Combining patient-level and summary-level data for Alzheimer’s disease modeling and simulation: a beta regression meta-analysis

Abstract

Our objective was to develop a beta regression (BR) model to describe the longitudinal progression of the 11 item Alzheimer’s disease (AD) assessment scale cognitive subscale (ADAS-cog) in AD patients in both natural history and randomized clinical trial settings, utilizing both individual patient and summary level literature data. Patient data from the coalition against major diseases database (3,223 patients), the Alzheimer’s disease neruroimaging initiative study database (186 patients), and summary data from 73 literature references (representing 17,235 patients) were fit to a BR drug-disease-trial model. Treatment effects for currently available acetyl cholinesterase inhibitors, longitudinal changes in disease severity, dropout rate, placebo effect, and factors influencing these parameters were estimated in the model. Based on predictive checks and external validation, an adequate BR meta-analysis model for ADAS-cog using both summary-level and patient-level data was developed. Baseline ADAS-cog was estimated from baseline MMSE score. Disease progression was dependent on time, ApoE4 status, age, and gender. Study drop out was a function of time, baseline age, and baseline MMSE. The use of the BR constrained simulations to the 0–70 range of the ADAS-cog, even when residuals were incorporated. The model allows for simultaneous fitting of summary and patient level data, allowing for integration of all information available. A further advantage of the BR model is that it constrains values to the range of the original instrument for simulation purposes, in contrast to methodologies that provide appropriate constraints only for conditional expectations.

Appendix: Linear approximation to the logit transformation

One may obtain linear approximations to the logit and inverse logit transformations using first-order Taylor expansion of logit(x) around x = 0.5, resulting in the approximation ${\rm logit}(\mathit{ADAS}/70) \approx -2 + 4 \times (\mathit{ADAS}/70)$ , or for the inverse relationship. This linear approximation is not invoked for modeling per se, although the justification of our “operational likelihood” for summary statistics does invoke the existence of a linear approximation. Rather, its primary utility is to enable an approximate interpretation of model parameters on the original scale, this being important for both prior specification and model (posterior) summary.

We tabulate the approximate transformations used for each parameter in Table 7. With the exception of the population average intercept, all conversions rely on the linear approximation described above (in the case of the population average intercept, one may apply the inverse logit transformation directly, since this is a location parameter; similar exact transformations are not possible for scale parameters). Where appropriate, a multiplicative factor of 52 is also used to convert weekly rates to annual rates.

Table 7 Approximate transformations between parametric scale and original scale