Average Effects Based on Regressions with a Logarithmic Link Function: A New Approach with Stochastic Covariates

Researchers often use regressions with a logarithmic link function to evaluate the effects of a treatment on a count variable. In order to judge the average effectiveness of the treatment on the original count scale, they compute average treatment effects, which are defined as the average difference between the expected outcomes under treatment and under control. Current practice is to evaluate the expected differences at every observation and use the sample mean of these differences as a point estimate of the average effect. The standard error for this average effect estimate is based on the implicit assumption that covariate values are fixed, i.e., do not vary across different samples. In this paper, we present a new way of analytically computing average effects based on regressions with log link using stochastic covariates and develop new formulas to obtain standard errors for the average effect. In a simulation study, we evaluate the statistical performance of our new estimator and compare it with the traditional approach. Our findings suggest that the new approach gives unbiased effect estimates and standard errors and outperforms the traditional approach when strong interaction and/or a skewed covariate is present.