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Single-Case Intervention Research Training Institute Madison, WI - June, 2019. James E. Pustejovsky pusto@austin.utexas.edu. Effect size measures for single-case designs: between-case standardized mean differences. Parametric between-case effect sizes.
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Single-Case Intervention Research Training Institute Madison, WI - June, 2019 James E. Pustejovsky pusto@austin.utexas.edu Effect size measures for single-case designs:between-case standardized mean differences
Parametric between-case effect sizes • Within-case standardized mean difference is not on the same scale as SMD from between-groups designs (e.g., between-subjects randomized trial). • Shadish, Rindskopf, & Hedges (2008) asked: Can we estimate a SMD based on the data from a single-case design that IS in the same metric a SMD from a between-groups design? • Why do this? (Shadish, Hedges, Horner, & Odom, 2015) • Translation of single-case research for researchers who work primarily with between-groups designs. • Comparison of results from single-case studies and between-groups studies, for purposes of understanding the utility and limitations of each type of design. • Synthesis involving both single-case and between-groups designs.
Between-case SMD • What is the SMD from a between-groups experiment? • These quantities can be estimated from single-case data using a hierarchical model that describes variation within and between participants. • But we’ll need to have a sample of multiple participants (bare minimum of 3, more for more complex models).
Estimating between-case SMDs: The broad strategy: • Develop a hierarchical model that describes a) the functional relationship for each case and b) how the outcome and functional relationship vary across cases. • Use the hierarchical model to imagine a hypothetical between-subjects experiment with the same population of participants, same treatment, same outcomes. • Calculate the between-case SMD for the hypothetical experiment.
Estimating BC-SMDs: Basic model Hedges, Pustejovsky, and Shadish proposed BC-SMD estimators for a basic hierarchical linear model • HPS (2012): Treatment reversal (ABAB) designs • HPS (2013): Multiple baseline/multiple probe designs Assumptions: • Baseline is stable (no baseline trend). • Intervention effect is immediate (no intervention-phase trend). • The outcome is normally distributed around mean level for each case, with variance σ2. • Within-case errors follow a first-order auto-regressive process (serial dependence). • The baseline level for each case is normally distributed, with variance τ2. • The treatment effect is constant across cases.
Estimating BC-SMDs: Basic model • Moment estimation • Web app: https://jepusto.shinyapps.io/scdhlm/ • R package: http://jepusto.github.io/getting-started-with-scdhlm • SPSS macro: http://faculty.ucmerced.edu/wshadish/software/software-meta-analysis-single-case-design/dhps-version-march-7-2015 • Shadish, Hedges, and Pustejovsky (2014) includes worked examples • Restricted maximum likelihood estimation (recommended) • Web app: https://jepusto.shinyapps.io/scdhlm/ • R package: http://jepusto.github.io/getting-started-with-scdhlm • Both methods produce estimates of BC-SMD (corrected for small-sample bias) and accompanying standard error.
More flexible models for BC-SMDs • Pustejovsky, Hedges, and Shadish (2014) extend the basic model to allow for less restrictive assumptions: • Variability of individual treatment effects • Baseline time trends (constant or varying across participants) • Time trends in treatment phase (constant or varying across participants) • More flexible models require more cases • For models with time trends, also need to specify a focal follow-up time
Barton-Arwood, Wehby, & Falk (2005). Reading instruction for elementary-age students with emotional and behavioral disorders: Academic and behavioral outcomes
Barton-Arwood, Wehby, & Falk (2005)Effect size calculations • Allow for baseline and intervention time trends, both varying across cases. • Focal follow-up time after 13 sessions. • BC-SMD estimate of 0.91 (SE = 0.93). • Substantial heterogeneity in intervention time trends.
Limitations of between-case SMD • Describes an average effect across a set of cases • Conceals potential individual heterogeneity • Inherent consequence of comparability with between-groups effect sizes. • Technical limitations • Only available for treatment reversal (ABAB) and multiple baseline/multiple probe across participant designs. • Requires at least 3 participants, preferably more. • Assumes normally distributed, interval-scale outcomes. • More work needed on evaluating model selection, model fit • Use between-case effect sizes as a complement to (not a replacement for) within-case effect size measures
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