Regression analysis based on conditional likelihood approach under semi-competing risks data

Jin-Jian Hsieh and Yu-Ting Huang have a new paper in Lifetime Data Analysis. This develops a conditional likelihood approach to fitting regression models with time-dependent covariate effects for the time to the non-terminal event in a semi-competing risks model. In line with some other recent treatments of competing and semi-competing risks (e.g. Chen, 2012), the authors use a copula to model the dependence between the times to the competing events. The authors express the data at a particular time point t in terms of indicator functions
and where refers to the time of the non-terminal event (or its censoring time) and refers to the terminal event (or its censoring time). The authors show that the likelihood for the data at t can be expressed in terms of a term relating solely to the , which depends only on the covariate function for the terminal event, and a term based on which contains all the information on the covariate function of interest for the non-terminal event. They therefore propose to base estimation on maximization on a conditional likelihood based on this conditional term only. The authors allow the copula itself to have a time specific copula dependence parameter. Solving the score function at a particular value of t gives consistent estimates of the parameters so the authors adopt a "working independence" model to obtain estimates across the sequence of times. The resulting estimates are step-functions that only change at observed event times.

Presumably allowing the copula dependence to be time varying could lead to situations where, for instance, , is not a decreasing function in s for fixed t and Z. So whilst allowing the copula dependence to vary is convenient computationally, it is unclear how the model would be interpreted if the dependence parameter was estimated to vary considerably (perhaps that the chosen copula family were inappropriate?).

As usual, with these models that ascribe an explicit dependence structure between the competing event times, one has to ask whether the marginal distribution of the non-terminal event is what we are really interested in and whether we should not instead be sticking to observable quantities like the cumulative incidence function?

Estimating treatment effects with treatment switching via semicompeting risks models: an application to a colorectal cancer study

Donglin Zeng, Qingxia Chen, Ming-Hui Chen and Joseph Ibrahim have a new paper in Biometrika. This considers the problem of comparing survival times in clinical trials in which there may be an intermediate disease progression event which may cause the treatment (if initially randomized to control) to be switched. Previous approaches to this problem have been proposed, through univariate survival methods. Here, the authors instead consider a multi-state (semi-competing risks) model. Interest lies in determining the survival distribution under a treatment given no switching.

In line with the other recent Biometrika paper, a pattern-mixture type parametrization is adopted in that a logistic regression component is defined for the probability of progression before death and separate conditional hazard function for time to death given no-progression and time to progression given progression, plus time to death from progression. Switching is assumed to be non-informative of outcome given that progression has occurred and the act of switching has a proportional effect on the hazard of death from progression.

Besides rigorous proofs of results, there doesn't seem to be any substantial conceptual advances in the paper, though it does seem to represent a better approach to the specific problem than the previous approaches.

Maximum likelihood analysis of semicompeting risks data with semiparametric regression models

Yi-Hau Chen has a new paper in Lifetime Data Analysis which extends his 2010 JRSS B paper from competing risks to semi-competing risks data. Essentially in both cases the main idea is to model the dependence between the competing risks by assuming their event time distributions are related via some family of copulas. Mathematically this approach is quite elegant as it allows regression models to be built on the marginal distributions of each failure time, with the inherent dependency in the censoring accounted for through the copula. From a practical perspective, particularly with semi-competing risks data and medical applications one has to question the sensibleness of the model and the objective of modelling marginal distributions.

It seems most useful to follow Xu, Kalbfleisch and Tai and view semi-competing risks as an illness-death model. After accounting for covariates, a patient's illness time and death time can be related either due to a shared frailty term, which it may be sensible to assume is determined from the outset, or through onset of illness causing death to occur sooner than it would have done. In the copula model these two distinct factors get pooled together. It is questionable how well the copula model would perform when the true process has a more event determined dependence.

More importantly the question has to be asked why you would want to try and estimate the "illness free" survival distribution? This breaks Andersen and Keiding's guideline to "Stick to this world". Illness (or relapse) is never going to be eliminated. More sensible measures like the cumulative incidence function of death (without illness having occurred) can of course be derived from Chen's copula model, although analogously to the case of semi-parametric models on cause-specific hazards, the effect of covariates on the CIF may be complicated.

A semi-competing risks model for data with interval-censoring and informative observation

Jessica Barrett, Fotios Siannis and Vern Farewell have a new paper in Statistics in Medicine. Essentially the paper uses similar methods as in Siannis et al 2006, to investigate informative loss-to-follow-up (LTF) in a study of aging and cognitive function.

LTF, refers to loss-to-follow-up of monitoring cognitive impairment - crucially survival continues to be monitored. LTF (from healthy) is modelled as a separate state in the process with its own transition intensity. Once LTF, the subject experiences different intensities of becoming cognitive impaired or dying. An unidentifiable parameter k determines the relative rate at which people who are lost to follow-up (before becoming cognitively impaired) proceed to the cognitively impaired state rather than the death state, compared to those not lost to follow-up.
k can be varied to see what impact assumptions about those LTF have on overall estimates. In the current study k has quite a large impact on estimates of cumulative incidence of cognitive impairment. This is in contrast to the Whitehall study where these methods were applied to right-censored data, where k had little effect.

A parametric Weibull intensities Markov model is used to model the data. Due to the interval censoring, computation of the likelihood requires numerical integration.

As an informal goodness-of-fit test the authors compare on Cox model of overall survival, with the corresponding survival estimates for the multi-state model with proportional intensity models on each intensity. The authors note that the Cox proportional hazards model for overall survival should be unbiased. Of course, it is only unbiased if the proportional hazards assumption holds on the overall hazard of death. If, however, the covariates are proportional on the individual intensities, as assumed in the multi-state model, the Cox model on overall survival will be biased. This approach is therefore more of a test of robustness to assumptions about the covariates than a goodness-of-fit test because the models aren't nested. The same method was used in Siannis et al 2006 and in Van den Hout et al, 2009.

Statistical Analysis of Illness-Death Processes and Semicompeting Risks Data

Xu, Kalbfleisch and Tai have a new paper in Biometrics. They make a compelling case against a latent failure times approach to the analysis of semi-competing risks data, advocating to instead take a classical cause-specific hazards approach. Moreover, they note that semi-competing risks is essentially just an illness-death model. They consider models where the intensities have a shared Gamma frailty and present methods for (Non-parametric) maximum likelihood estimation. In addition, covariates can be included acting proportionally on the conditional hazards (an alternative approach with proportionality on the marginal hazards is also outlined).