Power of a test

In statistics, the power of a binary hypothesis test is the probability that the test correctly rejects the null hypothesis () when a specific alternative hypothesis () is true. It is commonly denoted by , and represents the chances of a true positive detection conditional on the actual existence of an effect to detect. Statistical power ranges from 0 to 1, and as the power of a test increases, the probability of making a type II error by wrongly failing to reject the null hypothesis decreases. Type I and type II errors This article uses the following notation: β = probability of a Type II error, known as a "false negative" 1 − β = probability of a "true positive", i.e., correctly rejecting the null hypothesis. "1 − β" is also known as the power of the test. α = probability of a Type I error, known as a "false positive" 1 − α = probability of a "true negative", i.e., correctly not rejecting the null hypothesis For a type II error probability of β, the corresponding statistical power is 1 − β. For example, if experiment E has a statistical power of 0.7, and experiment F has a statistical power of 0.95, then there is a stronger probability that experiment E had a type II error than experiment F. This reduces experiment E's sensitivity to detect significant effects. However, experiment E is consequently more reliable than experiment F due to its lower probability of a type I error. It can be equivalently thought of as the probability of accepting the alternative hypothesis () when it is true – that is, the ability of a test to detect a specific effect, if that specific effect actually exists. Thus, If is not an equality but rather simply the negation of (so for example with for some unobserved population parameter we have simply ) then power cannot be calculated unless probabilities are known for all possible values of the parameter that violate the null hypothesis. Thus one generally refers to a test's power against a specific alternative hypothesis.