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 and latent trait , we would take

where the may be correlated random effects.

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## 2 comments:

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.

Thanks for the comments.

1. Perhaps the statement should have been clearer. What I am saying is time dependence in the

intensitiesis accounted for in a crude manner. Obviously it is still a continuous time model. The main problem is that your model can have two individuals with the same random effect for the growth model, but they will have different expected intensities for the multi-state model at age 60 if one was last observed aged 55 and the other aged 50. That may make things easier computationally but it is clearly unrealistic. The act of measuring cognition should not affect a patient's future hazard of stroke or death.2. My key criticism is that the model assumes a linear model for an individual patient's decline in cognition. If stroke causes a change in cognition the trajectories will not be linear and some of the estimated association could be an artifact of fitting straight lines to things which aren't linear. Rather than acknowledging this potential for bias the paper makes false claims about robustness to this type of model misspecification. By making the intensities depend on cognition directly the model is favouring the conclusion that a fall in cognition is followed by an increased risk of stroke (and not the other way around) - even if it doesn't rule out the possibility of a third process which affects both. So it is certainly implying Granger causation even if not true causation. In particular, using the model for dynamic prediction you would conclude someone was at a greater risk of a stroke if there has been a negative trend in cognition up to a point in time but would not expect a decline in cognition in someone who had suffered a stroke but had flat or increasing cognition scores previously.

The model I suggested is just your model with an additional dependence on I(X(t)=2). If X(t) were latent the the model would suffer identifiability problems but X(t) is generally observed when cognition is measured so if the extra parameter is non-zero it would manifest itself in a difference in means of the residuals of the longitudinal process for times where X(t)=2 compared to X(t)!=2. If the model were difficult to identify a further simplification would be to let the effect of I(X(t)=2) be fixed rather than a correlated random effect. The requirement that X(t) depends on the e_{ij} of the cognition trait seems unnecessary and dropping this would also help identifiability and would bring the model more in line with more mainstream joint longitudinal and survival models.

Feel free to give details of the other "incorrect" statements in my original post.

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