Meta-analytic methods provide a way to synthesize data across treatment evaluation studies. However, these well-accepted methods are infrequent with behavior analytic studies. Multilevel models may be a promising method to meta-analyze single-case data. This technical article provides a primer for how to conduct a multilevel model with single-case designs with AB phases using data from the differential-reinforcement-of-low-rate behavior literature. We provide details, recommendations, and considerations for searching for appropriate studies, organizing the data, and conducting the analyses. All data sets are available to allow the reader to follow along with this primer. The purpose of this technical article is to minimally equip behavior analysts to complete a meta-analysis that will summarize a current state of affairs as it relates to the science of behavior analysis and its practice. Moreover, we aim to demonstrate the value of analyses of this sort for behavior analysis.
Alderman, N., & Knight, C. (1997). The effectiveness of DRL in the management and treatment of severe behaviour disorders following brain injury. Brain Injury, 11, 79-101. https://doi.org/10.1080/026990597123683
Anglesea, M. M., Hoch, H., & Taylor, B. A. (2008). Reducing rapid eating in teenagers with autism: Use of a pager prompt. Journal of Applied Behavior Analysis, 41, 107-111. https://doi.org/10.1901/jaba.2008.41-107
Association of Professional Behavior Analysts (2019). Overview of state laws to license or otherwise regulate practitioners of applied behavior analysis. https://cdn.ymaws.com/www.apbahome.net/resource/resmgr/pdf/state_regulation_of_ba_feb20.pdf
Austin, J. L., & Bevan, D. (2011). Using differential reinforcement of low rates to reduce children's requests for teacher attention. Journal of Applied Behavior Analysis, 44, 451-461. https://doi.org/10.1901/jaba.2011.44-451
Baek, E. K., Moeyaert, M., Petit-Bois, M., Beretvas, S. N., Van den Noortgate, W., & Ferron, J. M. (2014). The use of multilevel analysis for integrating single-case experimental design results within a study and across studies. Neurpsychological Rehabilitation, 24, 590-606. https://doi.org/10.1080/09602011.2013.835740
Baer, D. M. (1977). Perhaps it would be better not to know everything. Journal of Applied Behavior Analysis, 10, 167-172. https://doi.org/10.1901/jaba.1977.10-167
Becraft, J. L., Borrero, J. C., Davis, B., Mendres-Smith, A. E., & Castillo, M. (2018). The role of signals in two variations of differential reinforcement-of-low-rate procedures. Journal of Applied Behavior Analysis, 51, 3-24. https://doi.org/10.1002/jaba.431
Becraft, J. L., Borrero, J. C., Mendres-Smith, A. E., & Castillo, M. (2017). Decreasing excessive bids for attention in a simulated early education classroom. Journal of Behavioral Education, 26, 371-393. https://doi.org/10.1007/s10864-017-9275-6
Biosoft (2004) UnGraph for Windows (Version 5.0) [Computer software]. Author.
Bonner, A. C., & Borrero, J. C. (2018). Differential reinforcement of low rate schedules reduce severe problem behavior. Behavior Modification, 42, 747-764. https://doi.org/10.1177/0145445517731723
Catania, A. C. (2013). Learning (5th ed.) Prentice Hall.
Cheung, M. W. L. (2015). metaSEM: An R package for meta-analysis using structural equation modeling. Frontiers in Psychology, 5, 1-7. https://doi.org/10.3389/fpsyg.2014.01521
Cooper, H. (2016). Research synthesis and meta-analysis: A step-by-step approach (5th ed.). Inc: Sage Publishing.
Dallery, J., & Raiff, B. R. (2014). Optimizing behavioral health interventions with single-case designs: From development to dissemination. Translational Behavioral Medicine, 4, 290-303. https://doi.org/10.1007/s13142-014-0258-z
Ferster, C. B., & Skinner, B. F. (1957). Schedules of reinforcement. Skinner Foundation: B. F.
Gadaire, D. M., Marshall, G., & Brissett, E. (2017). Differential reinforcement of low rate responding in social skills training. Learning and Motivation, 60, 34-40. https://doi.org/10.1016/j.lmot.2017.08.005
Ganz, J. B., & Ayres, K. M. (2018). Methodological standards in single-case experimental design: Raising the bar. Research in Developmental Disabilities, 79, 3-9. https://doi.org/10.1016/j.ridd.2018.03.003
Harrington, M., & Velicer, W. F. (2015). Comparing visual and statistical analysis in single-case studies using published studies. Multivariate Behavioral Research, 50, 162-183. https://doi.org/10.1080/00273171.2014.973989
Hedges, L. V., Pustejovsky, J. E., & Shadish, W. R. (2012). A standardized mean difference effect size for single case designs. Research Synthesis Methods, 3, 224-239. https://doi.org/10.1002/jrsm.1052
Hedges, L. V., Pustejovsky, J. E., & Shadish, W. R. (2013). A standardized mean difference effect size for multiple baseline designs across individuals. Research Synthesis Methods, 4, 324-341. https://doi.org/10.1002/jrsm.1086
Jamshidi, L., Heyvaert, M., Declercq, L., Fernandez-Castilla, B., Ferron, J. M., Moeyaert, M., … Van den Noortgate, W. (2018). Methodological quality of meta-analyses of single-case experimental studies. Research in Developmental Disabilities, 79, 97-115. https://doi.org/10.1016/j.ridd.2017.12.016
Jessel, J., & Borrero, J. C. (2014). A laboratory comparison of two variations of differential-reinforcement-of-low-rate procedures. Journal of Applied Behavior Analysis, 47, 314-324. https://doi.org/10.1002/jaba.114
Kahng, S., Chung, K., Gutshall, K., Pitts, S. C., Kao, J., & Girolami, K. (2010). Consistent visual analyses of intrasubject data. Journal of Applied Behavior Analysis, 43, 35-45. https://doi.org/10.1901/jaba.2010.43-35
Kratochwill, T. R., Hitchcock, J., Horner, R. H., Levin, J. R., Odom, S. L., Rindskopf, D. M., & Shadish, W. R. (2010). Single-case designs technical documentation. Retrieved from What Works Clearinghouse website: http://ies.ed.gov/ncee/wwc/pdf/wwc_scd.pdf
Lennox, D. B., Miltenberger, R. G., & Donnelly, D. R. (1987). Response interruption and DRL for the reduction of rapid eating. Journal of Applied Behavior Analysis, 20, 279-284. https://doi.org/10.1901/jaba.1987.20-279
Lipsey, M. W., & Wilson, D. B. (2000). Practical meta-analysis. Sage Publishing, Inc.
Manolov, R., Guilera, G., & Sierra, V. (2014). Weighting strategies in the meta-analysis of single-case studies. Behavior Research Methods, 46, 1152-1166. https://doi.org/10.3758/s13428-013-0440-0
McCormack, J. C., Elliffe, D., & Virues-Ortega, J. (2019). Quantifying the effects of the differential outcomes procedure in humans: A systematic review and a meta-analysis. Journal of Applied Behavior Analysis, 52, 870-892. https://doi.org/10.1002/jaba.578
Moeyaert, M., Ferron, J. M., Beretvas, S. N., & Van den Noortgate, W. (2014). From a single-level analysis to a multilevel analysis of single-case experimental designs. Journal of School Psychology, 52, 191-211. https://doi.org/10.1016/j.jsp.2013.11.003
Moher, D., Liberati, A., Tetzlaff, J., Altman, D. G., & The PRISMA Group. (2009). Preferred reporting items for systematic reviews and meta-analyses: The PRISMA statement. PLOS Medicine, 6(7), 1-6. https://doi.org/10.1371/journal.pmed.1000097
Package ‘metaSEM’ (2018). [Manual for the metaSEM package for R]. https://cran.r-project.org/web/packages/metaSEM/metaSEM.pdf
Parker, R. I., & Vannest, K. J. (2012). Bottom-up analysis of single-case research designs. Journal of Behavioral Education, 21, 254-265. https://doi.org/10.1007/s10864-012-9153-1
Parker, R. I., Vannest, K. J., & Davis, J. L. (2011). Effect size in single-case research: A review of nine nonoverlap techniques. Behavior Modification, 35, 303-322. https://doi.org/10.1177/0145445511399147
Piper, A., Borrero, J. C., & Becraft, J. L. (2020). Differential reinforcement-of-low-rate procedures: A systematic replication with students with autism spectrum disorder. Journal of Applied Behavior Analysis, 53(2), 1058-1070. http://doi.org/10.1002/jaba.631
Raudenbush, S. W., & Bryk, A. S. (2002). Hierarchical linear models: Applications and data analysis methods (2nd ed.). Sage Publications.
Reichow, B., Barton, E. E., & Maggin, D. M. (2018). Development and applications of the single-case design risk of bias tool for evaluating single-case design research study reports. Research in Developmental Disabilities, 79, 53-64. https://doi.org/10.1016/j.ridd.2018.05.008
Richman, D. M., Barnard-Brak, L., Grubb, L., Bosch, A., & Abby, L. (2015). Meta-analysis of noncontingent reinforcement effects on problem behavior. Journal of Applied Behavior Analysis, 48, 131-152. https://doi.org/10.1002/jaba.189
Rindskopf, D., & Ferron, J. (2014). Using multilevel models to analyze single-case design data. In T. R. Kratochwill & J. R. Levin (Eds.), Single-case intervention research: Statistical and methodological advances (pp. 221-246). American Psychological Association. https://doi.org/10.1037/14376-008
Rothstein, H. R., Sutton, A. J., & Borenstein, M. (Eds.). (2005). Publication bias in meta-analysis: Prevention, assessment, and adjustments. Ltd: John Wiley & Sons.
Scruggs, T. E., Mastropieri, M. A., & Casto, G. (1987). 'Meta-analysis for single-subject research: When does it clarify, when does it obscure?': Response to Salzberg, Strain, and Baer. RASE: Remedial & Special Education, 8, 49-52. https://doi.org/10.1177/074193258700800210
Shadish, W. R., Brasil, I. C. C., Illingworth, D. A., White, K. D., Galindo, R., Nagler, E. D., & Rindskopf, D. M. (2009). Using UnGraph to extract data from image files: Verification of reliability and validity. Behavior Research Methods, 41, 177-183. https://doi.org/10.3758/BRM.41.1.177
Shadish, W. R., Hedges, L. V., Pustejovsky, J. E., Rindskopf, D. M., Boyajian, J. G., & Sullivan, K. J. (2014). Analyzing single-case designs: d, G, hierarchical models, Bayesian estimators, generalized additive models, and the hopes and fears of researchers about analyses. In W. R. Shadish, L. V. Hedges, T. R. Kratochwill, J. R. Levin, T. R. Kratochwill, & J. R. Levin (Eds.), Single-case intervention research: Methodological and statistical advances. American Psychological Association. https://doi.org/10.1037/14376-009
Shadish, W. R., Kyse, E. N., & Rindskopf, D. M. (2013). Analyzing data from single-case designs using multilevel models: New applications and some agenda items for future research. Psychological Methods, 18, 385-405. https://doi.org/10.1037/a0032964
Shadish, W. R., & Sullivan, K. J. (2011). Characteristics of single-case designs used to assess intervention effects in 2008. Behavior Research Methods, 43, 971-980. https://doi.org/10.3758/s13428-011-0111-y
Shadish, W. R., Zelinsky, N. M., Vevea, J. L., & Kratochwill, T. R. (2016). A survey of publication practices of single-case design researchers when treatments have small or large effects. Journal of Applied Behavior Analysis, 49, 656-673. https://doi.org/10.1002/jaba.308
Sham, E., & Smith, T. (2014). Publication bias in studies of an applied behavior-analytic intervention: An initial analysis. Journal of Applied Behavior Analysis, 47, 663-678. https://doi.org/10.1002/jaba.146
Swaminathan, H., Rogers, H. J., & Horner, R. H. (2014). An effect size measure and Bayesian analysis of single-case designs. Journal of School Psychology, 52, 213-230. https://doi.org/10.1016/j.jsp.2013.12.002
Van den Noortgate, W., & Onghena, P. (2003). Hierarchical linear models for the quantitative integration of effect sizes in single-case research. Behavior Research Methods, Instruments, and Computers, 35, 1-10. https://doi.org/10.3758/BF03195492
Warton, D. I., & Hui, F. K. C. (2011). The arcsine is asinine: The analysis of proportions in ecology. Ecology, 92, 3-10. https://doi.org/10.1890/10-0340.1
Zelinsky, N. M., & Shadish, W. (2018). A demonstration of how to do a meta-analysis that combines single-case designs with between-groups experiments: The effects of choice making on challenging behaviors performed by people with disabilities. Developmental Neurorehabilitation, 21, 266-278. https://doi.org/10.3109/17518423.2015.1100690