# Why is the complement to Power not $\alpha$?

i) Wrongly rejecting $H_0$ is called a type I error (controlled by $\alpha$).

ii) Wrongly accepting $H_0$ is called a type II error (the probability of which is indicated by $\beta$).

iii) Power is the probability of correctly rejecting $H_0$ (equal to $1-\beta$). Then $1 = Power + \beta$, i.e. $\beta$ is the complement to Power.

So, if I had to verbalize the complement of Power given its definition in statement iii), I would arrive at "wrongly rejecting $H_0$" (simply by kind of flipping "correctly" to "wrongly").

But then that would be exactly the meaning of $\alpha$ and I would define $Power = 1 - \alpha$.

How come the complement to "correctly reject $H_0$" is "wrongly accept $H_0$"?

if I had to verbalize the complement of Power given its definition in statement iii), I would arrive at "wrongly rejecting H0" (simply by kind of flipping "correctly" to "wrongly").

The problem is that it's not that simple. Everyday English (at least when expressed that way) doesn't quite so simply take complements of conditional events. Let's look at the four events:

                     H0 true                H0 false

not reject    "correctly not reject"     Type II error

reject         Type I error          "correctly reject"


The column headings are the events being conditioned on. We don't change those when taking the complement. The row headings are the actions.

What the event "correctly rejecting $H_0$" means is "reject $H_0|H_0$ false".

(What "simply flipping correctly to wrongly" does is change the part after the "|".)

Taking the complementary event changes the part before "|" and leaves the conditioning event alone, which is "fail to reject $H_0|H_0$ false", which colloquially might be rendered as "wrongly failing to reject $H_0$*".

So the first thing is that it's the event "reject $H_0$" that is the thing that has to flip - that's what represents taking the complement. However, because we characterized it as "correctly" or "wrongly" instead of specifying the conditioning event (resulting in a a simpler English expression, but no longer one that's easily flipped), we have to also flip that characterization.

• You exactly understood my problem. Feb 2, 2015 at 18:24

I think if you form these statements into statements of probability it will make more sense. It is much harder for me to "take the compliment" of a sentence than it is to take the compliment of a conditional probability and then form it as a sentence.

Below I'm using the fact that the complement of $P(A|B)$ is $P(A^c|B)$.

i) Wrongly rejecting $H_0$ is called a type I error (controlled by $\alpha$).

Forming this as a probability statement, $P(\text{reject } H_0| H_0 \text{ is true})$. The complement of this is $P(\text{fail to reject } H_0| H_0 \text{ is true})$. In other words, correctly rejecting the null.

ii) Wrongly accepting $H_0$ is called a type II error (the probability of which is indicated by $\beta$).

Forming this as a probability statement, $P(\text{fail to reject } H_0| H_0 \text{ is false})$. The complement of this is $P(\text{reject } H_0| H_0 \text{ is false})$. This is power.

iii) Power is the probability of correctly rejecting $H_0$ (equal to $1-\beta$). Then $1 = Power + \beta$, i.e. $\beta$ is the complement to Power.

Forming this as a probability statement, $P(\text{reject } H_0| H_0 \text{ is false})$. The complement of this is $P(\text{fail to reject } H_0| H_0 \text{ is false})$. Verbalizing this, you would arrive at "wrongly failing to reject the null".

• "Compliment" is not the same as "complement" :) You should fix this in the answer. Feb 1, 2015 at 15:46