# Is the Chi-Square Test for Association more powerful than a Chi-Square Goodness of Fit Test?

Is the Chi-Square test for association (comparing two proportions) or a Fisher's exact test more powerful than a chi-square goodness of fit test (comparing multiple proportions to determine whether at least one proportion among several is different from the others)?

• What is the basis on which to compare power for tests of very different hypotheses? You could compare power of tests of the same null against the same alternative (apples with apples), but not for completely different kinds of test. It's a bit like asking who is better out of this novelist and that wrestler. Commented Aug 19, 2021 at 16:44

Power always depends on a specific alternative; most tests including these can have very low power against close alternatives (to the $$H_0$$) and high power against far alternatives. As the tests you are asking to compare here are for different types of problem (and therefore the alternatives to compare power on are also different), powers cannot be compared.
• In response to a question that seems to have been deleted in the meantime: As I wrote, power is computed on alternatives, so in the first place the alternatives have to be the same. However, it will be in most cases hard to argue that a comparison makes sense if the alternative is the same but the null hypothesis is different. (You could compare power against $\mu=1$, say, for a two-sided test with $H_0:\ \mu=0$, and for a one-sided test with $H_0:\ \mu\le 0$, although arguably these null hypotheses are still "pretty much the same".) Commented Aug 19, 2021 at 15:44