Positive association in a two-way contingency table: a numerical study

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Abstract
In a doubly-ordered two-way contingency table the rows and the columns both define ordinal variables. The idea of positive association between those variables may be formalised in many different ways. Each definition is a composite alternative against which the null hypothesis (no association) may be tested. Asymptotic tests based on 8 possible test statistics are studied using Monte Carlo sampling. For some tests the exact size is shown to be very sensitive to the configuration of the marginal probabilities. Power comparisons show that the choice of test statistic must depend on the form of positive association thought to be appropriate to the problem in hand. The choice is only clear-cut when a relatively strong definition is adopted. All comparisons are made over a reference set based on multinomial sampling, but excluding tables with very small marginal totals. Some reasons are given for not using the orthodox reference set in which all marginal totals are held fixed.