Statistical Analysis of Single Case Design Serial Dependence Is More than Needing Cheerios for Breakfast

Goal of Presentation Review concept of effect size Describe issues in using effect size concept for single case design Describe different traditional approaches to calculating effect size for single case design Illustrate one recent approach

Behavioral Intervention Research 3

Putting A Coat on Your Dog Before Going For a Walk

How We Have Gotten To This Meeting Long history of statistical analysis of SCD Criticism of quality of educational research (Shalvelson & Towne, 2002) Establishment of IES – Initial resistance to SCD Influence of professional groups IES willingness to fund research on statistical analysis

Concept of Effect Size of Study Effect size is a statistic quantifying the extent to which sample statistics diverge from the null hypotheses (Thompson, 2006)

Types of ESs for Group Design Glass Δ = (Me – Mc) / SDc Cohen’s d = (Me – Mc)/ Sdpooled Interpretation – Small =.20 – Medium =.50 – Large =.80 R 2 Eta 2 = SSeffect/SStotal

Statistical Analysis Antithetical To Single Case Design (SCD)? Original developers believed that socially important treatment effects have to be large enough to be reliably detected by visual inspection of the data.

Kazdin (2011) proposes Visual inspection less trustworthy when effects at not crystal clear Serial dependence may obscure visual analysis Detection of small effects may lead to understanding that could in turn lead to large effects Statistical analysis may generate ESs that allow one to answer more precise questions Effects for different types of individuals Experimenter effects

Example: PRT and Meta-Analysis (Shadish, 2012) Pivotal Response Training (PRT) for Childhood Autism 18 studies containing 91 SCD’s. For this example, to meet the assumptions of the method, the preliminary analysis: – Used only the 14 studies with at least 3 cases (66 SCDs). – Kept only the first baseline and PRT treatment phases, eliminating studies with no baseline After computing 14 effect sizes (one for each study), he used standard random effects meta- analytic methods to summarize results:

Results Distribution Description N Min ES Max ES Wghtd SD Fixed & Random Effects Model Mean ES -95%CI +95%CI SE Z P Fixed Random Random Effects Variance Component v = Homogeneity Analysis Q df p Random effects v estimated via noniterative method of moments. The results are of the order of magnitude that we commonly see in meta-analyses of between groups studies I 2 = 67.5%

Studies done at UCSB (=0) or elsewhere (=1) Analog ANOVA table (Homogeneity Q) Q df p Between Within Total Q by Group Group Qw df p Effect Size Results Total Mean ES SE -95%CI +95%CI Z P k Total Effect Size Results by Group Group Mean ES SE -95%CI +95%CI Z P k Maximum Likelihood Random Effects Variance Component v = se(v) = Of course, we have no idea why studies done at UCSB produce larger effects: different kinds of patients? different kinds of outcomes? But the analysis does illustrate one way to explore heterogeneity

Search for the Holy Grail of Effect Size Estimators No single approach agreed upon: (40+ have been identified, Swaminathan et al., 2008) Classes of approaches – Computational approaches – Randomization test – Regression approaches – Tau-U (Parker et al., 2011) as combined approach

Computational Approaches Percentage of Nonoverlapping Datapoints (PND) (Scruggs, Mastropieri, & Casto, 1987) Percentage of Zero Data (Campbell, 2004) Improvement Rate Difference (Parker, Vannest, & Brown, 2009)

ABAB 6 Level of Experimental Control No Exp Control Publishable Strong Exp Control

ABAB 7 Level of Experimental Control No Exp Control Publishable Strong Exp Control Evaluate for LEVEL Evaluate for TREND Evaluate for LEVEL Evaluate for TREND

Problem with phases

Randomization Test Edgington (1975, 1980) advocated strongly for use of nonparametric randomization tests. – Involves selection of comparison points in the baseline and treatment conditions – Requires random start day for participants (could be random assignment of participants in MB design, Wampold & Worsham, 1986) Criticized for SDC – Large Type I Error rate (Haardofer & Gagne, 2010) – Not robust to independence assumption and sensitivity low for data series < 30 to 40 datapoints (Manolov & Solanas, 2009)

ABAB 7 Level of Experimental Control No Exp Control Publishable Strong Exp Control Evaluate for LEVEL Evaluate for TREND Evaluate for LEVEL Evaluate for TREND

Regression (Least Squares Approaches) ITSACORR (Crosbie, 1993) – Interrupted time series analysis – Criticized for not being correlated with other methods White, Rusch, Kazdin, & Hartmann (1989) Last day of Treatment Comparison (LDT) – Compares two LDT for baseline and treatment – Power weak because of lengthy predictions

Regression Analyses Mean shift and mean-plus-trend model (Center, Skiba, & Casey; ) Ordinary least squares regression analysis (Allison & Gorman, 1993) – Both approaches attempt to control for trends in baseline when examining the performances in treatment d-Estimator (Shadish, Hedges, Rinscoff, 2012) GLS with removal of autocorrelaiton (Swaminathan, Horner, Rogers, & Sugai, 2012)

Tau-U (Parker, Vannest, Javis, & Sauber, 2011) Mann-Whitney U a nonparametric that compares individual data point in groups (AB comparisons) Kendal’s Tau does these same thing for trend within groups Tau-U – Tests and control for trend in A phase – Test for differences in A and B phases – Test and adjust for tend in B phase

ABAB 7 Level of Experimental Control No Exp Control Publishable Strong Exp Control Evaluate for LEVEL Evaluate for TREND Evaluate for LEVEL Evaluate for TREND

Tau-U Calculator Vannest, K.J., Parker, R.I., & Gonen, O. (2011). Single Case Research: web based calculators for SCR analysis. (Version 1.0) [Web-based application]. College Station, TX: Texas A&M University. Retrieved Sunday 15th July Combines nonoverlap between phases with trend from within intervention phases – Will detect and allow researcher to control for undesirable trend in baseline phase Data are easily entered on free website Generates a d for effects with trend withdrawn when necessary

Themes: An accessible and feasible effect size estimator As end users, SCD researchers need a tool that we can use without having to consult our statisticians – Utility of the hand calculated trend line analysis – Example of feasible tool, but criticized (ITSACORR< Crosbie, 1993). Parker, Vannest, Davis, & Sauber (2011) – Tau-U calculator

Theme: What is an effect—a d that detect treatment or/and level effect If a single effect size is going to be generated for an AB comparison: should the d be reported separately for level (intercept) or trend (slope)? – If so, problematic for meta-analysis ES estimators here appear to provide a combined effect for slope and intercept Parker et al. (2011) incorporate both

Theme: What comparisons get included in the meta-analysis Should we only use the initial AB comparison in ABAB Designs?

Theme: What comparisons get included in the meta-analysis Should we only include points at which functional relationship is established?

Theme: How many effects sizes per study? 30 Study Study 1 Study 2 … Study K Subject Subj 1 Subj 2 Subj 1 Subj 1 Subj 2 Subj 3 Subj4 Moments m mm m m m m m m mm mm m

Challenge: How do you handle zero baselines or treatment phases?

Heterogeneity In SCD: A reality SCD Researchers used a range of different designs and design combinations A look at current designs – Fall, 2011 Issue of Journal of Applied Behavior Analysis

Comparison of SDC and Group Design ESs: The Apples and Oranges Issues Logic of casual inferences different – Groups: Means differences between groups – SCD: Replication of effects either within or across all “participants” – Generally d represents different comparison Data collected to document an effect different – Group designs collect data before treatment and after treatment – SCDs collect data throughout treatment phase, so for treatments that build performance across time, they may appear less efficacious because they are including “acquisition” phase effects in analysis

ABAB 4 Level of Experimental Control No Exp Control Publishable Strong Exp Control

Conclusions I learned a lot Sophistication of analyses is increasing Feasibility of using statistical analysis is improving Can use statistical analysis as supplement to visual inspection (Kazdin, 2011) Statistical analysis may not be for everybody, but it is going to foster acceptability in the larger education research community, and for that reason SCD researchers should consider it.