Linear-rank testing of a non-binary, responder-analysis, efficacy score to evaluate pharmacotherapies for substance use disorders

Abstract

The design of pharmacological trials for management of substance use disorders is shifting toward outcomes of successful individual-level behavior (abstinence or no heavy use). While binary success/failure analyses are common, McCann and Li (CNS Neurosci Ther 2012; 18: 414-418) introduced "number of beyond-threshold weeks of success" (NOBWOS) scores to avoid dichotomized outcomes. NOBWOS scoring employs an efficacy "hurdle" with values reflecting duration of success. Here, we evaluate NOBWOS scores rigorously. Formal analysis of mathematical structure of NOBWOS scores is followed by simulation studies spanning diverse conditions to assess operating characteristics of five linear-rank tests on NOBWOS scores. Simulations include assessment of Fisher's exact test applied to hurdle component. On average, statistical power was approximately equal for five linear-rank tests. Under none of conditions examined did Fisher's exact test exhibit greater statistical power than any of the linear-rank tests. These linear-rank tests provide good Type I and Type II error control for comparing distributions of NOBWOS scores between groups (e.g. active vs. placebo). All methods were applied to re-analyses of data from four clinical trials of differing lengths and substances of abuse. These linear-rank tests agreed across all trials in rejecting (or not) their null (equality of distributions) at ≤ 0.05.

Keywords: Linear-rank tests; Type I error; pharmacotherapy development; randomized clinical trial; statistical power; substance use disorders.

Figure 1.

Mapping of the NOBWOS score W on the joint distribution of time to start of success T₁ and duration of success T₂ for an example where the total duration of τ = 84 days is partitioned into r =12 (i.e., weekly) intervals of which the last s = 2 weeks comprise the end-of-study threshold period. The vertical axis extends, in theory, to infinity. The nine quadrilaterals designate where the joint distribution of T₁ and T₂ map to the integer sequence of W for which 0 < W ≤ 9. A score of W = 10 only obtains at T₁ = 0 and is confined to the vertical line segment extending toward infinity from {0, τ}. The six-sided concave polygon defines the continuous region wherein the joint distribution of T₁ and T₂ maps to W = 0, which represents individuals without any continuous periods of success that fully include the end-of-study threshold period (and individuals without a period of success). For example, consider an individual whose success begins on day 64. Even though this start is prior to threshold period, the full tenth week is not a success; so the individual receives W = 0. This explains why the vertical line on the right side of the W = 1 region is T₁ = τ(r – s −1)/r. The region above the diagonal line (T₂ = τ – T₁, upper right triangle) is after the end of the active-intervention period (lower left triangle). If success is only defined during the active intervention period (Winsorization), NOBWOS scores greater than zero are mapped to the solid diagonal line.

Figure 2.

Frequency of NOBWOS scores by condition for bupropion for the treatment of methamphetamine use disorder, the COMBINE trial for the treatment of alcohol use disorder, and the varenicline trials for treatment of tobacco use disorder.^, Tabulations are for an end-of-study threshold duration of 1 week. Placebo condition is black bar and active condition is gray bar. Cognitive behavioral intervention = CBI.