A generalized least squares regression approach for computing effect sizes in single-case research: application examples

A new method for deriving effect sizes from single-case designs is proposed. The strategy is applicable to small-sample time-series data with autoregressive errors. The method uses Generalized Least Squares (GLS) to model the autocorrelation of the data and estimate regression parameters to produce an effect size that represents the magnitude of treatment effect from baseline to treatment phases in standard deviation units. In this paper, the method is applied to two published examples using common single case designs (i.e., withdrawal and multiple-baseline). The results from these studies are described, and the method is compared to ten desirable criteria for single-case effect sizes. Based on the results of this application, we conclude with observations about the use of GLS as a support to visual analysis, provide recommendations for future research, and describe implications for practice.     

A Gradual Effects Model for Single-Case Designs

Single-case designs are a class of repeated measures experiments used to evaluate the effects of interventions for small or specialized populations, such as individuals with low-incidence disabilities. There has been growing interest in systematic reviews and syntheses of evidence from single-case designs, but there remains a need to further develop appropriate statistical models and effect sizes for data from the designs. We propose a novel model for single-case data that exhibit nonlinear time trends created by an intervention that produces gradual effects, which build up and dissipate over time. The model expresses a structural relationship between a pattern of treatment assignment and an outcome variable, making it appropriate for both treatment reversal and multiple baseline designs. It is formulated as a generalized linear model so that it can be applied to outcomes measured as frequency counts or proportions, both of which are commonly used in single-case research, while providing readily interpretable effect size estimates such as log response ratios or log odds ratios. We demonstrate the gradual effects model by applying it to data from a single-case study and examine the performance of proposed estimation methods in a Monte Carlo simulation of frequency count data.     

The Single-Case Design in Family Therapy Evaluation Research

This paper discusses the need for the development of new research designs for family therapy evaluation, and a manner for meeting that need. The single-case research design has gained acceptance as bona fide experimental designs for evaluating the effectiveness of intervention techniques in drug and psychotherapy outcome research. Although the need for more outcome studies on the effectiveness of family therapy has been frequently noted, there has been virtually no use of the single-case design in family therapy outside of the behavior modification literature on families. This paper presents potential benefits of the application of single case-design for the practice of family therapy. Preliminary guidelines are suggested for the application of these designs.     

Using response ratios for meta-analyzing single-case designs with behavioral outcomes

Methods for meta-analyzing single-case designs (SCDs) are needed to inform evidence-based practice in clinical and school settings and to draw broader and more defensible generalizations in areas where SCDs comprise a large part of the research base. The most widely used outcomes in single-case research are measures of behavior collected using systematic direct observation, which typically take the form of rates or proportions. For studies that use such measures, one simple and intuitive way to quantify effect sizes is in terms of proportionate change from baseline, using an effect size known as the log response ratio. This paper describes methods for estimating log response ratios and combining the estimates using meta-analysis. The methods are based on a simple model for comparing two phases, where the level of the outcome is stable within each phase and the repeated outcome measurements are independent. Although auto-correlation will lead to biased estimates of the sampling variance of the effect size, meta-analysis of response ratios can be conducted with robust variance estimation procedures that remain valid even when sampling variance estimates are biased. The methods are demonstrated using data from a recent meta-analysis on group contingency interventions for student problem behavior.     

Characteristics of single-case designs used to assess intervention effects in 2008

This article reports the results of a study that located, digitized, and coded all 809 single-case designs appearing in 113 studies in the year 2008 in 21 journals in a variety of fields in psychology and education. Coded variables included the specific kind of design, number of cases per study, number of outcomes, data points and phases per case, and autocorrelations for each case. Although studies of the effects of interventions are a minority in these journals, within that category, single-case designs are used more frequently than randomized or nonrandomized experiments. The modal study uses a multiple-baseline design with 20 data points for each of three or four cases, where the aim of the intervention is to increase the frequency of a desired behavior; but these characteristics vary widely over studies. The average autocorrelation is near to but significantly different from zero; but autocorrelations are significantly heterogeneous. The results have implications for the contributions of single-case designs to evidence-based practice and suggest a number of future research directions.     

A Comparison of Rubrics for Identifying Empirically Supported Practices with Single-Case Research

The use of single-case research methods for validating academic and behavioral interventions has gained considerable attention in recent years. As such, there has been a proliferation of methods for evaluating whether, and to what extent, primary research reports provide evidence of intervention effectiveness. Despite the recent interest in harnessing single-case research to identify empirically supported strategies, examination of these tools has revealed that there is a lack of consistency in the methodological criteria sampled and scoring procedures used to evaluate primary research reports. The present study examined the extent to which various evidence rubrics addressed specific methodological features of single-case research and classified studies into similar evidence categories. Results indicated that the methodological criteria included within rubrics tended to vary, particularly for criteria related to determining the generality of the intervention under study. Moreover, there was substantial discordance observed in the evidence classifications assigned to reviewed studies. These findings are discussed in the context of the still-developing nature of single-case evidence reviews. Recommendations for both research and practice are provided.     

Robust regression for single-case data analysis: How can it help?

This study examined the degree to which outliers were present in a convenience sample of published single-case research. Using a procedure for analyzing single-case data Allison &; Gorman (Behaviour Research and Therapy, 31, 621–631, 1993), this study compared the effect of outliers using ordinary least squares (OLS) regression to a robust regression method and attempted to answer four questions: (1) To what degree does outlier detection vary from OLS to robust regression? (2) How much do effect sizes differ from OLS to robust regression? (3) Are the differences produced by robust regression in more or less agreement with visual judgments of treatment effectiveness? (4) What is a typical range of effect sizes for robust regression versus OLS regression for data from “effective interventions”? Results suggest that outliers are common in single-case data. The effects of outliers in single-case data are explored, and the implications for researchers and practitioners using single-case designs are discussed.     

Single-Case Experimental Designs in Family Therapy Research: Limitations and Considerations

Recent authors have encouraged the use of single-case experimental design in family therapy research. However, several issues related to the application of these designs have not been addressed. The present article suggests that the applicability of single-case methods to family therapy research may be limited. The first issue raised is the general lack of adequate dependent measures of family interaction that fulfill the requirements of the single-case experimental designs. Suggestions for development of appropriate measures are given. Second, the necessary reliance on relatively weak single-case designs often allows only relatively weak conclusions. Finally, the strategy of beginning the single-case study of family functioning and family therapy in laboratory settings is suggested.     

Examining Estimates of Effect Using Non-Overlap of All Pairs in Multiple Baseline Studies of Academic Intervention

The inclusion of single-case design (SCD) studies in meta-analytic research is an important consideration in identifying effective evidence-based practices. Various SCD effect sizes have been previously suggested; non-overlap of all pairs (NAP) is a recently introduced effect size. Preliminary field tests investigating the adequacy of NAP are promising, but no analyses have been conducted using only multiple baseline designs (MBDs). This preliminary study investigated typical values of NAP in MBDs, investigated agreement with visual analysis, and suggested cut scores for interpreting a NAP effect size. Typical values of NAP in MBDs were larger compared to a previous meta-analysis of studies using AB, MBD, or ABAB withdrawal designs, and agreement of suggested cut scores and visual analysis was moderate.     

Intervention-based school psychology services: Training for child-level accountability; preparing for program-level accountability

This research evaluated the outcomes of a school psychology training practicum by replicating intervention-based service delivery procedures established in prior research. The key components include describing a service delivery model, teaching the model, deriving practice guidelines that fit the model, supporting trainees in carrying out the steps, and evaluating the outcomes. Procedures to determine outcomes were based on single-case design facets including accountability design (A-B), visual analysis of graphic data, and social validity ratings. Meta-analysis techniques included calculation of effect sizes and percent of nonoverlapping data (PND). Goal attainment scaling (GAS) also was used to summarize outcomes. The analyses indicated that the interventions led to positive changes for most children. For example, the median effect size was 1.42 across cases. Social validity evidence showed that consumers judged the outcomes positively. Achieving ideal baseline and technical adequacy checks (e.g., observer agreement, intervention adherence) represented challenges for many consultations. The contributions of the study include describing methods for child- and program-level accountability in training and areas for improvement including further supporting the completion of technical checks for intervention services.     

Le protocole individuel en psychologie : tutoriel à l’usage des psychologues praticiens

Although single-case design seems to be an appropriate method for practicing psychologists as it fits perfectly with the evidence-based practice framework, their use remains quasi-anecdotal in psychology. The aim of the present article is to present the advantages of single-case protocols and to demonstrate the ease of their implementation. It provides practitioners with a tutorial of a free open-source statistical program (SCDA) to analyze single-case data. The main results from a single-case protocol are presented and illustrated through an example. Strategies for integrating these protocols in the clinical practice are discussed. This tutorial could encourage the use of the single-case design in research and practice settings.     

Visual analysis of single-case time series: Effects of variability, serial dependence, and magnitude of intervention effects

Visual analysis is the dominant method of analysis for single-case time series. The literature assumes that visual analysts will be conservative judges. We show that previous research into visual analysis has not adequately examined false alarm and miss rates or the effect of serial dependence. In order to measure false alarm and miss rates while varying serial dependence, amount of random variability, and effect size, 37 students undertaking a postgraduate course in single-case design and analysis were required to assess the presence of an intervention effect in each of 27 AB charts constructed using a first-order autoregressive model. Three levels of effect size and three levels of variability, representative of values found in published charts, were combined with autocorrelation coefficients of 0, 0.3 and 0.6 in a factorial design. False alarm rates were surprisingly high (16% to 84%). Positive autocorrelation and increased random variation both significantly increased the false alarm rates and interacted in a nonlinear fashion. Miss rates were relatively low (0% to 22%) and were not significantly affected by the design parameters. Thus, visual analysts were not conservative, and serial dependence did influence judgment.     

Regression-based techniques for statistical decision making in single-case designs

The present study evaluates the performance of four methods for estimating regression coefficients used to make statistical decisions about intervention effectiveness in single-case designs. Ordinary least square estimation is compared to two correction techniques dealing with general trend and a procedure that eliminates autocorrelation whenever it is present. Type I error rates and statistical power are studied for experimental conditions defined by the presence or absence of treatment effect (change in level or in slope), general trend, and serial dependence. The results show that empirical Type I error rates do not approach the nominal ones in the presence of autocorrelation or general trend when ordinary and generalized least squares are applied. The techniques controlling trend show lower false alarm rates, but prove to be insufficiently sensitive to existing treatment effects. Consequently, the use of the statistical significance of the regression coefficients for detecting treatment effects is not recommended for short data series.     

Meta- and statistical analysis of single-case intervention research data: Quantitative gifts and a wish list

In this commentary, we add to the spirit of the articles appearing in the special series devoted to meta- and statistical analysis of single-case intervention-design data. Following a brief discussion of historical factors leading to our initial involvement in statistical analysis of such data, we discuss: (a) the value added by including statistical-analysis recommendations in the What Works Clearinghouse Standards for single-case intervention designs; (b) the importance of visual analysis in single-case intervention research, along with the distinctive role that could be played by single-case effect-size measures; and (c) the elevated internal validity and statistical-conclusion validity afforded by the incorporation of various forms of randomization into basic single-case design structures. For the future, we envision more widespread application of quantitative analyses, as critical adjuncts to visual analysis, in both primary single-case intervention research studies and literature reviews in the behavioral, educational, and health sciences.     

Current Perspectives on Interventions for Self-Injurious Behavior of Children with Autism Spectrum Disorder: A Systematic Review of the Literature

Our purpose in this systematic review was to identify, evaluate, and summarize the single-case research design studies examining behavioral interventions to address self-injurious behavior for young children with autism spectrum disorder. We identified 24 studies published between 2000 and 2016 that met the inclusion criteria. We reviewed and coded each study across five content areas using systematic procedures. These content areas consisted of the methodological quality of single-case research designs, children’s demographic information, conditions under which the intervention was implemented, characteristics of intervention, and outcomes. Our findings indicate that procedural integrity data were reported for only one study and social validity for two studies. The majority of studies were implemented by researchers or therapists in the school setting. For 24 of 30 children included in the studies reviewed, the authors taught a replacement behavior selected on the basis of an assessment conducted prior to intervention. All authors reported a decrease in the self-injurious behavior post-intervention. A limited number of authors examined generalization and maintenance of low rates of self-injurious behavior. Limitations of the present review and future directions and implications for research and practice are discussed.     

A Meta-analysis of the Taped Problems Intervention

Taped problems is an intervention strategy for addressing mathematics fluency that has been evaluated in multiple single-case design studies. Although its efficacy has been supported in individual studies, no comprehensive quantitative synthesis has been conducted on taped problems. The purpose of this study was to synthesize the literature that examined this intervention using parametric and nonparametric effect size computations. We examined (a) the effectiveness of taped problems across effect size comparisons, (b) maintenance over time, (c) the degree to which studies met single-case design research standards, and (d) potential moderating variables. Parametric and nonparametric analyses yielded moderate and large effect sizes and indicated that taped problems appears to be an efficacious intervention.     

Hierarchical linear models for the quantitative integration of effect sizes in single-case research

In this article, the calculation of effect size measures in single-case research and the use of hierarchical linear models for combining these measures are discussed. Special attention is given to meta-analyses that take into account a possible linear trend in the data. We show that effect size measures that have been proposed for this situation appear to be systematically affected by the duration of the experiment and fail to distinguish between effects on level and slope. To avoid these flaws, we propose to perform a multivariate meta-analysis on the standardized ordinary least squares regression coefficients from the study-specific regression equations describing the response variable.     

Evaluating single-case research data for systematic review: A commentary for the special issue

The purpose of this commentary is to provide observation on the statistical procedures described throughout this special section from the perspective of researchers with experience in conducting systematic reviews and meta-analyses of single-case research to address issues of evidence-based practice. It is our position that both visual and statistical analyses are complimentary methods for evaluating single-case research data for these purposes. Given the recent developments regarding the use of single-case research to inform evidence-based practice and policy, the developments described in the present issue will be contextualized within the need for a widely accepted process for data evaluation to assist with extending the impact of single-case research. The commentary will, therefore, begin with providing an overview of the conceptual underpinnings of a systematic review of single-case research and will be followed by a discussion of several features that are essential to the development of a conceptually sound and widely used statistical procedure for single-case research. The commentary will conclude with recommendations and guidelines for the use of both visual and statistical analyses within primary research reports and recommendations for future research.     

Randomization tests for multiple-baseline designs: An extension of the SCRT-R package

Multiple-baseline designs are an extension of the basic single-case AB phase designs, in which several of those AB designs are implemented simultaneously to different persons, behaviors, or settings, and the intervention is introduced in a staggered way to the different units. These designs are well-suited for research in the behavioral sciences. We discuss the advantages and limitations for valid inferences, and suggest a statistical technique—randomization tests—for use with multiple-baseline data, to complement visual analysis. In addition, we provide an extension of our SCRT-R package (which already contained means for conducting randomization tests on single-case phase and alternation designs), for multiple-baseline AB data.     

Visual aids and structured criteria for improving visual inspection and interpretation of single-case designs

Because behavior analysis is a data-driven process, a critical skill for behavior analysts is accurate visual inspection and interpretation of single-case data. Study 1 was a basic study in which we increased the accuracy of visual inspection methods for A-B designs through two refinements of the split-middle (SM) method called the dual-criteria (DC) and conservative dual-criteria (CDC) methods. The accuracy of these visual inspection methods was compared with one another and with two statistical methods (Allison & Gorman, 1993; Gottman, 1981) using a computer-simulated Monte Carlo study. Results indicated that the DC and CDC methods controlled Type I error rates much better than the SM method and had considerably higher power (to detect real treatment effects) than the two statistical methods. In Study 2, brief verbal and written instructions with modeling were used to train 5 staff members to use the DC method, and in Study 3, these training methods were incorporated into a slide presentation and were used to rapidly (i.e., 15 min) train a large group of individuals (N = 87). Interpretation accuracy increased from a baseline mean of 55% to a treatment mean of 94% in Study 2 and from a baseline mean of 71% to a treatment mean of 95% in Study 3. Thus, Study 1 answered basic questions about the accuracy of several methods of interpreting A-B designs; Study 2 showed how that information could be used to increase the accuracy of human visual inspectors; and Study 3 showed how the training procedures from Study 2 could be modified into a format that would facilitate rapid training of large groups of individuals to interpret single-case designs.