing them. The result is always a positive number (because variances are always positive). The equation for comparing two variances with the f-test is: F = s 21 / s 22

The "general linear F-test" involves three basic steps, namely:Define a larger full model. (By "larger," we mean one with more parameters.) Define a smaller reduced model. (By "smaller," we mean one with fewer parameters.) Use an F-statistic to decide whether or not to reject the smaller reduced model in favor of the larger full model.; As you can see by the wording of the third step, the null ...

The Brown-Forsythe test is conceptually simple. Each value in the data table is transformed by subtracting from it the median of that column, and then taking the absolute value of that difference. One-way ANOVA is run on these values, and the P value from that ANOVA is reported as the result of the Brown-Forsythe test. How does it work.

If the p-value is less than the significance level (usually 0.05), then the null hypothesis is rejected in favor of the alternative hypothesis. Post-hoc tests: These are follow-up tests conducted after an ANOVA when the null hypothesis is rejected, to determine which specific groups' means (levels) are different from each other. Examples ...

The null hypothesis (H 0) of ANOVA is that there is no difference among group means. The alternative hypothesis (H a) is that at least one group differs significantly from the overall mean of the dependent variable. ... ANOVA uses the F test for statistical significance.

Statistical sentence: F (df) = = F-calc, p>.05 (fill in the df and the calculated F) This page titled 11.3: Hypotheses in ANOVA is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Michelle Oja. With three or more groups, research hypothesis get more interesting.

The null hypothesis in statistics states that there is no difference between groups or no relationship between variables. It is one of two mutually exclusive hypotheses about a population in a hypothesis test. When your sample contains sufficient evidence, you can reject the null and conclude that the effect is statistically significant.

The actual test begins by considering two hypotheses. They are called the null hypothesis and the alternative hypothesis. These hypotheses contain opposing viewpoints. H0, the — null hypothesis: a statement of no difference between sample means or proportions or no difference between a sample mean or proportion and a population mean or proportion.

The null hypothesis, H0: States the assumption (numerical) to be tested Begin with the assumption that the null hypothesis is TRUE Always contains the '=' sign The alternative hypothesis, Ha: Is the opposite of the null hypothesis Challenges the status quo Never contains just the '=' sign

Null hypothesis for ANOVA for regression - Cross Validated Null hypothesis for ANOVA for regression Asked 3 months ago Modified 3 months ago Viewed 444 times 7 Context: I know the "classic" ANOVA framework: we have $n$ groups and $k$ measurements of a variable $X$ for each group. Let $µ_1$, $µ_2$, ..., $µ_n$ be the mean of $X$ in each group.

F-Value = 14.664807616379283, P-Value = 7.275769568989161e-06. The results of the one-way ANOVA on our sample data are as follows: F-Value: Approximately 14.665; P-Value: Approximately 7.28e-06; Since the P-value is significantly less than 0.05, we can reject the null hypothesis.