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effective treatment is effective for only a subset of the population.
A locally most powerful test is given in Conover and Salsburg
[1988].”
Dependent Observations
The preceding statistical methods are not applicable if the observations are
interdependent. There are five cases in which, with some effort, analysis
may still be possible: repeated measures, clusters, known or equal pairwise
dependence, a moving average or autoregressive process,
4
and group
randomized trials.
Repeated Measures.
Repeated measures on a single subject can be dealt
with in a variety of ways including treating them as a single multivariate
observation. Good [2001, Section 5.6] and Pesarin [2001, Chapter 11]
review a variety of permutation tests for use when there are repeated
measures.
Another alternative is to use one of the standard modeling approaches
such as random- or mixed-effects models or generalized estimating equa-
tions (GEEs). See Chapter 10 for a full discussion.
Clusters.
Occasionally, data will have been gathered in clusters from fami-
lies and other groups who share common values, work, or leisure habits.
If stratification is not appropriate, treat each cluster as if it were a single
observation, replacing individual values with a summary statistic such as
an arithmetic average (Mosteller and Tukey, 1977).
Cluster-by-cluster means are unlikely to be identically distributed,
having variances, for example, that will depend on the number of individu-
als that make up the cluster. A permutation test based on these means
would not be exact.
If there are a sufficiently large number of such clusters in each treatment
group, the
bootstrap
defined in Chapter 3 is the appropriate method of
analysis.
With the bootstrap, the sample acts as a surrogate for the population.
Each time we draw a pair of bootstrap samples from the original sample,
we compute the difference in means. After drawing a succession of such
samples, we'll have some idea of what the distribution of the difference in
means would be were we to take repeated pairs of samples from the popu-
lation itself.
As a general rule, resampling should reflect the null hypothesis, accord-
ing to Young [1986] and Hall and Wilson [1991]. Thus, in contrast to
the bootstrap procedure used in estimation (see Chapter 3), each pair of
bootstrap samples should be drawn from the
combined sample
taken from
4
For a discussion of these latter, see Brockwell and Davis [1987].
