Fractional Brownian motion and multivariate-t models for longitudinal biomedical data, with application to CD4 counts in HIV-positive patients.

Longitudinal data are widely analysed using linear mixed models, with 'random slopes' models particularly common. However, when modelling, for example, longitudinal pre-treatment CD4 cell counts in HIV-positive patients, the incorporation of non-stationary stochastic processes such as Brownian motion has been shown to lead to a more biologically plausible model and a substantial improvement in model fit. In this article, we propose two further extensions. Firstly, we propose the addition of a fractional Brownian motion component, and secondly, we generalise the model to follow a multivariate-t distribution. These extensions are biologically plausible, and each demonstrated substantially improved fit on application to example data from the Concerted Action on SeroConversion to AIDS and Death in Europe study. We also propose novel procedures for residual diagnostic plots that allow such models to be assessed. Cohorts of patients were simulated from the previously reported and newly developed models in order to evaluate differences in predictions made for the timing of treatment initiation under different clinical management strategies. A further simulation study was performed to demonstrate the substantial biases in parameter estimates of the mean slope of CD4 decline with time that can occur when random slopes models are applied in the presence of censoring because of treatment initiation, with the degree of bias found to depend strongly on the treatment initiation rule applied. Our findings indicate that researchers should consider more complex and flexible models for the analysis of longitudinal biomarker data, particularly when there are substantial missing data, and that the parameter estimates from random slopes models must be interpreted with caution.