1. Randomization: A Missing Component of the Single-Case Research Methodological Standards Adapted from Kratochwill, T. R., & Levin, J. R. (2010). Enhancing the scientific credibility of single-case intervention research: Randomization to the rescue. Psychological Methods, 15, 122- 144.

2. Why Randomization? The single-case designs and analyses to be promoted here receive high marks with respect to two critical research validity criteria: Internal Validity (a research design issue) Elevates the status of single-case research by increasing the scientific credibility of its methodology Statistical-Conclusion Validity (a data-analysis issue) Legitimizes the conduct of various statistical tests and one’s interpretation of the results

3. Traditional Single-Case Designs Basic design (AB) and various extensions Reversal (or withdrawal or “operant”) design (ABAB) Alternating and simultaneous treatment designs; also ABABAB…AB (or AB k ) Multiple-probe design Changing criterion design Multiple-baseline design

4. Four Single-Case Design-and-Analysis Randomization Variations 1.Within-Case Intervention Randomization 2.Between-Case Intervention Randomization 3.Case Randomization 4.Intervention Start-Point Randomization

5. 1. Within-Case Intervention Randomization With within-case (or phase) randomization, the order in which the A and B phases are administered is randomly determined for each case (e.g., participant, pair, small group, classroom).

6. Days/Weeks/Months/Sessions Traditional Basic Design (AB)

7. AB Design “[I]nstead of automatically administering the two phases in an AB order, one could randomly determine which phase should come first.” (Levin, Marascuilo, & Hubert, 1978) Problems with randomizing phases? Problems with not randomizing phases? Levin, J. R., Marascuilo, L. A., & Hubert, L. J. (1978). N = nonparametric randomization tests. In T. R. Kratochwill (Ed.), Single subject research: Strategies for evaluating change (pp. 167-196). New York: Academic Press.

8. Reversal Design (ABAB)

9. Randomized Phase Designs Traditional ABAB Design Addresses maturation and carryover effects to some extent “[H]owever, systematic assignment (A preceding B in each pair) is not the same as random assignment (either within each pair or within the entire experiment).” (Levin et al., 1978) – other problems include Hawthorne/novelty effects Edgington, E. S. (1992). Nonparametric tests for single-case experiments. In T. R. Kratochwill & J. R. Levin (Eds.), Single-case research design and analysis (pp.133-157). Hillsdale, NJ: Erlbaum. Onghena, P. (1992). Randomization tests for extensions and variation of ABAB single-case experimental designs: A rejoinder. Behavioral Assessment, 14, 153-171.

10. Randomized Phase Designs For ABAB…AB and alternating-treatment designs, there are two basic types of phase randomization: simple and blocked. With simple randomization, the only constraint is that there be equal numbers of A and B phases in the design. With blocked randomization, additional constraints are imposed to control for order effects.

11. Randomized ABAB... AB Design With One Case, Two Within-Case Conditions, and 10 Time Periods 5 A and 5 B)

12. Replicated Randomized ABAB... AB Design With Four Cases, Two Within-Case Conditions, and 10 Time Periods (5 A and 5 B)

13. One Potentially Palatable Solution for the True Baseline (A) Situation? Prior to initiating the formal AB intervention study, include one or more mandatory baseline (adaptation, warmup) observations (A'). The administration order of the subsequent A and B phases is then randomly determined. The A' phase is not considered to be part of the actual study’s design or analysis.

14. Randomized Alternating Treatment Design (ATD) With One Case, Two Within-Case Conditions, and 13 Time Periods (Seven Mornings and Six Afternoons)

15. Randomized Alternating Intervention Design With Three Units, Two Within-Series Conditions, and 13 Time Periods (Seven Mornings and Six Afternoons)

16. 2. Between-Case Intervention Randomization In some “between-case” multiple-intervention single- case designs, Treatment X (a control or intervention condition) is administered to one or more cases and Treatment Y (an alternative intervention condition) is administered to other cases. With between-case intervention randomization, which cases receive Treatment X and which receive Treatment Y is randomly determined.

17. A Two-Intervention (Between Cases) Example

18. 3. Case Randomization With case randomization, cases are randomly assigned to the different replication positions within the design. Multiple-baseline designs, with their systematically staggered intervention start points, are uniquely suited to this type of randomization.

19. Traditional Multiple-Baseline Design Across Participants

20. Wampold, B. E., & Worsham, N. L. (1986). Randomization tests for multiple-baseline designs. Behavioral Assessment, 8, 135-143.

21. Time Out for an Introduction to “Intervention Start Points” Historically, the intervention start point [and subsequent transition points] has [have] been “response guided” – preferred by many traditional SCD researchers. The intervention start point [and subsequent transition points] is [are] designated on an a priori basis by the researcher – preferred by traditional methodologists. The intervention start point [and subsequent transition points] is [are] randomly selected from a set of potential points that are designated as “acceptable” by the researcher – preferred by “new-age” SCD methodologists.

22. 4. Intervention Start-Point Randomization With intervention start-point randomization, the actual A-to-B transition (“intervention start point”) is randomly selected from a set of researcher- designated “acceptable” (or “potential”) start points. This type of randomization can be implemented in single-case designs where A and B are either baseline and intervention conditions or two different intervention conditions.

23. AB Design With One Case (“Unit”), Two Within- Series Intervention Conditions, 20 Time Periods, and 13 Potential Intervention Start Points

24. Replicated AB Design With Three Cases (“Units”), Two Within-Series Intervention Conditions, 20 Time Periods, and 13 Potential Intervention Points for Each Case Marascuilo, L. A., & Busk, P. L. (1988). Combining statistics for multiple-baseline AB and replicated ABAB designs across subjects. Behavioral Assessment, 10, 1- 28.

25. Multiple-Baseline Design With 4 Randomized Cases (“Units”), Two Within-Series Conditions, 15 Time Periods, 3, 3, 2, and 2 Potential Intervention Start Points for Cases 1, 3, 2, and 4, Respectively, and a Staggered Intervention Introduction of at Least One Time Period

26. “Fascinating” Issue to Contemplate It can be argued that under certain conditions the Marascuilo and Busk (1988) replicated AB design, with randomly determined intervention start points for each case, should be considered very nearly equivalent (in terms of its scientific credibility) to a multiple-baseline design with random assignment of cases to the different multiple-baseline positions.

27. “Fascinating” Issue to Contemplate (cont.) Even better, the original Marascuilo-Busk procedure can be adapted to fit directly into a multiple-baseline structure. “Restricted” versions of the original procedure have recently been examined by Levin, Ferron, and Gafurov (2015). Levin, J. R., Ferron, J. F., & Gafurov, B. S. (2015). Comparison of randomization-test procedures for single-case multiple-baseline designs. Unpublished manuscript. University of Arizona, Tucson.

28. Other Randomized Start-Point Possibilities and Combinations Random Assignment of Interventions (Between Cases), Intervention Orders (Within Cases), and/or Intervention Start Points –when multiple interventions are included in the study

29. Levin, Ferron, & Gafurov’s (2014) Single-Case Randomized Intervention Start-Point, Random-Order Crossover Design Week 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Student 1 A A A A A A A B B B B B B B B Student 2 B B B B B B A A A A A A A A A Student 3 B B B B B B B B B A A A A A A Student 4 A A A A A B B B B B B B B B B Note: There are two interventions, A and B. Half of the students are randomly selected to receive an AB order of intervention administration and half to receive a BA order. With 15 sessions and a minimum of 5 sessions required for each intervention, each student receives a crossover start point randomly selected between Week 6 and Week 10 inclusive. This is a nice design because it can separate time/sequence effects and intervention effects by controlling for intervention order, Levin, J. R., Ferron, J. M., & Gafurov, B. S. (2014). Improved randomization tests for a class of single- case intervention designs. Journal of Modern Applied Statistical Methods, 13(2), 2-52; retrievable from http://digitalcommons.wayne.edu/jmasm/vol13/iss2/2

30. Levin & Wampold’s (1999) Replicated Simultaneous Start-Point Model Time Period 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Pair 1 X A A A A A A A A A A A B* B B B B B B B B Pair 1 Y A A A A A A A A A A A B* B B B B B B B B Pair 2 X A A A A A A A A A B* B B B B B B B B B B Pair 2 Y A A A A A A A A A B* B B B B B B B B B B Note: X and Y are two different intervention conditions, each of which is randomly assigned to a pair member in each pair. Potential intervention start points are between Time Periods 5 and 17 inclusive. *Randomly selected intervention start point for each pair of units

31. In Addition… Two interesting adaptations of the preceding design: 1. One can incorporate a mixture of a randomized component (start-point randomization) and a nonrandomized component (pair member classifications) to address either individual or group interaction/moderation hypotheses. In that situation, the X and Y members of each pair would represent different variable classifications of interest (e.g., gender, developmental level, classroom achievement).

32. In Addition… 2. X and Y could also be two different outcome measures, associated either with a single intervention or with two different interventions. For example, X could be a reading performance measure and Y an arithmetic performance measure. a.With a single intervention, A = Baseline, B = Intervention (e.g., B = a reading intervention), one could test the hypothesis that the intervention has comparable effects on the two measures. b.With two different interventions, A = Intervention 1, B = Intervention 2 (e.g., A = a reading intervention and B = an arithmetic intervention), one could test the hypothesis that the two interventions have comparable effects on their respective outcome measures.

33. Take-Home Message (to be Returned to Later in this Institute) Through various randomization schemes, it is possible to design single-case intervention studies that possess the same or similar scientific credibility characteristics as those of conventional randomized group intervention studies ‒ “best possible design” philosophy as a recurring theme With the additional inclusion of a sufficient number of replication components, who’s to say that a superbly implemented randomized single-case intervention study is less “valued” than a superbly implemented conventional randomized group intervention study?