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.