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ROBUST TESTS IN GROUP SEQUENTIAL ANALYSIS:

ONE- AND TWO-SIDED HYPOTHESES IN THE LINEAR MODEL

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MERVYN J. SILVAPULLE^{1} AND PRANAB K. SEN^{2}

^{1} *School of Agriculture, La Trobe University, Bundoora, Victoria 3083, Australia*

^{2} *Department of Biostatistics, University of North Carolina,*

Chapel Hill, NC 27599-7400, U.S.A.
(Received August 5, 1991; revised December 24, 1991)

**Abstract.**
Consider the linear model *Y* = *X**theta* + *E* in the
usual matrix notation where the errors are independent and identically
distributed. We develop robust tests for a large class of one- and
two-sided hypotheses about *theta* when the data are obtained and tests
are carried out according to a group sequential design. To illustrate
the nature of the main results, let
^{^}*theta* and ^{~}*theta*
be an *M*- and the
least squares estimator of *theta* respectively which are
asymptotically normal about *theta* with covariance matrices
*sigma*^{2}(*X*^{t}*X*)^{-1} and *tau*^{2}(*X*^{t}*X*)^{-1} respectively. Let the
Wald-type statistics based on
^{^}*theta* and ^{~}*theta*
be denoted by *RW* and
*W* respectively. It is shown that *RW* and *W* have the same asymptotic
null distributions; here the limit is taken with the number of groups
fixed but the numbers of observations in the groups increase
proportionately. Our main result is that the asymptotic Pitman
efficiency of *RW* relative to *W* is (*sigma*^{2} / *tau*^{2}). Thus, the
asymptotic efficiency-robustness properties of
^{^}*theta*
relative to ^{~}*theta*
translate to asymptotic power-robustness of *RW* relative to *W*.
Clearly, this is an attractive result since we already have a large
literature which shows that
^{^}*theta*
is efficiency-robust compared to
^{~}*theta*. The results of a simulation study show that with realistic
sample sizes, *RW* is likely to have almost as much power as *W* for
normal errors, and substantially more power if the errors have long
tails. The simulation results also illustrate the advantages of group
sequential designs compared to a fixed sample design, in terms of sample
size requirements to achieve a specified power.

*Key words and phrases*:
Clinical trial, comparison of two
treatments, composite hypothesis, inequality tests, interim analysis,
long tailed distribution, *M*-estimator, Pitman efficiency,
power-robustness, repeated tests, Wald-type statistics.

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