Constrained parametric model for simultaneous inference of two cumulative incidence functions

Haiwen Shi, Yu Cheng and Jong-Hyeon Jeong have a new paper in Biometrical Journal. This paper is somewhat similar in aims to the pre-print by Hudgens, Li and Fine in that it is concerned with parametric estimation in competing risks models. In particular, the focus is on building models for the cumulative incidence functions (CIFs) but ensuring that the CIFs sum to less than 1 at the asymptote as time tends to infinity. Hudgens, Li and Fine dealt with interval censored data but without covariates. Here, the data are assumed to be observed up to right-censoring but the emphasis is on simultaneously obtaining regression models directly for each CIF in a model with two risks.

The approach taken in the current paper is to assume that the CIFs will sum to 1 at the asymptote, to model the cause 1 CIF using a modified three-parameter logistic function with covariates via an appropriate link function. The CIF for the second competing risk is assumed to also have a three-parameter logistic form, but covariates only affect this CIF through the probability of this risk ever occurring.

When a particular risk in a competing risks model is of primary interest, the Fine-Gray model is attractive because it makes interpretation of the covariate effects straightforward. The model of Shi et al seems to be for cases where both risks are considered important, but still seems to require that one risk be considered more important. The main danger of the approach seems to be that the model for the effect of covariates on the second risk may be unrealistic, but will have an effect on the estimates for the first risk. If we only care about the first risk the Fine-Gray model would be a safer bet. If we care about both risks it might be wiser to choose a model based on the cause-specific hazards, which are guaranteed to induce a model with well behaved CIFs albeit at the expense of some interpretability of the resulting CIFs.

Obtaining a model with a direct CIF effect for each cause seems an almost impossible task because, if we allow a covariate to effect the CIF in such a way that a sufficiently extreme covariate leads to a CIF arbitrarily close to 1, it must be having a knock-on effect on the other CIF. The only way around this would be to have a model that assigns maximal asymptote probabilities to the CIFs at infinity that are independent of any covariates e.g. where are increasing functions taking values in [0,1] and . The need to restrict the to be independent of covariates would make the model quite inflexible however.

Semiparametric transformation models for current status data with informative censoring

Chyong-Mei Chen, Tai-Fang Lu, Man-Hua Chen and Chao-Min Hsu have a new paper in Biometrical Journal. In common with the recent paper by Wang et al, this considers the estimation of current status data under informative censoring. Here, rather than using a copula to describe the dependence between censoring and failure time distributions, a shared frailty is assumed between the censoring intensity and the failure intensity. The frailty is assumed to be log-normal such that and Covariate effects are also allowed for via a general class of transformation models. For estimation, the authors approximate the semi-parametric maximum likelihood estimate by assuming that the conditional intensities, for the censoring and failure events, are piecewise constant functions with an arbitrarily chosen set of change points. Since maximization of the likelihood requires estimation of a large number of unknown parameters and integration of the frailty distribution, the authors propose an EM algorithm. The method attempts to non-parametrically estimate the failure and censoring time distributions and also the variance of the frailty term. While the assumed dependence between T & C is reasonably restrictive, ie. the frailty could feasible have appeared as within the intensity for C allowing other types of dependence. Nevertheless, even with the restrictions it is not clear how the overall model is identifiable. We can only observe where is an indicator for whether the failure has occurred by censoring time . Log-normal frailties are not particularly nice computationally, whereas a Gamma frailty would allow some tractability. In the case of a shared frailty it can be shown that the marginal distribution of censoring times is and the conditional probability of a failure by time X is given by The problem is that we can vary the value of v and find new cumulative intensity functions which will result in the same distribution functions. The addition of covariates via a particular model facilitates some degree of identifiability but, in a similar way to frailty terms in univariate Cox-proportional hazard models, this could just as easily be describing misspecification of the covariate model rather than true dependence.

Non-homogeneous Markov process models with informative observations with an application to Alzheimer's disease

Baojiang Chen and Xiao-Hua Zhou have a new paper in Biometrical Journal. The methodological development of the paper is to extend the methods of Chen, Yi and Cook (Stat Med, 2010) to the case of a non-homogeneous Markov model using the time transformation model of Hubbard et al (Biometrics, 2008) rather than the piecewise constant intensities used in Chen, Yi and Cook. The authors claim that the time transformation model is more appealing than piecewise constant intensities because it requires fewer parameters. However, this parsimony is at the cost of flexibility, as the time transformation model assumes the same temporal trend for all intensities.

The assumption of non-informative examination times is an ever present spectre for multi-state models from panel data. Chen et al's method provides some methods when a complete set of planned examination times is known and it is simply the case that some examination times are missed, meaning the problem can be dealt within the Rubin framework of MAR/MNAR. A more general situation would be where the multi-state process and the process that generates the examination times are dependent. Here the only option seems to be to jointly model the two processes explicitly. A starting model might be one where the intensities of the counting process generating the examination times and the multi-state model are linked through a joint frailty, e.g. something analogous to models for joint modelling of longitudinal and (informative) drop-out (survival) processes.

Application of multistate models in hospital epidemiology: Advances and challenges

Jan Beyersmannm, Martin Wolkewitz, Arthur Allignol, Nadine Grambauer and Martin Schumacher have a new paper in Biometrical Journal. The Freiburg group have been very active in promoting multi-state modelling to a wider audience in recent years. This paper looks at the advantages adopting a multi-state model approach can bring to modelling processes related to hospital epidemiology.

Multiple imputation for estimating the risk of developing dementia and its impact on survival

Yu, Saczynski and Launer have a new paper in Biometrical Journal. This proposes an approach to fitting Cox-Markov or Cox-semi-Markov models to illness-death models subject to interval censoring via the use of iterative multiple imputation.

In essence this paper is a generalization of the method for interval censored survival data proposed by Pan (2000; Biometrics). The idea is to start by imputing the mid-point of the interval for unknown event times, calculate the parameter estimates treating these times as known. Then based on the new estimates, impute e.g. 20 different complete datasets, estimate the parameters based on these complete datasets, and the re-impute. This process continues until the parameter estimates converge.

The authors consider data regarding mortality and the onset of dementia. They model post-dementia hazard as being proportional to pre-dementia hazard and additionally allow onset age to affect hazard (for the semi-Markov model).

A drawback of the approach, which was mentioned in the paper by Pan, but not seemingly referred to by Yu et al, is that the support points for the baseline intensities will depend entirely on the initial values chosen, i.e. the final baseline intensities will have support at a subset of the original support points. For small datasets, there may be a decay in the number of support points down to 1. Pan suggested using a Link estimate of the baseline hazard (i.e. linear smoothed Breslow estimate) and to impute estimates from this to avoid the problem.

A further issue is determining what exactly constitutes convergence of the parameter estimates. After all, at each iteration a random imputation of M datasets takes place. Hence, we can expect the new parameter estimates to have a certain degree of random variation compared to the previous iteration.

The authors should be commended for having provided code and a subset of the full data in the Supporting Material of the paper. However, there seems to be an error in the code which results in an overestimation of the cumulative disease onset hazard. The code obtains the Breslow estimate of the cumulative baseline hazard. It then estimates the hazard between event times to be But instead of then simulating new event times from a piecewise-exponential distribution, they ascribe h(t) to all event times (e.g. previously imputed onset times and death times) between t_i and t_{i+1}. The coding error seems to also result in considerable bias in the other estimates as corrected code (assuming support for the hazard only at imputed onset times) seems to give quite different results.

The analysis seems to be incorrect in that it treats all patients as left-truncated (i.e. healthy) at age 50 (presumably time 0 in the data is age 50). No deaths occur before age 71.6 (or 21.6 on the time scale used in the data). Even prevalent cases enter the study conditional on survival until their entry age.

A comparison with existing methods can be made by computing the NPMLE under a Markov assumption. For this we use methods developed by Gauzere (2000; PhD Thesis) and Frydman and Szarek (2009; Biometrics). For data removing the prevalent cases, a corrected version of the imputation code gives estimates that are reasonably similar to the NPMLE (see figure), whereas the uncorrected version substantially overestimates the cumulative incidence of dementia.



The results of the analysis of the data in the paper are close to those using the subsample data with the code, suggesting the same code was used for the full data. It is unclear whether the same code was used for the simulation data.

Finally, it should be noted that the method, while undoubtedly more straightforward to implement than either a semi-parametric maximum likelihood estimate or the penalized likelihood approach of Joly and Commenges, is nevertheless still quite slow. Moreover, as with the other methods, the complexity increases rapidly as the number of states in the model increases.

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.

Competing risks and time-dependent covariates

Cortese and Andersen have a paper available as a research report from the Department of Biostatistics, University of Copenhagen. This concerns the problem of prediction in competing risks models where there are internal time-dependent covariates, meaning the trajectory of the covariate is not predictable, nor is it independent of the development of the disease/mortality process. The authors focus on the case where the time-dependent covariate is binary, and once it has value 1, cannot revert to value 0. They investigate three approaches. The first expands the state space of the competing risks model, having two alive states: 'alive and cov=0' and 'alive and cov=1' and applies standard methods based on Nelson-Aalen and Aalen-Johansen estimators. The two other methods are based on landmarking.

Update: This paper is now published in Biometrical Journal.