A semi-Markov model for stroke with piecewise-constant hazards in the presence of left, right and interval censoring

Venediktos Kapetanakis, Fiona Matthews and Ardo van den Hout have a new paper in Statistics in Medicine. This develops a progressive three-state illness-death model for strokes with interval-censored data. The proposed model is a time non-homogeneous Markov model. The main approach to computation is to assume a continuous effect of age (time) on transition intensities but to use a piecewise constant approximation to actually fit it. The intensity to death from the stroke state additionally depends on the time since entry into the state (i.e. age at stroke) and since the exact time is typically unknown, it is necessary to numerically integrate over the possible range of transition times (here using Simpson's rule).

The data include subjects whose time to stroke is left-censored because they have already suffered a stroke before the baseline measurement. The authors state that they cannot integrate out the unknown time of stroke because the left-interval (i.e. the last age at which the subject is known to have been healthy) is unknown. They then proceed to propose a seemingly unnecessary ad-hoc EM-type approach based on estimating the stroke age for these individuals, which requires the arbitrary choice of an age at which it can be assumed the subject was stroke free. However, surely if we can assume a , we can just use as the lower limit in the integral for the likelihood?

The real issue seems to be that all subjects are effectively left-truncated at the time of entry into the study (in the sense that they are only sampled due to not having died before their current age). For subjects who are healthy at baseline this left-truncation is accounted for by just integrating the hazards of transition out of state 1 from their age at baseline rather than age 0. For subjects who have already had a stroke things are more complicated because the fact they have survived provides information on the timing of the stroke (e.g. if stroke increases hazard of death, the fact they have survived implies the stroke occurred sooner than one would assume if no information on survival were known). Essentially the correct likelihood is conditional on survival to time and so the unconditional probability of the observed data needs to be divided through by the unconditional probability of survival to time . For instance, in their notation, a subject in state 2 at baseline censored at time should have likelihood contribution: The authors claim that their convoluted EM-approach has "bypassed the problem of left truncation". In reality, they have explicitly corrected for left-truncation (because the expected transition time is conditional on being in state 2 at baseline) but in a way that is seemingly much more computationally demanding than directly computing the left-truncated likelihood would be.

Modeling Left-truncated and right-censored survival data with longitudinal covariates

Yu-Ru Su and Jane-Ling Wang have a new paper in the Annals of Statistics. This considers the problem of modelling survival data in the presence of intermittently observed time varying covariates when the survival times are both left truncated as well as right-censored. They consider a joint model which involves assuming there exists a random effect which influences both the longitudinal covariate values (which are assumed to be a function of the random effects plus Gaussian error) and the survival hazard. Considerably work has been done in this area in cases where the survival times are merely right-censored (e.g. Song, Davidian and Tsiatis, Biometrics 2002). The authors show that the addition of left-truncation complicates inference quite considerably; firstly because the parameters affecting the longitudinal component may not be identifiable and secondly because the score equations for the regression and baseline hazard parameters become much more complicated than in the right-censoring case. To alleviate this problem, the authors propose to use a modified likelihood rather than either the full or conditional likelihood. The full likelihood can be expressed in terms of an integral over the conditional distribution of the random effect, given the event time occurred after the truncation time. The proposed modification is to instead integrate over the unconditional random effect distribution. Heuristically this is justified by noting that
and
where is the random effect. The authors also show inference based on this modified likelihood gives consistent and asymptotically efficient estimators of the regression parameters and the baseline survival hazard.

An EM algorithm to obtain the MMLE is outlined, in which the E-step involves a multi-dimensional integral which the authors evaluate through Monte Carlo approximation. The implementation of the EM algorithm is simplified if the random effect is assumed to have a multivariate Normal distribution.

A proportional hazards regression model for the subdistribution with right-censored and left-truncated competing risks data

Xu Zhang, Mei-Jie Zhang and Jason Fine have a new paper in Statistics in Medicine. This covers the same ground as the paper by Geskus in Biometrics, in developing an approach to fitting the Fine-Gray proportional subdistribution hazard model for competing risks data with left truncated and right censored observations by using inverse probability weights (IPW). Bizarrely, the paper makes no reference at all to the Geskus paper. Presumably this is because the paper was first submitted in 2009 before Geskus's work was published (April 2010). However, it is strange that neither the authors nor the referees became aware of the work in the interim (i.e. acceptance of the paper wasn't until March 2011).

What is interesting is the differences between the approach taken in this paper compared to Geskus. The authors work on the basis that since X = min(T,C) is only observable if X > L, where T is the time of failure, L the time of left truncation and C the time of right censoring, the IPW should be calculated conditional on L < X. Zhang et al use a stabilised weight rather than the IPW to reduce the variability in the original weight. The weights they derive seem quite different to Geskus's as they depend on an estimate of overall survival, which will have to depend on the covariates if the semi-parametric model for the subdistribution hazard is to apply.
The authors suggest using Aalen additive hazard models for the overall survival (thus allowing for time varying covariate effects that can ensure the weights are consistent with the proportional subdistribution hazard model).

Zhang et al start from the general case where the truncation and censoring distributions depend on covariates (but are independent conditional on these covariates), though they only detail non-parametric estimates of the weights. Geskus argued that even if the censoring/truncation distribution depended on covariates that didn't imply it was necessary to include these covariates in the weightings.

Given these discrepancies it would be of interest to contrast and compare the two approaches to the same problem. If both approaches are effective, Geskus's seems preferable because the weights are much easier to calculate.

mstate: An R Package for the Analysis of Competing Risks and Multi-State Models

The sixth paper in the special issue of Journal of Statistical Software is by Liesbeth de Wreede, Marta Fiocco and Hein Putter and is about the R package mstate. A journal article on the package already exists in Computer Methods and Programs in Biomedicine. However, while that paper primarily dealt with theoretical aspects, the current paper is largely a case-study example based on a 6 state model for leukemia patients after bone marrow transplantation.

mstate uses the existing survival package to fit the Cox proportional hazards models. Much of the infrastructure of mstate is in functions for preparing data to be in the correct form. To use mstate for a Cox-Markov or Cox-semi-Markov model, a single coxph() object is required. This results in somewhat messy commands being required because separate strata are required and the same covariate needs to appear multiple times to allow it to have different effects for each transition intensity. For instance the call to coxph in the paper requires 16 lines. While there is perhaps some pedagogical advantage to this complication, in ensuring the user really understands what they are fitting, there is surely scope to allow some automation to this process so that the user only need specify which covariates are required for each transition intensity and this could be passed to coxph behind the scenes.

Finally, as has been noted elsewhere, mstate currently does virtually all calculations within R itself. As a result computation times are sometimes disappointing, especially for models on large datasets.

Empirical Transition Matrix of Multi-State Models: The etm Package

The third paper in the special issue of Journal of Statistical Software is by Arthir Allignol, Martin Schumacher and Jan Beyersmann and concerns the etm package. etm has been mentioned here before. It's a fairly straightforward package that computes the Aalen-Johansen estimator of the transition probabilities (and their standard errors) for general right-censored and left-truncated Markov models. mstate also has the same functionality (but also allows Cox regression). The niche of etm is that if only the Aalen-Johansen estimator is required it provides a faster implementation than mstate since etm incorporates C code, whereas mstate does most of its computation in R only.

Accounting for bias due to a non-ignorable tracing mechanism in a retrospective breast cancer cohort study

Titman, Lancaster, Carmichael and Scutt have a new paper in Statistics in Medicine. This applies methods developed by Copas and Farewell (Biostatistics, 2001) to data from a retrospective cohort study where patients were traced conditional on survival up to a certain time. The authors note that the resulting observed process can be viewed as a purged process (Hoem, 1969). In addition to the pseudo-likelihood method of Copas and Farewell (which requires specification of the entry time distribution of patients), a full likelihood approach based on piecewise constant intensities under a Markov assumption is also applied. The term to take into account the conditional survival involves a transition probability, so estimation has similar difficulties to interval-censored data. For the breast cancer study considered, the two methods give very similar results.

Two Pitfalls in Survival Analyses of Time-Dependent Exposure: A Case Study in a Cohort of Oscar Nominees

Wolkewitz, Allignol, Schumacher and Beyersmann have a new paper in The American Statistican. This uses data on the survival outcomes of Oscar nominees as a nice illustrative example of the possibility of length bias and time dependent bias in survival analysis problems. This was originally considered by Redelmeier and Singh in an Annals of Internal Medicine paper where it is was claimed that winning an Oscar significantly increased survival prospects. Possible pitfalls of such an analysis could be assuming everyone is at risk of death from birth rather than when they enter the study (e.g. at nomination), assuming people who win an Oscar have the hazard relating to the win from the time of first nomination rather than the time they actually won, and so forth. The authors show the correct model, as well as a series of possible incorrect models, in terms of multi-state models.

Cause-specific cumulative incidence estimation and the Fine and Gray model under both left truncation and right censoring

Ronald Geskus has a new paper in Biometrics. This extends the Fine and Gray model, for regression of the subdistribution hazards for competing risks models, to the case of left truncated (and right censored) data. Essentially it is shown that the standard estimator of the cumulative incidence function (CIF) can be derived as a inverse-probability-weighting estimator. Thus the Fine-Gray model for left-truncated and right-censored data can be obtained as a weighted Cox-model where individuals who are censored or experience a competing risk at time s, have weights at time t>s given by where G(t) is the empirical CDF of the censoring distribution and H(t) is the empirical survivor distribution of the truncation distribution. This form appears to imply that independence is assumed between the censoring and truncation distributions. However, the equivalence of estimates of the CIF in the no-covariates case holds regardless of the relationship between censoring and truncation - only independence with failure times is required.

A Note on Variance Estimation of the Aalen-Johansen Estimator of the Cumulative Incidence Function in Competing Risks

Allignol, Schumacher and Beyersmann have a new paper in Biometrical Journal. This considers variance estimation for the cumulative incidence functions in a cause-specific hazards competing risks analysis. They compare a Greenwood-type estimator of variance with the counting process derived estimator for left-truncated and right-censored data. They find that the Greenwood-type estimator is generally preferable for finite sample sizes.