Author Institution: Department of Guidance and Educational Psychology, Southern Illinois University; Department of Educational Foundations, University of AkronWhen analyzing data which deals with repeated testing, one may find that extreme scores are regressing away from the mean, contrary to what one would expect based on the regression effect. This paper discusses the regression effect and presents the argument that when these contrary results occur, they are indicative of the violation of the underlying assumption of rectilinearity. One should be required to look for non-linear relationships when interpreting such data. In addition, three methods for determining whether or not non-linear relationships exist in data are suggested and briefly discussed

Newman, Isadore

Pohlman, John T.

The Ohio Journal of Science

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REGRESSION EFFECTS WHEN THE ASSUMPTION OFRECTILINEARITY IS NOT TENABLE1JOHN T. POHLMAN, Department of Guidance and Educational Psychology, Southern IllinoisUniversity, Cardonbale, IL G2901ISADORE NEWMAN, Department of Educational Foundations, University of Akron, Akron,OH 44325Abstract. When analyzing data which deals with repeated testing, one may find thatextreme scores are regressing away from the mean, contrary to what one would expectbased on the regression effect. This paper discusses the regression effect and pre-sents the argument that when these contrary results occur, they are indicative of theviolation of the underlying assumption of rectilinearity. One should be required tolook for non-linear relationships when interpreting such data. In addition, threemethods for determining whether or not non-linear relationships exist in data aresuggested and briefly discussed.OHIO J. SCI. 78(2): 96, 1978Introductory texts in measurement andstatistics typically present the topic ofregression effect by examples of bivariatedistributions in which extreme scorers onthe independent variable (X) tend toscore closer to the mean on the dependentvariable (Y). In the limiting case,where there is no systematic relationshipbetween X and Y, the predicted Zv (Zy)values for all values of Zy equal the meanof Zv. On the other hand, when the cor-relation between X and Y is perfect, theregression effect is, by definition, absent.This implies that knowledge of X allowsperfect prediction of Y. When both Xand Y have been transformed to Z scores,Zy = Zx for all values of Zx. PredictedZ scores, then, are assumed to take onabsolute values that are either equal toor less than the X value from which theywere derived (fig. 1). In this discussion,raw scores have been transformed tostandard scores because differences inmeans and/or variances between X andY can camouflage the regression phe-nomenon.This conception of the regression phe-nomenon has implications in many areaswhere correlation is employed in analyz-ing and inferring from data. In covari-ance analyses, control is exerted on acovariate while comparing the perform-xManuscript received February 17, 1976 andin revised form July 6, 1977 (#76-18).ance of treatment groups against a cri-terion. The scores and groups meansare adjusted in accordance with regres-sion notions. When extreme groups inpre-test/post-test designs are studied, thez. - 2 - 1 0 1 2-2 -1- 2 - 1 0 1 2- 2 - 1 0 1 2r = ,50= ,oo-2 -1 0FIGURE 1. The projection of Zx to Zy un-der various degrees of association. Relation-ship between Z and predicted Z (Z) as a functionof R values.use of a control group is recommended sothat change scores are not falsely attrib-uted to intervening treatments, when theregression phenomenon is the real sourceof change. In repeated testing para-digms where change scores are the ob-ject of interest, expectations aboutchanges by extreme scorers are often96Ohio J. Sci. REGRESSION AND LINEAR VIOLATIONSbased on the regression phenomenon. Astudy reported by Vernon (1954) pro-vides an example of expectations basedon the regression phenomenon failing tohold in a repeated testing paradigm. Inthat study, changes in IQ were beingstudied as a function of repeated testing,and the expectations were that highscorers would score lower and low scorerswould improve. Similar expectationshave been suggested by Campbell andStanley (1969), Kerlinger (1965), Borgand Gall (1971) and others. Vernonfound, however, that high scorers scoredeven higher upon re-testing and gainedmore than low scorers.When rxy does not equal one, there iserror about the least squares regressionline relating X to Y. This error has 2possible sources: pure error and lack offit error. Pure error is error that has amean of zero and is normally distributedin the population. Lack of fit error ischaracterized by having mean valuesthat are significantly different from zeroover certain ranges of X. That is, withincertain ranges of the independent vari-able X, Zy will be below the actual Zyvalues and for other ranges of X, Zy willbe above the corresponding Zy values.It should be stressed here that these de-partures must be significant at some spec-ified alpha level before the presence oflack of fit error can be entertained.The fitted regression line has been cal-culated using a model that assumes alinear relationship between X and Y. Ifthe assumption of linearity is tenable, theerror about the regression line is pureerror. If, on the other hand, the as-7 8 9 10 11 12 13 14 15 16 17PRE TEST Z SCORESFIGURE 2. Comparison of linear (model 1) and quadratic (model 2) regression lines fitted todata reflecting a curved relationship between X and Y.98 J. T. POHLMAN AND I. NEWMAN Vol. 78sumption of linearity is not correct, errorabout the regression line is composed ofboth pure error and lack of fit error.If a straight line is the best fitting re-gression line, and if rxy is not unity, thenregression to the mean of Y must occur.On the other hand, if the relationship be-tween X and Y can best be representedby a curved line of any sort, the standardphenomenon of regression effects towardthe mean will not result. For example,in figure 2, data points have been drawnsuch that the relationship between Zxand Zy is clearly non-linear. Regressionline 1 has been drawn through the scatterplot using the least squares criterion, anda certain amount of error has beenrealized. The model that reflects line 1in figure 2 is as follows:Model 1: Y = a0U + a1X1+E1where: Y is the dependent variableU is a unit vector whichwhen multiplied by a0yields the regression con-stantai is the regression weightEx is the error vectorXi is the independent vari-ableAdmittedly, the magnitude of error inthis model is small, but note what has oc-curred in our data. The subjects whohad a Z score of 8 on X have an averageZ score of 2.5 on Y. The Zy value of 2.5is greater than the Zx value of 8. Inother words, these subjects have "re-gressed" away from the mean of Y insteadof regressing toward the mean of Y, aswould be expected according to the regres-sion phenomenon. The reason for this,of course, is that the relationship be-tween X and Y is best represented by aregression model which allows for cur-vature. Model 2 is just such a model.Model 2: Y = a0U+a1Xi+a2X?+E2where: Y is the criterion scoreU is the unit vectorXi is the X valueXj is the X values squaredai, a2, and a3 are the regres-sion weights assigned toU, X, and X2, respec-tivelyE2 is the vector of residualsSince the relationship between X andY that is depicted in figure 2 is curved,the assumption of rectilinearity is not cor-rect. Consequently, expectations basedupon the regression phenomenon wouldbe in error and might mislead the re-searcher in interpreting his results. Thisexample is particularly noteworthy be-cause the linear model reflected in Model1 did an excellent job of accounting forvariance in Y, but Model 2 did a muchbetter job.The fact that the scores did not dis-tribute themselves in the bivariate dis-tribution as the regression phenomenonpredicted should have served as a cue forthe researcher to attempt a non-linearfit to the data. By determining the kindof curvilinear relationship that existed,inferences might be made as to the under-lying causes for that relationship.METHODS FOR DETECTINGCURVATUREVarious procedures exist for screeningpaired observations for possible non-linear trends. Oftentimes, inspection ofthe scatter plot is sufficient to note thepresence of curvature in the relationship.Inspection methods lack certainty whenthere are few data points or if the corre-lation is rather small. The followingmethods are offered as suggestions to as-sist in determining if a non-linear trendis present in a set of data.Method I. If there are only a fewobservations (fewer than 25) the X vari-able can be divided into quintiles and themean Y values can be calculated for eachquintile group on X. The plot of thesemeans could then be inspected for cur-vature. The researcher should recognizethat for a set of data this small, only dis-tinctly non-linear trends will approachsignificance. If the investigator suspects,as a result of this inspection, that a non-linear trend is present he may want totest the significance of that trend (Ed-wards 1960).Method II. Calculation of the corre-lation ratio (Eta2) for a set of data.The Eta2 could then be compared to r2 totest, if a significant non-linear trendexists in the data. An F-ratio could begenerated as follows:Ohio J. Sci. REGRESSION AND LINEAR VIOLATIONSF = -(Etax1.0-Eta2)/df2where: EtaXy is the Eta2 for ourdatarXy is the square of the Pear-son r for the same datadfi is the number of linearlyindependent variables orscores on X used to cal-culate Eta2, minus 2 and,df2 is the number of pairedobservations minus thenumber of linearly in-dependent variables orscores used to calculateEta2.If this F-ratio is found to be significant,one may entertain the assumption that anon-linear trend is present in the data dueto the global nature of Eta2, one cannot,from these analyses, describe the type ofrelationship that exists. Eta2 could besignificantly greater than r2 because ofany departure from linearity.Method III. The most direct methodof ascertaining if the relationship be-tween X and Y is significantly non-linear, and describing the nature of therelationship is a curve fitting approachemploying multiple regression methods.Various functional relationships can betested and the partial regression weightsmay be used to describe the nature of therelationship. The use of a multiple re-gression approach which allows the in-vestigator to try various data transforma-tions and possible linear combinations isparticularly useful (McNeil et al 1975).CONCLUSIONOne of the underlying assumptionsnecessary for the regression phenomenonto be accurate is that there is a linearrelationship between X and Y. If therelationship is linear, all error will, resultin extreme scores on X, scoring on theaverage closer to the mean of Y. If therelationship between X and Y is non-linear, the regression phenomenon willnot represent the true state of affairs inthe data. Indeed, if researchers notethat expectations based on the regressionphenomenon are not supported by theirdata, it would be advantageous for themto seek out the nature of the curved re-lationship that exists. In fact, the na-ture of the curvature may serve as a cluein determining why the regression phe-nomenon was insufficient for explainingthe data.LITERATURE CITEDBorg, W. R. and M. D. Gafl 1971 Educa-tional Research: An Introduction. DavidMcKay Co., New York. 533 p.Campbell, D. T. and J. C. Stanley 1969 Ex-perimental and Quasi-Experimental Designfor Research. Rand McNally, Chicago. 84 p.Edwards, A. L. 1960 Statistical Methods forBehavioral Sciences. Rinehart, New York.542 p.Kerlinger, F. N. 1965 Foundations of Be-havioral Research. Holt, Rinehart & Win-ston, New York. 739 p.McNeil, K., F. J. Kelly and J. McNeil 1975Testing Research Hypotheses Using MultipleLinear Regression. Southern Illinois Univ.Press, Carbondale, IL. 587 p.Vernon, P. E. 1954 Practice and coachingeffects in intelligence testing. In: The Edu-cational Forum 18: 269-280.

Regression Effects When the Assumption of Rectilinearity is Not Tenable

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Regression Effects When the Assumption of Rectilinearity is Not Tenable

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