A class of partially linear single‐index survival models

The authors define a class of "partially linear single-index" survival models that are more flexible than the classical proportional hazards regression models in their treatment of covariates. The latter enter the proposed model either via a parametric linear form or a nonparametric single-index form. It is then pos sible to model both linear and functional effects of covariates on the logarithm of the hazard function and if necessary, to reduce the dimensionality of multiple covariates via the single-index component. The par tially linear hazards model and the single-index hazards model are special cases of the proposed model. The authors develop a likelihood-based inference to estimate the model components via an iterative algorithm. They establish an asymptotic distribution theory for the proposed estimators, examine their finite-sample behaviour through simulation, and use a set of real data to illustrate their approach. Une classe de modeles de survie partiellement lineaires a indice simple Rgsum=: Les auteurs d6finissent une classe de modeles de survie dits "partiellement lineaires a indice simple" qui s'averent plus flexibles que les modeles de regression a risques proportionnels classiques dans le traitement des covariables. Ces dernieres entrent dans le modele propose soit sous une forme lin6aire param6trique, soit sous une forme a indice simple non parametrique. I1 est alors possible de modeliser a la fois des effets lineaires et fonctionnels de covariables sur le logarithme du risque, et de reduire au besoin la dimension de covariables multiples par l'intermediaire de la composante a indice simple. Les modeles a risques partiellement lineaires et a indice simple sont des cas speciaux du modele propose. Les auteurs developpent une methode d'inf6rence vraisemblantiste pour l'estimation des composantes du modele au moyen d'un algorithme it6ratif. Ils d6terminent la loi asymptotique des estimateurs proposes, en etudient le comportement a taille finie par voie de simulation et illustrent leur approche a l'aide de donnees reelles.

[1]  J. Lawless.  Statistical Models and Methods for Lifetime Data , 2002 .

[2]  D. Ruppert,et al.  Penalized Spline Estimation for Partially Linear Single-Index Models , 2002 .

[3]  Dorota M. Dabrowska,et al.  Smoothed Cox regression , 1997 .

[4]  Jianqing Fan,et al.  Generalized Partially Linear Single-Index Models , 1997 .

[5]  P. Sasieni,et al.  Non-orthogonal projections and their application to calculating the information in a partly linear Cox model , 1992 .

[6]  Jian Huang.  Efficient estimation of the partly linear additive Cox model , 1999 .

[7]  D. Pollard.  Asymptotics for Least Absolute Deviation Regression Estimators , 1991, Econometric Theory.

[8]  D. Harrington,et al.  Counting Processes and Survival Analysis , 1991 .

[9]  G Heller,et al.  The Cox Proportional Hazards Model with a Partly Linear Relative Risk Function , 2001, Lifetime data analysis.

[10]  P. Sasieni,et al.  Information Bounds for the Conditional Hazard Ratio in a Nested Family of Regression Models , 1992 .

[11]  D. Dabrowska.  Non-parametric regression with censored survival time data , 1987 .

[12]  D. Cox.  Regression Models and Life-Tables , 1972 .

[13]  Niels Keiding,et al.  Statistical Models Based on Counting Processes , 1993 .

[14]  R. Tibshirani,et al.  Generalized additive models for medical research , 1986, Statistical methods in medical research.

[15]  O. Linton,et al.  Kernel estimation in a nonparametric marker dependent hazard model , 1995 .

[16]  Peter J. Bickel,et al.  On a semiparametric survival model with flexible covariate effect , 1998 .

[17]  Irène Gijbels,et al.  Local likelihood and local partial likelihood in hazard regression , 1997 .

[18]  J. Fleiss,et al.  Risk stratification and survival after myocardial infarction. , 1983, The New England journal of medicine.

[19]  Anthony F. Desmond,et al.  A kernel smoothed semiparametric survival model , 2001 .

[20]  Paul H. C. Eilers,et al.  Flexible smoothing with B-splines and penalties , 1996 .

[21]  W. Wong,et al.  Profile Likelihood and Conditionally Parametric Models , 1992 .

[22]  Ker-Chau Li,et al.  Sliced Inverse Regression for Dimension Reduction , 1991 .

[23]  W. Härdle,et al.  Semi-Parametric Estimation of generalized Partially Linear Single-Index Models , 2002 .

[24]  David R. Cox,et al.  Regression models and life tables (with discussion , 1972 .