Optimality

90 D. G. Mayo and D. R. Cox
some theory being contemplated for general applicability, consistency with the null
hypothesis may be regarded as some additional evidence for the theory, especially
if the test and data are sufficiently sensitive to exclude major departures from the
theory. An aspect is encapsulated in Fisher’s aphorism (Cochran [3]) that to help
make observational studies more nearly bear a causal interpretation, one should
make one’s theories elaborate, by which he meant one should plan a variety of
tests of different consequences of a theory, to obtain a comprehensive check of its
implications. The limited result that one set of data accords with the theory adds
one piece to the evidence whose weight stems from accumulating an ability to refute
alternative explanations.
In the first type of example under this rubric, there may be apparently anomalous
results for a theory or hypothesisT , whereT has successfully passed appreciable
theoretical and/or empirical scrutiny. Were the apparently anomalous results forT
genuine, it is expected that H0 will be rejected, so that when it is not, the results
are positive evidence against the reality of the anomaly. In a second type of case,
one again has a well-tested theoryT,and a rival theoryT ∗ is determined to conflict
withT in a thus far untested domain, with respect to an effect. By identifying the
null with the prediction fromT , any discrepancies in the direction ofT ∗ are given
a very good chance to be detected, such that, if no significant departure is found,
this constitutes evidence forT in the respect tested.
Although the general theory of relativity, GTR, was not facing anomalies in the
1960s, rivals to the GTR predicted a breakdown of the Weak Equivalence Principle
for massive self-gravitating bodies, e.g., the earth-moon system: this effect, called
the Nordvedt effect would be 0 for GTR (identified with the null hypothesis) and
non-0 for rivals. Measurements of the round trip travel times between the earth and
moon (between 1969 and 1975) enabled the existence of such an anomaly for GTR
to be probed. Finding no evidence against the null hypothesis set upper bounds to
the possible violation of the WEP, and because the tests were sufficiently sensitive,
these measurements provided good evidence that the Nordvedt effect is absent, and
thus evidence for the null hypothesis (Will [31]). Note that such a negative result
does not provide evidence for all of GTR (in all its areas of prediction), but it does
provide evidence for its correctness with respect to this effect. The logic is this:
theoryT predicts H0 is at least a very close approximation to the true situation;
rival theoryT ∗ predicts a specified discrepancy from H0, and the test has high
probability of detecting such a discrepancy fromT wereT ∗ correct. Detecting no
discrepancy is thus evidence for its absence.
3.6. Confidence intervals
As noted above in many problems the provision of confidence intervals, in principle
at a range of probability levels, gives the most productive frequentist analysis. If
so, then confidence interval analysis should also fall under our general frequentist
principle. It does. In one sided testing of µ = µ0 against µ > µ0, a small p-value
corresponds to µ0 being (just) excluded from the corresponding (1−2p) (two-sided)
confidence interval (or 1−p for the one-sided interval). Were µ = µL, the lower
confidence bound, then a less discordant result would occur with high probability
(1−p). Thus FEV licenses taking this as evidence of inconsistency with µ = µL (in
the positive direction). Moreover, this reasoning shows the advantage of considering
several confidence intervals at a range of levels, rather than just reporting whether
or not a given parameter value is within the interval at a fixed confidence level.