# Should we use the NHST framework when we want to accept the null hypothesis?

Given that what a p-value tells us is the probability of observing the data assuming the null is true, should we ever use null hypothesis significance testing (NHST) when we would rather accept the null hypothesis?

Take the Shapiro-Wilk test through which we might want to establish the normality of our data. Given a p-value of .36 and a conventional $\alpha$ of .05, we would fail to reject the null hypothesis. And the typical follow up is the suggestion that our data are normal.

However, the specific interpretation of $p=.36$ is: assuming our data are normal, the probability of observing these data is 36%. Considering this, it does not seem like good practice to act like failure to reject the null is equivalent to accepting the null. Am I correct?

Absence of evidence is not evidence of absence. Why then is the NHST framework often used when we would want to accept the null?

• You are right, they are not equivalent and "fail to reject the null hypothesis" is preferable (as noted in other related questions -- see the right pane). If a researcher uses "accept the null hypothesis" in NHST framework, I think it is an unfortunate misunderstanding or bad habit. – T.E.G. Jul 12 '17 at 4:40
• But there are times when we want to accept the null hypothesis such as in tests for multivariate normality, not just fail to reject. Because absence of evidence - fail to reject - is not evidence of absence - accept the null. – Heteroskedastic Jim Jul 13 '17 at 1:15
• I don't think in tests like the one you gave as an example (Shapiro-Wilk test) the null hypothesis can be accepted (see this answer). Maybe the answers to this question would provide some different perspectives. – T.E.G. Jul 13 '17 at 2:01
• I believe I know this already. The Shapiro-Wilk cannot in any way be used to accept a null hypothesis. But I guess my question is why do prominent statisticians create such tests, when what we really want to do is accept the null hypothesis? – Heteroskedastic Jim Jul 13 '17 at 2:03