Generalized additive models with flexible response functions

Abstract

Common generalized linear models depend on several assumptions: (i) the specified linear predictor, (ii) the chosen response distribution that determines the likelihood and (iii) the response function that maps the linear predictor to the conditional expectation of the response. Generalized additive models (GAM) provide a convenient way to overcome the restriction to purely linear predictors. Therefore, the covariates may be included as flexible nonlinear or spatial functions to avoid potential bias arising from misspecification. Single index models, on the other hand, utilize flexible specifications of the response function and therefore avoid the deteriorating impact of a misspecified response function. However, such single index models are usually restricted to a linear predictor and aim to compensate for potential nonlinear structures only via the estimated response function. We will show that this is insufficient in many cases and present a solution by combining a flexible approach for response function estimation using monotonic P-splines with additive predictors as in GAMs. Our approach is based on maximum likelihood estimation and also allows us to provide confidence intervals of the estimated effects. To compare our approach with existing ones, we conduct extensive simulation studies and apply our approach on two empirical examples, namely the mortality rate in São Paulo due to respiratory diseases based on the Poisson distribution and credit scoring of a German bank with binary responses.

Appendix

Algorithm 1

(FlexGAM1)

The inner iteration is done until the convergence of $\varvec{\gamma }^{(k)}$, meaning $\frac{\left| \left| \varvec{\gamma }^{(k)} - \varvec{\gamma }^{(k-1)} \right| \right| }{\left| \left| \varvec{\gamma }^{(k)} \right| \right| }~<~\varepsilon _1$. Then, the outer iteration is repeated. The inner and outer loops are iterated until the coefficients of $\varPsi ^{(m)}$ are constant, meaning $\frac{\left| \left| \varvec{\nu }^{(m)} - \varvec{\nu }^{(m-1)} \right| \right| }{\left| \left| \varvec{\nu }^{(m)} \right| \right| } < \varepsilon _2$.

Algorithm 2

(FlexGAM2)

Again the inner iteration is done until the convergence of $\varvec{\gamma }^{(k)}$. Then, the outer iteration is repeated. The inner and outer loops are iterated until the coefficients of $\hat{h}^{(m)}$ are constant, meaning $\frac{\left| \left| \varvec{\nu }^{(m)} - \varvec{\nu }^{(m-1)} \right| \right| }{\left| \left| \varvec{\nu }^{(m)} \right| \right| }~<~\varepsilon _2$.