# How to make biased test unbiased?

#### Intro to question

I am trying to understand sense of unbiasedness in hypothesis testing - here is the definition:

Consider a statistical hypothesis test of size level $$0<\alpha<1$$ for testing a null hypothesis $$H_0:\theta \in \Theta_0 \subset \Theta$$ against an alternative hypothesis $$H_1:\theta \in \Theta_1=\Theta - \Theta_0$$. The test is considered "unbiased" if its power function $$\beta$$ satisfies the condition:

$$\beta(\theta)\leq\alpha \hspace{0.3cm}\text{if}\hspace{0.3cm}\theta \in \Theta_0,$$ $$\beta(\theta)\geq\alpha \hspace{0.3cm}\text{if}\hspace{0.3cm}\theta \in \Theta_1.$$

Answer to this question: "Unbiased" hypothesis test --- what does it mean actually? provides an example of biased test:

Now, suppose you were to use that rule (reject if $$t>1.645$$) to test $$\theta=0$$ against $$\theta\neq0$$. The probability that the test will reject will decrease the more negative the true $$\theta$$, as we shall rarely observe large positive t-ratios in that case. In particular, this test is be biased, as $$\beta(\theta)<\alpha$$ when $$\theta\in\Theta_1\cap(-\infty,0)$$.

and graphical explanation which is a plot of power function for this particular test:

#### The question

In the comment someone asked what can we do about the fact that the test is biased. I think, that the solution is to use test statistic $$|T|$$ instead of $$T$$. Then power function will be symmetrical around $$0$$, something like this:

Can anyone confirm if it is true?

• You should specify the new rejection rule but yes, that approach will work. Commented Dec 9, 2021 at 5:44
• As far as understanding test bias goes, does this help any? stats.stackexchange.com/q/541570/805 Commented Dec 9, 2021 at 5:52
• @Glen_b thanks, your answer to linked question is really useful Commented Dec 9, 2021 at 15:11

## 1 Answer

Although this is a good theoretical knowledge, but I have to say you will never design a study to have a $$\text{power}<\alpha$$ ($$=0.05$$ conventionally). Another thing in this explanation, I would like to mention whether $$\beta(\theta)$$ means type-II error rate or the actual power (although in the text above it says power). Usually $$\beta(\theta)$$ is used to denote type-II error rate; If $$\beta(\theta)$$ is the type-II error rate, then $$\text{power}=1-\beta(\theta)$$.

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Commented Jan 4 at 13:56