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## Generalized Smoothing Spline Models

Suppose that data have the form , where 's are independent observations and . The distribution of is from an exponential family with density function

 (25)

where , is a monotone transformation of known as the canonical link, and is a dispersion parameter. Assume that where is given in (). The penalized likelihood estimate of is the minimizer of
 (26)

where is the log-likelihood of . Again, the solution to () has the form () (Wahba et al., 1995), and and are solved by minimizing (). Usually the coefficients cannot be solved directly. If all 's are strictly concave, the Newton-Raphson iterative procedure can be used to calculate and for fixed smoothing parameters. The smoothing parameters can be estimated at each iteration using GCV, GML and UBR methods (Gu, 1990; Wahba et al., 1995; Gu, 1992). It was found that when the dispersion parameter is known, the UBR method works better than the GCV and GML methods (Wang et al., 1995). For binary, binomial, Poisson and gamma data, this procedure was implemented in GRKPACK (Wang, 1997). In our ASSIST package, the functions gdsidr and gdmudr serve as intermediate interface between S and several drivers in GRKPACK.

The argument family in ssr specifies the distribution of as in glm. Families supported are binary'', binomial'', poisson'', gamma'' and gaussian'' for Bernoulli, binomial, Poisson, gamma and Gaussian distributions respectively. The default is Gaussian.

Laplace approximations to the posterior mean and variance can be calculated by the predict function (Wahba et al., 1995). Then Bayesian confidence intervals can be constructed.

For example, one may fit a cubic spline to binary data with the UBR choice of the smoothing parameter and compute approximate posterior means and variances by

a <- ssr(y~t, rk=cubic(t), family=binary'', spar=u'', varht=1)
predict(a)
where varht specifies fixed variance (dispersion) parameter as 1 for the UBR function.

Next: Other Options in ssr Up: Smoothing Spline Regression Models Previous: Spline Smoothing with Correlated
Yuedong Wang 2004-05-19