The effect of marginal disuniformity on the χ 2 approximation to the distribution of Pearson's X 2 in sparse contingency tables

Abstract

Let n k be a sequence of contingency tables with an increasing number of cells q k , n k be the sample size and π k be the vector of cell probabilities generating n k . Theoretical papers, like Morris ( Annals of Statistics 3, 1975) and Haberman ( Annals of Statistics 5, 1977), suggest that the distribution of test-statistics for testing parametric models for π k depends on n k , on some function of the ratio n k / q k and also on the ‘disuniformity’ of the expected frequencies in the table. In the context of testing independence in two-way contingency tables, this paper investigates the χ 2 approximation to the conditional distribution of X 2 given the margins. Attention is focussed on the effect of disuniformity on this approximation, as n k and q k increase, for different levels of n k / q k . This investigation is first carried out by analytically studying the ratio VR = var( X 2)/var(χ 2) as a function of q k , of n k / q k and of a pair of disuniformity measures for the marginal totals that enter directly the formula for the exact variance of X 2. Moreover, the χ 2 approximation is studied by means of an extensive numerical work, involving several simulations for various combinations of q k / q k and of the pair of disuniformity measures mentioned above. The main outcome is that marginal disuniformity greatly affects the χ 2 approximation, mainly by affecting the X 2 variance. Hence, the variance ratio VR may be used to assess the appropriateness of the traditional χ 2 test; this criterion is many circumstances less severe than the traditional ones.