The Number of Events per Confounder for Valid Estimation of Risk Difference Using Modified Least-Squares Regression

Risk difference is a relevant effect measure in epidemiologic research. Although it is well known that when there are few events per confounder, logistic regression is not suitable for confounding control, it is not clear how many events per confounder are required for valid estimation of risk difference using linear binomial models. Because the maximum likelihood method has a convergence problem, we investigated the number of events per confounder necessary to validly estimate risk difference using modified least-squares regression in a simulation. We simulated 864 scenarios, according to the number of confounders (2-20), the number of events per confounder (2-12), marginal risk (0.5%-40%), exposure proportion (20% and 40%), and 3 sizes of risk difference. Our simulation showed that modified least-squares regression provided unbiased risk difference-regardless of the number of events per confounder-and reliable confidence intervals when more than 5 events were expected in the exposed and in the unexposed, irrespective of the number of events per confounder. We illustrated the modified least-squares regression analysis using perinatal epidemiologic data. Modified least-squares regression is considered to be a useful analytical tool for rare binary outcomes relative to the number of confounders.