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We calculate power in hypothesis testing as 1-beta (probability that null hypothesis is false given that alternative hypothesis is true). As this probability increases, power also increases.

By similar relationship can't we say that power could also be calculated as equal to alpha( probability that null hypothesis is rejected given that null hypothesis is correct) as alpha and beta are inversely related? So why did we choose the first definition?

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alpha and beta are inversely related

does not mean that $\alpha + \beta = 1$. It just means that, holding everything else constant (sample size, the effect size you wish to detect), if you increase $\alpha$ then $\beta$ will decrease. This decrease could happen in a arbitrarily complicated way.

So why did we define power in terms of beta instead of alpha as calculating alpha requires lesser steps

Power is defined as

$$ P(\text{Rejecting } H_0 \mid H_0 \text{ is false}) $$

usually with an assumption about the effect size of $H_a$. This simply cannot be reduced to a function of $\alpha$ alone.

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  • $\begingroup$ So why did we define power in terms of beta instead of alpha as calculating alpha requires lesser steps $\endgroup$ – user1825567 Jun 19 '17 at 13:38

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