A new variance estimator for parameters of semiparametric generalized additive models

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Generalized additive models (GAMs) have become popular in the air pollution epidemiology literature. Two problems, recently surfaced, concern implementation of these semiparametric models. The first problem, easily corrected, was laxity of the default convergence criteria. The other, noted independently by Klein, Flanders, and Tolbert, and Ramsay, Burnett, and Krewski concerned variance estimates produced by commercially available software. In simulations, they were as much as 50% too small. We derive an expression for a variance estimator for the parametric component of generalized additive models that can include up to three smoothing splines, and show how the standard error (SE) estimated by this method differs from the corresponding SE estimated with error in a study of air pollution and emergency room admissions for cardiorespiratory disease. The derivation is based on asymptotic linearity. Using Monte Carlo experiments, we evaluated performance of the estimator in finite samples. The estimator performed well in Monte Carlo experiments, in the situations considered. However, more work is needed to address performance in additional situations. Using data from our study of air pollution and cardiovascular disease, the standard error estimated using the new method was about 10% to 20% larger than the biased, commercially available standard error estimate.