Semi-Parametric Non-Proportional Hazard Model With Time Varying Covariate

Journal of Modern Applied Statistical Methods Volume 14 | Issue 2 Article 9 11-1-2015 Semi-Parametric Non-Proportional Hazard Model With Time Varying Covariate Kazeem A. Adeleke Obafemi Awolowo University, Alfred A. Abiodun University of Ilorin, Kwara State, R. A. Ipinyomi University of Ilorin, Kwara State, Follow this and additional works at: http://digitalcommons.wayne.edu/jmasm Part of the Applied Statistics Commons, Social and Behavioral Sciences Commons, and the Statistical Theory Commons Recommended Citation Adeleke, Kazeem A.; Abiodun, Alfred A.; and Ipinyomi, R. A. (2015) "Semi-Parametric Non-Proportional Hazard Model With Time Varying Covariate," Journal of Modern Applied Statistical Methods: Vol. 14 : Iss. 2 , Article 9. DOI: 10.22237/jmasm/1446350880 Available at: http://digitalcommons.wayne.edu/jmasm/vol14/iss2/9 This Regular Article is brought to you for free and open access by the Open Access Journals at DigitalCommons@WayneState. It has been accepted for inclusion in Journal of Modern Applied Statistical Methods by an authorized editor of DigitalCommons@WayneState. Semi-Parametric Non-Proportional Hazard Model With Time Varying Covariate Cover Page Footnote I sincerely acknowledge the contributions of Dr Abdul-Raheem AKINDELE, a Lecturer in the department of Psychology, Olabisi Onabanjo University, Ago-Iwoye for his immense contribution and counselling during data collection stage. This regular article is available in Journal of Modern Applied Statistical Methods: http://digitalcommons.wayne.edu/jmasm/vol14/ iss2/9 Journal of Modern Applied Statistical Methods November 2015, Vol. 14, No. 2, 68-87. Copyright © 2015 JMASM, Inc. ISSN 1538 − 9472 Semi-Parametric Non-Proportional Hazard Model with Time Varying Covariate Kazeem A. Adeleke Alfred A. Abiodun R. A. Ipinyomi Obafemi Awolowo University Ile-Ife, Nigeria University of Ilorin, Kwara State Ilorin, Nigeria University of Ilorin, Kwara State Ilorin, Nigeria The application of survival analysis has extended the importance of statistical methods for time to event data that incorporate time dependent covariates. The Cox proportional hazards model is one such method that is widely used. An extension of the Cox model with time-dependent covariates was adopted when proportionality assumption are violated. The purpose of this study is to validate the model assumption when hazard rate varies with time. This approach is applied to model data on duration of infertility subject to time varying covariate. Validity is assessed by a set of simulation experiments and results indicate that a non proportional hazard model performs well in the phase of violated assumptions of the Cox proportional hazards. Keywords: Survival time, non-proportional hazards, time-dependent covariate, semi parametric model. Introduction In survival or life testing experiments, the assumption of Cox model (1972), may not hold. Example of this is when effect of a treatment on survival diminishes in the course of time to event. Different systems have different prognostic factors, some are time fixed although some are time varying. One advantage of Cox proportional regression models is the ability to incorporate time varying coefficients and time varying covariates (Cox, 1972, Therneau & Grambsch, 2000). The former refers to a variable that is measured at baseline and whose values remain fixed to a variable whose value remains fixed over the duration of follow-up. Although, its effects on hazards is allowed to change over the follow-up period. The later refers to a variable whose value itself varies over time of follow-up. Example of time varying covariate includes the exposure of a pharmaceutical agent to cumulative dosage of radiation, duration of relationship Kazeem A. Adeleke is a lecturer in the Mathematics Department. Email him at: . Alfred A. Abiodun is a lecturer in the Department of Statistics. R. A. Ipinyomi is an Professor of Statistics. Email him at: . 68 ADELEKE ET AL. as a measure of duration of infertility in marriage, the receipt of an organ transplant. The natures of time varying covariate are very important and take major role of this work. In the above example, the first and second are continuous time variates whose value is non-decreasing over the time, the third example which is the receipt of an organ is also a time varying covariate but dichotomous in nature because the subject may be exposed or unexposed to the treatment. Recently a number of studies have been directed towards modelling time varying covariates as well as stratification which are semi-parametric nonproportional hazard models (Austin, 2012, Lehr, 2004, Abrahamowicz, 2007, Bender, Augustin, & Blettner, 2005, Ata & Sozer, 2007, Austin, 2012, Zhou, 2001). A more advanced method of generating time varying covariate is the work of Zhou (2001) where the use of an exponential distribution was examined in conjunction with a transformation to the Cox model including time varying covariate. A piecewise exponential distribution was used to obtain a dichotomous or step function covariate which was in turn incorporated into the Cox model and analysed through a semi-parametric approach. Bender et al. (2005) generated survival data that follows Cox proportional hazard model using three parametric distributions namely: exponential, Weibull and Gompertz and limited his study to only time fixed covariate. New extensions of Cox model with time varying covariate have been developed by Sylvestre and Abrahmowicz (2007) due to an undiscovered and complicated nature of longitudinal data structure where validation is made through simulation. They described and evaluated two alternatives for generation of survival times conditional on time varying covariate. Applications of Cox model with time varying covariate are likely to continue to become increasingly important in medical research. The methods put forth by Sylvester and Abrahmowicz are however not presented in a close form. Leemis (1987), Leemis, Shih and Ryertson (1990), and Shih and Leemis, (1993) have offered different frameworks for generation of survival time that follow a Cox model with time varying following accelerated life and proportional hazards models where his procedures adopted one time varying covariate and no time fixed covariates. A recent study on Cox regression model in the presence of nonproportional hazards was carried out by Ata and Sozer (2007), where they worked on alternative different models in the violation of proportional assumption. Our study extend the work of Bender et al. (2005), and Zhou (2001), with an additional argument that allows for a fixed covariate, continuous time varying covariate and a step function covariate using exponential model see Austin (2012). 69 SEMI-PARAMETRIC NON-PROPORTIONAL HAZARD MODEL Non-proportional hazards models Recall the Cox proportional hazards model with time fixed covariate x hi  t   hi  t , x   h0  t  exp  x (1) w (...truncated)