Testing delayed, gradual, and temporary treatment effects in randomized single-case experiments: A general response function framework

Randomization tests represent a class of significance tests to assess the statistical significance of treatment effects in randomized single-case experiments. Most applications of single-case randomization tests concern simple treatment effects: immediate, abrupt, and permanent changes in the level of the outcome variable. However, researchers are confronted with delayed, gradual, and temporary treatment effects; in general, with “response functions” that are markedly different from single-step functions. We here introduce a general framework that allows specifying a test statistic for a randomization test based on predicted response functions that is sensitive to a wide variety of data patterns beyond immediate and sustained changes in level: different latencies (degrees of delay) of effect, abrupt versus gradual effects, and different durations of the effect (permanent or temporary). There may be reasonable expectations regarding the kind of effect (abrupt or gradual), entailing a different focal data feature (e.g., level or slope). However, the exact amount of latency and the exact duration of a temporary effect may not be known a priori, justifying an exploratory approach studying the effect of specifying different latencies or delayed effects and different durations for temporary effects. We provide illustrations of the proposal with real data, and we present a user-friendly freely available web application implementing it.