Semiparametric Estimation And Variable Selection Under Length Biased Sampling With Heavy Censoring PDF Download

"Semiparametric estimation procedures under Cox proportional hazards model and length-biased sampling have been developed using the weighted estimating equation method and the likelihood-based approaches over the past decade \citep{shenetal2017}. The common feature of the procedures is that they are driven by risk sets just prior to failure times. Under length-biased sampling, however, censoring is informative and failing to incorporate the information on censored data into the estimation mechanismcan lead to a substantial loss of efficiency when length-biased data are subject to heavy censoring; i.e. more than 50\% of the data are censored. We compute the likelihood contribution for uncensored and censored data separately and propose maximum approximate partial likelihood estimation (MAPLE). The procedure is further improved by exploiting the additional information for uncensored data under length-biased sampling. We call this procedure maximum approximate composite partial likelihood estimation (C-MAPLE). The asymptotic properties of the estimator from C-MAPLE are established using the functional delta method. It is shown in a simulation study that C-MAPLE and MAPLE outperform other procedures under the Cox proportional hazards model and length-biased sampling with heavy censoring. We also apply the proposed procedures to the International Stroke Trial (IST) data collected in Argentina. We next develop a unified class of penalized estimating functions which encompasses any estimation procedure under the Cox proportional hazards model and length-biased sampling. We solve the penalized estimating function by slightly perturbing the penalty function in the Minorize-Maximization algorithm \citep{hunterli2005}.We then investigate the asymptotic properties of the penalized estimators. It is shown that the penalized estimators are $\sqrt n$-consistent and with a proper choice of the tuning parameter and the penalty function, they possess the same asymptotic properties as if the true model were known a priori which is termed as the oracle property. Two simulation studies are conducted to compare the performance of the penalized estimators and confirm our theoretical results. The procedure is also used for variable selection under the Cox proportional hazards model for the IST data collected in Argentina. We further study tuning parameter selections in the penalized estimating functions via generalized information criteria (GIC). We first demonstrate the asymptotic behaviour of the loss function under C-MAPLE and then investigate the consistency of the GIC.Finally, we use bias-adjusted risk set sampling to introduce the Schoenfeld residuals for length-biased data.We conduct a simulation study to illustrate the performance of the Schoenfeld-type residuals in verifying the proportionality assumption and highlighting the trend of nonproportionality"--