Analyzing bivariate continuous data that have been grouped into categories defined by sample quantiles of the marginal distributions

Abstract

Epidemiologists sometimes study the association between two measures of exposure on the same subjects by grouping the data into categories that are defined by sample quantiles of the two marginal distributions. Although such grouped data are presented in a twoway contingency table, the cell counts in this table do not have a multinomial distribution. We use the term bivariate quantile distribution (BQD) to describe the joint distribution of counts in such a table. Blomqvist (1950) gave an exact BQD theory for the case of only 4 categories based on division at the sample medians. The asymptotic theory he presented was not valid, however, except in special cases. We present a valid asymptotic theory for arbitrary numbers of categories and apply this theory to construct confidence intervals for the kappa statistic. We show by simulations that the confidence interval procedures we propose have near nominal coverage for sample sizes exceeding 90, both for 2 x 2 and 3 x 3 tables. These simulations also illustrate that the asymptotic theory of Blomqvist (1950) and the methods given by Fleiss, Cohen and Everitt (1969) for multinomial sampling can yield subnominal coverage for BQD data, although in some cases the coverage for these procedures is near nominal levels.