Fitting Smoothing Spline ANOVA Models with Non-Gaussian Responses

Description

Fit smoothing spline ANOVA models in non-Gaussian regression. The symbolic model specification via formula follows the same rules as in lm and glm.

Usage

gssanova(formula, family, type=NULL, data=list(), weights, subset,
         offset, na.action=na.omit, partial=NULL, alpha=NULL, nu=NULL,
         id.basis=NULL, nbasis=NULL, seed=NULL, random=NULL,
         skip.iter=FALSE)

Arguments

Details

The model specification via formula is intuitive. For example, y~x1*x2 yields a model of the form

y = C + f_{1}(x1) + f_{2}(x2) + f_{12}(x1,x2) + e

with the terms denoted by "1", "x1", "x2", and "x1:x2".

The model terms are sums of unpenalized and penalized terms. Attached to every penalized term there is a smoothing parameter, and the model complexity is largely determined by the number of smoothing parameters.

Only one link is implemented for each family. It is the logit link for "binomial", and the log link for "poisson", and "Gamma". For "nbinomial", the working parameter is the logit of the probability p; see NegBinomial. For "weibull", "lognorm", and "loglogis", it is the location parameter for the log lifetime.

The selection of smoothing parameters is through direct cross-validation. The cross-validation score used for family="poisson" is taken from density estimation as in Gu and Wang (2003), and those used for other families are derived following the lines of Gu and Xiang (2001).

A subset of the observations are selected as "knots." Unless specified via id.basis or nbasis, the number of "knots" q is determined by max(30,10n^{2/9}), which is appropriate for the default cubic splines for numerical vectors.

Value

gssanova returns a list object of class c("gssanova","ssanova").

The method summary.gssanova can be used to obtain summaries of the fits. The method predict.ssanova can be used to evaluate the fits at arbitrary points along with standard errors, on the link scale. The method project.gssanova can be used to calculate the Kullback-Leibler projection for model selection. The methods residuals.gssanova and fitted.gssanova extract the respective traits from the fits.

Responses

For family="binomial", the response can be specified either as two columns of counts or as a column of sample proportions plus a column of total counts entered through the argument weights, as in glm.

For family="nbinomial", the response may be specified as two columns with the second being the known sizes, or simply as a single column with the common unknown size to be estimated through the maximum likelihood.

For family="weibull", "lognorm", or "loglogis", the response consists of three columns, with the first giving the follow-up time, the second the censoring status, and the third the left-truncation time. For data with no truncation, the third column can be omitted.

Note

For simpler models and moderate sample sizes, the exact solution of gssanova0 can be faster.

The results may vary from run to run. For consistency, specify id.basis or set seed.

In gss versions earlier than 1.0, gssanova was under the name gssanova1.

Author(s)

Chong Gu, chong@stat.purdue.edu

References

Gu, C. and Xiang, D. (2001), Cross validating non Gaussian data: generalized approximate cross validation revisited. Journal of Computational and Graphical Statistics, 10, 581–591.

Gu, C. and Wang, J. (2003), Penalized likelihood density estimation: Direct cross-validation and scalable approximation. Statistica Sinica, 13, 811–826.

Examples

## Fit a cubic smoothing spline logistic regression model
test <- function(x)
        {.3*(1e6*(x^11*(1-x)^6)+1e4*(x^3*(1-x)^10))-2}
x <- (0:100)/100
p <- 1-1/(1+exp(test(x)))
y <- rbinom(x,3,p)
logit.fit <- gssanova(cbind(y,3-y)~x,family="binomial")
## The same fit
logit.fit1 <- gssanova(y/3~x,"binomial",weights=rep(3,101),
                       id.basis=logit.fit$id.basis)
## Obtain estimates and standard errors on a grid
est <- predict(logit.fit,data.frame(x=x),se=TRUE)
## Plot the fit and the Bayesian confidence intervals
plot(x,y/3,ylab="p")
lines(x,p,col=1)
lines(x,1-1/(1+exp(est$fit)),col=2)
lines(x,1-1/(1+exp(est$fit+1.96*est$se)),col=3)
lines(x,1-1/(1+exp(est$fit-1.96*est$se)),col=3)

## Fit a mixed-effect logistic model
data(bacteriuria)
bact.fit <- gssanova(infect~trt+time,family="binomial",data=bacteriuria,
                     id.basis=(1:820)[bacteriuria$id%in%c(3,38)],random=~1|id)
## Predict fixed effects
predict(bact.fit,data.frame(time=2:16,trt=as.factor(rep(1,15))),se=TRUE)
## Estimated random effects
bact.fit$b

## Clean up
## Not run: rm(test,x,p,y,logit.fit,logit.fit1,est,bacteriuria,bact.fit)
dev.off()
## End(Not run)