In many studies, it is known that one or more of the covariates have a monotonic effect on the response variable. In these circumstances, standard fitting methods for generalized additive models (GAMs) generate implausible results. A fitting procedure is proposed that incorporates monotonicity assumptions on one or more smooth components within a GAM framework. The algorithm uses the monotonicity restriction for B-spline coefficients and provides componentwise selection of smooth components. Stopping criteria and approximate pointwise confidence bands are derived. The method is applied to the data from a study conducted in the metropolitan area of São Paulo, Brazil, where the influence of several air pollutants like SO(2) on respiratory mortality is investigated.
Common approaches to monotonic regression focus on the case of a unidimensional covariate and continuous response variable. Here a general approach is proposed that allows for additive structures where one or more variables have monotone influence on the response variable. In addition the approach allows for response variables from an exponential family, including binary and Poisson distributed response variables. Flexibility of the smooth estimate is gained by expanding the unknown function in monotonic basis functions. For the estimation of coefficients and the selection of basis functions a likelihood-based boosting algorithm is proposed which is simple to implement. Stopping criteria and inference are based on AIC-type measures. The method is applied to several datasets.
A novel concept for estimating smooth functions by selection techniques based on boosting is developed. It is suggested to put radial basis functions with different spreads at each knot and to do selection and estimation simultaneously by a componentwise boosting algorithm. The methodology of various other smoothing and knot selection procedures (e.g. stepwise selection) is summarized. They are compared to the proposed approach by extensive simulations for various unidimensional settings, including varying spatial variation and heteroskedasticity, as well as on a real world data example. Finally, an extension of the proposed method to surface fitting is evaluated numerically on both, simulation and real data. The proposed knot selection technique is shown to be a strong competitor to existing methods for knot selection.
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