NONPARAMETRIC REGRESSION TECHNIQUES IN SURVIVAL ANALYSIS - Quantitative Medical Data Analysis Using Mathematical Tools and Statistical Techniques - page 235

of H(tjx) and h(tjx). The above approaches are quite dierent from the
local full likelihood procedure by Gentleman and Crowley 58

that used an

iterative procedure.
When there is more than one covariate, one could use a multivariate Tay-
lor's expansion to approximate () locally with a pth-degree polynomial.
This would lead to a straightforward generalization of the above results.
However, a serious problem in multivariate situations is the curse of dimen-
sionality, which was coined by Bellman 15 . A possible approach to tackling
this problem is to consider, e.g., additive modeling 68;69;70;71 , hazard regres-
sion models (low-order interaction models) 82;83 , adaptive regression spline
Cox models 91 that used the multivariate adaptive regression spline (MARS)
technique 53 , and functional analysis of variance (ANOVA) modeling 133;75 .
Fan and Gijbels 49 have also used the local polynomial tting proce-
dure on the transformed censored data to estimate the mean regression
function. Kim and Truong 81 used the local linear tting to estimate the
conditional survival, cumulative hazard, mean, and median functions by
modifying the procedure of Beran 17 , who employed local constant tting
to estimate the conditional survival and cumulative hazard functions as an
alternative to the PH model. Wu and Tuma 155 considered a general class of
local hazard models. Betensky, et al. 18;19 used the local likelihood method
to estimate the baseline hazard function in the PH model for right- and
interval-censored data.

3.2. Additive PH Models
As Friedman and Stuetzle 54 , among others, pointed out, dimensionality
problems incurred when using multidimensional smoothers. Friedman and
Stuetzle proposed the projection pursuit regression technique as an alterna-
tive to multidimensional smoothing. An additive model 68;69;71 is a special
case of a projection pursuit regression model in which exactly p directions
are xed at the coordinate directions. The additive model is less general
than the projection pursuit model, but it is more easily interpretable. There-
fore, Hastie and Tibshirani 70 proposed an additive model

X

p

(x; G) = g j (x j ) (24)

j=1

for (x) in the model (15), where G = (g 1 ;:::;g p ); the g j ()s are unspecied
smooth functions; and g j 2Q j that is the space of functions with square
integrable second derivatives on j that is the domain of the jth covariate,