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### 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

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 P

Fixed .4878 .3719 .6037 .0591 8.2485 .0000

Random .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.

I2 = 67.5%

The results are of the order of magnitude that we commonly see in meta-analyses of between groups studies

Studies done at UCSB (=0) or elsewhere (=1)

------- Analog ANOVA table (Homogeneity Q) -------

Q df p

Between 3.8550 1.0000 .0496

Within 16.8138 12.0000 .1567

Total 20.6688 13.0000 .0797

------- Q by Group -------

Group Qwdf 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 = .05453

se(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 NonoverlappingDatapoints (PND) (Scruggs, Mastropieri, & Casto, 1987)
- Percentage of Zero Data (Campbell, 2004)
- Improvement Rate Difference (Parker, Vannest, & Brown, 2009)

Evaluate for TREND

ABAB 7Level of Experimental Control

No Exp Control Publishable Strong Exp Control

1 2 3 4 5 6 7

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)

Evaluate for TREND

ABAB 7Level of Experimental Control

No Exp Control Publishable Strong Exp Control

1 2 3 4 5 6 7

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

Evaluate for TREND

ABAB 7Level of Experimental Control

No Exp Control Publishable Strong Exp Control

1 2 3 4 5 6 7

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?

Study Study 1Study 2 … Study K

Subject Subj1 Subj2 Subj1Subj1 Subj2 Subj3 Subj4

Moments m mm m mmmmm mm mm m

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.

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