Statistical analysis in Small-N Designs: using linear mixed-effects modeling for evaluating intervention effectiveness

Advances in statistical methods and computing power have led to a renewed interest in addressing the statistical analysis challenges posed by Small-N Designs (SND). Linear mixed-effects modeling (LMEM) is a multiple regression technique that is flexible and suitable for SND and can provide standardized effect sizes and measures of statistical significance.Our primary goals are to: 1) explain LMEM at the conceptual level, situating it in the context of treatment studies, and 2) provide practical guidance for implementing LMEM in repeated measures SND.We illustrate an LMEM analysis, presenting data from a longitudinal training study of five individuals with acquired dysgraphia, analyzing both binomial (accuracy) and continuous (reaction time) repeated measurements.The LMEM analysis reveals that both spelling accuracy and reaction time improved and, for accuracy, improved significantly more quickly under a training schedule with distributed, compared to clustered, practice. We present guidance on obtaining and interpreting various effect sizes and measures of statistical significance from LMEM, and include a simulation study comparing two p-value methods for generalized LMEM.We provide a strong case for the application of LMEM to the analysis of training studies as a preferable alternative to visual analysis or other statistical techniques. When applied to a treatment dataset, the evidence supports that the approach holds up under the extreme conditions of small numbers of individuals, with repeated measures training data for both continuous (reaction time) and binomially distributed (accuracy) dependent measures. The approach provides standardized measures of effect sizes that are obtained through readily available and well-supported statistical packages, and provides statistically rigorous estimates of the expected average effect size of training effects, taking into account variability across both items and individuals.