Consider the question whether a variable x1 at time t is a predictor of variable x2 at time t+1.

If the estimated mean difference between them is ~ 0, then we have evidence that the answer is yes.

I have seen research articles that uses this to frame the following hypothesis test:

H0: the expected difference between x1_t and x2_{t+1} is zero, that is the former is an unbiased predictor of the latter.

H1: the said quantity is nonzero

The research question is interesting only if H0 cannot be rejected. That is, in conventional framing, p value being the highest.

Is this a poor framing? Because, in standard framing, rejecting H0 is the challenge, I.e. getting a low p-value. Unable to reject H0 is usually disappointing.

In the framing I described above, the opposite is true.

  • $\begingroup$ Sometimes in goodness-of-fit tests one is hoping for evidence of randomness or equi-distribution. Whether that exactly fits the meaning of your title is not clear.// If you're looking for a useful pseudorandom number generator, it's easy to assume a particular generator is OK because it has passed many tests---until the day when an obvious flaw is encountered. (Consider the lore of RANDU.) $\endgroup$ – BruceET Mar 31 at 17:08
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    $\begingroup$ If testing whether the difference is small is your purpose, you may want to look into equivalence testing. $\endgroup$ – Do not reinstate Monica Apr 6 at 15:11
  • $\begingroup$ This is exactly what I was looking for. Equivalence tests vs t-test literature gives good answer to my question. $\endgroup$ – learning Apr 7 at 6:59

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