Application of the Within- and Between-Series Estimators to Non-normal Multiple-Baseline Data: Maximum Likelihood and Bayesian Approaches

Abstract

In single-case research, multiple-baseline (MB) design provides the opportunity to estimate the treatment effect based on not only within-series comparisons of treatment phase to baseline phase observations, but also time-specific between-series comparisons of observations from those that have started treatment to those that are still in the baseline. For analyzing MB studies, two types of linear mixed modeling methods have been proposed: the within- and between-series models. In principle, those models were developed based on normality assumptions, however, normality may not always be found in practical settings. Therefore, this study aimed to investigate the robustness of the within- and between-series models when data were non-normal. A Monte Carlo study was conducted with four statistical approaches. The approaches were defined by the crossing of two analytic decisions: (a) whether to use a within- or between-series estimate of effect and (b) whether to use restricted maximum likelihood or Markov chain Monte Carlo estimations. The results showed the treatment effect estimates of the four approaches had minimal bias, that within-series estimates were more precise than between-series estimates, and that confidence interval coverage was frequently acceptable, but varied across conditions and methods of estimation. Applications and implications were discussed based on the findings.

Keywords:

• Linear mixed model
• single case
• non-normality
• restricted maximum likelihood estimation
• Markov chain Monte Carlo

Notes

1 The thinning interval (25th draw) is not necessarily this high in general MCMC estimation. Researchers should make decisions that fit their particular modeling situation.

2 This pattern was not expected because the treatment effect estimate was not systematically biased as the number of measurement occasions increased. Although the decrement of the coverage rate is relatively small, more detailed investigation regarding the number of measurement occasions and the coverage rate is needed.