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TESTING HOMOGENEITY WITH AN ORDERED ALTERNATIVE

IN A TWO-FACTOR LAYOUT BY COMBINING *p*-VALUES

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RAMAL MOONESINGHE^{ 1} AND F. T. WRIGHT^{ 2}

^{1} *National Research Council, 2101 Constitution Avenue,
Washington, DC 20418, U.S.A.*

^{2} *Department of Statistics, University of Missouri,
Columbia,
MO 65211, U.S.A.*
(Received March 29, 1993; revised September 9, 1994)

**Abstract.**
Two-factor experiments in which
both factors are ordinal are considered. If it is believed
*apriori* that the mean response is
nondecreasing in each factor with the other held fixed,
then one may test for a treatment effect by testing
homogeneity with the appropriate ordered alternative.
The likelihood ratio test has been developed in the
literature, but the level probabilities needed to
implement the test have only been determined in a few
special cases by Monte Carlo techniques. A test
obtained by combining the *p*-values from a test
concerning the rows and a test concerning the columns
is studied. Fisher's method of combining *p*-values is
recommended. It is shown that the likelihood ratio test
is more powerful, but if one does not want to obtain
Monte Carlo estimates of the level probabilities, then
the procedure proposed here should be considered.

*Key words and phrases*:
Bivariate trends,
combining *p*-values, Fisher's method, likelihood
ratio tests, matrix ordering, order restricted tests,
two-moment approximations.

**Source**
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