Rank-Based Directional Test in k-Sample Multivariate Problems
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Date
2021-07-01
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Addis Ababa University
Abstract
Data from several response variables, potentially measured on different scales, occur
naturally in various practical settings such as clinical trials. Treatment differences
with respect to these response variables are usually analyzed using parametric methods
under the assumptions of multivariate normality and covariance homogeneity.
In many situations, however, these assumptions are not fulfilled, in particular when
the response variables under consideration are ordered categorical. Thus, a rankbased
(nonparametric) approach which disregards the above assumptions is desirable.
More importantly, an investigator may be interested to test a global hypothesis
of no treatment difference versus the alternative that treatment effects are monotonically
increasing (decreasing) with respect to all responses. A number of studies
have been conducted to address the issue of testing directional alternatives in two or
more multivariate samples both in the parametric as well as nonparametric framework.
Given that the response variables are measured in a mix of metric and ordered
categorical scales, the parametric methods are not suitable. In turn, most of the
contributions in the area of nonparametric statistics base their inferences on several
pairwise comparisons of treatments in such a way that the ranks are being computed
pairwise only, that is, only between those two levels that are compared at each step.
This reduces the amount of available information and is well known to potentially
lead to paradoxical situations. In order to incorporate more information from multivariate
data for testing directional hypotheses which involve variables measured
both in metric as well as ordered categorical scales in a unified manner we propose
a new rank-based test statistic. The statistic we have derived is a multivariate generalization
based on a coordinate-wise approach of a univariate test statistic proposed
by Bathke (2009) for alternative patterns within a nonparametric framework. Separate
ranking for different variables is employed in order to ensure invariance under
monotone transformations of the responses as well as the weights describing alternative
patters. Unlike most methods available in the literature, the newly introduced
test handles data with ties, in particular, ordered categorical data as the underlying
distribution is not required to be continuous. The test statistic introduced in
this dissertation is proved to be accurate in detecting pre-specified equi-directional
alternative patterns across two or more multivariate samples trough extensive simulation
studies. A comparison is also made with that of the rank-sum type test for
directional multivariate problems proposed by O’Brien (1984) in which the newly
developed test is in par and sometimes better than the test by O’Brien. Applications
to several datasets obtained from clinical trials are presented and potential extensions
in different directions are discussed.
The other more interesting practical issue in directional multivariate problems is
to test conjectured alternative patterns in which treatments effects are monotonically
increasing for some of the responses and monotonically decreasing for others. So
long as treatment effects can be specified on a priori basis, we suggest interchanging
the signs of different responses and making the anticipated direction of treatment
effects similar. Following this, we employ the newly developed test statistic to test
treatment effects in opposite directions in two or more multivariate samples. Furthermore,
we employ the closed testing principle in conjunction with the test we
have proposed in order to identify on which specific responses or sets of responses
the effects are actually observed. An application of this procedure is demonstrated
by re-analyzing a dose-response dataset. In summary, the test developed in this
dissertation can handle monotone trends based on a complete case multivariate data.
Developing a test statistic which can handle umbrella alternatives, and/or incomplete
cases is differed to future research.
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Keywords
Rank-Based, Directional Test, k-Sample, Multivariate Problems