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.     

The conditional power of randomization tests for single-case effect sizes in designs with randomized treatment order: A Monte Carlo simulation study

The conditional power (CP) of the randomization test (RT) was investigated in a simulation study in which three different single-case effect size (ES) measures were used as the test statistics: the mean difference (MD), the percentage of nonoverlapping data (PND), and the nonoverlap of all pairs (NAP). Furthermore, we studied the effect of the experimental design on the RT’s CP for three different single-case designs with rapid treatment alternation: the completely randomized design (CRD), the randomized block design (RBD), and the restricted randomized alternation design (RRAD). As a third goal, we evaluated the CP of the RT for three types of simulated data: data generated from a standard normal distribution, data generated from a uniform distribution, and data generated from a first-order autoregressive Gaussian process. The results showed that the MD and NAP perform very similarly in terms of CP, whereas the PND performs substantially worse. Furthermore, the RRAD yielded marginally higher power in the RT, followed by the CRD and then the RBD. Finally, the power of the RT was almost unaffected by the type of the simulated data. On the basis of the results of the simulation study, we recommend at least 20 measurement occasions for single-case designs with a randomized treatment order that are to be evaluated with an RT using a 5% significance level. Furthermore, we do not recommend use of the PND, because of its low power in the RT.     

Combining nonoverlap and trend for single-case research: Tau-U

A new index for analysis of single-case research data was proposed, Tau-U, which combines nonoverlap between phases with trend from within the intervention phase. In addition, it provides the option of controlling undesirable Phase A trend. The derivation of Tau-U from Kendall's Rank Correlation and the Mann-Whitney U test between groups is demonstrated. The equivalence of trend and nonoverlap is also shown, with supportive citations from field leaders. Tau-U calculations are demonstrated for simple AB and ABA designs. Tau-U is then field tested on a sample of 382 published data series. Controlling undesirable Phase A trend caused only a modest change from nonoverlap. The inclusion of Phase B trend yielded more modest results than simple nonoverlap. The Tau-U score distribution did not show the artificial ceiling shown by all other nonoverlap techniques. It performed reasonably well with autocorrelated data. Tau-U shows promise for single-case applications, but further study is desirable.     

Reconsidering the use of the general linear model with single-case data

Using a low point estimate of autocorrelation to justify analyzing single-case data with the general linear model (GLM) is questioned. Monte Carlo methods are used to examine the degree to which bias in the estimate of autocorrelation depends on the complexity of the linear model used to describe the data. A method is then illustrated for determining the range of autocorrelation parameters that could reasonably have led to the observed autocorrelation. The argument for using a GLM analysis can be strengthened when the GLM analysis functions appropriately across the range of plausible autocorrelations. For situations in which the GLM analysis does not function appropriately across this range, a method is provided for adjusting the confidence intervals to ensure adequate coverage probabilities for specified levels of autocorrelation.     

Meta-analysis for single-case consultation outcomes: Applications to research and practice

The application of meta-analysis holds much appeal for single-case consultation outcome research. We review a meta-analytic method for using within-study treatment effect sizes in reporting consultation outcomes. The strengths and limitations of traditional group design meta-analysis are examined. Various methods for analyzing single-case outcomes are discussed briefly, followed by an examination of the use of meta-analysis in single-case reviews across independent studies. Within-study meta-analytic results are presented that were derived from treatments implemented in consultations in natural settings. To conclude the article, an illustration is offered of a single-case data analysis display that incorporates meta-analytic results along with other indices of treatment outcome. Recommendations are provided for using meta-analytic methods to evaluate outcomes of single-case consultation treatment.     

Hierarchical models for relational event sequences

Interaction within small groups can often be represented as a sequence of events, each event involving a sender and a recipient. Recent methods for modeling network data in continuous time model the rate at which individuals interact conditioned on the previous history of events as well as actor covariates. We present a hierarchical extension for modeling multiple such sequences, facilitating inferences about event-level dynamics and their variation across sequences. The hierarchical approach allows one to share information across sequences in a principled manner—we illustrate the efficacy of such sharing through a set of prediction experiments. After discussing methods for adequacy checking and model selection for this class of models, the method is illustrated with an analysis of high school classroom dynamics from 297 sessions.     

Interpreting effect sizes when controlling for stability effects in longitudinal autoregressive models: Implications for psychological science

Effect sizes in longitudinal studies often are dramatically smaller than effect sizes in cross-sectional studies. Indeed, autoregressive models (which are often used in longitudinal studies but not in cross-sectional studies) control for past levels on the outcome (i.e., stability effects) in order to predict change in levels of the outcome over time and thus may greatly reduce the magnitude of the effect of a predictor on the outcome. Unfortunately, however, there have been no attempts to differentiate guidelines for interpreting effect sizes for longitudinal studies versus cross-sectional studies. Consequently, longitudinal effect sizes that fall below the universal guidelines for “small” may be incorrectly dismissed as trivial, when they might be meaningful. In the current paper, we first review the present guidelines for interpreting effect sizes. Next, we discuss several examples of how controlling for stability effects can dramatically attenuate effect sizes of other predictors, in order to support our argument that the current guidelines may be misleading for interpreting longitudinal effects. Finally, we conclude by making recommendations for researchers regarding the interpretation of effect sizes in longitudinal autoregressive models.     

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.     

Hierarchical diffusion models for two-choice response times

Two-choice response times are a common type of data, and much research has been devoted to the development of process models for such data. However, the practical application of these models is notoriously complicated, and flexible methods are largely nonexistent. We combine a popular model for choice response times-the Wiener diffusion process-with techniques from psychometrics in order to construct a hierarchical diffusion model. Chief among these techniques is the application of random effects, with which we allow for unexplained variability among participants, items, or other experimental units. These techniques lead to a modeling framework that is highly flexible and easy to work with. Among the many novel models this statistical framework provides are a multilevel diffusion model, regression diffusion models, and a large family of explanatory diffusion models. We provide examples and the necessary computer code.     

Expanding the single-case study: a proposed psychoanalytic research program

This paper offers a proposed program of research using single-case time-series methods that can be used by practicing clinicians. The paper is written for psychodynamically oriented clinicians who want to get involved in psychotherapy research and make contributions to the scientific literature. How to measure treatment outcomes and psychodynamic constructs are discussed. With few exceptions, conducting single-case time-series research using psychodynamic psychotherapy has far more advantages than disadvantages.     

A brief account of statistical tests for single-case research with persons with developmental disabilities

Education and rehabilitation research with persons with developmental disabilities is often based on single-case designs (with small numbers of ordinal data points collected at irregular intervals) and relies upon graphic display and visual inspection of the data. This paper (a) provides a brief account of some statistical tests, which may serve to supplement the visual inspection process and (b) underlines some of their strengths and limitations to help education and rehabilitation personnel make a reasonable choice among them.     

The effect of skewness and kurtosis on the robustness of linear mixed models

This study analyzes the robustness of the linear mixed model (LMM) with the Kenward–Roger (KR) procedure to violations of normality and sphericity when used in split-plot designs with small sample sizes. Specifically, it explores the independent effect of skewness and kurtosis on KR robustness for the values of skewness and kurtosis coefficients that are most frequently found in psychological and educational research data. To this end, a Monte Carlo simulation study was designed, considering a split-plot design with three levels of the between-subjects grouping factor and four levels of the within-subjects factor. Robustness is assessed in terms of the probability of type I error. The results showed that (1) the robustness of the KR procedure does not differ as a function of the violation or satisfaction of the sphericity assumption when small samples are used; (2) the LMM with KR can be a good option for analyzing total sample sizes of 45 or larger when their distributions are normal, slightly or moderately skewed, and with different degrees of kurtosis violation; (3) the effect of skewness on the robustness of the LMM with KR is greater than the corresponding effect of kurtosis for common values; and (4) when data are not normal and the total sample size is 30, the procedure is not robust. Alternative analyses should be performed when the total sample size is 30.     

An effect size measure and Bayesian analysis of single-case designs

This article describes a linear modeling approach for the analysis of single-case designs (SCDs). Effect size measures in SCDs have been defined and studied for the situation where there is a level change without a time trend. However, when there are level and trend changes, effect size measures are either defined in terms of changes in R² or defined separately for changes in slopes and intercept coefficients. We propose an alternate effect size measure that takes into account changes in slopes and intercepts in the presence of serial dependence and provides an integrated procedure for the analysis of SCDs through estimation and inference based directly on the effect size measure. A Bayesian procedure is described to analyze the data and draw inferences in SCDs. A multilevel model that is appropriate when several subjects are available is integrated into the Bayesian procedure to provide a standardized effect size measure comparable to effect size measures in a between-subjects design. The applicability of the Bayesian approach for the analysis of SCDs is demonstrated through an example.     

Hierarchical models of behavior and prefrontal function

The recognition of hierarchical structure in human behavior was one of the founding insights of the cognitive revolution. Despite decades of research, however, the computational mechanisms underlying hierarchically organized behavior are still not fully understood. Recent findings from behavioral and neuroscientific research have fueled a resurgence of interest in the problem, inspiring a new generation of computational models. In addition to developing some classic proposals, these models also break fresh ground, teasing apart different forms of hierarchical structure, placing a new focus on the issue of learning and addressing recent findings concerning the representation of behavioral hierarchies within the prefrontal cortex. In addition to offering explanations for some key aspects of behavior and functional neuroanatomy, the latest models also pose new questions for empirical research.     

Three-level meta-analysis of dependent effect sizes

Although dependence in effect sizes is ubiquitous, commonly used meta-analytic methods assume independent effect sizes. We describe and illustrate three-level extensions of a mixed effects meta-analytic model that accounts for various sources of dependence within and across studies, because multilevel extensions of meta-analytic models still are not well known. We also present a three-level model for the common case where, within studies, multiple effect sizes are calculated using the same sample. Whereas this approach is relatively simple and does not require imputing values for the unknown sampling covariances, it has hardly been used, and its performance has not been empirically investigated. Therefore, we set up a simulation study, showing that also in this situation, a three-level approach yields valid results: Estimates of the treatment effects and the corresponding standard errors are unbiased.     

Relations between linear models: Implications for the lens model

Brunswik's lens model has been widely used in the modeling and analysis of judgment tasks and the lens model equation has been an important part of most analyses. In such analyses, researchers often have based arguments and interpretations upon the magnitude of various components of the lens model equation. One component is G, an index of agreement between a linear model of the subject's judgments and a linear model of the task environment. This paper addresses the index G and shows that while in principle −1 G 1, the distribution of G is often skewed toward high values so that caution must be used in the interpretation of its magnitude. Moreover, the results reported here apply to the relation between linear models in general.     

Hierarchical single- and dual-process models of recognition memory

Recognition memory is commonly modeled as either a single, continuous process within the theory of signal detection, or with two-process models such as Yonelinas’ dual-process model. Previous attempts to determine which model provides a better account of the data have relied on fitting the models to data that are averaged over items. Because such averaging distorts conclusions, we develop and compare hierarchical versions of competing single and dual-process models that account for item variability. The dual-process model provides a superior account of a typical data set when models are compared with the deviance information criterion. Parameters of the dual-process model are highly correlated, however, suggesting that a single-process model may exist that can provide a better account of the data.