Modeling Nonlinear Effects in Risk Ratio and Risk Difference Using Poisson and Gaussian Additive Regression Models

The logistic additive regression model has been a standard method in modeling nonlinear effects for multivariate analyses of binary outcomes in the generalized additive model (GAM) framework. However, the resultant nonlinear estimate of the smooth function is interpreted as a nonproportional increment of the odds ratio in the increment of the explanatory variable. The odds ratio cannot be interpreted as an effect measure by itself; it is only interpretable as an approximation of the risk ratio when the frequency of events is low. In this article, we propose alternative nonlinear regression methods to estimate the risk ratio and risk difference directly. We propose extending Zou’s modified Poisson regression (Am J Epidemiol 159: 702–6) and Cheung’s modified least squares (Gaussian) regression (Am J Epidemiol 166: 1337–44) to the GAM framework and fitting the Poisson and Gaussian additive regression models to binary outcome data. We show that valid nonlinear effects estimates are obtained using these approaches and that they can be easily implemented using existing GAM statistical packages. We also provide valid computational methods for obtaining the standard errors and confidence intervals using a bootstrap method. We illustrate these proposed methods through applications to a breast cancer clinical study.