Likelihood ratio confidence bands in non-parametric regression with censored data

Let (X, Y ) be a random vector, where Y denotes the variable of interest possibly subject to random right censoring, and X is a covariate. We construct confidence intervals and bands for the conditional survival and quantile function of Y given X using a nonparametric likelihood ratio approach. This approach was introduced by Thomas and Grunkemeier (1975), who estimated confidence intervals of survival probabilities based on right censored data. The method is appealing for several reasons: it always produces intervals inside [0, 1], it does not involve variance estimation, and can produce asymmetric intervals. Asymptotic results for the confidence intervals and bands are obtained, as well as simulation results, in which the performance of the likelihood ratio intervals and bands is compared with that of the normal approximation method. We also propose a bandwidth selection procedure based on the bootstrap and apply the technique on a real data set.

[1]  Song Xi Chen,et al.  Empirical Likelihood Confidence Intervals for Linear Regression Coefficients , 1994 .

[2]  Hani Doss,et al.  An Approach to Nonparametric Regression for Life History Data Using Local Linear Fitting , 1995 .

[3]  Myles Hollander,et al.  Likelihood Ratio-Based Confidence Bands for Survival Functions , 1997 .

[4]  S. Murphy.  Likelihood Ratio-Based Confidence Intervals in Survival Analysis , 1995 .

[5]  Dorota M. Dabrowska,et al.  Variable Bandwidth Conditional Kaplan-Meier Estimate , 1992 .

[6]  Art B. Owen,et al.  Nonparametric Likelihood Confidence Bands for a Distribution Function , 1995 .

[7]  Song Xi Chen,et al.  Confidence intervals based on local linear smoother , 2002 .

[8]  E. Mammen.  The Bootstrap and Edgeworth Expansion , 1997 .

[9]  I. McKeague,et al.  An Omnibus Test for Independence of a Survival Time from a Covariate , 1995 .

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

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

[12]  J. Lawless,et al.  Empirical Likelihood and General Estimating Equations , 1994 .

[13]  Peter Hall,et al.  Smoothed empirical likelihood confidence intervals for quantiles , 1993 .

[14]  DavidR . Thomas,et al.  Confidence Interval Estimation of Survival Probabilities for Censored Data , 1975 .

[15]  E. Kaplan,et al.  Nonparametric Estimation from Incomplete Observations , 1958 .

[16]  I. Keilegom,et al.  Bootstrapping quantiles in a fixed design regression model with censored data , 1998 .

[17]  Song Xi Chen,et al.  On the accuracy of empirical likelihood confidence regions for linear regression model , 1993 .

[18]  Erling B. Andersen,et al.  Asymptotic Properties of , 1970 .

[19]  O. Kardaun,et al.  Statistical survival analysis of male larynx‐cancer patients ‐ a case study , 1983 .

[20]  P. Hall.  An Edgeworth View of the Bootstrap , 1992 .

[21]  Noël Veraverbeke,et al.  Estimation and Bootstrap with Censored Data in Fixed Design Nonparametric Regression , 1997 .

[22]  Somnath Datta,et al.  A Bootstrap Approach to Nonparametric Regression for Right Censored Data , 2001 .

[23]  Bing-Yi Jing,et al.  Empirical likelihood for partial linear models with fixed designs , 1999 .

[24]  Jon A. Wellner,et al.  Confidence bands for a survival curve from censored data , 1980 .

[25]  Art B. Owen,et al.  Empirical Likelihood for Linear Models , 1991 .

[26]  Gang Li,et al.  On nonparametric likelihood ratio estimation of survival probabilities for censored data , 1995 .

[27]  M. Akritas.  Nearest Neighbor Estimation of a Bivariate Distribution Under Random Censoring , 1994 .

[28]  Song Xi Chen,et al.  Empirical likelihood confidence intervals for local linear smoothers , 2000 .

[29]  Jing Qin,et al.  Empirical Likelihood in Biased Sample Problems , 1993 .

[30]  C. J. Stone,et al.  Consistent Nonparametric Regression , 1977 .

[31]  Vijayan N. Nair,et al.  Confidence Bands for Survival Functions With Censored Data: A Comparative Study , 1984 .

[32]  Yuichi Kitamura,et al.  Empirical likelihood methods with weakly dependent processes , 1997 .

[33]  Wenceslao González-Manteiga,et al.  Asymptotic properties of a generalized kaplan-meier estimator with some applications , 1994 .

[34]  P. Hall.  Edgeworth expansions for nonparametric density estimators, with applications , 1991 .

[35]  Ian W. McKeague,et al.  CONFIDENCE TUBES FOR MULTIPLE QUANTILE PLOTS VIA EMPIRICAL LIKELIHOOD , 1999 .

[36]  TOWARDS AN OMNIBUS DISTRIBUTION-FREE GOODNESS-OF-FIT TEST FOR THE COX MODEL , 1996 .

[37]  Myles Hollander,et al.  Nonparametric likelihood ratio confidence bands for quantile functions from incomplete survival data , 1996 .

[38]  A. Owen.  Empirical likelihood ratio confidence intervals for a single functional , 1988 .

[39]  Klaus J. Utikal,et al.  Inference for a Nonlinear Counting Process Regression Model , 1990 .

[40]  A. Owen.  Empirical Likelihood Ratio Confidence Regions , 1990 .