Beyond Breslow-Day: Homogeneity Across R x C Tables

In the epidemiological world, we often encounter the following analytical question: is the relationship between exposure and outcome different for different strata? The Breslow-Day test (with or without the Tarone adjustment) can be used in PROC FREQ to assess homogeneity across a series of 2 x 2 tables, but what if your tables are not 2 x 2? One alternative is to fit a log-linear model to a k x R x C table and test the fit with the three-way interaction removed. This can be done easily in PROC CATMOD or, with some additional effort, in PROC GENMOD and PROC GLIMMIX. A real-world data example will demonstrate the details of different methods to analyze k x R x C tables and tests for homogeneity. INTRODUCTION One frequent question that epidemiologists come across is whether the relationship between exposure and outcome is different for different strata. For example, in the study of treatment responses, we may want to test the hypothesis that the distribution between the study drug and a traditional therapy dose change is the same at each level of disease severity. In this case, the distribution we are interested in encompasses not only an exposure-outcome relationship but also the homogeneity across different strata. Homogeneity is defined as: the conditional relationship between any pair of variables given a third one is the same at each level of that third variable. Many of us are familiar with PROC FREQ as a tool for assessing homogeneity in a contingency table using count data. Specifically, the Breslow-Day (BD) Statistic produced using the Cochran Mantel Haenzel (CMH) option in PROC FREQ is commonly used for this purpose. If we add the BDT option, which requests the Breslow Day Statistic with the Tarone adjustment, SAS will give an adjusted, asymptotically chi-squared result. If we are interested in studying the distribution between exposures and outcomes that have more than two levels, we cannot use PROC FREQ to assess homogeneity. The Breslow-Day Statistic has not been generalized for this kind of k x R x C table and can only be used for k x 2 x 2 tables. Fitting a log-linear model can solve this problem. On one hand, the GENMOD procedure and the GLIMMIX procedure can derive the homogeneity statistic in this situation with pooled data. On the other hand, PROC CATMOD is set up to achieve the goal without pooling the data first. In addition, the PROC CATMOD method does not require manual calculation of the likelihood ratio P values. We will provide examples to demonstrate the details of adopting different methods for analyzing k x r x c tables. SAMPLE DATA In this paper, we will be using a sample dataset containing patients with medication information at baseline and 1 year follow-up, all of whom are receiving traditional therapy. Our exposure component is the treatment group: placebo, newly on study drug, and continuing on study drug. The outcome is the traditional therapy dose increase, decrease, or remaining the same from baseline to one-year follow-up. Finally, all the patients were stratified by physicians into moderate or severe disease categories. Our hypothesis is that moderate patients who start the study drug will decrease their traditional therapy use to a greater extent than severe patients who start the study drug. K × 2 × 2 TABLES In order to compare the PROC CATMOD output to the Breslow-Day output, we first collapsed the treatment into 2 groups: placebo and study drug groups. In addition, we grouped the ‘same’ and ‘increase’ traditional dose together. Figure 1 shows the collapsed data. A slight difference was shown between the distribution among the ‘decrease’ and ‘same/increase’ group in the placebo patients, i.e. 42% and 58% versus 46% and 54%. However, the row percents for the medicated patients are similar.