Assuming a test where p > alpha and n is large enough for power > 95% at effect size d, what is the exact interpretation of the test regarding the relationship between the observed data, the real effect, and power for d?
Some details: if my p is larger than my alpha, that means the observed data are not surprising from the perspective of a nil-null hypothesis. But according to Cohen (1990, p. 1309), the failed test, in combination with an estimate of power given a d, also allows me to estimate something similar to, but not actually the following statement: based on my sample, the real population effect is likely (where "likely" is somehow related to my 95% power) as close or closer to no effect than the d I have calculated my power for (not the d I have measured). However, I am not aware of a precise definition, and this statement is definitely false since it interprets the data from the perspective of p(H|D) and not p(D|H)…
I am looking for a statement comparable to “given a p value below alpha, assuming a zero effect, observing data as or more extreme than the evaluated sample obtained has a lower probability than alpha”, but from the other direction.
One perspective on this comes from a CI: if my CI is narrow (due to high power) and includes zero, I know that only for a small range of hypotheses centred around and including zero, the probability of obtaining the current (or a smaller) measure would be less surprising than my alpha; conversely, for all hypotheses assuming an effect outside of my CI, the data would be surprising. But I am not sure how to phrase this in reference to power.
Admittedly, this question could probably be answered by reading Cohen 1988, however, I do not have the book with me. Also, I assume this problem is commonly enough misunderstood. I would be happy with a pointer to an authoritative source, too.