Assessing age-at-onset risk factors with incomplete covariate current status data under proportional odds models

Chi-Chung Wen and Yi-Hau Chen have a new paper in Statistics in Medicine. This considers estimation of a proportional odds regression model for current status data in cases where a subset of the covariates may be missing at random for a subset of the patient population.

It is assumed that the probability that a portion of the covariates is missing depends on all the other observable outcomes (the failure status, the survey time and the rest of the covariate vector). The authors propose to fit a logistic regression model, involving all subjects in the dataset, for this probability of missingness. To fit the regression model for the current status data itself, they propose to use what they term a "validation likelihood estimator." This involves only working with the subset of patients with complete data but maximizing a likelihood that conditions on the fact that the whole covariate vector was observed. An advantage of using the proportional odds model over other candidate models (e.g. proportional hazards) is that the resulting likelihood remains of the proportional odds form.

Clearly a disadvantage of this "validation likelihood estimator" is that the data from subjects who have incomplete covariates is not used directly in the regression model. As a result the estimator is likely to be less efficient than approaches that effectively attempt to impute the missing covariate values. The authors argue that the validation likelihood approach will tend to be more robust since it is not necessary to make (parametric) assumptions about the conditional distribution of the missing covariates.

Nonparametric estimation of the cumulative intensities in an interval censored competing risks model

Halina Frydman and Jun Liu have a new paper in Lifetime Data Analysis. This concerns non-parametric estimation for competing risks models under interval censoring. The problem of estimating the cumulative incidence functions (or sub-distribution functions) under interval censoring has been considered by Hudgens et al (2001) and involves an extension of the NPMLE for standard survival data under interval censoring.

The resulting estimates of the cumulative incidence functions are only defined up to increments on intervals. Moreover, the intervals by which the CIFs are defined are not the same for each competing risk. This causes problems if one wants to convert the CIFs into estimates of the cumulative cause-specific hazards. Frydman and Liu propose estimating the cumulative cause-specific hazards by first constraining the NPMLEs of the CIFs to have the same intervals of support (NB: this is just a sub-set of the set of all NPMLEs involving sub-dividing the intervals) and adopting a convention to distribute the increment within the resulting sub-intervals (they assume an equal distribution across sub-interval).

In addition they show that an ad-hoc estimator of the cumulative hazards based on the convention that the support of each interval of the NPMLE for each CIF occurs at its midpoint leads to biased results. They also show that their estimator has standard N^0.5 convergence when the support of the observation time distribution is discrete and finite.

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.

Practicable confidence intervals for current status data

Byeong Yeob Choi, Jason P. Fine and M. Alan Brookhart have a new paper in Statistics in Medicine. Essentially the paper clarifies the practical implications of results relating to the asymptotic theory for current status data. In particular, it is known that the nonparametric bootstrap is inconsistent for current status data when the distribution of sampling times is continuous. The authors note that the most reliable method considered previously is a previous study by Ghosh et al concluded that construction of confidence intervals based on inversion of the likelihood ratio statistic (as original proposed in Banerjee and Wellner (2001)) gave the best results particularly for smaller sample sizes. However, they also note that this approach is difficult to implement (e.g. lack of available software). Here they therefore pursue approaches based on using the limiting Chernoff distribution to construct Wald type confidence intervals, but using cloglog or logit transformations to get better coverage, and also look at the performance of simple non-parametric bootstrap.

Perhaps unsurprisingly they find that, when sample sizes are relatively small, the performance of all methods is dependent on the quantile of the failure time distribution at which the confidence interval is computed (e.g. it performs much better when t is close to the median) and the observation density at the time point considered (performance is poorer at times with a lower observation density). Using cloglog or logit transformations is found to improve coverage, but the non-parametric bootstrap tended to outperform this approach for smaller sample sizes, suggesting the admissibility of using the non-parametric bootstrap.

An apparent omission in the paper is any mention of the altered asymptotics in the case where the observation distribution has support at a finite set of time points (or indeed where the rate of increase of points is less than ). This issue is most comprehensively discussed in Tang et al which didn't come out until after the paper was apparently submitted. However, the basic issue of standard asymptotics (and by implication a consistent bootstrap) when there is a finite set of observation points is discussed in Maathuis and Hugdens (2011) which the authors cite. For instance, in the Hoel and Walburg mice dataset used as illustration, the resolution of the data is to the nearest day. It is therefore reasonable to assume that in this case were the sample size to increase, the number of observations would either be bounded by a fixed value (e.g. ~1000) or else the number of unique points would increase at a rate much less than .

Effect of vitamin A deficiency on respiratory infection: Causal inference for a discretely observed continuous time non-stationary Markov process

Mingyuan Zhang and Dylan Small have a paper to appear in The Canadian Journal of Statistics, currently available here. The paper uses a multi-state model approach to obtain estimates of the causal effects of vitamin D deficiency on respiratory infection.

The observed data consist of observations of respiratory infection status, vitamin D deficiency status and whether the child is stunted at time t. Each of these is a binary variable, leading to 8 possible observation patterns.

The data are assumed to be generated from an underlying non-homogeneous Markov chain on 32 states, consisting of a latent 4-level definition of vitamin D deficiency, the stunting variable, the observed respiratory infection status and additionally a counter-factual respiratory infection status defined as the status at time t hat would have occurred had the child maintained the lowest level of vitamin D deficiency from time 0 to time t.

Let represent the observed infection status, the underlying vitamin deficiency and the counter-factual infection status, the assumed relationship between them is given by , and for j>0. then measures the additional risk of having a respiratory infection at a particular time, given a current vitamin D deficiency at level j>0.

The overall model is a pretty innovative use of a hidden Markov model structure to obtain those causal estimates. The true process is assumed to occur in continuous time. However, it is desired that the underlying transition intensities are not time constant. As a result, the authors choose to approximate the process by one in discrete time (with some similarities to the approach of Bacchetti et al 2010).

In practical terms, the weakness of the model seems to be the assumption that the relative effect of a current vitamin deficiency compared to a perfect record of vitamin D levels, is both constant in time and does not depend on the past history of vitamin D deficiency. The latter assumption is essentially the Markov assumption and would be quite difficult to relax. The observed infection status is effectively a misclassified version of the counter-factual infection status. As a result the former assumption could be relaxed by letting the depend on time, perhaps in a piecewise constant fashion.

The definition of the process as initially being continuous is a little artificial and ill-specified in places. For instance, the transition intensities are defined in terms of a logit transformation from the outset. Also, it is stated that the underlying 32 state process (including both of ) is a continuous-time Markov process. However if is defined only through the misclassification equations, there is no limiting intensity for as . Once the process is in discrete time this is not a problem, but it would be more sensible to define the underlying process in continuous time to be 16 states not including and then specify that the observed are observations in a hidden Markov model.

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.