Estimating the Relative Excess Risk Due to Interaction in Clustered-Data Settings

Abstract

The risk difference scale is often of primary interest when evaluating public health impacts of interventions on binary outcomes. However, few investigators report findings in terms of additive interaction, probably because the models typically used for binary outcomes implicitly measure interaction on the multiplicative scale. One measure with which to assess additive interaction from multiplicative models is the relative excess risk due to interaction (RERI). The RERI measure has been applied in many contexts, but one limitation of previous approaches is that clustering in data has rarely been considered. We evaluated the RERI metric for the setting of clustered data using both population-averaged and cluster-conditional models. In simulation studies, we found that estimation and inference for the RERI using population-averaged models was straightforward. However, frequentist implementations of cluster-conditional models including random intercepts often failed to converge or produced degenerate variance estimates. We developed a Bayesian implementation of log binomial random-intercept models, which represents an attractive alternative for estimating the RERI in cluster-conditional models. We applied the methods to an observational study of adverse birth outcomes in mothers with human immunodeficiency virus, in which mothers were clustered within clinical research sites.

Figures

Figure 1.

Convergence and degenerate estimates for the standard deviation of random intercepts in the frequentist log binomial (FLB) and frequentist Poisson (FP) random-intercept models for scenarios described in Table 1. A) Scenario 3; B) scenario 4; C) scenario 5; D) scenario 6; E) scenario 7; F) scenario 8; G) scenario 9; H) scenario 10; I) scenario 11. See Table 1 for the full scenario descriptions. Light gray bars represent the number of simulations for which the model did not converge. Dark gray bars represent the number of simulations for which the model had a degenerate variance estimate. For example, the fifth bar in panel E for scenario 7 shows the results for the FLB random-intercept model for data sets simulated with 275 clusters: In 1,703 simulations, the model did not converge, and among the remaining models that did converge, 191 had a degenerate estimate for the variance of the random intercepts.

Figure 2.

Mean percentage of bias in the estimated relative excess risk due to interaction (RERI) across exposure/outcome scenarios and cluster sizes, by model type, for scenarios described in Table 1. A) Scenario 3; B) scenario 4; C) scenario 5; D) scenario 6; E) scenario 7; F) scenario 8; G) scenario 9; H) scenario 10; I) scenario 11. See Table 1 for the full scenario descriptions. The squares represent results for the Bayesian log binomial random intercept with a half-Cauchy(0, 5) prior distribution on the standard deviation (SD) for the random intercepts; the circles represent results for the Bayesian log binomial random intercept with a gamma(2, 0.1) prior distribution on the SD; the triangles represent results for the frequentist fit of the log binomial random intercept model; and the diamonds represent results for the frequentist fit of the Poisson random-intercept model. Note that some scenarios/clusters do not have symbols for the frequentist models because there were no models that converged and had nondegenerate SD estimates under these fits.

Figure 3.

Mean estimated standard deviation (SD) in random intercepts across exposure/outcome scenarios and cluster sizes, by model type, for scenarios described in Table 1. A) Scenario 3; B) scenario 4; C) scenario 5; D) scenario 6; E) scenario 7; F) scenario 8; G) scenario 9; H) scenario 10; I) scenario 11. See Table 1 for the full scenario descriptions. Note that some scenarios/clusters do not have markers for the frequentist models because there were no models that converged and had nondegenerate SD estimates under these fits. “BC” (squares) indicates a Bayesian log binomial random intercept with a half-Cauchy(0, 5) prior distribution on the SD; “BG” (circles) indicates a Bayesian log binomial random intercept with a gamma(2, 0.1) prior distribution on the SD; “FLB” (triangles) indicates results for the frequentist fit of the log binomial random-intercept model; and “FP” (diamonds) indicates results for the frequentist fit of the Poisson random-intercept model.