what is mean by composite hypothesis

What does "Composite Hypothesis" mean?

Definition of Composite Hypothesis in the context of A/B testing (online controlled experiments).

What is a Composite Hypothesis?

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.

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A composite hypothesis is a type of statistical hypothesis that includes a range of possible values for a parameter, rather than specifying a single value. This concept is crucial when dealing with null and alternative hypotheses, as it allows researchers to consider multiple scenarios or conditions under which the data may be analyzed, providing a more flexible approach to hypothesis testing.

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5 Must Know Facts For Your Next Test

• Composite hypotheses can test multiple values for a parameter, which allows for broader applicability in real-world scenarios.
• Unlike simple hypotheses that specify a single value, composite hypotheses acknowledge uncertainty in parameter estimation.
• In hypothesis testing, rejecting the null hypothesis often implies accepting a composite alternative hypothesis rather than pinpointing a specific value.
• The formulation of composite hypotheses is critical in experiments where parameters are not fixed and can vary based on different conditions.
• Composite hypotheses are common in many statistical tests, including t-tests and ANOVA, where researchers evaluate the effects across groups rather than focusing on individual outcomes.

Review Questions

• A composite hypothesis differs from a simple hypothesis by encompassing a range of values for the parameter being tested rather than just one specific value. While simple hypotheses make concrete assertions that can be clearly accepted or rejected, composite hypotheses allow researchers to explore multiple possibilities within their data. This flexibility is particularly useful in real-world applications where parameters are not easily defined.
• Using composite hypotheses in research studies has significant implications for how results are interpreted and understood. Since these hypotheses consider various potential outcomes, they provide a more comprehensive view of the data and its variability. This approach helps researchers avoid oversimplification and encourages deeper exploration of how different factors may influence results, ultimately leading to more robust conclusions.
• Employing composite hypotheses offers several advantages, such as greater flexibility and the ability to encompass a wider range of scenarios that can occur in real-life situations. They facilitate a better understanding of variability within data sets. However, this complexity can also introduce challenges; for example, it may complicate the interpretation of results and increase the likelihood of Type I and Type II errors due to multiple comparisons. Balancing these factors is essential when deciding which type of hypothesis to use in statistical analyses.

Related terms

A statement that assumes there is no effect or no difference in a given situation, serving as the starting point for statistical testing.

The hypothesis that proposes an effect or a difference exists, challenging the null hypothesis in statistical testing.

parameter : A numerical characteristic or measure of a population, such as the mean or standard deviation, that is estimated through statistical methods.

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Simple Hypothesis and Composite Hypothesis

A simple hypothesis is one in which all parameters of the distribution are specified. For example, the heights of college students are normally distributed with $${\sigma ^2} = 4$$, and the hypothesis that its mean $$\mu $$ is, say, $$62”$$; that is, $${H_o}:\mu = 62$$. So we have stated a simple hypothesis, as the mean and variance together specify a normal distribution completely. A simple hypothesis, in general, states that $$\theta = {\theta _o}$$ where $${\theta _o}$$ is the specified value of a parameter $$\theta $$, ($$\theta $$ may represent $$\mu ,p,{\mu _1} – {\mu _2}$$ etc).

A hypothesis which is not simple (i.e. in which not all of the parameters are specified) is called a composite hypothesis. For instance, if we hypothesize that $${H_o}:\mu > 62$$ (and $${\sigma ^2} = 4$$) or$${H_o}:\mu = 62$$ and $${\sigma ^2} < 4$$, the hypothesis becomes a composite hypothesis because we cannot know the exact distribution of the population in either case. Obviously, the parameters $$\mu > 62”$$ and$${\sigma ^2} < 4$$ have more than one value and no specified values are being assigned. The general form of a composite hypothesis is $$\theta \leqslant {\theta _o}$$ or $$\theta \geqslant {\theta _o}$$; that is, the parameter $$\theta $$ does not exceed or does not fall short of a specified value $${\theta _o}$$. 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 $${H_o}:\mu = 62$$ or $$p > 0.5$$. A hypothesis is called an inexact hypothesis when it indicates more than one possible value for the parameter, such as $${H_o}:\mu \ne 62$$ or $${H_o}:p = 62$$. A simple hypothesis must be exact while an exact hypothesis is not necessarily a simple hypothesis. An inexact hypothesis is a composite hypothesis.

One Comment

Etini August 5 @ 7:28 pm

How can i design a sequential test for the shape parameters of the beta distribution

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