Model selection and diagnostics for joint modeling of survival and longitudinal data with crossing hazard rate functions

Comparison of two hazard rate functions is important for evaluating treatment effect in studies concerning times to some important events. In practice, it may happen that the two hazard rate functions cross each other at one or more unknown time points, representing temporal changes of the treatment effect. Also, besides survival data, there could be longitudinal data available regarding some time-dependent covariates. When jointly modeling the survival and longitudinal data in such cases, model selection and model diagnostics are especially important to provide reliable statistical analysis of the data, which are lacking in the literature. In this paper, we discuss several criteria for assessing model fit that have been used for model selection and apply them to the joint modeling of survival and longitudinal data for comparing two crossing hazard rate functions. We also propose hypothesis testing and graphical methods for model diagnostics of the proposed joint modeling approach. Our proposed methods are illustrated by a simulation study and by a real-data example concerning two early breast cancer treatments.

[1]  H. Akaike.  A new look at the statistical model identification , 1974 .

[2]  Rupert G. Miller,et al.  Survival Analysis , 2022, The SAGE Encyclopedia of Research Design.

[3]  N. Hjort.  Goodness of fit tests in models for life history data based on cumulative hazard rates , 1990 .

[4]  Peihua Qiu,et al.  Confidence intervals for the first crossing point of two hazard functions , 2009, Lifetime data analysis.

[5]  David A. Schoenfeld,et al.  Chi-squared goodness-of-fit tests for the proportional hazards regression model , 1980 .

[6]  R. D'Agostino,et al.  Goodness-of-Fit-Techniques , 1987 .

[7]  S. Weisberg.  Applied Linear Regression: Weisberg/Applied Linear Regression 3e , 2005 .

[8]  Yuhong Yang.  Can the Strengths of AIC and BIC Be Shared , 2005 .

[9]  I. Grama,et al.  Asymptotic theory for the Cox model with missing time-dependent covariate , 2006, math/0607033.

[10]  Yingwei Peng,et al.  Crossing Hazard Functions in Common Survival Models. , 2009, Statistics & probability letters.

[11]  Jerald F. Lawless,et al.  Statistical Models and Methods for Lifetime Data. , 1983 .

[12]  David P. Harrington,et al.  Modified Kolmogorov-Smirnov Test Procedures with Application to Arbitrarily Right-Censored Data , 1980 .

[13]  J. Anderson,et al.  A Two-step Regression Model for Hazard Functions , 1982 .

[14]  J O'Quigley,et al.  The problem of a covariate-time qualitative interaction in a survival study. , 1991, Biometrics.

[15]  Xun Lin,et al.  A New Testing Approach for Comparing the Overall Homogeneity of Survival Curves , 2004 .

[16]  Clifford M. Hurvich,et al.  Regression and time series model selection in small samples , 1989 .

[17]  D. Hosmer,et al.  A Simplified Method of Calculating an Overall Goodness-of-Fit Test for the Cox Proportional Hazards Model , 1998, Lifetime data analysis.

[18]  Lee J. Bain.  Statistical analysis of reliability and life-testing models : theory and methods , 1992 .

[19]  Ørnulf Borgan,et al.  A method for checking regression models in survival analysis based on the risk score , 1996, Lifetime data analysis.

[20]  Ralph B. D'Agostino,et al.  Goodness-of-Fit-Techniques , 2020 .

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

[22]  John O'Quigley,et al.  On a Two‐Sided Test for Crossing Hazards , 1994 .

[23]  Z. Ying,et al.  Checking the Cox model with cumulative sums of martingale-based residuals , 1993 .

[24]  D. Zeng,et al.  Asymptotic results for maximum likelihood estimators in joint analysis of repeated measurements and survival time , 2005, math/0602240.

[25]  J. Shao.  AN ASYMPTOTIC THEORY FOR LINEAR MODEL SELECTION , 1997 .

[26]  O. Aalen,et al.  Further results on the non-parametric linear regression model in survival analysis. , 1993, Statistics in medicine.

[27]  Geert Verbeke,et al.  Multiple Imputation for Model Checking: Completed‐Data Plots with Missing and Latent Data , 2005, Biometrics.

[28]  Peihua Qiu,et al.  A two‐stage procedure for comparing hazard rate functions , 2007 .

[29]  A. Raftery,et al.  Bayesian Information Criterion for Censored Survival Models , 2000, Biometrics.

[30]  R. Peled,et al.  Breast cancer, psychological distress and life events among young women , 2008, BMC Cancer.

[31]  Juvêncio Santos Nobre,et al.  Residual Analysis for Linear Mixed Models , 2007, Biometrical journal. Biometrische Zeitschrift.

[32]  Peihua Qiu,et al.  Comparing two crossing hazard rates by Cox proportional hazards modelling , 2007, Statistics in medicine.

[33]  Geert Molenberghs,et al.  Multiple‐Imputation‐Based Residuals and Diagnostic Plots for Joint Models of Longitudinal and Survival Outcomes , 2010, Biometrics.

[34]  Nathan Mantel,et al.  The Crossing Hazard Function Problem , 1988 .

[35]  J O'Quigley,et al.  Score tests for homogeneity of regression effect in the proportional hazards model. , 1989, Biometrics.

[36]  Gordon Johnston,et al.  Statistical Models and Methods for Lifetime Data , 2003, Technometrics.

[37]  V. Muggeo,et al.  A flexible approach to the crossing hazards problem , 2010, Statistics in medicine.

[38]  M. Wulfsohn,et al.  A joint model for survival and longitudinal data measured with error. , 1997, Biometrics.

[39]  Catherine Huber,et al.  Weighted log rank statistics for comparing two distributions , 1992 .

[40]  Sanford Weisberg.  Computing Primer for Applied Linear Regression, Third Edition , 2005 .

[41]  Rizopoulos Dimitris,et al.  Joint Modeling of Longitudinal and Time-to-Event Data , 2014 .

[42]  L Edler,et al.  A two-sample censored-data rank test for acceleration. , 1984, Biometrics.

[43]  John P. Klein,et al.  Multivariate Survival Analysis , 1997 .