Modelling the likely effect of the increase of the upper age limit from 70 to 73 for breast screening in the UK National Programme

Duffy, Sasieni, Olsen and Cafferty have a new paper in Statistical Methods in Medical Research. This considers the possible effect of increasing the upper age of breast cancer screening from 70 years to 73 years. Two approaches are taken. Firstly, a 4-state time homogeneous Markov model is considered, with the states representing healthy, asymptomatic breast cancer (detectable by screening), symptomatic breast cancer and death. Secondly, a discrete-time model where incidence of breast cancer and mortality from other causes is assumed to be uniform in each year of age. Other rates are still considered to be time homogeneous. The benefit in life years gained up to 88 years of age is considered. The intensities are estimated from external data. Both approaches give similar results, with about 1 life-year gained per 1000 women screened.

Estimating Transition Probabilities for Ignorable Intermittent Missing Data in a Discrete-Time Markov Chain

Yeh, Chan, Symanski and Davis have a new paper in Communications in Statistics - Simulation and Computation. This considers data from a discrete-time Markov model where some observations are missing in an ignorable way. This is a very common situation in longitudinal data. If the transition matrix is P for a single time period, then it is just P^2, P^3 etc. for 2,3,... time periods. For some reason the authors think that not being able to have a closed form expression for the MLE is a problem. They consider a naive approach based on only considering the observed one-step transitions, an EM algorithm approach and what they term a non-linear equations method. This latter approach is essentially computing the maximum likelihood estimate. However, the authors have failed to consider the possibility of boundary maximums. Hence they have sought to find where the gradient is 0 w.r.t. transition probabilities lying between 0 and 1. They run into problems in some cases simply because the MLE is at p=0, meaning there is no solution to the gradient equation within the allowable limits so negative values are suggested.

A further problem with the paper is that they have incorrectly estimated the standard errors under missing data in the EM algorithm, plugging estimates of the marginal counts into the complete-case formula. This is equivalent to using the full likelihood information rather than the observed information. Hence the standard errors will be underestimates.

In topic the paper has some similarities to Deltour et al, Biometrics (1999). However, that paper was considering a non-trivial problem of non-ignorable missing data, where a Stochastic-EM algorithm was proposed.

A nonstationary Markov transition model for computing the relative risk of dementia before death

Yu et al have a new paper in Statistics in Medicine. This is concerned with estimating the risk of dementia before death. A 5 state multi-state model is used, with three transient states representing levels of cognitive impairment, plus two absorbing states dementia and death. Unlike other recent dementia studies using multi-state models, improvements, as well as deteriation, in cognitive ability are assumed possible. A discrete-time approach is taken. This has some advantages in terms of the flexibility in modelling possible in terms of incorporating non-homogeneity over time and a frailty term - the Markov assumption. A drawback of using a discrete-time approach in the current study was that the study did not have equally spaced observation times, but it was necessary to assume this was the case for the discrete-time model. This is likely to cause some bias.

The main theoretical development in the paper is expressions for the mean and variance of the time spent in each transient state before absorption in the non-homogeneous case.

Vertical modeling: A pattern mixture approach for competing risks modeling

Nicolaie, van Houwelingen and Putter have a new paper in Statistics in Medicine. This presents a new approach to modeling competing risks data. In essence this involves splitting the model in two parts: firstly model all cause survival, e.g. P(T >=t), secondly model P(D | T=t), the probability of a particular type of failure given a failure at time t, which they term as the relative hazard. The cause specific hazards and cumulative incidence functions can be retrieved under this formulation. A multinomial type model is applied to the relative hazards, with time dependency modelled using either piecewise constant functions or cubic splines. All cause survival can be modelled through any standard survival model e.g. a proportional hazards model. Vertical modeling provides a third approach particularly useful in cases where a proportional hazards assumption is not appropriate on either the cause-specific hazards (the classical approach) or on the sub-distribution hazard (Fine-Gray model).

Whilst it would be relatively straightforward to implement a vertical model using existing R packages, there are plans to include vertical modeling within the mstate package.

Estimating disease progression using panel data

Micha Mandel has a new paper in Biostatistics. This considers estimation for panel observed data relating to MS progression. The underlying process can have backward transitions, so reaching a higher state is not in itself considered as evidence of progression. Instead, the patient is required to have stayed in the higher state for some period of time (e.g. 6 months). The quantities of interest in the study is therefore the time taken to first have stayed in state 3 for 6 months. For a continuous time, time-homogeneous Markov process expressions for the mean time and the distribution function are obtained.

As noted by Mandel, the estimates obtained are strongly dependent on the Markov assumption and time homogeneity. This is demonstrated in a simulation study. Mandel notes that more methods for semi-Markov models are required, the recent paper on phase-type semi-Markov models may be of use. Indeed, in the simulations Mandel actually uses phase-type distributions to create a semi-Markov process. However, the MS dataset may be too small to reliably estimate a semi-Markov model.

Informative observation times are a potential problem in the MS study. Mandel shows that observations that occur away from the scheduled 26-week gap time, are more likely to involve a transition, suggesting these observation times are informative. As a result, only observations within a 4 week period of the scheduled visit time are included in the analysis.

A comparison with a discrete-time Markov model approach showed that estimates based on assuming a discrete-time process gave larger estimates of the hitting times.

Estimating life expectancy of demented and institutionalized subjects from interval-censored observations of a multi-state model

Joly, Durand, Helmer and Commenges have a new paper in Statistical Modelling. This concerns the estimation of life expectancy in patients with dementia. Unlike similar analysis by Van den Hout and Matthews, here there are 5 states rather than 3 since they consider instiutionalization as an additional event. Age at death is known up to right-censoring but other transitions are interval censored. Like previous papers by Joly and Commenges, a penalized likelihood approach is taken. The penalized likelihood is approximated by cubic M-splines, with the degree of penalization chosen via an approximate cross-validation score. This allows smooth, flexible non-homogeneous intensities. A Markov assumption is assumed but semi-Markov models can also be fitted provided the model is progressive.

Life expectancies are found by integrating the estimated transition probabilities. Like Van den Hout and Matthews, a parametric bootstrap approach based on simulating from the asymptotic normal distribution of the parameters is used to obtain confidence bands. However, these will typically underestimate variability because the penalization factor is taken to be fixed.

Patient death as a censoring event or competing risk event in models of nursing home placement

Szychowski et al have a new paper in Statistics in Medicine. This looks at competing risks data where the event of interest is placement in a nursing home with death the other competing event. They compare the classical cause-specific-hazards regression approach with that of the Fine-Gray proportional subdistribution hazards model. The data were from a RCT on the effectiveness of an enhanced counseling and support invervention. The estimate of the effect of the intervention was similar in both cases (some evidence of a benefit in delaying admission to a nursing home). The authors attribute the similarity to a lack of a significant effect of the intervention on CSH of death. They recommend that both CSH and proportional subdistribution hazards approaches to covariate effect modelling should be considered.