Cross-product Ratios Research Articles

The analysis of ratios and cross-product ratios of poisson variates with application to incidence rates

The incidence of most diseases is low enough that in. large populations the number of new cases may be considered a Poisson variate. This paper explores models and methods for analyzing such data Specific cases are the estimation and testing of ratios and the cross-product ratios, both simple and stratified* We assume the Poisson means are exponential functions of the relevant parameters. The resulting sets of sufficient statistics are partitioned into a test statistic and a vector of statistics related to the nuisance parameters . The methods derived are based on the conditional distribution of the test statistic given the other sufficient statistics. The analyses of stratified cross-product ratios are seen to be analogues of the noncentral distribution associated with theanalysis of the common odds ratio in several 2×2 tables. The various methods are illustrated in numerical examples involving incidence rates of cancer in two metropolitan areas adjusting for both age and sex.

INTERPRETATION OF GOODMAN'S LOG-LINEAR MODEL EFFECTS An Odds Ratio Approach

In the analysis of multidimensional contingency tables using Goodman's log-linear models, the sometimes complex nature of the problems encountered may lead to an interpretation of analytic results which is based solely upon statistical significance, and not upon substantive importance. In this paper, the well-known relationships between log-linear model effects and odds ratios ("cross-product ratios") are reexamined using previously analyzed sample survey data. The major thrust of the paper concerns the transformation of the log-linear model effects into odds ratios, which then yield simple and direct substantive interpretations.

Exact and asymptotic methods for the combination of 2 × 2 tables

A FORTRAN program for computing the conditional maximum likelihood estimate of the combined odds ratios (also called relative risks or cross-product ratios) from a set of 2 × 2 tables is given. Exact tests for the main effect and interation as well as exact confidence limits are given. Optionally, these results may be computed asymptotically. The program is illustrated with a numerical example.

Simultaneous Confidence Limits for Cross-Product Ratios in Contingency Tables

Summary 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 the r(r - 1) c(c - l)/4 cross-product ratios in an r × c table (or for a subset of them) and also for the relative differences between the corresponding cross-product ratios in K different r × c tables. 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 the r × c table. 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 in K r × c tables 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.