## Monday, 14 November 2011

### Bayesian inference for an illness-death model for stroke with cognition as a latent time-dependent risk factor

Van den Hout, Fox and Klein Entink have a new paper in Statistical Methods in Medical Research. This looks at joint modelling of cognition data measured through the Mini-Mental State Examination (MMSE) and stroke/survival data modelled as a three-state illness-death model. The MMSE produces item response type data and the longitudinal aspects are modelled through a latent growth model (with random slope and intercept). The three-state multi-state model is Markov conditional on the latent cognitive function. Time non-homogeneity is accounted for in a somewhat crude manner by pretending that age and cognitive function vary with time in a step-wise fashion.

The IRT model for MMSE allows the full information from the item response data to be used (e.g. rather than summing responses to individual questions to get a score). Similarly, compared to treating MMSE as an external time dependent covariate, the current model allows prediction of the joint trajectory of stroke and cognition. However, the usefulness of these predictions are constrained by how realistic the model actually is. The authors make the bold claim that the fact that a stroke will cause a decrease in cognition (which is not explicitly accounted for in their model) does not invalidate their model. It is difficult to see how this can be the case. The model constrains the decline in cognitive function to be linear (with patient specific random slope and intercept). If it actually falls through a one-off jump, then the model will still try to fit a straight line to a patient's cognition scores. Thus the cognition before the drop will tend to be down-weighted. It is therefore quite feasible that the result that lower cognition causes strokes is mostly spurious. One possible way of accommodating the likely effect of a stroke on cognition would be to allow stroke status to be a covariate in the linear growth model e.g. for multi-state process $\inline X(t)$ and latent trait $\inline \theta(t)$, we would take

$\theta(t_{ij}) = \beta_0 + \beta_1 t_{ij} + \beta_2 I(X(t_{ij})=2)$
where the $\inline \beta$ may be correlated random effects.

blogreaction said...

Some incorrect statements are made about the paper. We will react on two of them.

1.”Time non-homogeneity is accounted for in a somewhat crude manner by pretending that age and cognitive function vary with time in a step-wise fashion”

This is not true. Age and cognitive function are both considered to be continuous, only the limited number of measurement occasions leads to the discrete interpretation. The model does not imply a restriction on the number of measurement occasions, actually it is very lenient towards the data observations (e.g., missings, incomplete design, etc). It is growth modelling with a continuous time parameter. In section3.2, cognitive function is modelled as a continuous latent variable using subject-specific growth parameters, obviously the growth parameters are estimated using a limited (discrete) number of measurement occasions.

2. "The authors make the bold claim that the fact that a stroke will cause a decrease in cognition
(which is not explicitly accounted for in their model) does not invalidate their model."

This is an incorrect interpretation of the model and results and assumes that a causal relationship between cognitive function and a stroke has been shown. A regression method has been used to explain between and within-subject variability in transition rates in the three-state model using cognitive function as a time-varying predictor.

The suggestion to let the stroke status (i.e. multi-state information) influence cognitive function seems to lead to an identification problem, where cognitive function influences the transition rates and the states influence cognitive function. Then, in a latent variable framework, the intercept in the equation for transition rates is likely to interfere with the state effect on the latent variable in an unidentified way.

cauchy said...