On the maximum penalized likelihood approach for proportional hazard models with right censored survival data

Penalized likelihood estimation of the proportional hazards model for survival data with interval censoring

This paper considers the problem of semi-parametric proportional hazards model fitting where observed survival times contain event times and also interval, left and right censoring times. Although this is not a new topic, many existing methods suffer from poor computational performance. In this paper, we adopt a more versatile penalized likelihood method to estimate the baseline hazard and the regression coefficients simultaneously. The baseline hazard is approximated using basis functions such as M-splines. A penalty is introduced to regularize the baseline hazard estimate and also to ease dependence of the estimates on the knots of the basis functions. We propose a Newton-MI (multiplicative iterative) algorithm to fit this model. We also present novel asymptotic properties of our estimates, allowing for the possibility that some parameters of the approximate baseline hazard may lie on the parameter space boundary. Comparisons of our method against other similar approaches are made through an intensive simulation study. Results demonstrate that our method is very stable and encounters virtually no numerical issues. A real data application involving melanoma recurrence is presented and an R package 'survivalMPL' implementing the method is available on R CRAN.

Competing risks analysis with missing cause-of-failure-penalized likelihood estimation of cause-specific Cox models

Competing risks models are attractive tools to analyze time-to-event data where several causes of an event are competing. However, a complexity may arise when, for instance, some subjects experience the event of interest but the causes are not known. Assuming that unknown causes of events are missing at random, we developed a novel constrained maximum penalized likelihood method for fitting semi-parametric cause-specific Cox regression models. Here, penalty functions were used to smooth the baseline hazards. An appealing feature of this approach is that all the relevant estimands in competing risks models are estimated including cause-specific hazard ratios, cause-specific baseline hazards, and cumulative incidence functions. Asymptotic results for these estimators were also developed, allowing for direct inferences. The proposed method was compared with some existing methods through a simulation study. A real data example was analyzed using the new method to evaluate the association of age at diagnosis with melanoma-death and non-melanoma-death in patients diagnosed with thin melanoma (tumour thickness ≤1.0  mm). An R function for our proposed method is currently available on GitHub and will be included in the R package "survivalMPL" at CRAN.

On hazard-based penalized likelihood estimation of accelerated failure time model with partly interval censoring

In survival analysis, the semiparametric accelerated failure time model is an important alternative to the widely used Cox proportional hazard model. The existing methods for accelerated failure time models include least-squares, log rank-based estimating equations and approximations to the nonparametric error distribution. In this paper, we propose another fitting method for the accelerated failure time model, formulated from the hazard function of the exponential error term. Our method can handle partly interval-censored data which contains event time, as well as left, right and interval censoring time. We adopt the maximum penalized likelihood method to estimate all the parameters in the model, including the nonparametric component. The penalty function is used to regularize the nonparametric component of the accelerated failure time model. Asymptotic properties of the penalized likelihood estimate are developed. A simulation study is conducted to investigate the performance of the proposed method and an application of this method to an AIDS study is presented as an example.

Proportional hazard model estimation under dependent censoring using copulas and penalized likelihood

This paper considers Cox proportional hazard models estimation under informative right censored data using maximum penalized likelihood, where dependence between censoring and event times are modelled by a copula function and a roughness penalty function is used to restrain the baseline hazard as a smooth function. Since the baseline hazard is nonnegative, we propose a special algorithm where each iteration involves updating regression coefficients by the Newton algorithm and baseline hazard by the multiplicative iterative algorithm. The asymptotic properties for both regression coefficients and baseline hazard estimates are developed. The simulation study investigates the performance of our method and also compares it with an existing maximum likelihood method. We apply the proposed method to a dementia patients dataset.