# Confusion about confusion matrix, or why 'power of test' + $\beta$ = 1?

Consider a confusion table summarizing the outcomes of $n$ tests of hypothesis $H_0$ for $n$ independent experiments:

             accept H0    reject H0
H0 is true      a             b
H0 is false     c             d


Here $a$, $b$, $c$ and $d$ are number of times we accepted/rejected $H_0$ given that $H_0$ was true/false, and $a+b+c+d=n$.

If we divide numbers in the table by $n$, we will get estimates of $P(TN)$, $P(FN)$, $P(FP)$, $P(TP)$:

             accept H0    reject H0
H0 is true     P(TN)         P(FP)
H0 is false    P(FN)         P(TP)


Here, $P$ is probability, $TN$ - true negative decision, $FN$ - false negative decision, $FP$ - false positive decision, $TP$ - true positive decision.

Now, the power of test is defined as $P(TP)$; $P(FN)$ is designated by $\beta$. An article on statistical power in Wikipedia says that power = $1 - \beta$, so that $P(TP) + P(FN) = 1$, so that $d/n + c/n \approx 1$. I don't see how it is possible in this setup, so what did I get wrong? Similarly, $P(TN)$ + $P(FP)$ should not be 1, or should it?

• Power is actually defined as $1-\beta$, so it's not remotely surprising that power+$\beta$=1 (as per your title) - it's a tautology. Your confusion arises elsewhere. Dec 19, 2014 at 1:54
• We never accept the H0. We fail to reject. Jul 17, 2021 at 19:07

The confusion here is that you're mixing up conditional and unconditional probability.

Power is a conditional probability: P(reject H0|H0 false).

Similarly $\beta$ is P(fail to reject H0|H0 false).

These conditional probabilities add to 1. That is: $\frac{c}{c+d} + \frac{d}{c+d} =1$.

Similarly, for the top row: $\frac{a}{a+b} + \frac{b}{a+b} =1$ (where the probabilities represent $1-\alpha$ and $\alpha$).

This is NOT the same as saying $\frac{c}{a+b+c+d} + \frac{d}{a+b+c+d} =1$; that's plainly not true.

power of test is defined as $P(TP)$

Beware! For it to be a true positive (rather than just "a positive"), you've specified that the null is false.

The sensitivity, or true positive rate (TPR) = TP/P = TP/(TP+FN). That's what power is.

That is, if by P(TP) you mean P(reject|H0 false), then I agree that's the power, but then it isn't $\frac{d}{a+b+c+d}$.