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Copyright 2008 by the American Psychological Association 2008, Vol. 53, No. 3, 357–369 DOI: 10.1037/a0012973 Advanced Regression Methods for Single-Case Designs: Studying Propranolol in the Treatment for Agitation Associated With Traumatic Daniel F. Brossart Texas A&M University Wayne State University Richard I. Parker, James McNamara, and Timothy R. Elliott Texas A&M University Objective: The use of single-case designs in intervention research is discussed. Regression methods for
analyzing data from these designs are considered, and an innovative use of logistic regression to analyze
data from a double-blind, randomized clinical trial of propranolol for agitation among persons with
traumatic brain injury (TBI) is used. Method: Double-blind, randomized clinical trial performed in an
outpatient rehabilitation setting. Participants: Nine men and 4 women with TBI. Results: Logistic
models indicated that propranolol was not associated with less agitation for most participants (⌽ ⫽ .135;
90% exact confidence interval was ⫺.03 ⬍ .135 ⬍ .29). Four participants displayed a significant
response to propanolol. Two participants demonstrated significant improvement, and the other 2 expe-
rienced significantly more agitation in the treatment phase. Summary: Advanced regression methods can
be used to analyze data from single-case designs to obtain information of clinical and statistical
significance from a variety of psychological and medical treatments.
Keywords: single-case design, logistic regression, propranolol, brain injury, agitation In a thoughtful commentary, Aeschleman (1991) observed a informs otherwise: Many of the influential research programs in decreasing interest in single-case research (SCR) designs in the rehabilitation psychology first appeared in the literature in single- rehabilitation psychology literature: Between 1985 and 1989, Ae- case designs. Behavioral approaches— championed in the classic schleman found only 6 out of 402 empirical papers published in Behavioral Methods in Chronic Pain and Illness (Fordyce, Rehabilitation Psychology, Archives of Physical Medicine and 1976)—were based on earlier single-case studies. The potential of Rehabilitation, and Rehabilitation Counseling Bulletin used a sin- supported employment— arguably one of the few evidence-based gle-subject design (⬍1.5% of the total; Aeschleman, 1991, p. 43).
practices in rehabilitation psychology with considerable support A brief examination of the past 15 years of Rehabilitation Psy- from many randomized clinical trials (RCTs; Dunn & Elliott, in chology reveals one article that offered an innovative way to press)— appeared in a study using a single-case case design analyze single-case data (Callahan & Barisa, 2005) and another published in the Journal of Applied Behavior Analysis (Wehman et that was a true single-case study (Pijnenborg, Withaar, Evans, van al., 1989). And the ground-breaking extensions of Neal Miller's den Bosch, & Brouwer, 2007).
operant learning models to visceral, reflex, and motor responses We disagree with Aeschleman's bleak conclusion that SCR were achieved in single-case designs (Brucker & Ince, 1977; Ince, designs ". . have not made a methodological impact on research Brucker, & Alba, 1978). Clearly, SCR designs have played a in rehabilitation psychology" (Aeschleman, 1991, p. 47). History pivotal role in the rehabilitation psychology research base.
Unfortunately, SCR and case studies are often misconstrued as one in the same. An uncontrolled case study is a study of a singleclient, dyad, or group in which observations are made under Daniel F. Brossart, Richard I. Parker, James McNamara, and Timothy R.
Elliott, Department of Educational Psychology, Texas A&M University; uncontrolled and unsystematic conditions. The lack of experimen- Jay M. Meythaler, Department of Physical Medicine and Rehabilitation, tal control in such a study may have contributed to an overall Wayne State University and Rehabilitation Institute of Michigan, Detroit, suspicion or distrust of results based on a single subject in general.
Designs that add more experimental control include systematic, This study was funded in part by National Institute of Disability Re- repeated observations of a single client, dyad, or group and are search and Rehabilitation Grant H 133G000072 awarded to Jay M.
often called intensive single-case designs. For even more experi- Meythaler. Appreciation is expressed to Michael E. Dunn for sharing mental rigor, one could use a single-case experimental design, information and opinions about the history of single-case designs in reha- which is typically viewed as having greater control than intensive bilitation psychology research. Graphs of participant data not presented in single-case designs. These designs usually have behavioral goals or this article are available upon request from Daniel F. Brossart.
target behaviors that are the main focus of interest and function as the Correspondence concerning this study should be addressed to Daniel F.
Brossart, Department of Educational Psychology, 4225 TAMU, College dependent variable. They also have repeated measurements over time Station, TX, 77845. E-mail: [email protected] and at least two treatment phases (baseline and treatment). Some have BROSSART, MEYTHALER, PARKER, MCNAMARA, AND ELLIOTT stated that the core essence of single-case research is that "all (if not impossible) to attain due to the lack of services that negate dependent measures are collected repeatedly over the course of the a "usual treatment" scenario for a controlled, comparison group experiment, and these data are not combined with those from other (such that any attention to control participants would be above and participants to produce group averages for purposes of data anal- beyond the typical experience or "treatment-as-usual"; Elliott, ysis" (Morgan & Morgan, 2001, p. 122). Nevertheless, there are also instances in which evaluating single-case data across partic- The use of single-case designs also helps address the overuse of ipants is helpful because it can increase the internal validity of the cross-sectional methods so common in rehabilitation psychology.
Just as many introductory research design texts talk about the In this article, we begin by briefly discussing some present monomethod bias for a single research study, overuse of a single issues, past practice, and some misunderstandings regarding sin- design within a field creates a lopsided literature base that lacks the gle-case research. We then show how the application of single- advantages of triangulation with multiple research methodologies.
case research can be helpful in answering substantive questions.
Researchers across the health care fields have called for an ex- To illustrate this, we use data collected from a double-blind, panded evidence base, reflected in a broadened focus and a plu- crossover RCT to examine the effectiveness of drug therapy in rality of methodologies to answer questions regarding informed reducing agitation in individuals with a traumatic brain injury practice (Concato, Shah, & Horwitz, 2000; Spring et al., 2005).
(TBI). Furthermore, we introduce a new methodology for analyz- Single-case designs seem a ready way to add methodological ing single-case data and compare it with a more traditional regres- diversity to the literature base.
sion method.
SCR Compared With Traditional Cross-Sectional Why Consider SCR Now? Although others have urged for an increased use of single-case The more commonly applied cross-sectional research designs research, such calls for the use of single-case research appear to are, in general, nomothetic approaches: They "aim to establish have had little effect in changing the behavior of researchers lawful relations that apply across individuals" (Nesselroade, 1991, (Blampied, 2000; Goldfried & Wolfe, 1996; Hilliard, 1993; Mor- p. 96). Thus, two key characteristics of cross-sectional designs are gan & Morgan, 2001). SCR continues to be an underused research "static observations and multiple behavioral categories" (Baltes & Nesselroade, 1979, pp. 11–12). In contrast, SCR designs may be Several forces, however, do appear to be making an impact. One seen as a hybrid form of the longitudinal approach. Longitudinal is the present-day focus on effect sizes. Many journals now require designs have the ability to identify not only the processes and investigators to report effect size with contextual information for causes of intraindividual change but also the processes and causes their interpretation (Fidler, 2002). A similar trend toward account- of interindividual patterns of intraindividual change in behavioral ability, objectively measured outcomes, and greater scientific rigor development. Although single-case designs may be used to explore can be seen in policy statements by influential groups such as the patterns and processes, they typically focus on evaluating the National Research Council (Shavelson & Towne, 2002). The med- impact of a treatment on a client, student, or patient. Because ical profession's accountability reform has also played a part in the attention is given to collecting data before treatment begins, after movement for the broader use of effect sizes (Oakley, 2002).
treatment starts, and sometimes even after treatment ends, each Funding agencies, public and private, are increasingly requiring research participant may serve as their own control. Thus, SCR can empirical results and effect sizes. In addition, the call for greater be viewed as an alternative methodology for answering many of accountability and objective, defensible results (Shavelson & the same research questions as cross-sectional group research and Towne, 2002) in psychological and educational research has been as a methodology that is uniquely capable of answering different an important factor leading to greater scrutiny of how SCR is and new research questions.
When Should One Use SCR Designs? Recognizing the Limitations of RCTs SCR should be considered as a top candidate research design There appears to be an increasing recognition that RCTs are to use in several circumstances. It is ideally suited for studying ideal for answering some research questions but that the design low-incidence problems and conditions. Many behavioral issues itself is not able to answer all important questions and that its that accompany conditions such as TBI and spinal cord injury implementation has certain limitations. This has led some to con- (SCI) are difficult to study in designs that rely on large, repre- tinue to call for both efficacy and effectiveness studies (Tucker & sentative samples for randomization and treatment. For exam- Roth, 2006). Important questions about how any given single RCT ple, SCR has been used to study treatments to promote wheel- is conducted and the validity of the results gained have prompted chair pushups among men with SCIs (White, Mathews, & guidelines for registering RCTs for public scrutiny (Elliott, 2007).
Fawcett, 1989) and other attempts to prevent pressure sores The intention is that this requirement will address deficiencies in (Malament, Dunn, & Davis, 1975). These are significant clini- the quality control of RCTs. However, the validity of RCTs are cal issues that often challenge and confound clinicians; how- often compromised in many applications relevant to rehabilitation ever, they are not manifested in a sufficient number of individ- psychology by a low number of available participants (with low- uals required to attract the necessary attention and financial incidence disabilities) and because true control groups are difficult support for a large-scale (or multisite) RCT.
SPECIAL ISSUE: SINGLE-CASE RESEARCH For low-incidence problems, SCR designs are probably one of Our own experience highlights the importance of using both the few designs that researchers could use to expand the knowl- visual and statistical analysis. For example, in previous studies, edge base productively in a time-efficient manner. Cross-sectional we noticed large differences between visual analysis and the designs can take a considerable amount of time to obtain a sample output from ITSACORR (Crosbie, 1993, 1995). Further inves- of sufficient size for data analysis. SCR designs are also indicated tigation showed that ITSACORR was unrelated to other statis- for studies in which few participants are able to meet the inclusion tical techniques as well as to visual analysis of single-case data criteria for a study. In addition, SCR would be beneficial in any (Brossart & Parker, 2001; Parker & Brossart, 2003), which study in which participants are required to participate over an raised serious concerns about its viability as a useful technique.
extended period of time. Such studies often experience a fair Additional empirical studies have also highlighted its weak- amount of attrition. If an SCR design was used, for the data that nesses (Huitema, 2004). It is time for single-case researchers to was complete, although possibly much smaller than the number of abandon the sole use of visual analysis; the dogged refusal to participants the study began with, this would still allow important incorporate statistical analysis of single-case data will simply research questions to be answered. Because each participant serves result in various fields or lines of research being ignored as as their own control, the existing data would still allow one to irrelevant, archaic, and unsophisticated.
make important inferences (this is not to diminish the import of Some of the underuse of statistical methods has been due to the considerations one must make when interpreting results with high cautiousness of researchers in applying univariate parametric anal- levels of attrition). Multiple scenarios are presented in Appendix A yses because of well-placed concerns that the data fail to meet as examples of when SCR should be considered.
assumptions of homogeneity of variance, normality, and serialindependence. In fact, these assumptions are commonly violated Problems With Data Analyses by short, interrupted data series. Even greater concerns have beenvoiced about the use of more complex parametric analyses, such as In spite of the fact that SCR has played an important historical role repeated measures analysis of variance (ANOVA), as it makes in psychology and that there have been a number of replicable even stronger assumptions of the data (sphericity; Stevens, 2007).
empirical findings in differing domains, Morgan and Morgan (2001) Because of these stringent assumptions, multivariate analysis of stated that SCR "remains relatively obscure because of its disavowal variance (MANOVA) has sometimes been used to replace re- of the statistical machinery that defines psychological research in the peated measures ANOVA (RM-ANOVA). However, MANOVA 21st century" (p. 120). Furthermore, those involved in SCR have still has strict assumptions (homogeneity of variance-covariance historically relied on visual analysis (Busk & Marascuilo, 1992; matrices, absence of multicollinearity and singularity) and does not Kratochwill & Brody, 1978), which Kazdin (1982) defined as the provide output as useful as RM-ANOVA's partial effect sizes.
procedure (largely informal) for reaching a judgment about reliable or For simpler parametric analyses, concerns about unequal variance consistent intervention effects by examining graphed data visually.
and nonnormal distributions are reasonably well addressed by boot- Indeed, one of the most recent review articles on single-subject strapping, a resampling technique that sidesteps data assumptions by research in rehabilitation failed to acknowledge any of the available relying on an empirical sampling distribution (Davison & Hinkley, statistical procedures for analyzing data from these designs (Back- 1997; Good, 2001; Lunneborg, 2000; Simon, 1999). The bootstrap is man, Harris, Chisholm, & Monette, 1997).
attractive and is just beginning to be applied to SCR (Parker, 2006).
There is a continued and legitimate need for visual analysis. As Violation of the assumption of serial independence can be addressed recently noted by Parker, Cryer, and Byrns (2006), visual analysis through autoregressive integrated moving average backcasting plays at least seven important roles in SCR: (Parker et al., 2006). We take a different approach in the presentarticle; however, the use of nonparametric analyses is burdened by (a) to simultaneously consider multiple data attributes in complex only the minimal assumptions of nominal-level data.
graphs; (b) to identify cycles and other patterns embedded within and Advantages of nominal-level data analysis include its applica- across phases; (c) to distinguish between improvement and deterioration bility to any SCR data set, regardless of parametric assumptions, in effect sizes, and to interpret effect size magnitudes; (d) to validate and its greater ease of use, as remedial data transformations are not whether results (with predictions lines) are meaningful, by being withinscore-scale limits; (e) to select the best statistical analysis techniques needed. The main assumption made by nominal-level data analy- from multiple options; (f) to validate the procedures and results from ses is an adequate sample size for a 2 ⫻ 2 table of about five newer SCR analytic techniques, which lack a track record of successful expected data points per cell (total N of at least 20 –25). All published applications; (g) to judge whether SCR datasets meet para- nominal-level analyses based on the 2 ⫻ 2 table can produce two metric data assumptions (p. 420).
effect sizes: (a) Phi (⌽), which is Pearson's R for a 2 ⫻ 2 table, and Nevertheless, results on the basis of visual analysis have been (b) the clinical outcome index, the "risk difference" (medical shown to have low reliability even when judges are experienced terminology), here more appropriately named "improvement rate professionals, editors of single-case journals, or others provided difference" (IRD). Given a 2 ⫻ 2 table with balanced marginal with fully contextualized graphs with other design and measure- values, these two values are equal (⌽ ⫽ IRD). Standard output for ment improvements (Brossart, Parker, Olson, & Mahadevan, 2006; both indices includes confidence intervals around the obtained DeProspero & Cohen, 1979; Harbst, Ottenbacher, & Harris, 1991; values. For more complex single-case designs, these nominal-level Ottenbacher, 1990; Park, Marascuilo, & Gaylord-Ross, 1990).
indices can be obtained through logistic regression (LR).
Neither technique—visual analysis or statistical analysis—should Other concerns with using statistical analyses on SCR data are be used in isolation: "In single-case research it seems especially related to the lack of relevance of effect sizes to the traditional important to investigate how these two methods inform and sup- standard of visual analysis (Parsonson & Baer, 1992). An R2 (or R) port each other" (Brossart et al., 2006, p. 558).
effect size derived from ordinary least squares regression and BROSSART, MEYTHALER, PARKER, MCNAMARA, AND ELLIOTT interpreted as "percent of variance accounted for" does not re- produced. In addition, because Phase A-predicted values are gen- sound with more traditional SCR practitioners. A further advan- erated for Phase B, the technique may infrequently produce values tage of nominal-level 2 ⫻ 2 table-based analyses is that they are that extend beyond the range of the dependent variable (on the y based on nonoverlapping data between phases, a keystone of visual -axis). Such values should be constrained to fit within the limits of analysis. Depending on the particular method, the approach to the y-axis variable. An additional limitation of the regression measuring nonoverlapping data varies, but in all cases, the data model promoted by Allison is that one cannot graph the output for overlap can be confirmed visually.
visual analysis. The semipartialing performed by this methodchanges the data so much that visual analysis is difficult. Althoughtrend is removed, graphing the final output does not lend itself Comparison Method: Simple Mean Shift toward a straightforward interpretation. In an effort to improve the Allison technique, Parker et al. (2006) renamed the techniquemean and slope adjustment (MASAJ) and modified it so that it was Regression models have been used by single-case researchers visually interpretable and the question it addressed was slightly since at least 1983 (Gorsuch, 1983). Since that time, many differ- adjusted. The MASAJ technique now answers the question, "What ent models for analyzing single-case data have been proposed if phase A trend influence were eliminated or controlled in phase (e.g., Allison & Gorman, 1993; Center, Skiba, & Casey, 1985– B?" (Parker et al., 2006, p. 426). In contrast, the Allison technique 1986; Faith, Allison, & Gorman, 1996). One of the advantages of answers a similar but different question: "What phase differences regression models is that they are familiar to many because they would have been obtained if there had been no phase A trend in the are often covered in doctoral training programs in the behavioral entire dataset?" (p. 426).
sciences. They also produce a common effect size, R2, which can We used a regression model that looks at an SMS between the be converted to other effect sizes such as Cohen's d (Rosenthal, baseline and treatment phase for the present study to provide a 1991). Results from individual studies may also be summarized in comparison to the LR technique. Although it is one of the simplest meta-analytic studies. Additional advantages include the relative models and does not control for baseline trend, we felt it was ease of evaluating power and creating confidence intervals around important to provide a familiar comparison technique because it is the effect size. It is also fairly easy to bootstrap regression models, very different from the LR technique in terms of conceptual especially those models that entail a single step (as opposed to framework and output. This technique was also chosen because a those that involve multiple steps; e.g., Allison & Gorman, 1993).
few data sets contained the treatment drug in the first phase with Every statistical method has limitations, and one disadvantage the "baseline" or placebo phase following. We deemed it inappro- of the regression models is that the effect size, R2, is not easily priate to use a regression method that controlled for baseline trend interpreted in terms of treatment effectiveness. Another disadvan- when the treatment phase came prior to the "baseline" phase.
tage is that there are numerous regression models a single-caseresearcher may choose from. Some models try to control for trend in various ways, some across the entire data series similar to acovariate in analysis of covariance (e.g., Gorsuch, 1983), others In cases in which the investigator chooses to use a regression attempt to control for trend in the baseline phase only (Allison & technique, it is important to be aware that autocorrelation has been Gorman, 1993; Faith et al., 1996). The choice of model depends on an enduring problem. Data sets with levels of autocorrelation ⱖ ⫾ the question the investigator wants to answer. Furthermore, the .20 may be considered problematic regardless of statistical signif- effect sizes produced by these regression methods are not directly icance (Matyas & Greenwood, 1996). The presence of autocorre- comparable to those found in typical cross-sectional regression lation violates the assumption of data independence. To remove studies in terms of the characteristic range and magnitude seen in autocorrelation, one may use an autoregressive integrated moving SCR. Thus, the interpretive guidelines found in texts by Cohen average model with a lag-1 parameter for backcasting rather than (Cohen, 1988), for instance, are of little help in SCR. Investigators forecasting, as is typically done. The traditional cautions against have made some progress in trying to provide tentative interpretive using time series analysis for this application do not apply (see guidance, but guidelines per se are not available yet (see Brossart Parker et al., 2006).
et al., 2006; Parker & Brossart, 2003; Parker et al., 2005). Thus,the effect size coefficient does not directly communicate the de-gree of intervention effectiveness.
Addressing Threats to Validity Among the regression methods available, the one discussed by Allison and colleagues appears to be one of the more conceptually Among the strongest (in internal validity) and most flexible and empirically sound options (Allison & Gorman, 1994; Brossart SCR designs is the multiple baseline design (MBD) across et al., 2006; Parker & Brossart, 2003). This method involves subjects (Kazdin, 1982). The MBD permits an overall judgment multiple steps and effectively controls baseline trend, but it is not of intervention effectiveness from multiple (typically 3 or 4) without limitations. Because it controls for baseline trend, the data data series. Each data series represents one client. The most simple series needs to have enough data points to assess trend accurately.
data series is AB, that is, a baseline phase followed by an inter- Although one may draw a trend line through three data points, any vention phase. The strength of the MBD is in implementing the baseline based on only three data points should only be analyzed intervention at different times for the clients, thus reducing the by a regression method in which mean shift is examined, and even likelihood that the performance change is due to some event other then such analysis should be considered tentative. More data in than the intervention. Increasing the number of clients, each with each phase serves to increase the accuracy of any trend line staggered intervention onset, improves the control of "history" as SPECIAL ISSUE: SINGLE-CASE RESEARCH an alternative explanation for behavior change (Kazdin, 1982). For with a placebo control among patients who were more than 1 year history to be present, the external event would need to impact the postbrain injury (BI).
participants concurrently. Any history effect should be seen across Agitated behavior after BI can be very disruptive during acute all individuals at approximately the same time. Without such medical care, inpatient rehabilitation, and in the community. Pre- evidence, the threat of history can usually be ruled out. Maturation vious studies have reported agitated behavior in 11%–34% of is only a problem in special circumstances in which the length of patients with BI in the acute phase (Brooke, Questad, Patterson, & the study and the variables measured may, in fact, reflect devel- Bashak, 1992; Levin & Grossman, 1978; Reyes, Bhattacharyya, & opmental changes in the participants.
Heller, 1981). Although prevalence rates of agitation in the post- With MBD, each data series and client are viewed as an inde- acute phase are lacking, many patients seen in long-term follow-up pendent replication, contributing evidence to the omnibus judg- after severe BI demonstrate significant behavioral dyscontrol and ment. That judgment is easy to make when improvement is uni- agitation. Such sequelae have a devastating impact on family formly strong across clients, but when results vary, the overall relationships and overall functioning, considerably hampering judgment of intervention effectiveness is more difficult to make.
community reintegration of persons with BI.
That problem situation can be handled by calculating effect sizes.
Agitation is generally regarded as a disturbed behavioral pattern often accompanied by overactivity and an "explosive" (i.e., lack-ing goal direction), impulsive aggression among persons with BI Statistical Methods Have Improved who have regained cognitive awareness (Corrigan & Mysiw, 1988;Silver & Yudofsky, 1994). Historically, clinicians have relied on Recent innovations in SCR include the ability to calculate pharmacological treatments of agitated behavior (Cardenas & effects sizes, in most cases with confidence intervals (Parker et al., McLean, 1992; Rowland & DePalma, 1995). A recent Cochrane 2005; Parker & Hagan-Burke, 2007b), the use of phase contrasts review of these agents observed that beta-blockers (particularly (Parker & Brossart, 2006), controlling autoregression, controlling propanolol) appear to have the best evidence of effectiveness preexisting baseline trend (Parker et al., 2006), and the use of the (Fleminger, Greenwood, & Oliver, 2006). In spite of such reviews bootstrap (Parker, 2006). In the past 20 years, the number of supporting the use of beta-blockers, a recent survey indicates that analytic techniques available for short data series has easily tripled specialists seem to prefer anti-epileptics and atypical antipsychot- since the early 1980s (Barlow & Hersen, 1984; Kazdin, 1982). The ics (Francisco, Walker, Zasler, & Bouffard, 2007). The mechanism difficulty has been that few studies compared the statistical tech- of action for the anti-aggressive properties of propanolol is essen- niques with each other and with visual analysis. Thus, those who tially unknown, although it is unlikely to be due to propranolol's wanted to use these statistical techniques had little information in peripheral beta-blocking activity because the doses required to terms of how to interpret the output. Increasingly, researchers have manage agitated behavior often well exceed the doses required to recognized this deficiency in the literature base and have made saturate fully peripheral beta-adrenergic receptors (Coltart & some progress in addressing this need (e.g., Brossart et al., 2006; Shand, 1970; Yudofsky, Williams, & Gorman, 1981). Propranolol Parker & Brossart, 2003). Presently, it appears that effect sizes may likely exert its anti-aggressive properties via central antago- vary, depending on the statistical technique used to produce them, nism of noradrenergic and serotonergic neurotransmission at sev- and that the effect size magnitudes produced from cross-sectional eral subsets of receptors.
research are very different than those produced from SCR (e.g., For example, both the noradrenergic and serotonergic sys- Parker et al., 2005).
tems have been implicated as neurophysiologic substrates ofaggressive behavior in animal studies, though these systems probably subserve different types of aggressive behavior andseem to interact in a complex fashion (Cassidy, 1990; Eichel- To summarize, SCR designs should be used because they are man, 1987; Miczek, Weerts, Haney, & Tidey, 1994). The loca- ideally suited to address questions unanswerable by cross-sectional tions of noradrenergic and serotonergic cell bodies (the locus designs, they address the overuse of cross-sectional designs in the ceruleus and dorsal raphe nuclei, respectively), as well as their literature base, and it is no longer the case that there are few neuronal (white matter) projections, are particularly vulnerable statistical methods to analyze single-case data. In addition, the to injury within the brain as a result of acceleration/deceleration MBD is a powerful design that competes well against other de- injuries, the most common mechanism of BI (Morrison, Millier, signs in terms of internal validity. In the remainder of the present & Grzanna, 1979; Whyte & Rosenthal, 1993). Because propran- article, we present a small RCT that can be conceptualized as a olol has effects on both beta-adrenergic receptors as well as hybrid multiple baseline study. We then analyze these data using serotonin 5-HT1A and 5-HT1B receptors, its apparent effec- a statistical technique burdened by few assumptions, which is well tiveness in managing agitation may be related to modulation of suited for SCR.
neurotransmission in these damaged pathways.
However, the Cochrane review noted several problems that undermine our confidence in the evidence base that merit a closer Illustrative Study scrutiny of propanolol as preferred intervention for agitation. Thereviewers found very few RCTs to evaluate (only six were iden- To illustrate the usefulness of SCR and advanced regression tified, generally, in the pharmacological literature), a reliance on methods for analyzing data from these designs, we examined data small sample sizes and lack of a systematic reporting of all treated collected from a funded project (awarded to Jay Meythaler) to participants, and no replication studies and a lack of a global conduct a randomized, double-blind, crossover trial of propanolol outcome measure to assess the complexity of agitated behavior in BROSSART, MEYTHALER, PARKER, MCNAMARA, AND ELLIOTT this population (Fleminger et al., 2006). Although the reviewers ethnicity consisted of 12 Caucasians and 1 African American. The cited the need for further RCTs of the effectiveness of pharmaco- average age of the participants was 34 (SD ⫽ 9.78).
logical agents, researchers and clinicians were strongly advised torevisit the use of "N of 1 research methods" to analyze the effectiveness of the intervention in research projects and in clinicalcase management (Fleminger et al., 2006).
The ABS (Corrigan, 1989) was used to assess agitation. The As we observed earlier, these clinical realities and methodolog- ABS is a 14-item scale designed to assess agitation objectively ical issues often vex intervention research in rehabilitation. And as among persons with TBI. At the end of each observation period, we demonstrate, SCR designs and advanced regression techniques raters assign a number ranging from 1 (absent ) to 4 (present to an can be used efficiently to examine the effectiveness of clinical extreme degree ) for each item, representing the frequency of the interventions for grouped data (necessary for RCTs) and for clin- agitated behavior and/or the severity of a given incident. Total ical case management (to monitor individual response to treat- scores range from 14 (no agitation) to 56 (extremely severe agi- ment). In the remainder of this article, we demonstrate the use and tation). In previous studies, the ABS has demonstrated adequate implications of SCR and regression techniques in a randomized, reliability and validity (Corrigan, 1989). Factor analysis of the double-blind crossover trial of propanolol in the treatment of ABS yielded a three-factor solution: Aggression, Disinhibition, agitation among persons with postacute BI.
and Lability (Corrigan & Bogner, 1994).
The initial ABS was completed by a family member in an interview conducted by Timothy R. Elliott. This was used to determine sufficient level of agitation to qualify for the study. Atthe introductory evaluation prior to randomization, family mem- Twenty individuals with BI who were sequentially enrolled in bers met with Timothy R. Elliott to learn how they were to assess an outpatient brain injury clinic were invited to participate in the agitation each week of participation with the ABS. During this present study. Each potential participant and his or her family session, family members were instructed in the use of the ABS. An members were given a thorough explanation of the study together instructional videotape (depicting various agitated behaviors) was with a detailed informed consent document. Every effort was made played for the family members to rate the depictions of agitation to explain the purpose of the study and the risks and benefits of on the ABS. These ratings were reviewed and critiqued by the staff participation to the potential participant, and to obtain assent or member. Family members were given copies of the ABS and refusal. For individuals unable to provide informed consent, deci- instructed to rate the participant's agitation each week. Completed sions regarding participation fell to family members or the per- scales were mailed to the research team or returned in subsequent son's designated surrogate decision maker.
The inclusion criteria were as follows: (a) BI due to closed or penetrating head trauma and/or hypoxia greater than 1 year prior toentry into the study; (b) 14 years of age or older; and (c) a clinically significant level of agitated behavior, defined as that The study was designed to be a randomized, double-blinded, which interferes with activities of daily living or independent crossover trial. Upon enrollment in the study, each participant had living skills. In order to more carefully operationalize the level of a 2-week observation period during which placebo was adminis- agitated behavior necessary for inclusion, this study relied on the tered in a single-blind fashion. ABS observations began during this behavioral ratings by family members on the Agitated Behavior period. Pharmacy personnel used a double-blind randomization Scale (ABS; Corrigan, 1989) obtained by the staff member. Pro- procedure to assign participants to receive either the active agent spective individuals qualified for entry into the study if they obtain (propranolol) or placebo for the first arm of the study. The study at least two scores on the ABS (described in the Measures section) drug (propranolol or placebo) was prepared by the pharmacy and of 25 or greater in a 2-week period.
delivered to the clinic. A 2-week supply of study drug contained in The exclusion criteria were as follows: (a1) medical contrain- a blister pack and labeled with the dosage increment was provided dications to initiation of a beta-adrenergic blocker, including a at each clinic visit.
recent history of congestive heart failure, cardiac arrhythmia, atrio- Participants had pulse and blood pressure checked at each clinic ventricular conduction defect (2nd degree or higher), or asthma visit. Dose of the study drug was adjusted to a tolerated dosage requiring pharmacologic intervention; (b) clear medical indica- increment for supine blood pressures less than 55 diastolic or 95 tions for prescription of a beta-adrenergic blocker for reasons other systolic in patients under 50 years of age; less than 70 diastolic or than agitation; (c) demonstrated inability to tolerate propranolol 110 systolic in patients 50 years of age and over. Eight participants due to hypotension or bradycardia; (d) suspected development of were started at an initial dose of 60 mg of long-acting propranolol increased intracranial pressure requiring neurosurgical interven- (Inderal-LA) per day; 2 participants were started at an initial dose tion (e.g., placement or revision of ventricular-peritoneal shunt).
of 80 mg of propranolol (Participants 4 and 7). Dosages wereincreased for participants who demonstrated tolerance for thepreceding dosage. From this protocol, 1 participant received a maximum dosage of 180 mg (Participant 1), 1 received a maxi- The sample available for study consisted of 13 persons with TBI mum dosage of 120 mg (Participant 8), 6 participants received a (4 women, 9 men). Participants who had only two data points in maximum dosage of 80 mg (Participants 3, 4, 5, 6, 7, 10), and 2 either the baseline or treatment phase were excluded. The final were maintained at a dosage of 60 mg (Participants 2 and 9).
sample that was analyzed consisted of 10 participants. Sample Ratings of agitation for each individual were conducted weekly SPECIAL ISSUE: SINGLE-CASE RESEARCH from 6 to 14 weeks (average 10 weeks). Of the 10 clients, 7 were ANOVA or OLS regression. The dependent variable, PhaseAB, is assessed over 10 or more weeks. The design for each of 9 clients dichotomous (Levels A, B). A one-way (noninteraction) model is was a simple AB (baseline period of no treatment, followed by a specified. The output needed for the present study is only the 2 ⫻ 2 treatment period). For 1 participant, the treatment preceded the prediction accuracy table, which is ordinarily used for prediction baseline period, forming a BA design. Baseline phases ranged specificity and sensitivity (involving false negatives and false from 3 to 8 data points (mean 5.3 data points), and treatment positives). Through LR, an attempt is made to predict the phase to phases had the same range (mean 5.1 data points).
which a score belongs (baseline vs. treatment), based on its size.
The prediction is made on the basis of all participants' data, but the Data Analysis classification results also can be disaggregated by individualparticipant.
For many research designs, logistic regression (LR) is a close contender to ANOVA in power and sensitivity, while being bur- dened with fewer data assumptions (Fox, 2000; Menard, 2002;Pampel, 2000; Tabachnick & Fidell, 1996). LR is similar to Analysis of the propranolol data set results in a classification ordinary least squares (OLS) multiple regression but uses iterative table presented in Table 1. Table 1 indicates that the classification maximum likelihood estimation (MLE) rather than OLS. Like accuracy for these data is only about chance level, 50%. Any given multiple regression, LR can use any combination of categorical or data point has an equal chance of belonging to the baseline versus continuous predictors, but the dependent variable must be categor- treatment condition. These results represent an unsuccessful inter- ical. LR performs similarly to discriminant function analysis vention. From a total of 104 data points, only 57% were classified (DFA), but it is increasingly favored over the latter because of its correctly for phase membership, which is close to chance level.
fewer data assumptions (Press & Wilson, 1978). Unlike OLS Submitting this table (only the interior four scores) to a chi-square regression, LR does not assume (a) a linear relationship between analysis yields, ␹2 ⫽ 1.9. Phi is output directly as .135 or can be the independent variables and the dependent variable, (b) normally calculated as, ⌽ ⫽ 冑␹2/N ⫽ 冑1.9/104 ⫽ .135. Phi can be inter- distributed variables, or (c) equal variance per cell. LR is offered preted approximately as "prediction accuracy beyond chance." by most statistics packages, including NCSS (Hintze, 2007), SPSS, From the 2 ⫻ 2 table, we also can calculate phi from the Stata, S-Plus, SYSTAT, and SAS.
difference between two ratios: d/f ⫺ b/e ⫽ 30/51 ⫺ 24/53 ⫽ Although LR is burdened by few data assumptions, ideally it .5882 ⫺ .4528 ⫽ .135. A two proportions statistical module needs at least 10 observations for each predictor variable level provides a 90% exact confidence interval as: ⫺.03 ⬍ .135 ⬍ .29, (e.g., the smaller Phase A or B; Peduzzi, Concato, Kemper, Hol- and because it spans zero, we note that it could have been obtained fold, & Feinstein, 1996). In addition, all predictor cells should by chance alone. On the basis of all 10 participants, this phi effect have frequencies of at least 1, and no more than 20% of cells size of approximately .14 indicates the magnitude of change from should have less than 5 per cell.
baseline to intervention phases for this particular intervention.
LR does not yield a true effect size but rather one or more Guidelines for interpreting phi magnitudes were recently derived quasi-R2 approximations (e.g., Cox & Snell, 1989; Nagelkerke, from 165 analyses of published SCR data (Parker & Hagan-Burke, 1991). These quasi effect sizes must be interpreted cautiously (e.g., 2007a). LR results correlated .83 with visual judgments, and they do not represent "percent of variance explained" as do true studies judged to show small or negligible results had effect sizes R2s). A second LR output, and the one most important to this (interquartile range [IQR]) of .09 –.43. Studies judged as showing article, is a summary of LR prediction accuracy in a 2 ⫻ 2 table.
medium-size results had effect sizes (IQR) of .53–.72. And studies LR predicts membership of each data point in either baseline or judged as showing large results had effect sizes (IQR) of .82–1.0.
intervention phase, based on its relative magnitude. Chance level is This effect size does not indicate whether this change (or lack of 50% accuracy. The 2 ⫻ 2 agreement table, when analyzed by change) can be attributed to the intervention. Attributing change to using chi-square, yields the Pearson's phi index of association, a the intervention depends on strength of the design. The design of bona fide effect size. Pearson's ⌽ and ⌽2 are R2 family members, this example is a multiple-baseline design with 10 independent and familiar to many researchers (Cohen, 1988). Phi also can be client AB data series, and with treatment initiated at 10 different calculated from chi-square: ⌽ ⫽ 冑␹2/N (where N is the total times. Most single-case researchers would consider this a strong number of frequency counts in the 2 ⫻ 2 table).
design. Thus, our hypothesis that participants with agitation would In a balanced 2 ⫻ 2 table (from LR), phi also can be obtained have a significant reduction in ABS scores on propranolol as by submitting four internal scores to analysis in a "two propor- compared with placebo was not supported.
tions" statistical module. Phi approximates the difference betweenthe two proportions and is exactly the same in a balanced table. Anadvantage of using a "two proportions" module for analysis is that it commonly outputs confidence intervals.
Classification Accuracy Table In a single-case design, LR analysis requires two predictor variables, participant and scores, and the dependent measure, PhaseAB. Though not essential, a fourth serial sequence variable,time, should be added. Participant is a categorical predictor vari- able whose number of levels equals the number of clients (data series) (Levels I, II, III, etc.). Scores serve as a predictor ratherthan as a dependent or criterion variable, as is the case with Percent correctly classified ⫽ 56.7%.
BROSSART, MEYTHALER, PARKER, MCNAMARA, AND ELLIOTT Analyses by Participant tistical output needs to include visual analysis. These results showthat large effect sizes do not inform one as to the directionality of Besides obtaining an index of overall intervention effect, diag- the results. This study produced some high-correct classification nostic understanding can be gained from effect sizes for individual rates; however, half those participants did better on propranolol, participants. This is accomplished in LR by dropping the partici- and the other half did worse on propranolol.
pant predictor variable and entering the data only one participant ata time. Table 2 includes the 10 effect sizes for the individual participants, which is compoised of roughly two groups: a largergroup of "little or no effect" (⌽ ⫽ .04, .00, .00, .07, .33, .00) and Our main objective in this article was to present a discussion of a smaller group of "moderate to strong effect" (⌽ ⫽ .52, 1.0, .63, advanced regression methods for the analysis of data produced by .87). We include these effect sizes and confidence intervals for the SCR. We presented two very different methods, LR and OLS individual participants because clinicians involved in monitoring regression. This is the first attempt that we are aware of in which patient progress would focus on each unique client's progress, investigators have used LR to analyze SCR. The application to a whereas researchers would probably want to distill the results RCT with multiple-baseline data from a drug study of the effec- across multiple baselines in order to determine whether the treat- tiveness of propranolol to treat agitation among individuals with ment was effective. Generally, these results indicate that propran- BI was ideal for this demonstration because of the high internal olol was not effective in lowering agitation for the majority of validity and the multiple data sets available for analysis. Factors participants. The level of analysis one uses depends on the ques- suggesting a high degree of internal validity include multiple- tion one needs answered.
baseline design, double-blind features, and random assignment tothe ordering of treatment condition. Thus, although the overall Comparison to Regression sample size was small, the degree of experimental control for thepresent study appears to be rather high.
The results from the SMS regression model conducted on each Our analysis of the multiple-baseline data suggests that overall participant are included in Table 2. In general, there were three propranolol was not an effective treatment for agitation. The effect groups of participants. The largest group contained those partici- size based on these data was .14. This is a small and nonsignificant pants that demonstrated no effect while taking propranolol. These effect size and could have been obtained by chance alone. Yet, participants obtained R2 values of .02, .07, .04, .05, .22, and .02 when we analyzed participants separately, we found that there and classification rates of 54.5%, 50%, 50%, 54.5%, 66.7%, and were interesting differences among the participants. Six individu- 62.5%, respectively. A graphic depiction of the lack of effects als experienced little or no effect on propanolol. Four others observed for one participant in this group is presented in Figure 1.
evidenced a moderate to strong effect in response to propranolol: Two participants exhibited significantly elevated agitation dur- 2 of these participants improved, and the other 2 did worse. The ing the propranolol phase (a 33-year-old Caucasian woman and a individual variations in treatment response, which any analysis of 37-year-old Caucasian man). We obtained R2 values of .23 and .70 overall group performance cannot address, suggest that agitation with classification rates of 80% and 83.3%. The small R2 value for may be influenced by several factors that have yet to be isolated or the first participant seems to reflect the nonstatistically significant understood. The results of our demonstration, then, have implica- phi. Figure 2 depicts the ratings obtained for the 37-year-old man tions for clinical case management and for isolating other variables who exhibited significantly greater agitation on propranolol com- in future studies of propanolol in the treatment of agitation.
pared with placebo.
Clinical scientists are typically interested in their patient's re- Two other participants displayed significantly less agitation on sponse to treatment. The analysis of each participant's data sepa- propranolol than on placebo (a 51-year-old African American man rately is in line with the clinician's interest in patient progress. We and a 23-yearold Caucasian man). We obtained R2 values of .70 can see in these profiles that any particular client's response may and .73 and classification rates of 100% and 92.9%. Figure 3 vary markedly from the overall analysis (which suggested no depicts the significant improvement exhibited by the 51-year-old effect for the group). As seen in these results, a few participants man during the propanolol phase.
had notable results with propranolol. In the absence of contrain- There were 2 participants who obtained results with R2 values dications and troublesome side effects, some clinicians may of .23 and .22. Their classification rates and phi values were choose to prescribe propranolol for agitation because it was effec- 80%, .52 (p ⫽ .10) and 66.7%, .33 (p ⫽ .41), respectively. The tive for some clients. Such a choice would seem to require ade- case with the 80% classification rate is rather high, but the phi quate monitoring to determine whether continued administration value and examination of the confidence interval for the boot- was beneficial, worthwhile, and cost-effective. These observations strapped R2 value, which contains zero, suggest that such a high are consistent with other expert opinions concerning the use of classification rate should be interpreted with caution. Interpretabil- propanolol in the treatment of agitation (Fleminger et al., 2006).
ity would likely be increased if this case had one or two more data There are limitations with the techniques we have demon- points in the baseline phase, beyond the minimum of three. The strated in this study. One does not evaluate trends or curves other R2 value of .22 was not associated with a high classification with LR. In some cases, trend lines or curves may be of primary rate. This data series also obtained a non-significant phi, and the interest. In such cases, LR would not be an ideal analytic tool.
confidence interval from the bootstrapped R2 also contained zero.
LR also has a ceiling limitation. If a treatment obtains a 100% One can more confidently conclude that there is no treatment effect correct classification rate (a ⌽ of 1), then there is no way in which in these cases.
to evaluate any magnitude of difference with the technique beyond It is important to emphasize that the interpretation of any sta- the minimum required to obtain the 100% classification rate.
SPECIAL ISSUE: SINGLE-CASE RESEARCH Table 2Classification Tables for Each Individual Participant % correctly classified ⫽ 80.0 % correctly classified ⫽ 100.0 ␹2 ⫽ 2.74, ⌽ ⫽ .52, p ⫽ .097 ␹2 ⫽ 12, ⌽ ⫽ 1.0, p ⬍ .001 90% exact C.I. around difference between 2 proportions: ⫺.04 ⬍ .52 ⬍ .86 90% exact C.I. around difference between 2 proportions: .56 ⬍ 1.00 ⬍ 1.00 SMS R2 ⫽ .23, bootstrapped Mean SMS R2 ⫽ .28, 90% C.I. ⫽ 0, .45 SMS R2 ⫽ .70, bootstrapped Mean SMS R2 ⫽ .70, 90% C.I. ⫽ .48, .88 % correctly classified ⫽ 54.5 % correctly classified ⫽ 83.3 ␹2 ⫽ .02, ⌽ ⫽ .04, p ⫽ .89 ␹2 ⫽ 4.68, ⌽ ⫽ .625, p ⫽ .03 90% exact C.I. around difference between 2 proportions: ⫺.38 ⬍ .03 ⬍ .46 90% exact C.I. around difference between 2 proportions: .09 ⬍ .63 ⬍ .88 SMS R2 ⫽ .02, bootstrapped Mean SMS R2 ⫽ .11, 90% C.I. ⫽ 0, .39 SMS R2 ⫽ .70, bootstrapped Mean SMS R2 ⫽ .70, 90% C.I. ⫽ .45, .89 % correctly classified ⫽ 92.9 % correctly classified ⫽ 50.0 ␹2 ⫽ 10.5, ⌽ ⫽ .87, p ⫽ .001 ␹2 ⫽ 0, ⌽ ⫽ 0, p ⫽ 1.0 90% exact C.I. around difference between 2 proportions: .44 ⬍ .88 ⬍ .99 90% exact C.I. around difference between 2 proportions: ⫺.40 ⬍ .00 ⬍ .40 SMS R2 ⫽ .73, bootstrapped Mean SMS R2 ⫽ .75, 90% C.I. ⫽ 47, .93 SMS R2 ⫽ .07, bootstrapped Mean SMS R2 ⫽ .26, 90% C.I. ⫽ 0, .15 % correctly classified ⫽ 50.0 % correctly classified ⫽ 54.5 ␹2 ⫽ 0, ⌽ ⫽ 0, p ⫽ 1.0 ␹2 ⫽ .05, ⌽ ⫽ .07, p ⫽ .82 90% exact C.I. around difference between 2 proportions: ⫺.40 ⬍ .00 ⬍ .40 90% exact C.I. around difference between 2 proportions: ⫺.40 ⬍ .07 ⬍ .53 SMS R2 ⫽ .04, bootstrapped Mean SMS R2 ⫽ .17, 90% C.I. ⫽ 0, .56 SMS R2 ⫽ .05, bootstrapped Mean SMS R2 ⫽ .14, 90% C.I. ⫽ 0, .47 % correctly classified ⫽ 66.7 % correctly classified ⫽ 62.5 ␹2 ⫽ .67, ⌽ ⫽ .33, p ⫽ .41 ␹2 ⫽ .00, ⌽ ⫽ .00, p ⫽ 1.0 90% exact C.I. around difference between 2 proportions: ⫺.33 ⬍ .33 ⬍ .81 90% exact C.I. around difference between 2 proportions: ⫺.40 ⬍ .00 ⬍ .54 SMS R2 ⫽ .22, bootstrapped Mean SMS R2 ⫽ .32, 90% C.I. ⫽ 0, .81 SMS R2 ⫽ .02, bootstrapped Mean SMS R2 ⫽ .16, 90% C.I. ⫽ 0, .49 Prt ⫽ Participant; C.I. ⫽ confidence interval; SMS ⫽ simple mean shift.
Furthermore, additional work remains to determine how this LR tant to note that the ratings we obtained in this study were not procedure performs with a wide variety of single-case data sets.
complicated by patient self-report. The participants were rated by Although we have focused on the statistical results, it is impor- their family member. Thus, for any participant who improved on BROSSART, MEYTHALER, PARKER, MCNAMARA, AND ELLIOTT Example data set of noneffective treatment of agitation with propranolol. ABS ⫽ Agitated Behavior Scale. Solid circles represent datacollected in the baseline phase; solid triangles represent data from the Example data set of participant improvement while on pro- treatment phase.
pranolol for the treatment of agitation. ABS ⫽ Agitated Behavior Scale.
Solid circles represent data collected in the baseline phase; solid trianglesrepresent data from the treatment phase.
propranolol, it may not be necessary for statistical significance tobe achieved. Improved quality of life for the family may be a moreimportant consideration in some clinical scenarios.
Beyond the findings of this particular study, it should be noted key elements in place, any reluctance to publish such a study that with an appropriate measure of outcome and the implemen- probably reflects editorial bias more than a scholarly critique.
tation of a multiple-baseline design, we presented in this article a In many respects, the present controversies and needs in our statistical procedure that should be appreciated by peer reviewers research make for an exciting time for single-case researchers.
and peer-reviewed outlets. There is no longer an acceptable ratio- New statistical methods continue to be developed and refined. No nale for conducting SCR without statistical analysis. Single-case longer must the single-case researcher rely solely on visual anal- studies that feature a strong rationale, a multiple-baseline design, ysis: Regression methods such as those presented here provide two and appropriate statistical analyses deserve a place in the eviden- powerful yet very different methods for analyzing single-case data.
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SPECIAL ISSUE: SINGLE-CASE RESEARCH Scenarios In Which Single-Case Research Designs are Useful Sample or Client Characteristics 1. When participants are few and/or uniquely different, so pooling together may obscure important differences.
2. When clients are atypical, so are not well represented in normative samples of standardized assessments.
3. When clients have limited response repertoires or low attention abilities, so standardized assessment procedures are of questionable validity.
Clinical or Research Issue 1. When one extensive assessment may have doubtful validity, and periodic, shorter probes would be more credible.
2. When participant performance shows considerable variability over time, from day-to-day or week-to-week.
3. When short-term and medium-term client improvements are of interest and expected.
4. When the concern is about the process of learning or development, styles, etc., rather than outcomes alone.
5. When formative evaluation data are needed to inform further development of an intervention or program.
6. When the amount, intensity, or type of intervention can be varied to an optimum level to better meet individual client needs.
7. When participants are likely to respond to an intervention at different rates, or with different trajectories, curves, profiles, etc.
8. When the focus is on typical daily performance rather than on capacity or aptitude, e.g., habits, addictions, tolerances, social interactions, 9. When the "ecological validity" of measurement is very important.
10. When events or behaviors over the recent past are important, and yet their recall retrospectively would be inaccurate.
11. When the interest is in the relationship between two behavioral measures over time (cross-correlation).
12. When the interest is in the sequence of behavioral measures over time (lag-sequential conditional probability analysis).
13. When the interest is in contingent relationships between events and behaviors over time (ordinal contingency analysis).
14. When atypical results (e.g., amount of improvement or qualitative pattern of improvement) from individual subjects could be washed out in group dence Intervals box. The output will list the Phi under the columntitled Estimated Value, and the confidence intervals will be listed Within NCSS 2007, select Analysis, Regression/Correlation, next to it. Using the Confidence Intervals tab, one may set the Logistic Regression. Enter phase variable as dependent variable range of the confidence intervals the program produces. One may and the variable for treatment score as Numeric Independent also use the Chi-square Effect Size Estimator found under Anal- Variable (assuming it is continuous). Under the Response Analysis ysis, Descriptive Statistics, Contingency Tables. When the classi- Section in the output, you will find the % Correctly Classified, look fication table is entered in the cell boxes, the program produces the in the row titled Total, for the overall % of correctly classified data chi-square, effect size (Phi), and the probability level. Following points. In the output under the section titled Classification Table these directions should give one all the necessary output to report are the values you will enter into the Proportions –Two Indepen- one's results.
dent analysis, which is listed under Analysis, Proportions. Makesure you enter the values from the classification table correctly into Received December 31, 2007 the cells for the proportions test. Select Difference in the Statistics Revision received June 3, 2008 box and Exact (although a Bootstrap is available) in the Confi- Accepted June 5, 2008 䡲

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