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Composite Hypothesis:
A statistical hypothesis which does not completely specify the distribution of a random variable is referred to as a composite hypothesis.
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Definition of Composite Hypothesis in the context of A/B testing (online controlled experiments).
In hypothesis testing a composite hypothesis is a hypothesis which covers a set of values from the parameter space. For example, if the entire parameter space covers -∞ to +∞ a composite hypothesis could be μ ≤ 0. It could be any other number as well, such 1, 2 or 3,1245. The alternative hypothesis is always a composite hypothesis : either one-sided hypothesis if the null is composite or a two-sided one if the null is a point null. The "composite" part means that such a hypothesis is the union of many simple point hypotheses.
In a Null Hypothesis Statistical Test only the null hypothesis can be a point hypothesis. Also, a composite hypothesis usually spans from -∞ to zero or some value of practical significance or from such a value to +∞.
Testing a composite null is what is most often of interest in an A/B testing scenario as we are usually interested in detecting and estimating effects in only one direction: either an increase in conversion rate or average revenue per user, or a decrease in unsubscribe events would be of interest and not its opposite. In fact, running a test so long as to detect a statistically significant negative outcome can result in significant business harm.
Like this glossary entry? For an in-depth and comprehensive reading on A/B testing stats, check out the book "Statistical Methods in Online A/B Testing" by the author of this glossary, Georgi Georgiev.
One-tailed vs Two-tailed Tests of Significance in A/B Testing blog.analytics-toolkit.com
Statistical Methods in Online A/B Testing
Take your A/B testing program to the next level with the most comprehensive book on user testing statistics in e-commerce.
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In previous chapters, we have discussed two methods for estimating unknown parameters , namely point estimation and interval estimation . Estimating unknown parameters is an important area in statistical inference, and in this chapter we will discuss another important area, namely hypothesis testing , which is related to decision making . Indeed, the concepts of confidence intervals and hypothesis testing are closely related, as we will demonstrate.
Before discussing how to conduct hypothesis testing, and evaluate the "goodness" of a hypothesis test, let us introduce some basic concepts and terminologies related to hypothesis testing first.
Definition. (Hypothesis) A (statistical) hypothesis is a statement about population parameter(s).
There are two terms that classify hypotheses:
Definition. (Simple and composite hypothesis) A hypothesis is a simple hypothesis if it completely specifies the distribution of the population (that is, the distribution is completely known, without any unknown parameters involved), and is a composite hypothesis otherwise.
Sometimes, it is not immediately clear that whether a hypothesis is simple or composite. To understand the classification of hypotheses more clearly, let us consider the following example.
In hypothesis tests, we consider two hypotheses:
Example. Suppose your friend gives you a coin for tossing, and we do not know whether it is fair or not. However, since the coin is given by your friend, you believe that the coin is fair unless there are sufficient evidences suggesting otherwise. What is the null hypothesis and alternative hypothesis in this context (suppose the coin never land on edge)?
Now, we are facing with two questions. First, what evidences should we consider? Second, what is meant by "sufficient"? For the first question, a natural answer is that we should consider the observed samples , right? This is because we are making hypothesis about the population, and the samples are taken from, and thus closely related to the population, which should help us make the decision.
Let us formally define the terms related to hypothesis testing in the following.
Exercise. What is the type of this hypothesis test?
Right-tailed test.
As we have mentioned, the decisions made by hypothesis test should not be perfect, and errors occur. Indeed, when we think carefully, there are actually two types of errors, as follows:
We can illustrate these two types of errors more clearly using the following table.
Accept | Reject | |
---|---|---|
is true | Correct decision | Type I error |
is false | Type II error | Correct decision |
You notice that the type II error of this hypothesis test can be quite large, so you want to revise the test to lower the type II error.
To describe "control the type I error probability at this level" in a more precise way, let us define the following term.
Exercise. Calculate the type I error probability and type II error probability when the sample size is 12 (the rejection region remains unchanged).
0.01 | |
0.04 | |
0.06 | |
0.08 | |
0.1 |
After discussing some basic concepts and terminologies, let us now study some ways to evaluate goodness of a hypothesis test. As we have previously mentioned, we want the probability of making type I errors and type II errors to be small, but we have mentioned that it is generally impossible to make both probabilities to be arbitrarily small. Hence, we have suggested to control the type I error, using the size of a test, and the "best" test should the one with the smallest probability of making type II error, after controlling the type I error.
These ideas lead us to the following definitions.
Using this definition, instead of saying "best" test (test with the smallest type II error probability), we can say "a test with the most power", or in other words, the "most powerful test".
Neyman-pearson lemma.
For the case where the underlying distribution is discrete, the proof is very similar (just replace the integrals with sums), and hence omitted.
Now, let us consider another example where the underlying distribution is discrete.
Exercise. Calculate the probability of making type II error for the above test.
Previously, we have suggested using the Neyman-Pearson lemma to construct MPT for testing simple null hypothesis against simple alternative hypothesis. However, when the hypotheses are composite, we may not be able to use the Neyman-Pearson lemma. So, in the following, we will give a general method for constructing tests for any hypotheses, not limited to simple hypotheses. But we should notice that the tests constructed are not necessarily UMPT.
We have mentioned that there are similarities between hypothesis testing and confidence intervals. In this section, we will introduce a theorem suggesting how to construct a hypothesis test from a confidence interval (or in general, confidence set ), and vice versa.
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Lesson 27: likelihood ratio tests.
In this lesson, we'll learn how to apply a method for developing a hypothesis test for situations in which both the null and alternative hypotheses are composite. That's not completely accurate. The method, called the likelihood ratio test , can be used even when the hypotheses are simple, but it is most commonly used when the alternative hypothesis is composite. Throughout the lesson, we'll continue to assume that we know the the functional form of the probability density (or mass) function, but we don't know the value of one (or more) of its parameters. That is, we might know that the data come from a normal distrbution, but we don't know the mean or variance of the distribution, and hence the interest in performing a hypothesis test about the unknown parameter(s).
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So if we have
$H_0 :\theta=\theta_0$ vs $H_1 :\theta=\theta_1$
It is easy to see that this is a case of simple vs simple hypothesis (assuming that $\theta$ is the only unknown parameter of our distribution)
$H_0 :\theta\leq\theta_0$ vs $H_1 :\theta>\theta_0$
Is this composite vs composite or simple vs composite?
Since it is somewhat equivalent to
$H_0 :\theta=\theta_0$ vs $H_1 :\theta>\theta_0$
Which I guess it's a simple vs composite hypothesis
And last, if we have two unkown parameters, is $H_0 :\alpha=\alpha_0 , \beta\geq\beta_0$
Simple or composite?
$H_0 :\theta=\theta_0$ vs $H_1 :\theta>\theta_0$ is a composite hypothesis since for $H_1$ you can have many different $\theta$s.
You can check these links the explanations are pretty clear.
http://www.emathzone.com/tutorials/basic-statistics/simple-hypothesis-and-composite-hypothesis.html
http://isites.harvard.edu/fs/docs/icb.topic1383356.files/Lecture%2014%20-%20Intro%20to%20Hypothesis%20Testing%20-%204%20per%20page.pdf
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Hypothesis Testing > Uniformly Most Powerful (UMP) Test
A Uniformly Most Powerful (UMP) test has the most statistical power from the set of all possible alternate hypotheses of the same size α . The UMP doesn’t always exist, especially when the test has nuisance variables (variables that are irrelevant to your study but that have to be be accounted for). However, if the UMP does exist, you can use the Neyman-Pearson lemma (NPL) to find it.
A UMP test is usually defined in terms of a uniformly most powerful rejection region (UMPCR) (also called a “critical region”); A region C of size α is the UMPCR for testing a simple null hypothesis against a set of alternate hypotheses if it is the “best” critical region. The “best” critical region is one that minimizes the probability of making a Type I or a Type II error . In other words, the UMPCR is the region that gives the smallest chance of making a Type I or II error . It is also the region that gives a UMP test the largest (or equally largest) power function .
The Neyman-Pearson lemma can tell you the best hypothesis test if you have a simple null hypothesis and a simple alternative hypothesis. If you have multiple hypotheses (also called a composite hypothesis ), the NPL can be extended to all individual alternate hypotheses. Composite hypotheses have multiple options for solutions. For example, H 0 :σ 2 > 8 is a composite hypothesis because it doesn’t specify a value for σ 2 ; The solution could be anything over 8. This contrasts to H 0 : μ = 0, which specifies the single value of zero.
The basic idea is that you test each simple hypothesis in turn to see if it is the UMP out of all possibilities.
Casella and Berger (2002) define a UMP test as follows:
“Let C be a class of tests for testing H 0 : θ ∈ Θ 0 versus H 1 : θ ∈ Θ c 1 . A test in class C, with power function β(θ), is a uniformly most powerful (UMP) class C test if β(θ) ≥ β′(θ) for every θ ∈ Θ 0 c and every β′(θ) that is a power function of a test in class C.
If you have an entire set of possibilities (which would be the case with composite hypotheses ), each test should be tested individually using the above criteria.
Kotz, S.; et al., eds. (2006), Encyclopedia of Statistical Sciences , Wiley. Everitt, B. S.; Skrondal, A. (2010), The Cambridge Dictionary of Statistics , Cambridge University Press.
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A composite hypothesis test contains more than one parameter and more than one model. In a simple hypothesis test, the probability density functions for both the null hypothesis (H 0) and alternate hypothesis (H 1) are known. In academic and hypothetical situations, the simple hypothesis test works for most cases.
The concept of simple and composite hypotheses applies to both the null hypothesis and alternative hypothesis. Hypotheses may also be classified as exact and inexact. A hypothesis is said to be an exact hypothesis if it selects a unique value for the parameter, such as Ho: μ = 62 H o: μ = 62 or p > 0.5 p > 0.5. A hypothesis is called an ...
Introduction to the Science of Statistics Composite Hypotheses • For 2 ⇥ 0, ⇡( ) is the probability of making a type I error, i.e., rejecting the null hypothesis when it is indeed true. • For 2 ⇥ 1, 1 ⇡( ) is the probability of making a type II error, i.e., failing to reject the null hypothesis when it is false.
A statistical hypothesis test is a method of statistical inference used to decide whether the data sufficiently supports a particular hypothesis. A statistical hypothesis test typically involves a calculation of a test statistic. Then a decision is made, either by comparing the test statistic to a critical value or equivalently by evaluating a ...
then we have a simple hypothesis, as discussed in past lectures. When a set contains more than one parameter value, then the hypothesis is called a composite hypothesis, because it involves more than one model. The name is even clearer if we consider the following equivalent expression for the hypotheses above. H 0: X ˘p 0; p 0 2fp 0(xj 0)g 02 ...
STATS 200: Introduction to Statistical Inference Autumn 2016 Lecture 7 | Composite hypotheses and the t-test 7.1 Composite null and alternative hypotheses This week we will discuss various hypothesis testing problems involving a composite null hypothesis and a compositive alternative hypothesis. To motivate the discussion, consider
Definition: Simple and composite hypothesis. Definition: Let H H be a statistical hypothesis. Then, H H is called a simple hypothesis, if it completely specifies the population distribution; in this case, the sampling distribution of the test statistic is a function of sample size alone. H H is called a composite hypothesis, if it does not ...
Composite Hypothesis: A composite hypothesis specifies a range of values. Example: A company is claiming that their average sales for this quarter are 1000 units. This is an example of a simple hypothesis. Suppose the company claims that the sales are in the range of 900 to 1000 units. Then this is a case of a composite hypothesis.
Rejection and failure to reject the null hypothesis, critical regions, C, and type I and type II errors have the same meaning for a composite hypotheses as it does with a simple hypothesis. 1 Power Power is now a function ˇ( ) = P fX2Cg: that gives the probability of rejecting the null hypothesis for a given value of the parameter.
Introduction to Statistical Methodology Composite Hypotheses In reality, incorrect decisions are made. Thus, for 2 0, ˇ( ) is the probability of making a type I error; i. e., rejecting the null hypothesis when it is indeed true, For 2 1, 1 ˇ( ) is the probability of making a type II error; i.e., failing to reject the null hypothesis when it ...
Any hypothesis that is not a simple hypothesis is called a composite hypothesis. Example 26-1 Section . Suppose \(X_1 , X_2 , \dots , X_n\) is a random sample from an exponential distribution with parameter \(\theta\). ... Lesson 25: Power of a Statistical Test. 25.1 - Definition of Power; 25.2 - Power Functions; 25.3 - Calculating Sample Size ...
Composite Hypothesis. In subject area: Mathematics. Classically, composite hypotheses are used to determine if a point null is statistically distinguishable from the best alternative, or to determine if the best supported alternative lies on a specified side of the point null. From: Philosophy of Statistics, 2011.
Our mentors and academic advisors are standing by to help guide you towards the courses or program that makes the most sense for you and your goals. 300 W Main St STE 301, Charlottesville, VA 22903. (434) 973-7673. [email protected].
In hypothesis testing, the goal is to see if there is sufficient statistical evidence to reject a presumed null hypothesis in favor of a conjectured alternative hypothesis.The null hypothesis is usually denoted \(H_0\) while the alternative hypothesis is usually denoted \(H_1\). An hypothesis test is a statistical decision; the conclusion will either be to reject the null hypothesis in favor ...
The "composite" part means that such a hypothesis is the union of many simple point hypotheses. In a Null Hypothesis Statistical Test only the null hypothesis can be a point hypothesis. Also, a composite hypothesis usually spans from -∞ to zero or some value of practical significance or from such a value to +∞.
(Hypothesis) A (statistical) hypothesis is a statement about population parameter(s). There are two terms that classify hypotheses: Definition. (Simple and composite hypothesis) A hypothesis is a simple hypothesis if it completely specifies the distribution of the population ...
a statistical hypothesis that is not specific about all relevant features of a population or that does not give a single value for a characteristic of a population but allows for a range of acceptable values. For example, a statement that the average age of employees in academia exceeds 50 is a composite hypothesis, as there are a variety of ...
Definition: A interval on μ at X% confidence level is defined such that the true of value of μ is contained X% of the time in the interval. Note that the output is not a probabilistic statement on the true s value. The true μ is fixed but unknown - each observation will result in an estimated interval [μ-,μ+].
Lesson 27: Likelihood Ratio Tests. In this lesson, we'll learn how to apply a method for developing a hypothesis test for situations in which both the null and alternative hypotheses are composite. That's not completely accurate. The method, called the likelihood ratio test, can be used even when the hypotheses are simple, but it is most ...
Statistical hypothesis: A statement about the nature of a population. It is often stated in terms of a population parameter. ... A composite score from 0 to 10 points, representing both recall and persuasion, has been produced for each consumer in the sample. ... Definition 3.1. The null hypothesis is a statement about the values of one or more ...
The Neyman-Pearson Lemma is a way to find out if the hypothesis test you are using is the one with the greatest statistical power. The power of a hypothesis test is the probability that test correctly rejects the null hypothesis when the alternate hypothesis is true. The goal would be to maximize this power, so that the null hypothesis is ...
3. So if we have. H0: θ = θ0 H 0: θ = θ 0 vs H1: θ = θ1 H 1: θ = θ 1. It is easy to see that this is a case of simple vs simple hypothesis (assuming that θ θ is the only unknown parameter of our distribution) what about. H0: θ ≤ θ0 H 0: θ ≤ θ 0 vs H1: θ > θ0 H 1: θ > θ 0.
A test in class C, with power function β (θ), is a uniformly most powerful (UMP) class C test if β (θ) ≥ β′ (θ) for every θ ∈ Θ 0c and every β′ (θ) that is a power function of a test in class C. In more simple terms, this is really just telling you that the UMP test is the one with the biggest power function (out of all tests ...