## Sunday, 23 January 2011

### A comparison of non-homogeneous Markov regression models with application to Alzheimer's disease progression

Rebecca Hubbard and Andrew Zhou have a new paper in Journal of Applied Statistics. This considers panel data relating to the progression of Alzheimer's disease using non-homogeneous Markov models. The time transformation method proposed by Hubbard et al (Biometrics, 2008) is extended to allow for (fixed) covariates. In the time transformation model the generator matrix is $\inline \mathbf{Q}(t) = \mathbf{Q}_{0}dh(t)/dt$ for some increasing function $\inline h(t)$, which in Hubbard's method can be estimated non-parametrically (or at least flexibly using kernel weights centered on pre-specified time points). Covariates are incorporated as proportional intensities on the intensities in the homogeneous time generator $\inline \mathbf{Q}_{0}$.

A simulation study is presented which aims to compare the robustness of piecewise constant intensity models and time transformation models for estimating covariate effects on intensities. It is no secret that in proportional hazard type models, estimates are reasonably robust to misspecification of the shape of the baseline hazard (in contrast to relative lack of robustness to misspecification of the model for the effect of the covariate on the hazard). In general therefore, each model performs fairly well when the other is true. The simulation puts the piecewise intensities model at a bit of a disadvantage since it requires Q to be estimated separately for each time period (four extra parameters), whereas the time transformation model only requires 1 extra parameter over the homogeneous model. It might have been a fairer comparison if a time transformation model of similar complexity was used. The conclusion that time transformation models are more robust for small sample sizes is therefore not particularly convincing.

The argument against piecewise constant intensities in general is a bit weak as it revolves around the notion that one must estimate a separate generator matrix for each time period leading to many extra parameters. The obvious argument for piecewise constant intensities is that we can look at a particular intensity of interest and let that be time varying while leaving the others constant. In contrast the time transformation method requires the non-homogeneity to be the same for all intensities. Obviously the smoothness of the intensities in the time transformation model is an advantage though.

In the Alzheimer's example a 4-state model is fitted where patients can be in states Normal, Mildly Cognitively Impaired, Alzheimer's or Death. Since the observed data include patients who make transitions back from Alzheimer's to MCI and from MCI to Normal, the Markov models assume backward transitions are possible. In the conclusion it is noted that there is the possibility of misclassification. However, the possible remedy of a hidden Markov model is not mentioned.