# Power (statistics)

**statistical powerpowerpowerfulpoweredpower analysisunderpoweredpower functionstatistically powerfuladequately powereddetection power**

The power of a binary hypothesis test is the probability that the test rejects the null hypothesis (H 0 ) when a specific alternative hypothesis (H 1 ) is true.wikipedia

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### Type I and type II errors

**Type I errorfalse-positivefalse positive**

The statistical power ranges from 0 to 1, and as statistical power increases, the probability of making a type II error (wrongly failing to reject the null hypothesis) decreases.

The rate of the type II error is denoted by the Greek letter β (beta) and related to the power of a test (which equals 1−β).

### Effect size

**Cohen's deffect sizesmagnitude**

Power analysis can be used to calculate the minimum sample size required so that one can be reasonably likely to detect an effect of a given size.

Effect sizes complement statistical hypothesis testing, and play an important role in power analyses, sample size planning, and in meta-analyses.

### Statistical hypothesis testing

**hypothesis testingstatistical teststatistical tests**

Statistical tests use data from samples to assess, or make inferences about, a statistical population.

Unless a test with particularly high power is used, the idea of "accepting" the null hypothesis may be dangerous.

### Nonparametric statistics

**non-parametricnon-parametric statisticsnonparametric**

In addition, the concept of power is used to make comparisons between different statistical testing procedures: for example, between a parametric test and a nonparametric test of the same hypothesis.

The wider applicability and increased robustness of non-parametric tests comes at a cost: in cases where a parametric test would be appropriate, non-parametric tests have less power.

### Design of experiments

**experimental designdesignExperimental techniques**

The design of an experiment or observational study often influences the power.

Related concerns include achieving appropriate levels of statistical power and sensitivity.

### Sampling error

**sampling variabilitysampling variationless reliable**

The sample size determines the amount of sampling error inherent in a test result.

The likely size of the sampling error can generally be controlled by taking a large enough random sample from the population, although the cost of doing this may be prohibitive; see sample size determination and statistical power for more detail.

### Sample size determination

**sample sizeSampling sizessample**

Power analysis can be used to calculate the minimum sample size required so that one can be reasonably likely to detect an effect of a given size.

In practice, the sample size used in a study is usually determined based on the cost, time, or convenience of collecting the data, and the need for it to offer sufficient statistical power.

### Sensitivity and specificity

**sensitivityspecificitysensitive**

In the context of binary classification, the power of a test is called its statistical sensitivity, its true positive rate, or its probability of detection.

In the traditional language of statistical hypothesis testing, the sensitivity of a test is called the statistical power of the test, although the word power in that context has a more general usage that is not applicable in the present context.

### Analysis of variance

**ANOVAanalysis of variance (ANOVA)corrected the means**

In regression analysis and analysis of variance, there are extensive theories and practical strategies for improving the power based on optimally setting the values of the independent variables in the model.

Power analysis is often applied in the context of ANOVA in order to assess the probability of successfully rejecting the null hypothesis if we assume a certain ANOVA design, effect size in the population, sample size and significance level.

### P-value

**p''-valuepp''-values**

In particular, it has been shown that post-hoc "observed power" is a one-to-one function of the p-value attained.

The curve is affected by four factors: the proportion of studies that examined false null hypotheses, the power of the studies that investigated false null hypotheses, the alpha levels, and publication bias.

### Efficiency (statistics)

**efficientefficiencyinefficient**

How increased sample size translates to higher power is a measure of the efficiency of the test—for example, the sample size required for a given power.

For comparing significance tests, a meaningful measure of efficiency can be defined based on the sample size required for the test to achieve a given task power.

### Student's t-test

**t-testt''-testStudent's ''t''-test**

A hypothesis test may fail to reject the null, for example, if a true difference exists between two populations being compared by a t-test but the effect is small and the sample size is too small to distinguish the effect from random chance.

Paired t-tests are a form of blocking, and have greater power than unpaired tests when the paired units are similar with respect to "noise factors" that are independent of membership in the two groups being compared.

### Student's t-distribution

**Student's ''t''-distributiont-distributiont''-distribution**

The test statistic under the null hypothesis follows a Student t-distribution.

This distribution is important in studies of the power of Student's t-test.

### Statistical significance

**statistically significantsignificantsignificance level**

Furthermore, assume that the null hypothesis will be rejected at the significance level of Since n is large, one can approximate the t-distribution by a normal distribution and calculate the critical value using the quantile function \Phi^{-1}, the inverse of the cumulative distribution function of the normal distribution.

A two-tailed test may still be used but it will be less powerful than a one-tailed test, because the rejection region for a one-tailed test is concentrated on one end of the null distribution and is twice the size (5% vs. 2.5%) of each rejection region for a two-tailed test.

### Clinical trial

**clinical trialsclinical studiesclinical study**

Many clinical trials, for instance, have low statistical power to detect differences in adverse effects of treatments, since such effects may be rare and the number of affected patients small.

This ability is described as its "power," which must be calculated before initiating a study to figure out if the study is worth its costs.

### Uniformly most powerful test

**uniformly more powerfulKarlin–Rubin theoremmost powerful test**

In statistical hypothesis testing, a uniformly most powerful (UMP) test is a hypothesis test which has the greatest power 1 - \beta among all possible tests of a given size α.

### Neyman–Pearson lemma

**Neyman-Pearson lemmaabstractNeyman & Pearson**

Then, the Neyman–Pearson lemma states that the likelihood ratio, \Lambda(x), is the most powerful test at significance level α.

### Predictive probability of success

**predictive distribution**

To address this issue, the power concept can be extended to the concept of predictive probability of success (PPOS).

This is a frequentist statistical power.

### Null hypothesis

**nullnull hypotheseshypothesis**

The power of a binary hypothesis test is the probability that the test rejects the null hypothesis (H 0 ) when a specific alternative hypothesis (H 1 ) is true.

### Alternative hypothesis

**alternative hypothesesalternativealternatives**

The power of a binary hypothesis test is the probability that the test rejects the null hypothesis (H 0 ) when a specific alternative hypothesis (H 1 ) is true.

### Parametric statistics

**parametricparametric testparametric inference**

In addition, the concept of power is used to make comparisons between different statistical testing procedures: for example, between a parametric test and a nonparametric test of the same hypothesis.

### Binary classification

**binary classifierbinarybinary categorization**

In the context of binary classification, the power of a test is called its statistical sensitivity, its true positive rate, or its probability of detection.

### Sampling (statistics)

**samplingrandom samplesample**

Statistical tests use data from samples to assess, or make inferences about, a statistical population.

### Statistical inference

**inferential statisticsinferenceinferences**

Statistical tests use data from samples to assess, or make inferences about, a statistical population.

### Statistical population

**populationsubpopulationsubpopulations**