NONPARAMETRIC REGRESSION TECHNIQUES IN SURVIVAL ANALYSIS - Quantitative Medical Data Analysis Using Mathematical Tools and Statistical Techniques - page 237

because any additive constant can be absorbed into h(t). According to Gill,
Murray, and Wright 59 , if step size is optimized, the proposed algorithm is
globally convergent.
When the x values are tied, tted values will be required only at the
unique values of a given covariate, so the tied values will reduce the pa-
rameter space. The algorithm handles the tied values correctly as long as
the smoother returns the same estimated value for the same x values. This
is the case for the cubic spline smoother, and other reasonable smoothers.
Hastie and Tibshirani derived some approximate methods for inference and
smoothing parameter selection through heuristic arguments. The proposed
methodology may be applied in principle to time-dependent covariates,
though substantial computational diculties may arise.
O'Sullivan 107 proposed an algorithm for the PH model based on a con-
jugate gradient method in which the cubic B-spline representation was
used for (x); the proposed algorithm is globally convergent. Sleeper and
Harrington 129 have also used a liner combination of B-splines to approxi-
mate (x). Durrelman and Simon 41 used restricted cubic splines for (x).
In contrast to smooth additive functions for (x) by Hastie and
Tibshirani 70 , O'Sullivan 107 , and Sleeper and Harrington 129 , among others,
LeBlanc and Crowley 91 modeled (x) by using the MARS technique 53 . The
conditional hazard model is referred to as an adaptive regression spline PH
model. The technique can automatically t models with terms that rep-
resent nonlinear eects and interactions among covariates. LeBlanc and
Crowley's method is related to the method by Gray 63 who used xed knot
splines in the PH model. However, their method adaptively selects locations
and is restricted to piecewise linear functions of the covariates x.
To ameliorate the curse of dimensionality, Huang, Kooperberg, Stone,
and Truong 75 proposed a functional ANOVA model for (x) in which the
overall eect of the covariates is modeled as a specied sum of a constant
eect, main eects (functions of one covariate), and selected low-order in-
teractions (functions of a few covariates). At the same time, the functional
ANOVA model retains the exibility of nonparametric modeling. This ap-
proach also can deal with the situation of time-dependent covariates. Stone,
Hansen, Kooperberg, and Truong 133 gave a comprehensive review of one
approach to functional ANOVA modeling. In addition, Wahba, Wang, Gu,
Klein, and Klein 147 discussed ANOVA decompositions for smoothing spline
models in a general context.