Assessing environmentally significant effects: a better strength-of-evidence than a single P value?

Interpreting a P value from a traditional nil hypothesis test as a strength-of-evidence for the existence of an environmentally important difference between two populations of continuous variables (e.g. a chemical concentration) has become commonplace. Yet, there is substantial literature, in many disciplines, that faults this practice. In particular, the hypothesis tested is virtually guaranteed to be false, with the result that P depends far too heavily on the number of samples collected (the 'sample size'). The end result is a swinging burden-of-proof (permissive at low sample size but precautionary at large sample size). We propose that these tests be reinterpreted as direction detectors (as has been proposed by others, starting from 1960) and that the test's procedure be performed simultaneously with two types of equivalence tests (one testing that the difference that does exist is contained within an interval of indifference, the other testing that it is beyond that interval-also known as bioequivalence testing). This gives rise to a strength-of-evidence procedure that lends itself to a simple confidence interval interpretation. It is accompanied by a strength-of-evidence matrix that has many desirable features: not only a strong/moderate/dubious/weak categorisation of the results, but also recommendations about the desirability of collecting further data to strengthen findings.