I've been involved in off-list discussion with Duncan Murdoch. At one
stage there I was about to retire in disgrace. But sighs of relief... his
objection is Bayesian. OK. The p value is a device to put in a
publication to communicate something about precision of an estimate of an
effect, under the assumption of no prior knowledge of the magnitude of the
true value of the effect. If we assume no prior knowledge of the true
value, then my claim stands: the p value for a one-tailed test is the
probability of an opposite true effect--any true effect opposite in sign or
impact to that observed.
I can't see how a Bayesian perspective dilutes or invalidates this
interpretation. The same Bayesian perspective would make you re-evaluate
the p value under its conventional interpretation. In other words, if you
have some other reason for believing that the true value has the same sign
as the observed value, reduce the p value in your mind. Or if you believe
it has opposite sign, increase it.
If we are stuck with p values, then I believe we should start showing
one-tailed p values, along with 95% confidence limits for the
effect. Both these are far far easier to understand than hypothesis
testing and statistical significance. Put a note in the Methods saying
something like: "The p values, which were all derived from one-tailed
tests, represent the probability that the true value of the effect is
opposite in sign (correlations; differences or changes in means) or impact
(relative risks, odds ratios) to that observed."
Will
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