Coverage properties of confidence intervals for generalized additive model components

We study the coverage properties of Bayesian confidence intervals for the smooth component functions of generalized additive models (GAMs) represented using any penalized regression spline approach. The intervals are the usual generalization of the intervals first proposed by Wahba and Silverman in 1983 and 1985, respectively, to the GAM component context. We present simulation evidence showing these intervals have close to nominal ‘across-the-function’ frequentist coverage probabilities, except when the truth is close to a straight line/plane function. We extend the argument introduced by Nychka in 1988 for univariate smoothing splines to explain these results. The theoretical argument suggests that close to nominal coverage probabilities can be achieved, provided that heavy oversmoothing is avoided, so that the bias is not too large a proportion of the sampling variability. Otherwise, because the Bayesian intervals account for bias and variance, the coverage probabilities are surprisingly insensitive to the exact choice of smoothing parameter. The theoretical results allow us to derive alternative intervals from a purely frequentist point of view, and to explain the impact that the neglect of smoothing parameter variability has on confidence interval performance. They also suggest switching the target of inference for component-wise intervals away from smooth components in the space of the GAM identifiability constraints. Instead intervals should be produced for each function as if only the other model terms were subject to identifiability constraints. If this is done then coverage probabilities are improved.