### An Empirical Comparison between Partial Likelihood and Penalized Partial Likelihood Estimators for Semi-parametric Non-proportional Hazards Models with Frailty

#### Abstract

This study compared partial likelihood (PL) and penalized partial likelihood (PPL) estimators in non-proportional hazards model with dichotomous time-varying covariates and subject–specific frailty. We considered Gamma and Inverse Gaussian as frailty distributions. The methods were illustrated with a dataset on diabetes. Extensive numerical studies were conducted using Monte Carlo simulations to compare the efficacy of the methods in terms of Relative Bias (RB) and Root Mean Square Error (RMSE). A sensitivity analysis was carried out to assess the power of the estimators under misspecification of frailty distributions. It was found, that PPL estimator generally outperformed PL estimator in all scenarios considered. Efficiency was found to increase with increase in sample size, and decrease with increase in censoring proportion. The sensitivity analysis conducted to assess the effect of frailty misspecification revealed that sample size, proportion of censored observations and the shape of the frailty distribution (log-skewed) severely affected the power of the estimators.

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