Why is the P value not the probability that the null hypothesis is true?
A moment’s reflection should convince you that the P value could not be the probability that the null hypothesis is true. Suppose we got exactly the same value for the mean in two samples (if the samples were small and the observations coarsely rounded this would not be uncommon; the difference between the means is zero). The probability of getting the observed result (zero) or a result more extreme (a result that is either positive or negative) is unity, that is we can be certain that we must obtain a result which is positive, negative or zero. However, we can never be certain that the null hypothesis is true, especially with small samples, so clearly the statement that the P value is the probability that the null hypothesis is true is in error. We can think of it as a measure of the strength of evidence against the null hypothesis, but since it is critically dependent on the sample size we should not compare P values to argue that a difference found in one group is more “significant”
