Combining Nonoverlap and Trend for Single-Case …

1 Author's personal copyCombining Nonoverlap and Trend for Single-Case Research: Tau-URichard I. ParkerKimberly J. VannestJohn L. DavisStephanie B. SauberTexas A&M University at College StationA new index for analysis of Single-Case research data wasproposed, Tau-U, which combines Nonoverlap betweenphases with Trend from within the intervention phase. Inaddition, it provides the option of controlling undesirablePhase A Trend . The derivation of Tau-U from Kendall'sRank Correlation and the Mann-Whitney U test betweengroups is demonstrated. The equivalence of Trend andnonoverlap is also shown, with supportive citations fromfield leaders. Tau-U calculations are demonstrated forsimple AB and ABA designs. Tau-U is then field tested ona sample of 382 published data series. Controllingundesirable Phase A Trend caused only a modest changefrom Nonoverlap .

2 The inclusion of Phase B Trend yieldedmore modest results than simple Nonoverlap . The Tau-Uscore distribution did not show the artificial ceiling shownby all other Nonoverlap techniques. It performed reasonablywell with autocorrelated data. Tau-U shows promise forsingle-case applications, but further study is models versus regressionmodelsSingle-case research (SCR) has received renewedinterest in the behavioral sciences for its focus onchange within an individual rather than change inthe group aggregate (Borckardt et al., 2008).Statistical analysis for evaluating change in SCRdesigns are still in an early stage of least squares regression analysis (OLS)with a long history of use in largeNstudies, hasshown unequalled flexibilityand power whenapplied to SCR data (Allison & Gorman, 1993;Busk & Serlin, 1992; Parker & Brossart, 2003).

3 However, OLS has been criticized for failing toaddress the unique constraints of short time seriesdata that are typical in SCR (Parsonson & Baer,1992; Scruggs & Mastropieri, 1994). OLS is aparametric statistical test, and as such requires anormal score distribution, constant variance, andinterval level measurement. Applying OLS to SCRdata has been criticized because these data often donot meet OLS assumptions of constant variance,normality, and linearity of relationship, and thescaling assumption of at least an interval-level scale(Cohen & Cohen, 1983; Kutner, Nachtsheim &Neter, 2004). These problems notwithstanding,only OLS analysis has to date been able todemonstrate (a) control of undesirable positivebaseline Trend ; (b) sensitivity to improvement inlevel change trends ; (c) adequate power for shortdata series; and (d) the ability to discriminate wellamong published data sets, avoiding ceiling or flooreffects.

4 All Nonoverlap indices suffer from a ceilingeffect of 100%; they are insensitive to amount ofseparation of data contrasted between two phasesbeyond the point where there is no least four regression models have beendesigned to do those four things, which aresummarized in texts byFranklin, Allison, andGorman (1997), and Kratochwill and Levin(1992). They are (a)Crosbie's ITSACORR model(1993, 1995); (b) the Last Treatment Day predictiontechnique ofWhite, Rusch, Kazdin, and Hartmann(1989);(c)Center, Skiba, and Casey's (1985 1986)Available online at Therapy 42 (2011) 284 correspondence to Richard I. Parker, , Texas A&MUniversity, 604 Harrington Office Building, Mail Stop 4225, CollegeStation, TX 77843; 299/$ 2011 Association for Behavioral and Cognitive Therapies. Published byElsevier Ltd. All rights 's personal copymean-shift and mean-plus- Trend family of models;and (d) Allison et al.

5 's mean-shift and mean-plus- Trend models (Allison & Gorman, 1993; Faith,Allison, & Gorman, 1997).Crosbie's ITSACORR (1993, 1995) was posi-tively cited by several researchers for a decade, butused infrequently, and has suffered two majorsetbacks. First, the experience of several researchers,including ourselves, was that its results borelittle relationship to those from other , ITSACORR results were not sub-stantiated by visual analysis. Finally, the statisti-cian BradHuitema (2004)described fatal flaws in the model, in response to which Crosbieofficially retired it: I trust Brad's scholarship, soeffective immediately ITSACORR is Now it's dead, and will soon bereplaced (Southerly, 2006).The last treatment day (LTD) prediction tech-nique ofWhite et al. (1989)extended the baselinetrend clear to the end of the treatment period; the last treatment day.

6 The predicted value at LTDwas differenced from the LTD value predicted (as aYhatscore) from the Phase B Trend line alone, andthe two predicted values at LTD were standard error of the difference was calculatedfor the two predicted scores. A Cohen'sdeffect sizewas then calculated from their difference divided bythe pooled standard error term. Two flaws of thismodel were (a) linear prediction from Phase A tothe end of Phase B resulted in extreme scores andextreme differences, and therefore, extreme effectsizes; and (b) the statistical power of the techniquewas quite weak due to the large error involved inpredicting an individual score far into the future, tothe end of Phase B (Parker & Brossart, 2006).Applied regression texts commonly warn thatprediction of scores into the future is hazardous,even with large data sets and short-term predictions(Neter, Kutner, Nachtsheim, & Wasserman, 1996).

7 Center et al.'s (1985 1986) method marked anew level of sophistication, including both meanshift and trends in a single index, while controllingtrend. However, in attempting to control positivebaseline Trend , Center's method also undesirablycontrolled some Trend from the intervention 's method was critiqued and improved on byAllison, Faith, and colleagues (Allison & Gorman,1993; Faith et al., 1997), whose model controlledonly Phase A Trend , but extended through the entiredata series. The Allison et al. model is considered bymost to be the leading OLS approach. It has proveditself in several published studies and at least twometa-analyses (Allison & Gorman, 1993). Nonoverlap or dominance (Sprent & Smeeton,2007) indices of improvement are based oncomparisons of individual data points across twogroups (two phases). Nonoverlap does not summa-rize the difference between central tendency (mean,median, or mode), but rather the separation of thetwo data clouds, giving equal attention to alldata points.

8 The dominance of one data cloudover another is its degree of elevation above theother on a vertical score axis. Judging data overlapbetween phases has been a part of visual analysissince at least the 1960s, along with judging datatrend (Cooper, Heron, & Heward, 1987; Johnston& Pennypacker, 1993; Kazdin, 1982). Nonoverlapwas first measured statistically in the mid-1980s(Scruggs, Mastropieri, & Casto, 1987), and in thepast two decades Nonoverlap techniques haveincreased in number and refinement (Parker &Vannest, 2009; Parker, Vannest, & Brown, 2009). Nonoverlap methods vary mainly in how ties(across phases) are handled, and how overlappingversus nonoverlapping data pair counts are com-bined. However, allcompletenonoverlap indiceshave in common the pairwise comparison ofindividual data points across Phases A and B, todetermine dominance of one score set over theother (Cliff, 1993).

9 The most recently publishednonoverlap method, termed NAP ( Nonoverlap of allpairs) can be derived from Sommer'sd, or from areceiver operator curve (ROC) analysis as areaunder the Curve (AUC;Parker & Vannest, 2009).NAP equals percent of nonoverlapping data. Thenew method demonstrated in this paper, derivedfrom Kendall's Rank Correlation (Kendall &Gibbons, 1999), is the percent of nonoverlappingdata minus the percent of overlapping data. In otherrespects, NAP and this new method are its long use as part of visual analysis,and its user-friendliness (often carried out withpencil and ruler), Nonoverlap has other , Nonoverlap methods are distribution free, not requiring interval-level measurement or alinear relationship between time and scores, norrequiring constant variance or a normal distribu-tion (Armitage, Berry, & Matthews, 2002).

10 Non-overlap methods also are robust or resistive to theundue influence of outlier scores, a particularstrength in client-based research where bouncy scores are common. Furthermore, in some datasets the Nonoverlap or dominance of one phaseover another is a better, more sensitive summarythan is mean or even median difference (Cliff,1993; Huberty & Lowman, 2000). When scoresare severely skewed, are bi- or tri-modal, orotherwise lack central tendency, a mean or medianis not a good distribution summary (Wilcox, 2001).In those cases it makes more sense to consider alldata points equally, as a dominance summary Nonoverlap and Trend for Single-Case researchAuthor's personal copyAlthough Nonoverlap methods are distributionfree (Cliff, 1993), they are held to the standard ofserial independence or lack of autocorrelation(rauto), which applies to residual scores from ananalysis.