Extracting scalar measures from functional data with applications to placebo response

Abstract

In controlled and observational studies, outcome measures are often observed longitudinally. Such data are difficult to compare among units directly because there is no natural ordering of curves. This is relevant not only in clinical trials, where typically the goal is to evaluate the relative efficacy of treatments on average, but also in the growing and increasingly important area of personalized medicine, where treatment decisions are optimized with respect to a relevant patient outcome. In personalized medicine, there are no methods for optimizing treatment decision rules using longitudinal outcomes, e.g., symptom trajectories, because of the lack of a natural ordering of curves. A typical practice is to summarize the longitudinal response by a scalar outcome that can then be compared across patients, treatments, etc. We describe some of the summaries that are in common use, especially in clinical trials. We consider a general summary measure (weighted average tangent slope) with weights that can be chosen to optimize specific inference depending on the application. We illustrate the methodology on a study of depression treatment, in which it is difficult to separate placebo effects from the specific effects of the antidepressant. We argue that this approach provides a better summary for estimating the benefits of an active treatment than traditional non-weighted averages.

Keywords: Average tangent slope; Longitudinal data; Ordering curves; Placebo effects.

Figures

Figure 1.

Fitted quadratic trajectories of depression severity (measured on the HRSD) over time for drug (blue solid curves) and placebo (red dashed curves) treated subjects. The thick solid curves are the corresponding estimated mean parabolas.

Figure 2.

Nonparametric densities for the 4 ATS estimation methods (i) Fitting a quadratic model (solid black curve), (ii) fitting a straight line model (dashed red curve), (iii) using the formula (6) (dotted green curve) and (iv) the crude estimate (broken-dashed blue curve). The vertical dotted line marks the true average slope at −0.4. Left panel:No missing data;Middle panel: 30% MCAR data;Right panel: Dropout missing pattern with 50% no missing, 30% missing only the last observation, 10% missing exactly the last 2 observations, 5% missing exactly the last 3 observations, and 5% missing exactly the last 4 observations.

Figure 3.

Distribution of the average quadratic slope estimates from the antidepressant study for drug-treated (black curves) and placebo-treated (red curves). The solid curves are based on fitting a straight line to the data and the dashed curves are estimates using the average tangent slope formula (6) obtained from fitting a quadratic linear mixed-effects model.

Figure 4.

Three different longitudinal outcome trajectories of depression severity (HRSD) versus time (week) for an 8-week RCT. The three trajectories have equal average tangent slopes.

Figure 5.

Estimated mean quadratic trajectory for placebo-treated subjects (left panel) and the corresponding weight function w^(t) in the right panel.

Figure 6.

Four drug-treated subjects: subjects on the same row (a, b) and (c, d) have roughly equal (unweighted) average tangent slopes, but their weighted average tangent slopes differ substantially. Blue dots:observed data;Blue solid curve:estimated quadratic trajectory;Gray solid line:estimated linear trajectory;Red dashed curve:mean quadratic trajectory for placebo treated subjects.