We wish to answer this question: If you observe a 'significant' p-value after doing a single unbiased experiment, what is the probability that your result is a false positive? The weak evidence provided by p-values between 0.01 and 0.05 is explored by exact calculations of false positive risks. When you observe p = 0.05, the odds in favour of there being a real effect (given by the likelihood ratio) are about 3 : 1. This is far weaker evidence than the odds of 19 to 1 that might, wrongly, be inferred from the p-value. And if you want to limit the false positive risk to 5%, you would have to assume that you were 87% sure that there was a real effect before the experiment was done. If you observe p = 0.001 in a well-powered experiment, it gives a likelihood ratio of almost 100 : 1 odds on there being a real effect. That would usually be regarded as conclusive. But the false positive risk would still be 8% if the prior probability of a real effect were only 0.1. And, in this case, if you wanted to achieve a false positive risk of 5% you would need to observe p = 0.00045. It is recommended that the terms 'significant' and 'non-significant' should never be used. Rather, p-values should be supplemented by specifying the prior probability that would be needed to produce a specified (e.g. 5%) false positive risk. It may also be helpful to specify the minimum false positive risk associated with the observed p-value. Despite decades of warnings, many areas of science still insist on labelling a result of p < 0.05 as 'statistically significant'. This practice must contribute to the lack of reproducibility in some areas of science. This is before you get to the many other well-known problems, like multiple comparisons, lack of randomization and p-hacking. Precise inductive inference is impossible and replication is the only way to be sure. Science is endangered by statistical misunderstanding, and by senior people who impose perverse incentives on scientists.
Keywords: false positive risk; null hypothesis tests; reproducibility; significance tests; statistics.