The Aggregation of Single-Case Results Using Hierarchical Linear Models.

To investigate the generalizability of the results of single-case experimental studies, evaluating the effect of one or more treatments, in applied research various simultaneous and sequential replication strategies are used. We discuss one approach for aggregating the results for single-cases: the use of hierarchical linear models. This approach has the potential to allow making improved inferences about the effects for the individual cases, but also to estimate and test the overall effect, and explore the generality of this effect across cases and under different conditions. Keywords: single-case; hierarchical linear model; replication; aggregation ********** Single-case experimental designs are used to evaluate the effect of one or more treatments on a single case. The case may be a subject or another single entity that forms the research unit, such as a school or a family. This entity is repeatedly observed, over the levels of one or several manipulated independent variables (Onghena, 2005). In the most basic design, the AB-phase design or interrupted time series design, the case is observed repeatedly during a first phase (A), typically a baseline phase before an intervention takes place, and in a second phase (B) after or during an intervention. To evaluate the effect of the intervention, scores in both phases are compared. Single-case designs have a long history in behavioral science (Ittenbach & Lawhead, 1997), but the last decades, single-case methodology has further been elaborated, aiming at improving the internal validity of the conclusions. For instance, reversal phase designs (e.g., an ABAB-design) or alternation designs with rapidly alternating conditions (e.g., an AABBBABAABB-design) rather than a simple AB-phase design may be used in order to assess or control statistically for the effect of history, maturation or other time-related confounding variables. The effect of such confounding variables may further be controlled by means of randomization while setting up the study, for instance by randomly assigning measurement occasions over treatments or randomizing the time of intervention (Edgington, 1996). Although group designs receive much more attention in methodological courses and handbooks, in the last decades there has been renewed interest in single-case designs, especially in behavior modification and clinical psychology (Barlow & Hersen, 1984. Kazdin, 1982), neuropsychology (Caramazza, 1990), psychopharmacology (Cook, 1996), and educational research (Kratochwill & Levin, 1992). The popularity of the designs is also reflected in the relatively large number of articles published in the Behavior Analyst Today that discuss or apply a variety of single-case designs (about twenty between 2001 and 2006). Single-case designs indeed are very attractive in several situations (Franklin, Allison, Gorman, 1997; Onghena, 2005). Single-case studies may be relatively easy to set up and are much less expensive than large-scale group-comparison studies. This makes the designs also attractive for practitioners, who want to get a first insight into the effect of a treatment. An additional strength of single-case designs is that, in contrast to group designs that give insight into the average effect of a treatment, they give an in-depth insight into the behavior of one single case. Especially in clinical settings, the research indeed often focuses on the effect of a treatment for a specific case. Finally, since only a single case is investigated, the design often allows making a large number of repeated observations, enabling a detailed study of the evolution of the behavior. Single-case designs thus are (initially) aimed at drawing valid conclusions regarding one entity. Sometimes, for instance in applied clinical settings, the primary interest may indeed be in this single entity, since it concerns a case that presented itself with a problem to solve. …