Statistical Randomization Techniques for Single-Case Intervention Data. Joel R. Levin University of Arizona

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1 Statistical Randomization Techniques for Single-Case Intervention Data Joel R. Levin University of Arizona

Purpose of This Presentation (Dejà Vu?) To broaden your minds by introducing you to new and exciting, scientifically credible single-case intervention design-and-analysis possibilities.

2 Purpose of This Presentation (Dejà Vu?) To broaden your minds by introducing you to new and exciting, scientifically credible single-case intervention design-and-analysis possibilities. To let you know that these procedures are becoming increasingly acceptable to single-case intervention researchers and are beginning to appear in the SCD research literature. Whether YOU ever choose to adopt them in your own SCD research (after this week!) is entirely up to you.

3 Why Learn How to Count and to Calculate Probabilities? 1. To lose less the next time you're in Las Vegas playing blackjack. 2. To amaze your students with the birthday problem demonstration. 3. To count your chickens correctly.

4 Why Learn How to Count and to Calculate Probabilities? (New York Times, May 16, 2017, p. A13) A 13-year-old boy from Texas won a national math competition on Monday with an answer rooted in probabilities and a dash of farming. The boy, Luke Robitaille, took less than a second to buzz in at the Raytheon Mathcounts National Competition with the correct answer.

5 Why Learn How to Count and to Calculate Probabilities? The question: In a barn, 100 chicks sit peacefully in a circle. Suddenly, each chick randomly pecks the chick immediately to its left or right. What is the expected number of unpecked chicks?

6 Why Learn How to Count and to Calculate Probabilities? How about this one? A bag of coins contains only pennies, nickels and dimes with at least five of each. How many different combined values are possible if five coins are selected at random? (Give these two problems your best shot. Then, come back tomorrow and find out )

7 Why Learn How to Count and to Calculate Probabilities?

8 A Permutation-Test Primer (Based on Levin, 2007) Rationale and assumptions Samples and populations Individuals and groups Scores and ranks exact probabilities and sampling approximations for scores Levin, J. R. (2007). Randomization tests: Statistical tools for assessing the effects of educational interventions when resources are scarce. In S. Sawilowsky (Ed.), Real data analysis (pp ). Greenwich, CT: Information Age.

9 Is an Instructional Intervention Effective? From a total of 6 elementary school classrooms, 3 are randomly assigned to receive an instructional intervention that is designed to boost students academic performance (intervention classrooms), while students in the other 3 classrooms continue to receive their regular instruction (control classrooms). Following the instructional phase of the experiment, students in all classrooms are administered a 50-point achievement test and the average test performance within each classroom is calculated. Of interest in this study is the mean achievement-test difference between the 3 intervention classrooms and the 3 control classrooms.

10 Data-Analysis Rationale The obtained mean difference is examined in the context of the distribution of all possible mean differences that can be generated by assigning the 6 obtained classroom means to two instructional conditions, assuming that 3 classroom means must be assigned to each condition. A statistical test is then conducted by addressing the question: How (un)likely or (im)probable is what actually occurred (i.e., the obtained intervention-control mean difference) in relation to everything that could have occurred (i.e., the distribution of all possible intervention-control mean differences, given the study's design structure and the set of means produced)? Should the result of the foregoing test be deemed statistically improbable (e.g., p <.05), then the researcher would conclude that the two instructional methods differ with respect to students' average achievement-test performance.

11 How Do I Test These?: Let Me Count the Ways Let us systematically count the specific ways that these 3 scores could be assigned to Group 1. (Note: The order in which the 3 scores are listed is not important.) In how many different ways can 6 scores be assigned to 2 groups, if 3 scores must end up in each group? That is the same thing as asking how many different combinations of 3 objects are there if selected from a total of 6 objects. So as not to waste time attempting to express that quantity symbolically here: The answer, my friends, boils down to 6!/3!3!= 20. For example, consider the following 6 scores:

13 Example 1: Classroom Means and All Possible Assignments of Them to the Two Conditions (N 1 = N 2 = 3) Condition 1 Condition 2 M Difference (C1-C2) , 40.1, 39.6 (122.3, 40.8) 37.6, 36.7, 36.3 (110.6, 36.9) 3.9* p 1 = 1/20 = , 40.1, 37.6 (120.3, 40.1) 39.6, 36.7, 36.3 (112.6, 37.5) 2.6 p 2 = 2/20 = , 40.1, 36.7 (119.4, 39.8) 39.6, 37.6, 36.3 (113.5, 37.8) , 40.1, 36.3 (119.0, 39.7) 39.6, 37.6, 36.7 (113.9, 38.0) , 39.6, 37.6 (119.8, 39.9) 40.1, 36.7, 36.3 (113.1, 37.7) , 39.6, 36.7 (118.9, 39.6) 40.1, 37.6, 36.3 (114.0, 38.0) , 39.6, 36.3 (118.5, 39.5) 40.1, 37.6, 36.7 (114.4, 38.1) , 37.6, 36.7 (116.9, 39.0) 40.1, 39.6, 36.3 (116.0, 38.7) , 37.6, 36.3 (116.5, 38.83) 40.1, 39.6, 36.7 (116.4, 38.80) , 36.7, 36.3 (115.6, 38.5) 40.1, 39.6, 37.6 (117.3, 39.1) , 39.6, 37.6 (117.3, 39.1) 42.6, 36.7, 36.3 (115.6, 38.5) , 39.6, 36.7 (116.4, 38.80) 42.6, 37.6, 36.3 (116.5, 38.83) , 39.6, 36.3 (116.0, 38.7) 42.6, 37.6, 36.7 (116.9, 39.0) , 37.6, 36.7 (114.4, 38.1) 42.6, 39.6, 36.3 (118.5, 39.5) , 37.6, 36.3 (114.0, 38.0) 42.6, 39.6, 36.7 (118.9, 39.6) , 36.7, 36.3 (113.1, 37.7) 42.6, 39.6, 37.6 (119.8, 39.9) , 37.6, 36.7 (113.9, 38.0) 42.6, 40.1, 36.3 (119.0, 39.7) , 37.6, 36.3 (113.5, 37.8) 42.6, 40.1, 36.7 (119.4, 39.8) , 36.7, 36.3 (112.6, 37.5) 42.6, 40.1, 37.6 (120.3, 40.1) , 36.7, 36.3 (110.6, 36.9) 42.6, 40.1, 39.6 (122.3, 40.8) -3.9

14 What Summary Measure(s) Can/Should Be Analyzed by The To-Be-Presented Techniques? Means (Levels) of the phases Medians Truncated/Censored data based on a priori rules Even randomly selected observations? Slopes (Trends) of the phases Variances of the phases Any Predicted Pattern within or between phases Special considerations when groups are the units of intervention administration

15 Randomized Adaptations of Traditional Single-Case Designs Phase/Intervention Randomization (Within Cases) the order of the A and B phases/interventions is randomly determined for each case (e.g., participant, pair, group, classroom) Intervention Randomization (Between Cases) cases are randomly assigned to interventions Intervention Start-Point Randomization the transition point from one phase to the next is randomly determined for each case Case Randomization cases are randomly assigned to positions within the design

16 Why Randomization Statistical Tests? Conceptually and computationally straightforward easy to explain to others parsimonious Logically consistent connection to the design s randomization components implications for Type I error control Statistical power characteristics Underlying statistical assumptions Versatility and adaptability relative to other single-case statistical approaches

17 Background Eugene Edgington s randomization-test contributions Randomized phase/intervention designs Basic design (AB) and return to baseline design (ABA), including when A and B consist of two different interventions Reversal (or withdrawal or operant ) design (ABAB AB) and alternating treatment design Randomized intervention start-point designs Basic design (AB) and return-to-baseline design (ABA), including when A and B consist of two different interventions Reversal (or withdrawal or operant ) design (ABAB AB) and alternating treatment design Single-case crossover design Multiple-baseline design Replications (i.e., multiple units) and extensions of these

18 Bottom-Line Conclusions Based on Statistical Simulation Studies Tests of Change in Level 1. ABAB AB and Alternating Treatment Design (Levin, Ferron, & Kratochwill, 2012) 2. Variety of AB Designs (Levin, Ferron, & Gafurov, 2014) 3. Multiple-Baseline Designs (Levin, Ferron, & Gafurov, 2016; 2017) Tests of Change in Trend or Variability Multiple-Baseline Designs (Levin, Ferron, & Gafurov, in progress)

19 Top 10 List of Distinctions Between Actual and Simulated Single-Case Intervention Studies (With thanks to David Letterman, wherever you are!) 10. Participants 9. Planning and Design 8. Methods and Procedures 7. Institutional Review Boards 6. Investigator Personality Characteristics

20 Top 10 Distinctions (Continued) 5. Time and Hassle 4. Data Analysis 3. Rationalizations and Rationale for Future Research 2. Follow-Up 1. Publication Outlets and Impact Issues Bonus Question: Why did I ever bother?

21 Systematic vs. Randomized ABAB AB and Alternating-Treatment Designs (Levin, Ferron, & Kratochwill, 2012) 1. Claims that systematic ABAB AB designs produce inflated Type I error probabilities in series with positively autocorrelated observations are grossly inflated. 2. A systematic design consisting of alternating A and B individual observations yields very respectable statistical power for detecting larger effect sizes. From a methodological perspective, an even more appealing option is the randomized pairs design, perhaps with one or more mandatory initial A' observations in situations where a true baseline is desired (Kratochwill & Levin, 2010). The latter is not an issue if A and B represent two alternative interventions.

22 Effect-Size Alert for Single-Case Research Outcomes, or Don t Dis Large Effect Sizes Here! Marquis et al. (2000) noted in their meta-analysis of positive behavior support that [t]he smallest [conventional effect-size measure] for outcomes was 1.5, which would be considered quite large in a group study context (p. 165); and that their effect-size estimates ranged from 1.5 standardized units to 3.1 units (p. 167). Rogers and Graham (2008, p. 885) indicated that [W]hen we have used [the conventional method of effect-size calculation in metaanalyses of] single subject design studies in writing, the effect sizes are typically 3.0 and higher. In a single-case enuresis-treatment study conducted by Miller (1973), the conventional effect sizes calculated for the two participants were 5.98 and 6.41 (Busk & Serlin, 1992, p ). *BUT come back tomorrow for concerns and cautions!

23 References Busk, P. L., & Serlin, R. C. (1992). Meta-analysis for single-case research. In T. R. Kratochwill & J. R. Levin (Eds.), Single-case research design and analysis (pp ). Hillsdale, NJ: Erlbaum. Marquis, J. G., Horner, R. H., Carr, E. G., Turnbull, A. P., Thompson, M., Behrens, G. A., et al. (2000). A meta-analysis of positive behavior support. In R. Gersten, E. P. Schiller, S. Vaughn, (Eds.), Contemporary special education research: Syntheses of knowledge base on critical instructional issues (pp ). Mahwah, NJ: Erlbaum. Miller, P. M. (1973). An experimental analysis of retention control training in the treatment of nocturnal enuresis in two institutional adolescents. Behavior Therapy, 4, Rogers, L. A., & Graham, S. (2008). A meta-analysis of single subject design writing intervention research. Journal of Educational Psychology, 100,

24 Randomized Intervention Start-Point Designs and Analysis 1. Basic AB Design

25 Adapted from Levin, J. R., & Wampold, B. E. (1999). Generalized single-case randomization tests: Flexible analyses for a variety of situations. School Psychology Quarterly, 14,

27 Randomized Intervention Start-Point Designs and Analyses 1. Basic AB Design 2. Replicated AB Design

28 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.

29 Randomized Intervention Start-Point Designs and Analyses 1. AB Design 2. Replicated AB Design 3. Paired AB Design

30 Levin & Wampold s (1999) Simultaneous Start-Point Model Time Period Pair 1X A A A A A A A A A A A B B B B B B B B B Pair 1Y A A A A A A A A A A A B B B B B B B B B Note: Potential intervention start points are between Time Periods 5 and 17 inclusive. *Randomly selected intervention start point for the pair of units Levin, J. R., & Wampold, B. E. (1999). Generalized single-case randomization tests: Flexible analyses for a variety of situations. School Psychology Quarterly, 14,

31 From Levin, J. R., & Wampold, B. E. (1997, July) Single-case randomization tests for a variety of situations. Paper presented at the 10th European Meeting of the Psychometric Society, Santiago de Compostela, Spain.

32 Means and Mean Differences Associated With Each of the 9 Potential Intervention Startpoints, By Reinforcer Type Start Point Social Token T-S A B B-A A B B-A Diff Rank * *

33 Randomized Intervention Start-Point Designs and Analyses 1. AB Design 2. Replicated AB Design 3. Paired AB Design 4. Replicated Paired AB Design

34 Levin & Wampold s (1999) Replicated Simultaneous Start-Point Model Time Period Pair 1X A A A A A A A A A A A* B B B B B B B B B Pair 1Y A A A A A A A A A A A* B B B B B B B B B Pair 2X A A A A A A A A A* B B B B B B B B B B B Pair 2Y A A A A A A A A A* B B B B B B B B B B B Note: Potential intervention start points are between Time Periods 5 and 17 inclusive. *Randomly selected intervention start point for each pair of units Levin, J. R., & Wampold, B. E. (1999). Generalized single-case randomization tests: Flexible analyses for a variety of situations. School Psychology Quarterly, 14, 59 93

35 Randomized Intervention Start-Point Designs and Analyses 1. AB Design 2. Replicated AB Design 3. Paired AB Design 4. Replicated Paired AB Design 5. Dual Order AB Design

36 Dual-Order AB Designs A novel approach for improving the power of single-case randomization tests 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 Onghena, P., Vlaeyen, J. W. S., & de Jong, J. (2007). Randomized replicated singlecase experiments: Treatment of pain-related fear by graded exposure in vivo. In S. Sawilowsky (Ed.), Real data analysis (pp ). Greenwich, CT: Information Age.

37 In addition, the case receives a randomly selected intervention start point, with an a priori specification of 10 potential start points, from Observations 4 through 13 inclusive. Suppose that in a 16-observation design, A and B are either a baseline phase and an intervention phase or two different interventions that a single case is to receive. The case is assigned randomly to one of the two phase orders (AB or BA). [With a true baseline-intervention design this can be accomplished by including one or more mandatory baseline/adaptation observations (A') for both phase orders.] The random assignment of phase orders is required for the subsequent AB randomization test (modified Edgington test) to be valid.

38 AB Randomized Phase-Order Design (With Mandatory Initial A' Baseline Phase) With the original Edgington (1975) model, the study can be diagrammed as: A' A' A' A A A B B B* B B B B B B B B B B However, with the revised Edgington model, the opposite pretend ordering of As and Bs was also possible and therefore can be included in the randomization distribution: A' A' A' B B B A A A* A A A A A A A A A A

39 AB Randomized Phase-Order Design (With Mandatory Initial A' Baseline Phase) Single-order (AB) randomization: With 10 potential intervention points, even if the observed outcome were the most extreme the lowest onetailed significance probability would be p = 1/10 =.10. Dual-order (AB and BA) randomization: Even with only 10 potential intervention points it would be possible to attain a one-tailed significance probability of p = 1/20 =.05.

Randomized AB Order Designs:Summary and Conclusions (Levin et al., 2014) Note: The to-be-reported simulation results are typically based on a within-phase autocorrelation of ρ =.30.

40 Randomized AB Order Designs:Summary and Conclusions (Levin et al., 2014) Note: The to-be-reported simulation results are typically based on a within-phase autocorrelation of ρ =.30. For all AB design variations examined: 1. Dual-order randomization maintains satisfactory Type I error control. 2. Dual-order randomization exhibits power that is generally about points higher (and in some situations, even more) than that of single-order randomization.

41 Selected Single- Versus Dual-Randomization Power Comparisons of Longer and Shorter Series Simulations (SL = Series Length, NSP = Number of Potential Intervention Start Points) N d r Size (SL/NSP) Single Dual Difference Longer (15/5) Shorter (9/5) (Crossover Design as Special Case) Longer (15/5) Shorter (7/3) Longer (15/5) Shorter (8/2)

Randomized Intervention Start-Point Designs and Analyses 1. AB Design 2. Replicated AB Design 3. Paired AB Design 4. Replicated Paired AB Design 5.

42 Randomized Intervention Start-Point Designs and Analyses 1. AB Design 2. Replicated AB Design 3. Paired AB Design 4. Replicated Paired AB Design 5. Dual Order AB Design (Crossover Design as a Special Case) Potential difference in design credibility between a dual-order AB design and a systematic crossover design?: Don t lose your balance!

43 Randomized Intervention Start-Point Designs and Analyses 1. AB Design 2. Replicated AB Design 3. Paired AB Design 4. Replicated Paired AB Design 5. Dual Order AB Design (Crossover Design as a Special Case) 6. Multiple-Baseline Design, Including Restricted Replicated AB Design (Levin, Ferron, & Gafurov, 2014; 2016; 2017)

44 Multiple-Baseline Design Ferron, J., & Sentovich, C. (2002). Statistical power of randomization tests used with multiple-baseline designs. Journal of Experimental Education, 70, Revusky s between case procedure Revusky, S. H. (1967). Some statistical treatments compatible with individual organism methodology. Journal of the Experimental Analysis of Behavior, 10, Wampold & Worsham s and Koehler & Levin s within-case procedures Wampold, B. E., & Worsham, N. L. (1986). Randomization tests for multiplebaseline designs. Behavioral Assessment, 8, Type I error and power assessments have been made by Ferron and Sentovich (2002) of the Wampold-Worsham, Marascuilo- Busk, and Koehler-Levin procedures

45 Multiple-Baseline Design Updates: Summary and Conclusions (Levin et al., 2016, 2017) 1. Modifications (and improved) randomized startpoint versions of the Revusky and Marascuilo- Busk procedures have been developed. 2. Generally speaking, the Koehler-Levin and modified Marascuilo-Busk procedures are recommended, based on their superior power to detect immediate abrupt intervention effects, along with the modified Revusky procedure in special situations. But hold on, life is not so simple. Here s the rest of the story.

46 Multiple-Baseline Design Updates: Summary and Conclusions (Levin et al., 2016, 2017) 1. Modifications (improved) randomized start-point versions of the Revusky and Marascuilo-Busk procedures have been developed. 2. Generally speaking, the Koehler-Levin and modified Marascuilo-Busk procedures are recommended, based on their superior power to detect immediate abrupt intervention effects, along with the modified Revusky procedure in special situations.

47 (A)

48 Multiple-Baseline Design Updates: Summary and Conclusions (Levin et al., 2016, 2017) 1. Bad news: When the intervention effects are not immediate and abrupt, power is drastically reduced for all of the procedures (especially for 2-observation delayed abrupt effects).

49 (A)

50 Multiple-Baseline Design Updates: Summary and Conclusions (Levin et al., 2016, 2017) 1. Bad news: When the intervention effects are not immediate and abrupt, power is drastically reduced for all of the procedures (especially for 2-observation delayed abrupt effects). 2. Good news: With careful researcher forethought and planning (and good luck!), adjusted analyses can be formulated for all of the procedures to recapture much of the lost power. 3. Other news: Under certain alternate-effect conditions, the original fixed-point Wampold- Worsham procedure performs as well as (or better) the more recent randomized start-point procedures.

51 Multiple-Baseline Design Updates: Summary and Conclusions (Levin et al., 2016, 2017) And a few final comments on these randomization tests: 1. Slippery slopes and variability (Levin, Ferron, & Gafurov, in progress) 2. The power of increased sample size (specifically, the number of cases) 3. Systematic power comparisons of randomization tests and HLM procedures

52 References Levin, J. R., Ferron, J., M., & Kratochwill, T. R. (2012). Nonparametric statistical tests for single-case systematic and randomized ABAB AB and alternating treatment designs: New developments, new directions. Journal of School Psychology, 50, 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 Levin, J. R., Ferron, J. M., & Gafurov, B. S. (2016). Comparison of randomization-test procedures for single-case multiple-baseline designs. Developmental Neurorehabilitation, DOI: / Levin, J. R., Ferron, J. M., & Gafurov, B. S. (2017). Additional comparisons of randomization-test procedures for single-case multiple-baseline designs: Alternative effect types. Journal of School Psychology, 62, Levin, J. R., Ferron, J. M., & Gafurov, B. S. (in progress). Randomization tests of trend and variability for single-case multiple-baseline designs.

53 Some Randomization-Test Software Randomized Phase Designs Edgington & Onghena s (2007) SCRT (Single-Case Randomization Tests) program, in their book Levin, Ferron, & Kratochwill s (2012) SAS software for various ABAB AB and alternating treatment designs Randomized Intervention Start-Point Designs Edgington & Onghena s (2007) SCRT program, in their book (also Bulté & Onghena, 2008) Koehler s (2012) program for the Koehler-Levin (1998) multiple-baseline procedure ( Gafurov & Levin s ExPRT (Excel Package of Randomization Tests) Version 2.1, June 2017; downloadable at Other Borckhardt et al. s (2008) Simulation Modeling Analysis (SMA) program

54 Current and Projected Features of Gafurov & Levin s ExPRT Statistical Software Features of ExPRT s programs include: exact nonparametric statistical analyses based on some type of randomization options for either random or fixed intervention start points and intervention orders designs based on either one or multiple cases (replications) an unlimited number of total observations for up to 15 cases either individual or paired cases analyses conducted with either raw or standardized data

Current and Projected Features of the ExPRT Statistical Software Features of ExPRT s programs include: user-defined α levels (one- or two-tailed tests) statistical decisions (reject, do not reject)

55 Current and Projected Features of the ExPRT Statistical Software Features of ExPRT s programs include: user-defined α levels (one- or two-tailed tests) statistical decisions (reject, do not reject) and significance probabilites (p-values) statistical tests based on either mean (level), slope (trend), or variance (variability) output distribution of all possible outcomes graph of the outcomes for each case case-by-case and across-case summary measures and effect-size estimates (conventional d and rescaled NAP) a randomizing routine for planned studies

56 AB Design Basic time-series design Baseline/Control (A) vs. Intervention (B) Intervention A vs. Intervention B Intervention start-point randomization procedure (Edgington model; Marascuilo-Busk model; Levin- Wampold simultaneous start-point model for two different matched interventions: comparativeeffectiveness and general-effectiveness tests) Levin et al. s (2014) dual intervention-order option; single-case systematic crossover design: intervention-effect and time-effect.

separate two-phase tests Multiple-Baseline Design Within-case comparisons (Wampold-Worsham model) Within-case comparisons;

63 Additional ExPRT Randomization Tests ABA Design Intervention start-point randomization (Onghena model) Overall test and separate two-phase tests Reversal (ABAB) Design Intervention start-point randomization (Onghena model) Overall test and separate two-phase tests Multiple-Baseline Design Within-case comparisons (Wampold-Worsham model) Within-case comparisons; intervention start-point randomization (Koehler-Levin and restricted Marascuilo- Busk models) Between-case comparisons (modified Revusky model)

64 Additional References Bulté, I., & Onghena, P. (2008). An R package for single-case randomization tests. Behavior Research Methods, 40, Edgington, E. S., & Onghena, P. (2007). Randomization tests (4th ed.) Boca Raton, FL: Chapman & Hall/CRC. Koehler, M. J., & Levin, J. R. (1998). Regulated randomization: A potentially sharper analytical tool for the multiple-baseline design. Psychological Methods, 3, Levin, J. R., & Ferron, J. M. (2014). Review of Dugard, File, and Todman s Single-case and small-n designs: A practical guide to randomization tests (2 nd ed.). American Statistician, 68, Levin, J. R., Lall, V. F., & Kratochwill, T. R. (2011). Extensions of a versatile randomization test for assessing single-case intervention effects. Journal of School Psychology, 49, 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 ). New York, NY: Academic Press.

Improved Randomization Tests for a Class of Single-Case Intervention Designs

Journal of Modern Applied Statistical Methods Volume 13 Issue 2 Article 2 11-2014 Improved Randomization Tests for a Class of Single-Case Intervention Designs Joel R. Levin University of Arizona, jrlevin@u.arizona.edu