2018 Australian Statistical Publishing Association Inc. Published by John Wiley & Sons Australia Pty Ltd.
Aust. N. Z. J. Stat. 60(1), 2018, 132–139 doi: 10.1111/anzs.12188
A new permutation test statistic for K -sample multivariate designs
University of Sydney
We consider permutation tests based on a likelihood ratio like statistic for the one way or
k-sample design where the observations are l-dimensional vectors. This generalizes both
the univariate k-sample case and the two sample multivariate case used in the examples of
Kolassa and Robinson. We also give a saddlepoint approximation for the permutation test.
Numerical examples are given to illustrate the accuracy of the saddlepoint approximation
and the improvement in the power of the tests compared to the classical statistics in the
case of long tailed error distributions and no loss of power for normal error distributions.
Key words: MANOVA; Saddlepoint approximations.
We dedicate this paper to the memory of Peter Hall (1951–2016). Peter’s contribution
to asymptotic approximations used in nonparametric inference, in particular in considering
second order approximation for the bootstrap, are unparalleled, with his book, Hall (1992),
giving a detailed treatment based on Edgeworth and Cornish-Fisher expansions. Both the
bootstrap and permutation tests are known as resampling methods. Permutation tests predate
the bootstrap but are more limited in their application. However when they can be applied to
a problem they give exact conditional tests. This paper provides a permutation test statistic
which can give improved power over standard methods and whose distribution can be
approximated using a saddlepoint approximation giving second order relative error accuracy.
We consider observations, consisting of vectors in l dimensions, from an experiment in
which k treatments are allocated at random to subsets of size n
, or in which we draw
samples of sizes n
from k distributions F
in l dimensions. In the ﬁrst case we
wish to test the hypothesis that the treatments have identical effects and in the second case
we we wish to test the hypothesis that F
. In the permutation test we compare an
observed test statistic to the set of statistics obtained under permutations of the observations.
The choice of test statistic depends on some model for departures from the null hypothesis.
In both cases such a permutation test is valid, in the sense of providing correct size of test
or approximate uniformity of P-values under the null hypotheses, for any but trivial test
statistics. However, to get a test with appropriate power we need to consider the model for
alternative hypotheses. Here we will think of alternatives where treatments have different
additive effects in the ﬁrst case and different means of the distributions in the second case.
*Author to whom correspondence should be addressed.
School of Mathematics and Statistics, University of, Sydney, NSW 2006, Australia
Acknowledgement. Inga Samonenko was supported by University of Sydney Postgraduate Award and John
Robinson was supported by Australian Research Council Discovery Project 0773345.
Australian & New Zealand Journal of Statistics