Ties between event times and jump times in the Cox model

Xin, Horrocks and Darlington have a new paper in Statistics in Medicine. This considers approaches for dealing with ties in Cox proportional hazard models, not between event times but between event times and changes to time dependent covariates.

If a change in a time-dependent covariate coincides with a failure time there is ambiguity over which value of the time dependent covariate, z(t+) or z(t-), should be taken for the risk set at time t. By convention, it is usually assumed that z(t-) should be taken, i.e. that the change in the covariate occurs after the failure time. The authors demonstrate that for small sample sizes and/or a large proportion of ties, the estimates can be sensitive to the convention chosen. The authors also only consider cases where z(t) is a binary indicator that jumps from 0 to 1 at some point and cannot make the reverse jump. Obviously this will magnify the potential for bias because the "change after" convention will always underestimate the true risk whereas the "change before" will always overestimate the true risk.

The authors consider some simple adjustments for the problem: compute the "change before" and "change after" estimates and take their average or use random jittering. A problem with the averaging approach is estimating the standard error of the resulting estimator. An upper bound can be obtained by assuming the two estimators have perfect correlation. The jittering estimator obviously has the problem that different random jitters will give different results, though in principle the jittering could be repeated multiple times and combined in a fashion akin to multiple imputation.

It is surprising that the further option of adopting an method akin to the Efron method for ties. Essentially at each failure time there is an associated risk set. It could be argued that every tied covariate jump time had a 50% chance of occurring before or after the failure time. The expected contribution from a particular risk set could then be
It should also be possible to apply this approach using standard software, e.g. coxph() in R. It is simply necessary to replace any (start,stop) interval that ends with a tied "stop" with two intervals (start, stop - 0.00001) and (start, stop + 0.00001) each of which are associated with a weight of 0.5.

Survival analysis with time varying covariates measured at random times by design

Stephen Rathbun, Xiao Song, Benjamin Neustiftler and Saul Shiffman have a new paper in Applied Statistics (JRSS C). This considers estimation of a proportional hazards survival model in the presence of time dependent covariates which are only intermittently sampled at random time points. Specifically they are interested in examples relating to ecological momentary assessment where data collection may be via electronic devices like smart phones and the decision on sampling times can be automated. They consider a self-correcting point-process sampling design, where the intensity of the sampling process depends on the past history of sampling times, which allows an individual to have random sampling times that are more regular than would be achieved from a Poisson process.

The proposed method of estimation is to use inverse intensity weighting to obtain an estimate of an individual's integrated hazard up to the event time. Specifically the estimator is for a individual with sampling times at times and point process sampling intensity . This then replaces the integrated hazard in an approximate log-likelihood.

In part of the simulation study and in an application the exact point process intensity is unknown and taken from empirical estimates from the sample. Estimating the sampling intensity didn't seem to have major consequences on the integrity of the model estimates. This seems to suggest the approach might be applicable in other survival models where the covariates are sampled longitudinally in a subject specific manner, provided a reasonable model for sampling can be devised.

Drawbacks of the method seem to be that covariates measured at baseline (which is not a random time point) cannot be incorporated in the estimate and that it seems that the covariates must be measured at the event time which may not be the case in medical contexts. The underlying hazard also needs to be specified parametrically, but as stated flexible spline modelled can be used.

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.

Effect of an event occurring over time and confounded by health status: estimation and interpretation. A study based on survival data simulations with application on breast cancer

Alexia Savignoni, David Hajage, Pascale Tubert-Bitter and Yann De Ryckea have a new paper in Statistics in Medicine. This considers developing illness-death type models to investigate the effect of pregnancy on the risk of recurrence of cancer amongst breast cancer patients. The authors give a fairly clear account of different potential models with particular reference to the hazard ratio The simplest model to consider is a Cox model with a single time dependent covariate representing pregnancy, here . This can be extended by assuming non-proportional hazards which effectively makes the effect time dependent i.e. . Alternatively, an unrestricted Cox-Markov model could be fitted with separate covariate effects and non-parametric hazards from each pregnancy state, yielding: This model can be restricted by allowing a shared baseline hazard for giving either under a Cox model with a fixed effect or for a time dependent effect.

If we were only interested in and any of these models seems feasible, there doesn't actual seem that much point in formulating the model as an illness-death model. Note that the transition rate does not feature in any of the above equations but would be estimated in the illness-death model. The above models can be fitted by a Cox model with a time dependent covariate (representing pregnancy) that has an interaction with the time fixed covariates. The real power of a multi-state model approach would only become apparent if we were interested in the overall survival for different covariates, treating pregnancy as a random event.

The time dependent effects are represented simply via a piecewise constant time indicator in the model. The authors do acknowledge that a spline model would have been better. The other issue that could have been considered is whether the effect of pregnancy depends on time since initiation of pregnancy (i.e. a semi-Markov effect). An issue in their data example is that pregnancy is only determined via a successful birth meaning there may be some truncation in the sample (through births prevented due to relapse/death).

Applying competing risks regression models: an overview

Bernhard Haller, Georg Schmidt and Kurt Ulm have a new paper in Lifetime Data Analysis. This reviews approaches to building regression models for competing risks data. In particular, they consider cause specific hazard regression, subdistribution hazard regression (via both the Fine-Gray model and pseudo-observations), mixture models and vertical modelling. The distinction between mixture models and vertical modelling is the order of the conditioning. In mixture models, imply a separate time to event model is developed for each cause of death. Whereas in vertical modelling, meaning there is an overall "all cause" model for survival with a time dependent model for the conditional risk of different causes. Vertical modelling fits in much nearer to the standard hazard based formulation used in classical competing risks. Haller et al also prefer it to mixture modelling for computational reasons. The authors conclude however that vertical modelling's main purpose is as an exploratory tool to check modelling assumptions which may be made in a more standard competing risks model. They suggest that in a study, particularly a clinical trial, it would be more appropriate to use a Cox model either on the cause-specific hazards or on the sub-distribution hazard. The choice between these two models would depend on the particular research question of interest.

Bootstrap confidence bands for sojourn distributions in multistate semi-Markov models with right censoring

Ronald Butler and Douglas Bronson have a new paper in Biometrika. This develops approaches to non-parametric estimation of survival or hitting time distributions in multi-state semi-Markov models with particular emphasis on cases where (multiple) backward transitions are possible. The paper is an extension of the authors' previous JRSS B paper. Here the methodology additionally allows for right-censored observations.

The key property used throughout is that, under a semi-Markov model, the data can be partitioned into separate independent sets relating to the sojourn and destinations for each state.
A straightforward, but computationally intensive, approach to finding bootstrap confidence bands for the survival distribution of a semi-Markov process involves a nested re-sampling scheme, first sampling with replacement from the set of pairs of sojourn times and destination states for each state in the network, and for each set of data simulating a large number of walks through the network (starting from the initial state and ending at the designated 'hitting' state) from the empirical distributions implied from the first layer of re-sampling. An alternative, faster approach proposed involves replacing the inner step of the bootstrap with a step which computes saddlepoint approximations of the empirical sojourn distributions and then using flowgraph techniques to find the implied hitting time distributions. A drawback of both approaches is that they require all the mass of the sojourn distributions to be allocated. Hence, in the presence of right-censoring some convention for the mass at the tail of the distribution must be made. Here, the convention of "redistribute-to-the-right" is used, which effectively re-weights the probability mass of all the observed survival times so that it sums to 1. On the face of it this seems a rash assumption. However, right-censoring in the sample only occurs at the end of a sequence of sojourns in several states. As such, in most cases where this technique would be used, the censoring in any particular state is likely to be quite light, even if the overall observed hitting times are heavily censored. Alternative conventions on the probability mass (for instance assuming a constant hazard beyond some point) could be made, but in all cases would be arbitrary, but hopefully would have little impact on the overall estimates.

Unlike non-parametric estimates under a Markov assumption, for which the overall survival distribution will essentially equal the overall Kaplan-Meier estimate, with increasing uncertainty as time increases, under a (homogeneous) semi-Markov assumption the estimated hazard tends to a constant limit and can hence be estimated to a relatively large degree of precision at arbitrarily high times.

Absolute risk regression for competing risks: interpretation, link functions, and prediction

Thomas Gerds, Thomas Scheike and Per Andersen have a new paper in Statistics in Medicine. To a certain extent this is a review paper and considers models for direct regression on the cumulative incidence function for competing risks data. Specifically models of the form where is a known link function and is the cumulative incidence function for event 1 given covariates X. The Fine-Gray model is a special case of this class of models, where a complementary log-log link is adopted. Approaches to estimation based on inverse probability of censoring weights and jackknife based pseudo-observations are considered. Model comparison based on predictive accuracy as measured through Brier score and model diagnostics based on extended models allowing time dependent covariate effects are also discussed.
The discussion gives a clear account of the various pros and cons of direct regression of the cumulative incidence functions. In particular, an obvious, although perhaps not always sufficiently emphasized issue is that if, in a model with two causes, a Fine-Gray (or other direct model) is fitted to the first cause, and another to the second cause, the resulting predictions will not necessarily have the property that an issue that is not problematic if the second cause is essentially a nuisance issue, but obviously problematic if both causes are of interest. In such cases regression of the cause-specific-hazards is preferable even if it makes interpreting the effect on the cumulative intensity functions more difficult.

A multistate modelling approach for pancreatic cancer development in genetically high risk families

Kolamunnage-Dona, Vitone, Greenhalf, Henderson and Williamson have a new paper in Applied Statistics. This uses a competing risks model with shared frailties to model data on the progression of pancreatic cancer in the presence of clustering and informative censoring. Clustering is present due to data being available on patients from the same family groups. A patient having a resection causes censoring of the main event, time to pancreatic cancer, but is likely to be informative. This is dealt with by allowing the cause-specific hazards to depend on the same frailty term. The methodology in the paper is very similar to that of Huang and Wolfe (Biometrics, 2002), the only extension being that the current formulation allows for the possibility of time dependent covariates. It isn't clear what if any complication this adds to the original procedure in Huang and Wolfe. An MCEM algorithm is used where the E-step is approximated by using Metropolis-Hastings in order to calculate the required expected quantities. It's not clear what the authors mean when they say the Metropolis-Hastings step use "a vague prior for the frailty variance." Hopefully they mean an improper uniform prior as otherwise they would be pointlessly adding Bayesian features to an otherwise frequentist estimating procedure. On a related computational point, since the random effect for each cluster is one dimensional, I suspect using a Laplace approximation to compute the required integrals at each step would perform quite well and be a lot faster than using Metropolis-Hastings

In the analysis there does seem evidence for a shared frailty within clusters, but it appears that the parameter which links the frailty in the time to pancreatic cancer to the time to resection intensity is hard to identify having a very wide 95% confidence interval encompassing strong negative dependence through independence to strong positive dependence. The typical cluster size in the data is quite small (e.g. median is 3) and this is probably insufficient as you would ideally need some subjects to fail and some to be informatively censored in each cluster to assess their association. The point estimate is negative implying a counter intuitive negative association between resection and pancreatic cancer. The authors suggest as a model extension to allow a specific bivariate frailty linking competing risks (presumably within individuals?) - which is unlikely to be helpful.

Bayesian analysis of a disability model for lung cancer survival

Armero, Cabras, Castellanos, Perra, Quirós, Oruezábal and Sánchez-Rubio have a new paper in Statistical Methods in Medical Research. This develops a Bayesian three-state illness-death type model to the progression and survival of lung cancer patients. The data considered are assumed to be complete up to right censoring (in reality there may be some interval censoring but the authors argue the patients can be considered `quasi-continuously' followed. A Weibull semi-Markov model is assumed for the transition intensities and covariates are accommodated via an accelerated failure time model (it's worth noting that for a Weibull distribution the proportional hazard and accelerated failure time models are equivalent up to reparameterization). A feature of the dataset used is the rather small sample size (35 patients) which is perhaps the strongest reason for taking a parametric and Bayesian approach in this case.

Investigating hospital heterogeneity with a multi-state frailty model

Benoit Liquet, Jean Francois Timsit and Virginie Rondeau have a new paper in BMC Medical Research Methodology. They consider data on the evolution of patient's status for patients in intensive care units. The data originate from 16 distinct ICUs and there is therefore inherent clustering in the data. A progressive four-state model, with admission as state 0, ventilator-associated pneumonia infection (VAP) as state 1, death as state 2 and discharge as state 3 is considered. To account for the clustering, shared Gamma frailty terms are included in the intensities for particular transitions. To maintain simplicity of the method the authors consider cases where either each transition intensity has a ICU related frailty that is assumed independent of frailties for other intensities, or where there are individual level frailties that act in common across intensities. In each, there is only one level of clustering (i.e. either independent frailties on each intensity common to each centre or a subject specific intensity affecting multiple intensities). In the first case, a non-homogeneous Markov or semi-Markov allows separate models to be fitted for each transition using methods applicable to univariate survival analysis. In the second case, multiple intensities can be fitted in a single survival model by specifying separate "strata" for each intensity. In either case the presence of only one level of clustering makes estimation, at least if a Gamma frailty is assumed, relatively straightforward. The authors use the R package frailtypack which Rondeau maintains. This allows fully parametric models (via either Weibull or piecewise constant intensities) or semi-parametric models using spline intensities fitted using penalized likelihood. The emphasis on simple(r) approaches, for which existing software exist, is a good one. But some mention of the existing capacity of the more general survival package to fit Cox models with gamma frailties by using frailty() in the formula.

The wider issue of what to do in the case of nested frailties, i.e. where there are multiple layers of clustering, is an interesting one, but is not discussed which is surprising given that frailtypack has some capability for fitting such models. It is also questionable whether the assumption of independent frailties across intensities is a realistic one. To some extent this doesn't matter because if the data are Markov or semi-Markov conditional on the frailties, then even if the frailties are dependent assuming independence should produce unbiased estimates of the marginal frailty distribution (provided it is Gamma). Similarly estimates of the individual frailties should be reasonably unbiased (empirical Bayes estimates aren't fully unbiased anyway) but will be inefficient if frailty terms across intensities are actually correlated. It would be relatively straightforward to look at the correlation in the empirical Bayes estimates of the frailties associated with the intensities under the independence model as a semi-informal diagnostic. The difficult part would be fitting the multivariate frailty model if the independence model appeared inadequate.

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 note on the decomposition of number of life years lost according to causes of death

Per Kragh Andersen has a new paper available as a Department of Biostatistics, Copenhagen Research Report. He shows that the integral of the cumulative incidence function of a particular risk has an interpretation as the expected number of life years lost due to this cause, i.e.

It is argued that this is a more appropriate quantification of the effect of a cause-of-death than using a hypothetical estimate of life expectancy without a particular cause of death, which is reliant on an (untestable) assumption of independent competing risks.

Regression models based around expected "life years lost" are proposed using the pseudo-observations method.

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?

Modeling hospital length of stay by Coxian phase-type regression with heterogeneity

Xiaoqin Tang, Zhehui Luo and Joseph Gardiner have a new paper in Statistics in Medicine. This considers modelling right-censored length-of-stay data by using Coxian phase-type distributions, which are distributions defined by the times to absorption of a class of acyclic finite-state time homogeneous Markov processes. Coxian phase-type distributions have been used quite extensively to model right-censored survival data , particularly length-of-stay, see e.g. Marshall and Zenga. The novelty of the current paper is in the estimation of the model. Phase-type distributions suffer from being over parameterized and as a result suffer identifiability issues that in turn cause poor behaviour of optimization procedures. A further issue is the choice of the number of phases of the phase-type distribution. In the current paper a Bayesian reversible jump MCMC approach is taken.

Any acyclic Markov chain can be represented by a Coxian distribution in which the absolute values of the diagonal of the subgenerator matrix are decreasing. If parameters are unrestricted then there are inherent identifiability problems which would hamper the MCMC procedure. To avoid this the authors parameterize based on the first diagonal element and then the ratio (between 0 and 1) of the second element to the first, and so on, with the hazard of absorption from each phase being determined by a proportion (again between 0 and 1) of the diagonal element.

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.

Bayesian analysis of multistate event history data: beta-Dirichlet process prior

Yongdai Kim, Lancelot James and Rafael Weissbach have a new paper in Biometrika. This develops a conjugate prior process suitable for non-parametric and semi-parametric Bayesian modelling of right-censored multi-state Markov data. The model is parametrised in terms of the sum of the intensities out of each state

and instantaneous transition probabilities

A possible choice for a prior process is a Dirichlet distribution but this is not independent in the limit of a continuous time process. Instead the authors propose a new beta-Dirichlet process consisting of a beta distributed part which determines the increment in (between 0 and 1) and a Dirichlet part determining the instantaneous transition probabilities for each particular transition. The authors prove this prior process is conjugate in the continuous limit.

A semi-parametric regression model is proposed, which the authors term as a semi-proportional intensities model. This consists of a proportional intensities model for the all-cause hazard of exiting state h and a multinomial type model for the instantaneous transition probabilities out of state h and bears some resemblance to the vertical modeling parametrization for competing risks regression.

In an aside the authors claim that interval censoring can easily be dealt with by treating the unknown transition time as missing data that can be accounted for in the Gibbs sampling. This only works under the assumption that only one transition can have occurred between examination times. While other authors have made this assumption (e.g. Foucher et al 2007) it is dubious to say the least and likely to result in biased estimates. Similarly, the authors claim right-censoring can be dealt with by treating a censoring event as an additional state. While this will obviously allow the observed process to be modelled, it is not clear how this approach would allow the underlying process (without censoring) is estimated?

A dynamic model for the risk of bladder cancer progression

Núria Porta, M. Luz Calle, Núria Malats and Guadalupe Gómez have a new paper in Statistics in Medicine. This develops a model for progression of bladder cancer with particular emphasis on predicting future risk given events up to a certain point in time.

In many ways the paper is taking a similar approach to Cortese and Andersen in explicitly modelling a time dependent covariate (here recurrence) in order to obtain predictions.
They fit a semi-parametric Cox Markov multi-state model to the data and define a prediction process


where is the time of the second event, is the type of the second event where P denotes progression and represents the history of the process up to time t. Analogously to outcome measures like the cumulative incidence functions, this predictive process is a function of the transition intensities. They also consider time dependent ROC curves to assess the improvement in classification accuracy that can be achieved by taking into account past history in addition to baseline characteristics.

Frailties in multi-state models: Are they identifiable? Do we need them?

Hein Putter and Hans van Houwelingen have a new paper in Statistical Methods in Medical Research. This reviews the use of frailties within multi-state models, concentrating on the case of models observed up to right-censoring and primarily looking at models where the semi-Markov (Markov renewal) assumption is made. The paper clarifies some potential confusion over when frailty models are identifiable. For instance, for simple survival data with no covariates, if a non-parametric form is taken for the hazard then no frailty distribution is identifiable. A similar situation is true for competing risks data. Things begin to improve once we can observe multiple events for each patient (e.g. in illness-death models), although clearly if the baseline hazard is non-parametric, some parametric assumptions will be required for the frailty distribution. When covariates are present in the model, the frailty term has a dual role of modelling dependence between transition times but also soaking up lack of fit in the covariate (e.g. proportional hazards) model.

The authors consider two main approaches to fitting a frailty; assuming a shared Gamma frailty which acts as a multiplier on all the intensities of the multi-state model and a two-point mixture frailty, where there are two mixture components with a corresponding set of multiplier terms for the intensities for each component. Both approaches admit a closed form expression for the marginal transition intensities and so are reasonably straightforward to fit, but the mixture approach has the advantage of permitting a greater range of dependencies, e.g. it can allow negative as well as positive correlations between sojourn times.

In the data example the authors consider the predictive power (via K-fold cross-validation) of a series of models on a forward (progressive) multi-state model. In particular they compare models which allow sojourn times in later states to depend on the time to first event, with frailty models and find the frailty model performs a little better. Essentially, these two models do the same thing and as the authors note, a more flexible model for the effect of time of first event on the second sojourn time may well give a better fit than the frailty model.

Use of frailties in models with truly recurrent events seems relatively uncontroversial. However for models where few intermediate events are possible their use, rather than some non-homogeneous model allowing dependence both on time since entry to the state and time since initiation, the choice is largely related to either what is more interpretable in terms of the application at hand or possibly what is more computationally convenient.

Isotonic estimation of survival under a misattribution of cause of death

Jinkyung Ha and Alexander Tsodikov have a new paper in Lifetime Data Analysis. This considers the problem of estimation of the cause specific hazard of death from a particular cause in the presence of competing risks and misattribution of cause of death. They assume they have right-censored data for which there is an associated cause of death, but that there is some known probability r(t) of misattributing the cause of death from a general cause to a specific cause (in this case pancreatic cancer) at time t.

The authors consider four estimators for the true underlying cause-specific hazards. Firstly they consider a naive estimator which obtains Nelson-Aalen estimates of the observed CSHs and transforms them to true hazards by solving the implied equations



This estimator is unbiased but has the drawback that there are negative increments to the cause-specific hazards.
The second approach is to apply a (constrained) NPMLE estimate for instance via an EM algorithm. The authors show that, unless the process is in discrete time (such that the number of failures at a specific time point increases as the sample size increases), this estimator is asymptotically biased.
The third and fourth approaches take the naive estimates and apply post-hoc algorithms to ensure monotonicity of the cumulative hazards, by using the maximum observed naive cumulative hazard up to time t (sup-estimator) or by applying the pool-adjacent-violators algorithm to the naive cumulative hazard. These estimators have the advantage of being both consistent and guaranteed to be monotonic.

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.