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\name{sscox}
\alias{sscox}
\title{Estimating Relative Risk Using Smoothing Splines}
\description{
Estimate relative risk using smoothing spline ANOVA models. The
symbolic model specification via \code{formula} follows the same
rules as in \code{\link{lm}}, but with the response of a special
form.
}
\usage{
sscox(formula, type=NULL, data=list(), weights=NULL, subset,
na.action=na.omit, partial=NULL, alpha=1.4, id.basis=NULL,
nbasis=NULL, seed=NULL, random=NULL, prec=1e-7, maxiter=30,
skip.iter=FALSE)
}
\arguments{
\item{formula}{Symbolic description of the model to be fit, where
the response is of the form \code{Surv(futime,status,start=0)}.}
\item{type}{List specifying the type of spline for each variable.
See \code{\link{mkterm}} for details.}
\item{data}{Optional data frame containing the variables in the
model.}
\item{weights}{Optional vector of counts for duplicated data.}
\item{subset}{Optional vector specifying a subset of observations
to be used in the fitting process.}
\item{na.action}{Function which indicates what should happen when
the data contain NAs.}
\item{partial}{Optional symbolic description of parametric terms in
partial spline models.}
\item{alpha}{Parameter defining cross-validation score for smoothing
parameter selection.}
\item{id.basis}{Index of observations to be used as "knots."}
\item{nbasis}{Number of "knots" to be used. Ignored when
\code{id.basis} is specified.}
\item{seed}{Seed to be used for the random generation of "knots."
Ignored when \code{id.basis} is specified.}
\item{random}{Input for parametric random effects (frailty) in
nonparametric mixed-effect models. See \code{\link{mkran}} for
details.}
\item{prec}{Precision requirement for internal iterations.}
\item{maxiter}{Maximum number of iterations allowed for
internal iterations.}
\item{skip.iter}{Flag indicating whether to use initial values of
theta and skip theta iteration. See \code{\link{ssanova}} for
notes on skipping theta iteration.}
}
\details{
A proportional hazard model is assumed, and the relative risk is
estimated via penalized partial likelihood. The model specification
via \code{formula} is for the log relative risk. For example,
\code{Suve(t,d)~u*v} prescribes a model of the form
\deqn{
log f(u,v) = g_{u}(u) + g_{v}(v) + g_{u,v}(u,v)
}
with the terms denoted by \code{"u"}, \code{"v"}, and \code{"u:v"};
relative risk is defined only up to a multiplicative constant, so
the constant term is not included in the model.
\code{sscox} takes standard right-censored lifetime data, with
possible left-truncation and covariates; in
\code{Surv(futime,status,start=0)~...}, \code{futime} is the
follow-up time, \code{status} is the censoring indicator, and
\code{start} is the optional left-truncation time.
Parallel to those in a \code{\link{ssanova}} object, 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.
The selection of smoothing parameters is through a cross-validation
mechanism designed for density estimation under biased sampling,
with a fudge factor \code{alpha}; \code{alpha=1} is "unbiased" for
the minimization of Kullback-Leibler loss but may yield severe
undersmoothing, whereas larger \code{alpha} yields smoother
estimates.
A subset of the observations are selected as "knots." Unless
specified via \code{id.basis} or \code{nbasis}, the number of
"knots" \eqn{q} is determined by \eqn{max(30,10n^{2/9})}, which is
appropriate for the default cubic splines for numerical vectors.
}
\note{
The function \code{Surv(futime,status,start=0)} is defined and
parsed inside \code{sscox}, not quite the same as the one in the
\code{survival} package. The estimation is invariant of monotone
transformations of time.
The results may vary from run to run. For consistency, specify
\code{id.basis} or set \code{seed}.
}
\value{
\code{sscox} returns a list object of class \code{"sscox"}.
The method \code{\link{predict.sscox}} can be used to evaluate the
fits at arbitrary points along with standard errors. The method
\code{\link{project.sscox}} can be used to calculate the
Kullback-Leibler projection for model selection.
}
\author{Chong Gu, \email{chong@stat.purdue.edu}}
\references{
Gu, C. (2013), \emph{Smoothing Spline ANOVA Models (2nd Ed)}. New
York: Springer-Verlag.
Chong Gu (2014), Smoothing Spline ANOVA Models: R Package gss.
\emph{Journal of Statistical Software}, 58(5), 1-25. URL
http://www.jstatsoft.org/v58/i05/.
}
\examples{
## Relative Risk
data(stan)
fit.rr <- sscox(Surv(futime,status)~age,data=stan)
est.rr <- predict(fit.rr,data.frame(age=c(35,40)),se=TRUE)
## Base Hazard
risk <- predict(fit.rr,stan)
fit.bh <- sshzd(Surv(futime,status)~futime,data=stan,offset=log(risk))
tt <- seq(0,max(stan$futime),length=51)
est.bh <- hzdcurve.sshzd(fit.bh,tt,se=TRUE)
## Clean up
\dontrun{rm(stan,fit.rr,est.rr,risk,fit.bh,tt,est.bh)}
}
\keyword{smooth}
\keyword{models}
\keyword{survival}
|
