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

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Behavioral Intervention Research

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

• R2

• Eta2 = 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 14.000 .181 2.087 .374

------- Fixed & Random Effects Model ----------------------------- Mean ES -95%CI +95%CI SE Z PFixed .4878 .3719 .6037 .0591 8.2485 .0000Random .6630 .4257 .9002 .1210 5.4774 .0000

------- Random Effects Variance Component ------------------------v = .112554

------- Homogeneity Analysis ------------------------------------- Q df p 39.9398 13.0000 .0001

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

I2 = 67.5%

Studies done at UCSB (=0) or elsewhere (=1)------- Analog ANOVA table (Homogeneity Q) ------- Q df pBetween 3.8550 1.0000 .0496Within 16.8138 12.0000 .1567Total 20.6688 13.0000 .0797

------- Q by Group ------- Group Qw df p .0000 1.9192 3.0000 .5894 1.0000 14.8947 9.0000 .0939

------- Effect Size Results Total ------- Mean ES SE -95%CI +95%CI Z P k Total .6197 .0980 .4277 .8118 6.3253 .0000 14.0000

------- Effect Size Results by Group ------- Group Mean ES SE -95%CI +95%CI Z P k .0000 1.0228 .2275 .5769 1.4686 4.4965 .0000 4.0000 1.0000 .5279 .1086 .3151 .7407 4.8627 .0000 10.0000

------- Maximum Likelihood Random Effects Variance Component -------v = .05453se(v) = .04455

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 7

Level of Experimental Control

No Exp Control Publishable Strong Exp Control 1 2 3 4 5 6 7

Evaluate for LEVELEvaluate 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 1 2 3 4 5 6 7

Evaluate for LEVELEvaluate 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; 1985-86)

• 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 1 2 3 4 5 6 7

Evaluate for LEVELEvaluate for TREND

Tau-U Calculatorhttp://www.singlecaseresearch.org/

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 2012.

• 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?

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Study Study 1 Study 2 … Study K

Subject Subj1 Subj2 Subj1 Subj1 Subj2 Subj3 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

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.