# What is the meaning of a large p-value?

I understand that the $p$-value is the conditional probability of observing the test statistic or something more extreme given that the null hypothesis is true. I have read the great explanation by @user28 in this post: What is the meaning of p values and t values in statistical tests? However, do large $p$-values say anything? Does a larger $p$-value lend greater support to the null hypothesis? If I set rejection region to be $<0.05$, then does it make a difference if I get $p$-value $0.06$ or $0.99$? (After all, $0.05$ is arbitrary, and $0.06$ is so close to being rejected that if I arbitrarily set $0.05$ as $0.1$ instead, the null hypothesis would have been rejected.) Can one make any statistical use of a non-rejecting $p$-value?

• One example is mentioned at the end of this answer: Fisher's re-examination of Mendel's pea experiments. – whuber Jun 23 '14 at 14:37
• You cannot conclude that a test statistic supports the null hypothesis when the calculation of that statistic is conditional on the assumption that the null is true. That is to say, if $E$ and $H_0$ are events, you cannot say $H_0$ is true if $\Pr[E \mid H_0]$ is "large." – heropup Jun 23 '14 at 16:04

How you should 'use' the p-value depends on how you have designed your study with regard to the analyses you will run. I discuss two different philosophies about p-values in my answer here: When to use Fisher and Neyman-Pearson framework? You may find it helpful to read that. If you have, for example, run a power analysis and intend to use the p-value to make a final decision, you should not use close to the line ('marginally significant') as a meaningful category. It is fine to use a different alpha than $0.05$ (such as $0.10$), but once you decided on it and set your study up accordingly, you should stick with it.