Estimating net transition probabilities from cross-sectional data with application to risk factors in chronic disease modeling

van de Kassteele, Hoogenveen, Engelfriet, van Baal and Boshuizen have a new paper in Statistics in Medicine. This considers the estimation of the transition probabilities in a non-homogeneous discrete-time Markov model, when the only available information is cross-sectional data, i.e. for each time (or age) we have only a sample of individuals and their state occupancy from which the prevalence at that time can be estimated. Note this type of observation is more extreme than aggregate data, considered for instance by Crowder and Stephens, where we only have prevalences at a series of times but the state occupation counts correspond to the same set of subjects.

The authors take a novel, if slightly quirky approach, to estimation. They firstly use P-splines to smooth the observed prevalences. Having obtained these they then need to translate them into transition probabilities. This is not straightforward since there are more parameters to estimate than degrees of freedom. To get around this problem the authors restrict their estimate to be the values that minimize a transportation problem. Essentially this assigns a "cost" to transitions, penalizing those to further apart states and giving zero cost to remaining in the same state. So gives a solution that aims to maximize the diagonals of the transition probability matrices whilst constraining the prevalences to take their P-spline smoothed values.

What is absent from the paper is formal justification for the approach. Presumably a similar outcome could be achieved by applying a penalized likelihood approach, possibly formulating the problem in continuous time and setting the penalty to be the magnitude of the transition intensities (and possibly their derivatives). However, this would require some calibration to choose the penalty weights and it is not clear how this would be done (the usual approach of cross-validation would not work here).

Accelerated failure time regression for backward recurrence times and current durations

Keiding, Fine, Hansen and Slama have a new paper in Statistics & Probability Letters. This considers regression models for time to event data in which only cross-sectional data is available. In this situation the process is assumed to be a stationary renewal process. The observed times are then taken to be backward recurrence times (i.e. time since last renewal to a given observation time). Here it is noted that when the inter-arrival times f(x) are subject to an accelerated failure time model, the same accelerated failure time model will apply to the backward recurrence times (a result apparently first found by Yamaguchi in the social science literature). As a consequence, an AFT model can be fitted to the observed backward recurrence times to give estimates of the AFT model for the inter-arrival times.

Keiding et al consider modelling time-to-pregnancy using data on the current duration spent attempting to become pregnant by modelling the backward recurrence times as Pareto or generalised Gamma distributed within an accelerated failure time model with frequency of sexual intercourse as a covariate in the AFT model.

The equivalence of backward and forward recurrence times (the latter being the time to next event given observation from some fixed time) means that the same approach could be applied to prevalent cohort studies with an unknown initiation time.