Bayesian evidence synthesis for a transmission dynamic model for HIV among men who have sex with men

Presanis, De Angelis, Goubar, Gill and Ades have a new paper in Biostatistics. The paper attempts to use several disparate sources of data on prevalence and incidence to estimate transition rates and prevalences in a multi-state model of transmission of HIV among men who have sex with men (ironically abbreviated as MSM) via Bayesian evidence synthesis. The multi-state model has 4 transient states relating to "Eligible (non MSM)", "Susceptible (MSM)", "Undiagnosed" and "Diagnosed". Subjects enter aged 15 (or later due to migration) and may exit due to death or reaching age 45.

It should be noted that the multi-state model is deterministic and not stochastic, i.e. the proportions in each state at a given time conditional on the parameters and starting conditions are the solution of an ordinary differential equation. While this is unproblematic in terms of getting estimates of the expected proportions in each state at a given time, there must be some level of under representation of the uncertainty. In particular the likelihood assumes that the number of deaths from HIV from the SOPHID dataset at time is where is the number in the diagnosed state at time and is the probability of death in one year from HIV for the year . In reality D(t) should be stochastic rather than deterministic and as such we would expect the number of deaths to have a higher variance than assumed by the binomial model.

In principle the covariance matrix of the numbers in each state for each year could be derived, i.e. given the Markov model, we have N independent individuals each of whom's state occupancy at time is multinomial given the occupancy at time . A more realistic model would then assume that conditional on the parameters, the state occupation counts have a multivariate normal distribution (e.g. analogous to the approaches taken in the estimation of aggregate Markov data) and, for instance, the number of deaths from HIV is binomial with a denominator that is itself normally distributed. Quite possibly this extra source of uncertainty is negligible compared to the vast existing uncertainties but it nevertheless ought to be explored.

Estimating the distribution of the window period for recent HIV infections: A comparison of statistical methods

Michael Sweeting, Daniela De Angelis, John Parry and Barbara Suligo have a new paper in Statistics in Medicine. This considers estimating the time to a biomarker crossing a threshold from seroconversion in doubly-censored data (denoted T). Such methods have been used in the past to model time to seroconversion from infection. Sweeting et al state in the introduction that a key motivation is to allow the prevalence of recent infection to be estimated. This is defined as





where S(t) is the time from seroconversion to biomarker crossing. Here we see there is a clear assumption that T is independent of calendar time d. This is consistent with the framework first proposed by De Gruttola and Lagakos (Biometrics, 1989) where the time to first event is assumed to be independent of the time between first and second events, the data are essentially a three-state progressive semi-Markov model where interest lies in estimating the sojourn distribution in state 2. Alternatively, the methods of Frydman (JRSSB, 1992) are based on a Markov assumption. Here the marginal distribution of the sojourn in state 2 could be estimated by integrating over the estimated distribution of times to entry into state 2. Sweeting et al however treat it as standard bivariate survival data (where X denotes the time to serocoversion and Z denotes time to biomarker crossing) using MLEcens for estimation. It's not clear whether this is sensible if the aim is to estimate P(d) as defined above. Moreover, Betensky and Finklestein (Statistics in Medicine, 1999) suggested an alternative algorithm would be required for doubly-censored data because of the inherent ordering (i.e. Z>X). Presumably as long as the intervals for seroconversion and crossing are disjoint the NPMLE for S is guaranteed to have support only at non-negative times.

Sweeting et al make a great deal about the fact that since the NPMLE of (X,Z) is non-unique, e.g. the NPMLE can only ascribe mass to regions of the (X,Z) plane, and that T = Z - X, there will be huge uncertainty over the distribution of S between the extremes of assuming mass only in the lower left corner or only in the upper right corner of the support rectangles. They provide a plot to show this in their data example. The original approach to doubly-censored data of De Gruttola and Lagakos assumes a set of discrete time points of mass for S (the generalization to higher order models was given by Sternberg and Satten). This largely avoids any problems of non-uniqueness (though some problems remain see e.g. Griffin and Lagakos 2010).

For the non-parametric approach, Sweeting et al seem extremely reluctant to make any assumptions whatsoever. In contrast, they go completely to town on assumptions once they get onto their preferred Bayesian method. It should firstly be mentioned that the outcome time is time to crossing of a biomarker and there is considerable auxiliary data available in the form of intermediate measurements. Thus, under any kind of monotonicity assumptions, we can see that extra modelling of the growth process of the biomarker has merit. Sweeting et al use a parametric, mixed effects growth model, to model the growth of the biomarker. The growth is measured in terms of time since seroconversion, which is unknown. They consider two methods: a naive method that assumes seroconversion occurs at the midpoint of the time interval and a uniform prior method that assumes a priori that the seroconversion time is distributed uniformly within the times at which it is interval censored (i.e. last time before seroconversion and the first biomarker measurement time). The first method is essentially like imputing the interval midpoint for X. The second method is like assuming the marginal distribution for X is uniform.

Overall, the paper comes across as a straw man argument against non-parametric methods appended onto a Bayesian analysis that makes multiple (unverified) assumptions.

Estimation of overall survival in an "illness-death" model with application to vertical transmission of HIV-1

Halina Frydman and Michael Szarek have a new paper in Statistics in Medicine. This presents an additional application to their previous work on non-parametric estimation in a Markov illness-death model under interval censoring. Specifically, they use estimates of survival pre and post disease, to provide insights into the effect on overall survival. Sub-sampling is used to obtain estimates of standard errors (rather than a simple bootstrap).

A multistate approach for estimating the incidence of human immunodeficiency virus by using HIV and AIDS French surveillance data

Sommen, Alioum and Commenges have a new paper in Statistics in Medicine. This applies the penalized likelihood approach to multi-state models, used extensively by the INSERM group, to the area of back-calculation for estimating HIV incidence. The penalization parameters are chosen by minimizing an approximate cross-validation score.