As with linear models and multicollinearity, concurvity leads to variance inflation for the fitted parameters in a GAM. This effect differs from that of multicollinearity, however, in that the variance inflation is not reflected in the standard error estimates produced by S-Plus.
In theory, for any GAM in which the back-fitting algorithm converges, there exists a matrix R such that the fitted values [mu] are given by the relation [mu] = Rz, where z = g(y) is the vector of observations transformed by the link function.
In the wake of the discovery of the impact of concurvity on the standard error estimates of the gam functions, Dominici et al.
The second stage employed a spatial GAM to relate mortality to air pollution by modeling the log-relative risks as the sum of a nonparametric function of location and a linear function of (average) airborne sulfate particle concentration.
(2001) used the gam function in S-Plus to fit a smooth function f and a linear parameter [beta] = 0.0087 to the data, with an estimated standard error of 0.00277.
Because the mechanical details of the two studies are similar, we begin by giving the details of our method for simulating GAM models and data sets with random concurvity.
In Figure 1B, the sample standard deviation in [beta] as a function of the concurvity coefficient is in contrast to the S-Plus estimate of the standard deviation of [beta], and indicates that the true (empirical) standard error of the estimate of [beta] increases substantially with increasing concurvity, whereas the estimated standard error produced by the S-Plus gam function is increased by comparatively little.
The performance of the alternative standard error estimate was superior to that of the estimate provided by the S-Plus gam function; at each level of concurvity, the test based on the S-Plus estimator resulted in a larger overall type I error.