Optimal Partitioning for Linear Mixed Effects Models: Applications to Identifying Placebo Responders

Abstract

A long-standing problem in clinical research is distinguishing drug treated subjects that respond due to specific effects of the drug from those that respond to non-specific (or placebo) effects of the treatment. Linear mixed effect models are commonly used to model longitudinal clinical trial data. In this paper we present a solution to the problem of identifying placebo responders using an optimal partitioning methodology for linear mixed effects models. Since individual outcomes in a longitudinal study correspond to curves, the optimal partitioning methodology produces a set of prototypical outcome profiles. The optimal partitioning methodology can accommodate both continuous and discrete covariates. The proposed partitioning strategy is compared and contrasted with the growth mixture modelling approach. The methodology is applied to a two-phase depression clinical trial where subjects in a first phase were treated openly for 12 weeks with fluoxetine followed by a double blind discontinuation phase where responders to treatment in the first phase were randomized to either stay on fluoxetine or switched to a placebo. The optimal partitioning methodology is applied to the first phase to identify prototypical outcome profiles. Using time to relapse in the second phase of the study, a survival analysis is performed on the partitioned data. The optimal partitioning results identify prototypical profiles that distinguish whether subjects relapse depending on whether or not they stay on the drug or are randomized to a placebo.

Figures

Figure 1

Quadratic curves from a linear mixed effects model with a single binary covariate. The top panels show k = 4 principal point curves obtained using the parametric k-means algorithm for the true model. The bottom panel shows n = 50 parabolas from a simulated data set: the solid and dashed curves correspond to the two levels of the binary covariate.

Figure 2

Nonparametric densities obtained from the 100 simulated data sets of the average squared L2 distances to the nearest principal point curve estimate (k = 4) using: the true principal points (solid curve), the principal point curves estimated using the OPME approach (dashed curve), and principal point curves estimated using the filtering approach (dotted curve).

Figure 3

Top panels: k = 5 principal point profile curves fit to depression severity during the open treatment phase of the study using a B-spline basis. Bottom panel: a scatterplot of the best linear unbiased predictors (solid = melancholia, open = no melancholia) in the principal component subspace spanned by the first two principal components of the coefficient distribution: The “A” and the “P” mark the estimated fixed effects for absence and presence respectively of the melancholia features.

Figure 4. Predominantly Non-specific Responders

Left panels show the principal point profile curves estimated from the open phase of the study. The right panels show the Kaplan-Meier survivorship curves for subjects associated to these two principal point trajectories broken down by treatment (drug versus placebo) in the discontinuation phase.

Figure 5. Predominantly Specific Responders

Left panels show the principal point profile curves estimated from the open phase of the study. The right panels show the Kaplan-Meier survivorship curves for subjects associated with these three principal point trajectories broken down by treatment (drug versus placebo) in the discontinuation phase.