There is the following value to analyzing the power of a test when not
being able to reject the null hypothesis. This is useful when one does
not have explicit confidence intervals. A power analysis can reveal the
following insights on where the true data generating process may lie.
The set of all dgps such that the probability of rejecting the null is
below 95% under the given test is a 95% confidence region for the true dgp.
More specifically, assume one is interested in a parameter tau and has a
test for the null hypothesis that tau <= 7. Assume that the null
hypothesis is rejected in the sense that the p value is above 5%. Assume
that a rigorous power analysis reveals that the probability of rejecting
the null hypothesis is above 95% whenever tau >16. Then tau=16 is a 95%
upper confidence bound on tau.
In other words, a rejection at 5% means that tau=7 is a 95% lower
confidence bound while no rejection at 5% means that tau = 16 is an
upper confidence bound. The formal proof uses the standard construction
of confidence regions from tests.
Another example could be a noninferiority test where one is not able to
reject that mean1-mean2 <= -0.2. If one finds that the power is greater
than 1-beta whenever mean1 >=mean2 then the p value associated to
equivalence (in terms of equality of the two means) is equal to beta. So
if beta is small then when not being able to reject inferiority then one
in fact has evidence against equivalence.
Karl
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Karl Schlag
Professor Tel: +34 93 542 1493
Department of Economics and Business Fax: +34 93 542 1746
Universitat Pompeu Fabra email: [log in to unmask]
Ramon Trias Fargas 25-27 www.iue.it/Personal/Schlag/
Barcelona 08005, Spain room: 20-221 Jaume I
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