The reproducibility of research and the misinterpretation of P values

Abstract

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 rates.

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 rate 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 rate would still be 8% if the prior probability of a real effect was only 0.1. And, in this case, if you wanted to achieve a false positive rate 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 rate. It may also be helpful to specify the minimum false positive rate 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 “significant”. This practice must account for a substantial part of the lack of reproducibility in some areas of science. And this is before you get to the many other well-known problems, like multiple comparisons, lack of randomisation and P-hacking. Science is endangered by statistical misunderstanding, and by university presidents and research funders who impose perverse incentives on scientists.