Quantile Generalized Additive Model a Robust Alternative to Generalized Additive Model

Abstract

Nonparametric regression is an approach used when the structure of the relationship between the response and the predictor variable is unknown. It tries to estimate the structure of this relationship since there is no predetermined form. The generalized additive model (GAM) and quantile generalized additive (QGAM) model provides an attractive framework for nonparametric regression. The QGAM focuses on the features of the response beyond the central tendency, while the GAM focuses on the mean response. The analysis was done using gam and qgam packages in R, using data set on live-births, fertility-rate and birth-rate, where, live-birth is the response with fertility-rate and birth-rate as the predictors. The spline basis function was used while selecting the smoothing parameter by marginal loss minimization technique. The result shows that the basis dimension used was sufficient. The QGAM results show the effect of the smooth functions on the response variable at 25th, 50th, 75th and 95th quantiles, while the GAM showed only the effect of the predictors on the mean response. The results also reveal that the QGAM have lower Akaike information criterion (AIC) and Generalized cross-validation (GVC) than the GAM, hence producing a better model. It was also observed that the QGAM and the GAM at the 50th quantile had the same R2adj(77%), meaning that both models were able to explain the same percentage of variation in the models, this we attribute to the fact that mean regression and median regression are approximately the same, hence the observation is in agreement with existing literature. The plots reveal that some of the residuals of the GAM were seen to fall outside the confidence band while in QGAM all the residuals fell within the confidence band producing a better smooth.