Power Function - StatLect

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In statistics, the power function is a function that links the true value of a parameter to the probability of rejecting a null hypothesis about the value of that parameter.

Table of Contents

Table of contents

  1. Definition

  2. Example

  3. Terminology

  4. Power and size

  5. Graph of the power function

  6. How to derive the power function

  7. Dependence on sample size

  8. More details

  9. Keep reading the glossary

Definition

Here is a more formal definition.

Definition In a test of hypothesis about a parameter $ heta $, let the null hypothesis be[eq1]The power function [eq2] is a function that gives, for any $ heta $, the probability of rejecting the null hypothesis when the true parameter is equal to $ heta $.

Note that the power function depends on the null hypothesis: if we change $ heta _{0}$, also the power function changes.

Example

Suppose that we are testing the null hypothesis that the true parameter is equal to zero: [eq3]

Suppose that the value of the power function at $ heta =1$ is [eq4]

What does this mean? It means that if the true parameter is equal to 1, then there is a 50% probability that the test will reject the (false) null hypothesis that the parameter is equal to 0.

Terminology

The parameter $ heta $ is often called alternative hypothesis and [eq5] is called power against the alternative $ heta $.

Power and size

The size of a test is the probability of rejecting the null hypothesis when it is true.

Therefore, when[eq6]the power function evaluated at $ heta _{0}$ gives the size $lpha $ of the test:[eq7]

Graph of the power function

We plot below the graph of a typical power function.

Graph of the power function of a z-test for the mean of a normal distribution.

It plots the probability of rejecting an alternative $ heta $ in a z-test for the mean of a normal distribution, in which:

  • $ heta $ is the unknown mean of the distribution;

  • the variance of the distribution is known: $sigma ^{2}=1$;

  • the null is $H_{0}:\theta =0$;

  • the size of the test is equal to 5%;

  • the sample is made of 100 independent draws from the distribution.

Note that the minimum of the graph corresponds to the null and it is equal to the size of the test.

The power function, known in closed form, is[eq8]where F is the cumulative distribution function of the normal distribution, $z=1.96$ is the critical value corresponding to a 5% size, and $n=100$ is the number of draws.

How to derive the power function

For examples of how to derive the power function, see the lectures:

  • Hypothesis testing about the mean (z-test and t-test);

  • Hypothesis testing about the variance (Chi-square test).

Dependence on sample size

Usually, the power of a test is an increasing function of sample size: the more observations we have, the more powerful the test.

More details

You can find a more exhaustive explanation of the concept of power function in the lecture entitled Hypothesis testing.

Some related concepts are found in the following glossary entries:

  • alternative hypothesis;

  • Type I error;

  • Type II error.

Keep reading the glossary

Previous entry: Posterior probability

Next entry: Precision matrix

How to cite

Please cite as:

Taboga, Marco (2021). "Power function", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix. https://www.statlect.com/glossary/power-function.

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