Type-II Error Rate

Definition

Let $\varphi$ be a test, the probability of a Type II Error under a given probabilistic model is central to the quality of a test. The probability of a Type II Error is defined as

$p_{{\Theta }_1}(\varphi (y)=0),$

and referred as Type II error rate of the test. Its complementary probability is calculated as

\begin{equation}\tag{test power}\beta:=p_{\Theta_1}(\phi(y)=1)\;.\end{equation}

and called as the power of a test.

Interpretation

The power of a test is thus the probability of accepting the alternative hypothesis (rejecting the null hypothesis), if $\theta \in {\Theta }_1$, i.e., if the alternative hypothesis is true.

Note

Basic introductions to test|test error probabilities often denote the probability of a Type II error by $\beta \in [0,1]$ and thus define power by $1-\beta$.

Reference

D. Ostwald - The general linear model 20_21 pp 117