So if I set size at 5%, collect the data, and run the test, and repeat the whole experiment with fresh data multiple times, should I expect that, if the null hypothesis is true, the test accepts exactly %5 of times, or at most 5% of times?
If the null hypothesis is simple (that is, if it picks out a single point in the hypothesis space), and the model assumptions are true blah blah blah, then the test (falsely) rejects the null with exactly 5% probability. If the null is composite (comprises a non-singleton subset of parameter space), and there is no nice reduction to a simple null via mathematical tricks like sufficiency or the availability of a pivot, then the test falsely rejects the null with at most 5% probability.
But that’s all very technical; somewhat less technically, almost always, a bootstrap procedure is available that obviates these questions and gets you to “exactly 5%”… asymptotically. Here “asymptotically” means “if the sample size is big enough”. This just throws the question onto “how big is big enough,” and that’s context-dependent. And all of this is about one million times less important than the question of how well each study addresses systematic biases, which is an issue of real, actual study design and implementation rather than mathematical statistical theory.
You want size), not p-value. The difference is that size is a “pre-data” (or “design”) quantity, while the p-value is post-data, i.e., data-dependent.
Thanks.
So if I set size at 5%, collect the data, and run the test, and repeat the whole experiment with fresh data multiple times, should I expect that, if the null hypothesis is true, the test accepts exactly %5 of times, or at most 5% of times?
If the null hypothesis is simple (that is, if it picks out a single point in the hypothesis space), and the model assumptions are true blah blah blah, then the test (falsely) rejects the null with exactly 5% probability. If the null is composite (comprises a non-singleton subset of parameter space), and there is no nice reduction to a simple null via mathematical tricks like sufficiency or the availability of a pivot, then the test falsely rejects the null with at most 5% probability.
But that’s all very technical; somewhat less technically, almost always, a bootstrap procedure is available that obviates these questions and gets you to “exactly 5%”… asymptotically. Here “asymptotically” means “if the sample size is big enough”. This just throws the question onto “how big is big enough,” and that’s context-dependent. And all of this is about one million times less important than the question of how well each study addresses systematic biases, which is an issue of real, actual study design and implementation rather than mathematical statistical theory.