Simultaneous Confidence Limits for Cross-Product Ratios in Contingency Tables

Abstract

Fisher (1962) has proposed a method for obtaining confidence limits for the cross-product ratio in a 2 × 2 table which requires that the user solve a quartic equation. Based on a different mathematical derivation, Cornfield (1956) had proposed a similar method. In the present article, we shall present simpler methods for obtaining confidence limits for the cross-product ratio in a 2 × 2 table, and we shall extend these methods to obtain simultaneous confidence intervals for ther(r -1)c(c -l)/4 cross-product ratios in anr × ctable (or for a subset of them) and also for the relative differences between the corresponding cross-product ratios inKdifferentr × ctables. In addition, we shall present a modification of a method suggested by Gart (1962a) for the 2 × 2 table, and we shall extend the modified method to ther × ctable. The methods presented herein are easier to apply than those given in the earlier literature. For the 2 × 2 table, the confidence limits presented herein are asymptotically equivalent to the limits given earlier. When max (r, c, K) > 2, the simultaneous confidence intervals presented herein for the cross-product ratios and for the relative differences between cross-product ratios are asymptotically shorter than the corresponding intervals given by Cornfield (1956), for the usual probability levels. The method proposed herein for studying the relative differences between cross-product ratios inK r × ctables can also be used to supplement the earlier methods of analysis given by Plackett (1962) and Goodman (1963b) for testing the null hypothesis that these differences are all nil.