Why are lower p-values not more evidence against the null? Arguments from Johansson 2011 Johansson (2011) in "Hail the impossible: p-values, evidence, and likelihood" (here is also link to the journal) states that lower $p$-values are often considered as stronger evidence against the null. Johansson implies that people would consider evidence against the null to be stronger if their statistical test outputted a $p$-value of $0.01$, than if their statistical test outputted a $p$-value of $0.45$. Johansson lists four reasons why the $p$-value cannot be used as evidence against the null:

  
*
  
*$p$ is uniformly distributed under the null hypothesis and can therefore never indicate evidence for the null. 
  
*$p$ is conditioned solely on the null hypothesis and is therefore unsuited to quantify evidence, because evidence is always
  relative in the sense of being evidence for or against a hypothesis
  relative to another hypothesis.
  
*$p$ designates probability of obtaining evidence (given the null), rather than strength of evidence.
  
*$p$ depends on unobserved data and subjective intentions and therefore implies, given the evidential interpretation, that the
  evidential strength of observed data depends on things that did not
  happen and subjective intentions.
  

Unfortunately I cannot get an intuitive understanding from Johansson's article. To me a $p$-value of $0.01$ indicates there is less chance the null is true, than a $p$-value of $0.45$. Why are lower $p$-values not stronger evidence against null?  
 A: Adding to @Momo's nice answer:
Do not forget multiplicity. Given many independent p-values, and sparse non-trivial effect sizes, the smallest p-values are from the null, with probability tending to $1$ as the number of hypotheses increases.
So if you tell me you have a small p-value, the first thing I want to know is how many hypotheses you have been testing.
A: My personal appraisal of his arguments:


*

*Here he talks about using $p$ as evidence for the Null, whereas his thesis is that $p$ can't be used as evidence against the Null. So, I think this argument is largely irrelevant.

*I think this is a misunderstanding. Fisherian $p$ testing follows strongly in the idea of Popper's Critical Rationalism that states you cannot support a theory but only criticize it. So in that sense there only is a single hypothesis (the Null) and you simply check if your data are in accordance with it.

*I disagree here. It depends on the test statistic but $p$ is usually a transformation of an effect size that speaks against the Null. So the higher the effect, the lower the p value---all other things equal. Of course, for different data sets or hypotheses this is no longer valid.   

*I am not sure I completely understand this statement, but from what I can gather this is less a problem of $p$ as of people using it wrongly. $p$ was intended to have the long-run frequency interpretation and that is a feature not a bug. But you can't blame $p$ for people taking a single $p$ value as proof for their hypothesis or people publishing only $p<.05$.   


His suggestion of using the likelihood ratio as a measure of evidence is in my opinion a good one (but here the idea of a Bayes factor is more general), but in the context in which he brings it is a bit peculiar: First he leaves the grounds of Fisherian testing where there is no alternative hypothesis to calculate the likelihood ratio from. But $p$ as evidence against the Null is Fisherian. Hence he confounds Fisher and Neyman-Pearson. Second, most test statistics that we use are (functions of) the likelihood ratio and in that case $p$ is a transformation of the likelihood ratio. As Cosma Shalizi puts it: 

among all tests of a given size $s$ , the one with the smallest miss
  probability, or highest power, has the form "say 'signal' if
  $q(x)/p(x) > t(s)$, otherwise say 'noise'," and that the threshold $t$
  varies inversely with $s$. The quantity $q(x)/p(x)$ is the likelihood
  ratio; the Neyman-Pearson lemma says that to maximize power, we should
  say "signal" if it is sufficiently more likely than noise.

Here $q(x)$ is the density under state "signal" and $p(x)$ the density under state "noise". The measure for "sufficiently likely" would here be $P(q(X)/p(x) > t_{obs} \mid H_0)$ which is $p$. Note that in correct Neyman-Pearson testing $t_{obs}$ is substituted by a fixed $t(s)$ such that $P(q(X)/p(x) > t(s) \mid H_0)=\alpha$.  
A: The reason that arguments like Johansson's are recycled so often seem to be related to the fact that P-values are indices of the evidence against the null but are not measures of the evidence. The evidence has more dimensions than any single number can measure, and so there are always aspects of the relationship between P-values and evidence that people can find difficult.
I have reviewed many of the arguments used by Johansson in a paper that shows the relationship between P-values and likelihood functions, and thus evidence: http://arxiv.org/abs/1311.0081
Unfortunately that paper has now been three times rejected, although its arguments and the evidence for them have not been refuted. (It seems that it is distasteful to referees who hold opinions like Johansson's rather than wrong.)
A: Is Johansson talking about p-values from two different experiments?  If so, comparing p-values may be like comparing apples to lamb chops.  If experiment "A" involves a huge number of samples, even a small inconsequential difference may be statistically significant.  If experiment "B" involves only a few samples, an important difference may be statistically insignificant.  Even worse (that's why I said lamb chops and not oranges), the scales may be totally incomparable (psi in one and kwh in the other). 
