NONPARAMETRIC REGRESSION TECHNIQUES IN SURVIVAL ANALYSIS - Quantitative Medical Data Analysis Using Mathematical Tools and Statistical Techniques - page 242

When the covariates are absent, the model (38) reduces to

q

X

(tj) = j B j (t): (40)

j=1

Kooperberg, Stone, and Truong 82 developed the approach hazard estima-
tion with exible tails (HEFT) to estimate the log-hazard function by us-
ing cubic splines. To allow for greater exibility in the extreme tails, they
incorporated two additional log terms into the tted model for the log-
hazard function. With inclusion of these two basis functions, HEFT can t
Weibull and Pareto distributions exactly; HEFT is useful as a preprocessor
of HARE. They wrote programs in C for implementing HARE and HEFT
and developed interfaces based on S 14;29 ; the software is available from
statlib [[e-mail]] by requesting hare from S or heft from S.
Under suitable conditions, Kooperberg, Stone, and Truong 83 obtained
the L 2 rate of convergence for a nonadaptive version of the proposed
methodology. Kooperberg and Clarkson 84 extended the HARE method-
ology to accommodate interval-censored data, time-dependent covariates,
and cubic splines. Gu 66 formulated a general procedure for penalized like-
lihood hazard estimation. When a covariate is present, the class of the
conditional hazard models constructed via tensor-product splines includes
the PH model and the model of Zucker and Karr 160 as special cases, and
in the absence of the covariate, the estimate of the hazard function reduces
to that of [106]. Gu's methodology is similar to the HARE methodology.

4. Discussion

We have reviewed some nonparametric regression techniques for estimation
of the hazard or log-hazard functions. We also have discussed functional
forms of the eects of the covariates in the PH model and some semi-
parametric or nonparametric regression models for the conditional hazard
function as alternatives to the PH model. Although we have focused on
nonparametric modeling of time-independent covariate eects in the PH
model (Sec. 3), examining the PH assumption and modeling nonpropor-
tional hazards are also very important issues that have generated an exten-
sive literature. See [115, 138, 124, 61, 56, 13, 113, 141, 62, 122, 72, 108, 6,
37] and those mentioned in the previous sections for details. In addition, the
topic 140 provided a detailed discussion of model building, testing for the PH
models, and using SAS and S-Plus for these methodologies. The propor-
tional odds regression model 16;158 is also an alternative to the PH model.