Regressing toward the Mean vs Progressing toward Individual Differences Miriam Cherkes-Julkowski, Ph.D. Educational Diagnostician and Consultant Abstract Although formulas that use regression toward the mean are currently touted as a statistically informed and improved method for identifying learning disabilities, they in fact appear to be particularly ill suited to the job of identification. Regression formulas assume typical, normative and linear relationships between intelligence and achievement in defiance of what is essential to any learning disability, i.e., atypical brain organization and therefore idiosyncratic achievement-intelligence relationships. This article enumerates the problems with regression approaches, attempts to make them more transparent and concludes that they suppress individual differences, making regression approaches inappropriate as a tool for identification of learning disabilities. Although there is an aura of scientific credibility surrounding the mathematical intricacy of regression formulas, their validity as a basis for assessing the presence or absence of a learning disability is seriously in question. At the heart of the issue is whether a model that assumes drift toward the mean (average) is suitable in the context of those atypicalities of learning disability that are, by nature, in defiance of the mean. To the extent that learning disabilities are characterized by atypical brain organization with specific processing profiles, relationships between intelligence and achievement are likely to evolve in nonstandard ways. Regression formulas use intelligence-achievement score correlations to preselect a number of conditions. Full scale IQ scores are selected as the basis for predicting achievement levels that are then qualified based on their proximity to the mean. The inherent bias toward the mean is the one in question. It seems more than reasonable to expect that intelligence (usually defined by IQ) is not a perfect predictor of achievement, that the intelligence-achievement relationship needs to be qualified in some number of ways. Common sense would tell you so as do empirical findings (Baldwin, Baldwin, & Cole,1990). Issues such as motivation, background knowledge, health, emotional status, environmental stressors are all variables that will intervene to affect knowledge acquisition. So are inherent differences in brain organization. In the case of atypical children, in this case children with learning disabilities, there are specific issues that can be expected to mediate the relationship between intelligence and achievement, those that revolve around specified deficits along with their competing strengths. Rather than preordaining drift toward the mean, imperfect intelligence-achievement relationships more cautiously might yield a range of achievement possibilities. For example, an IQ of 120, compared to an achievement test whose correlation with IQ is .9, might predict an achievement score to fall within the range of 118 and 122; rather than be pinpointed regressively at 118 as preordained in regressive formulas. Rather than solve the tricky statistical and theoretical problems of determining intelligence-achievement discrepancies and therefore eligibility for identification under the auspices of IDEA, formulas based on regressing expected achievement scores toward the mean simply import those same problems and compound them while making them less transparent. Since the determination of discrepancy provides the gateway to entitlements under IDEA it is critical that parents, teachers and other professionals are informed thoroughly about the benefits and liabilities of proposed approaches. Regression Formulas Prohibited under the condition of Test Discrepancies The problems with regression formulas are manifold but most center around a number of steps that force out individual differences, denying the essence of a learning disability. The first sacrifice of individual differences would be the use of full scale IQ scores in the many cases of children with learning disabilities who have significant discrepancies either among individual test scores (i.e., WISC III (Wechsler, 1991) similarities vs arithmetic; picture arrangement vs block design) or between the scale scores, (performance vs verbal IQs). The state of Washington, as one of many examples, prohibits the use of regression formulas under such a condition (Bergeson, Heuschel, Harmon & Gill, 2003): The severe discrepancy tables were developed using correlation coefficients between the Full Scale, or Composite, intellectual ability and achievement test scores. Only Full Scale or overall Composite scores may be used to enter the severe discrepancy tables. If the evaluation group determines that the Full Scale score (emphasis mine) or the overall Composite score (emphasis mine) does not accurately reflect the student’s intellectual ability, then a data-based professional judgment must be made regarding the existence of a severe discrepancy... (page 5). As the Washington state guidelines point out, the entire process is predicated on the use of full scale intelligence and composite achievement scores. Since full scale and composite scores lose their validity when there are significant differences among the individual contributing scores, such global scores are only useable, as the above guidelines meticulously specify, when significant differences do NOT exist at the individual test level (Hale & Fiorello, 2002). Given the definition of learning disabilities, “disorder in one or more of the basic psychological processes...that may manifest itself in an imperfect ability...” (Federal Register 1999), it is highly unlikely that full scale or composite scores would ever pass the qualifying standard. In fact, failure to conform to consistent, within-test scores has been and is considered an identifying feature of LD. Any mathematical procedure that washes out these essential inconsistencies would be specifically designed to make LDs undiscoverable. Regression formulas do just that mathematically when they use full scale or composite scores unjustifiably and when they insist on forcing scores toward the mean. The Washington state guidelines for identification of students with specific learning disabilities presages the next issue, i.e., composite scores. Just as combining significantly discrepant subscale IQ scores obscures important differences, so would combining significantly different subscale achievement scores. As with the full scale IQ, available correlation coefficients are for broad, cluster or composite scores. When they are rendered unusable by virtue of internal discrepancies within the composite scores, regression formulas could not apply. Of course the same is true for any of the IQ-achievement discrepancy analyses. If full/composite/global/cluster scores must be abandoned for the purposes of regression formulas, they must also be abandoned in any other form of calculation. This pertains to simple subtraction formulas as well, i.e., discrepancy computed by simply subtracting equivalent standard scores for achievement from intelligence. However, simple subtraction computations have the advantage of not being beholden to global scores for their validity. In their simplicity and transparency, simple subtraction computations can be used more flexibly to reflect the idiosyncrasies of those children whose idiosyncrasies are in fact the issue under observation. For example, a severely phonologically impaired child might do very poorly on a test of pseudoword reading and significantly better when reading real, high frequency words. A composite of the two tests would be invalid due to the significant difference between the two contributing scores. No comparison then could be made between IQ and the composite score. However, providing that the full scale IQ can be computed validly, comparisons could be made between IQ and pseudoword reading using the subtraction formula. There would be no prohibition against doing so as long as the pseudoword reading test met sufficient test standards for reliability and validity. Since learning disabilities are meant to reflect specific deficits within a context of otherwise average or above average functioning, most true instances of learning disabilities must be characterized by exactly those irregularities that prohibit the use of composite and full scale scores. Neuropsychological evaluations of those particular brain functions contributing to intelligence or otherwise classified cognitive functions are especially sensitive to the need to avoid composite scores: “composite scores of any kind have no place in neuropsychological assessment” (Lezak, 1995). Rather than aggregate divergent individual scale scores, neuropsychological approaches rely on using the highest subscale scores on an IQ test as the basis for estimating intelligence (Lezak, 1995). The current import of neuropsychology into the field of learning disabilities would necessitate importing this fundamental principle as well. Do Atypicalities Regress toward the Mean? A joint neuropsychological and learning disability issue is one of regression toward the mean in cases where the point of interest is to uncover an atypicality. The first part of the regression calculation modifies predicted achievement based on the imperfect relationship between IQ and achievement. As an idea, this seems to be a worthy one. However, it devolves rapidly at the next computational step which forces the predicted score in the direction of the mean. The standard formula is as follows: Expected Achievement = (SDa/SDi)r(IQ-100) + 100 The first part of the procedure provides for the correction of the achievement score based on its correlation (r) with IQ and expressed in terms of standard deviations (sd) of the achievement (a) and intelligence (i) tests. So far so good. The next step requires the subtraction of 100, i.e., the mean. This is the step that forces the score toward the mean without any better justification than the claim that regression is what one might expect under normal conditions (note the tautology). The expectation that scores will migrate (regress) in the direction of the mean seeks justification on the statistical assumption that events occur on a random basis rather than being biased by one or more preexisting conditions (internal or external to the learner). On that assumption, on the basis of sheerly random causation (i.e., statistically), it is most probable that any given event will be found among the most frequently occurring events, those that hover most tightly around the mean. It is highly suspect that this assumption holds in the case of individual learners, most especially those whose learning is most atypical. It is highly suspect that the randomness/statistical assumption holds at all in nonidealized, lawful, real world dynamics (Swenson, 2000). As a matter of theory, regression formulations ignore the impact of atypical brain organization. In their application, regression formulas fail to recognize differential dynamics within and between aspects of learning, reading, math and writing. For example differential brain organization has been documented clearly in dyslexic and nonimpaired readers (Shaywitz, Shaywitz, Pugh, Fulbright, Constable, Mencl, Shankweiler, Liberman, Skudlarski, Fletcher, Katz, Marchione, Lacadie, Gatenby & Gore, 1998). Dyslexics manifest less activation in posterior regions when they read, more activation in inferior frontal gyrus. As a matter of speculation it is at least possible and potentially probable that, for the dyslexic group, achievement predictions based on intelligence would progress to a level above the mean for memorized words, based on probable activation of frontal areas supporting executive functions including working memory. The same brain dynamic might then suggest that dyslexics would do more poorly than the mean on tasks requiring the pattern recognition needed for orthography-phonology connections (Compton, 2002) and supported in part by posterior areas. This bias holds true regardless of intelligence and regardless of the normative relationship between intelligence and achievement. Of course, subtle differences of the kind that neuropsychological investigations such as those conducted by the Shaywitz group are hopelessly lost when global scores are used. For example, imagine a test that derives a composite reading score by aggregating the results of two tests: one of reading high frequency, specifically memorized words; and one of reading novel, pseudowords (see the TOWRE for a test that does just that; Torgesen, Wagner & Rashotte, 1999). Students could achieve the same composite score by having very different contributing profiles. As an extreme example, one student could earn a total score by having 0 on the test for pseudowords, 10 on the test of high frequency words. The other could have the opposite, 10 for pseudowords, 0 for high frequency words. Their composite scores would be virtually the same; their very discrepant profiles masked and undetected. On an individual case basis, according to regression formulations, a highly discrepant cognitive profile characterized by high verbal and low performance IQ’s would predict identical scores for reading and math (given the usual similarity in full scale IQ-math and full scale IQ-reading correlations) despite the fact that there is ample evidence to indicate that such a learner should be expected to have higher reading than math performance (Rourke, 1995). The dynamics that resulted in a given IQ score distribution, for example a performance IQ of 120 and a verbal IQ of 80, would also act upon those dynamics influencing achievement. Suppose, for example, the low verbal IQ score reflected a specific learning deficit in phonological processing as manifested in poor word finding with generally delayed and awkward verbal communication. This essential deficit is likely to deflate individual tests on the verbal scale of the WISC where fluid and well organized expression earns credit on tests such as vocabulary, similarities, and comprehension. Since phonological deficits also affect working memory (Shankweiler, Smith & Mann, 1984), the arithmetic and certainly the digit span scores would also be affected. The same underlying phonologic deficit would also drag down achievement in reading skill, reading comprehension, spelling and prose writing. It would not be expected, on this view, for the language arts achievement scores to regress, in this case elevate, to the mean at all. It would be expected for them to be systematically compromised, in the same negative direction as verbal IQ. Regression Goes Both Ways Correlation relationships are completely symmetrical. If they can be used to predict achievement from intelligence, they can be used equally to predict intelligence from achievement. And, all other rules would apply symmetrically as well, i.e., the predicted score would, according to regression formula rationale, regress toward the mean. To see how this would work in predicting IQ from achievement, the terms of the above formula would be: Expected Intelligence = (SDi/SDa)r(ACHIEVEMENT-100) + 100. Using the same numbers as in the examples above, assume that achievement is 120, the intelligence-achievement correlation .9. Intelligence would be forced toward the mean, 118. Or, if achievement were 80, we would be forced to expect that it had been driven by a higher intelligence, 82. If you run the “predictions” both ways, you are forced to accept that two conflicting realities are equally as valid at the same time: 1) an intelligence-achievement correlation of .9 would substantiate an intelligence score of 120, achievement 118; and 2) an intelligence-achievement correlation of .9 would substantiate an intelligence score of 118, achievement 120. The difference can be critical in making the decision about the presence or absence of a learning disability. The defense that it doesn’t matter (the numbers are within the standard error of measurement, the differences insignificant) does no good since if that is so, fine points in the discrepancy analysis created by the standard application of the regression computation don’t matter either and you are back to the subtraction formula. If you make achievement the independent variable, the one from which intelligence is predicted, you could never find a learning disability except in the indefensible case of situations where there is a nearly nonexistent correlation between intelligence and achievement. Measuring Achievement in What Next is the problem of what achievement areas are measured and how they might correlate with intelligence. As the Washington state guidelines so aptly point out, all correlated scores available on regression tables are at the most global level, least sensitive to specific deficits. For example, there are tables for IQ-broad score and IQ-composite score correlations. Remember, however, that global scores are aggregates of contributing individual scores and thus potentially disguise important differences revealed by those differences. Take, for example, the broad reading score on the Woodcock-Johnson (2001). The broad score is a composite of three contributing subscales: passage comprehension, reading fluency, letter-word identification. None of these is a measure of the most critical skill in reading; in the most often affected area for children; with the most common and detrimental deficit. The most significant reading skill which is at one and the same time most negatively impacted by phonological deficits and most related to the ultimate goal of reading comprehension, is the reading of pseudowords. Shankweiler, Lundquist, Katz, Stuebing, Fletcher, Brady, Fowler, Dreyer, Marchione, Shaywitz & Shaywitz (1999) found the very substantial correlation of .79 between the reading of pseudowords and reading comprehension. On the Woodcock-Johnson - III (Woodcock, McGrew & Mather, 2001) pseudoword reading is measured with the word attack test, glaringly missing from the broad reading score. It is the pseudoword-deprived broad reading score that can be used in regression formulas as a defining measure to qualify a student as having a learning disability or not, despite its omission of the most critical information in the identification of most learning disabilities. To what composite score does the word attack test contribute? Word attack in combination with letter-word identification forms the basic reading score. But, why would any diagnostic procedure want to dilute an already highly informative data point (word attack) with one that is less informative, i.e., real word reading as measured by letter-word identification? Furthermore, why would one want to confound the very brain dynamics that seem to differentiate LDs from nonimpaired readers, as suggested by the findings of the Shaywitz group and mentioned above (Shaywitz, et. al., 1998)? If an informed and diligent diagnostician insists on maintaining the integrity of the word attack test or a test of pseudoword reading like it, s/he is locked out of the regression formula since empirically established correlations with IQ are rarely provided. Despite the restrictions upheld by the state of Washington guidelines, some promoters of the regression approach allow estimation of intelligence-achievement score correlations by multiplying the reliabilities of each. Two tests, each with reliabilities of .9 would be estimated to have a correlation of .81. There are examples that show how counterintuitive this approach can be. Suppose there is an instrument of .9 reliability that measures career preference and one of the same reliability, .9., that measures color preference. Should it be concluded that the correlation between color preference and career preference is .81? A strictly computational approach leads to the same untenable position. Multiplying by a decimal/fraction (as each correlation coefficient is with the exception of a perfect correlation of 1 or a complete noncorrelation of 0) sets up a proportion, a colinearity which in fact “smuggles in” (term credited to R. Swenson, personal communication) a forced belief in and “conclusion” of a linear relationship. Forcing two tests to have a linear relationship simultaneously forces out the possibility of uncovering other, perhaps more atypical and idiosyncratic relationships among test areas; those that would be more probable and more information in cases of learning disabililty. Instability of Correlations with Age and Diagnostic Category The tables available for use with regression formulas rarely, if ever, provide intelligence-achievement correlations broken down by age or diagnostic category groups. The relationship between IQ and achievement has been shown to vary across age and grade ( Kline, 1991). Similarly, the relationships among processing capacity and achievement vary with age and grade (Carlisle, Gugisberg, Strasser & Patton, 2002). Across different diagnostic categories, different patterns of correlations reveal different relationships among intelligence, cognitive processes and achievement. This is especially true of reading when there are known neuropsychologically distinct profiles such as with William's Syndrome (Levy, Smith & Tager-Flusberg, 2003), ADHD (Rapport, Scanlan, & Denney, 1999) and learning disabilities (Shaywitz, et al, 1998). Differential effects of underlying abilities on achievement are no less characteristic of the subdivisions of learning disability, mainly verbal and nonverbal learning disabilities. Consider two hypothetical individuals with the same full scale IQ’s, one with language based learning disability (LD) and the other with nonverbal learning disability (NLD). The LD student with right hemisphere strengths in interaction with left hemisphere weaknesses is predicted to have an achievement profile with strengths in nonverbal areas, weaknesses in more language dependent tasks. In contrast, the individual with NLD is expected to have strong achievement in language, particularly phonological areas, poor achievement in those areas related to spatial reasoning, i.e., math and global awareness (Pelletier, Ahmad, & Rourke, 2001). Despite these highly relevant individual differences, regression formulas would predict achievement in all skill and content areas to be at the same level provided their correlations with full scale IQ were the same or highly similar (they generally are). Neuropsychology or Regression Formulas but not Both Learning disabilities are characterized by a specific deficit in the context of otherwise average or above average functioning. In their conceptualization, learning disabilities represent a different quality of brain organization rather than increases or decreases along a standard continuum. As such, neuropsychological models of assessment offer themselves as a more appropriate model for test interpretation than standard regression models. Neuropsychological models stress the importance of preserving the information conveyed in subscale scores (Lezak, 1995) so as not to wash out the effects of strengths and weaknesses with special consideration given to using high points to project intellectual potential. Because of their atypical brain organization (Shaywitz, et. al., 1998), individuals with learning disability idiosyncratically organize their responses to environmental, in our case school, challenges. The probability of finding their behavior around the central tendency (mean/average) for normative groups, is far less likely, i.e., regression toward the mean (as defined normatively) is not a necessary or likely given for atypical groups, LDs prominently included. A more productive framework for understanding behavior, especially atypical behavior, would be the idea of attractors: patterned, dynamical, neuronally based propensities for capturing input which allow for a moderate degree of response stability while maintaining enough flexibility to adapt, grow and learn. Attractor states are formed idiosyncratically based on the experience of the learner (Freeman, 1991; Cherkes-Julkowski & Mitlina, 1999) in combination with already preexistent attractor states, one type of which would be initial brain organization. While there is a general tendency for individual neuronal and behavioral elements to be recruited into conformity with more macroscopic attractor structures (Swenson, 2000), this is not the same as regression toward the mean. At the level of the individual, there are global determinants of behaviors that tend to bring more microscopic “defectors” back into alignment with those attractor states (Cherkes-Julkowski, 2003). At the level of group analysis, expectations act in the same way, i.e., to draft defectors back into conformity with the more dominant structures. These dynamics, however, do not describe a linear continuum or a gradual progression toward the dominant structure, in contradiction to the critical assumption of linearity underlying any regression model. Instead they reflect spontaneous ordering. At critical thresholds, a current nonmember is either dichotomously recruited in or left out. Sample Applications Examples might serve best to show how regression formulas fail to work. Assume a fictional but fairly illustrative case of a student whose verbal IQ score is 80, performance IQ 120, full scale 1001.. To use the regression formula to predict calculation achievement level on the Woodcock-Johnson - III (Woodcock, McGrew & Mather, 2001), it would be necessary to estimate IQ-achievement correlations since calculation is an individual test aggregated within the Woodcock-Johnson III broad math score. The IQ-calculation correlation would be estimated by multiplying the reliabilities of the two tests. With a full scale WISC IQ reliability of .96 and a calculation test reliability of .92, the computed correlation between the two tests would be .88. In any case, the predicted calculation achievement score would be 100 (see formula above). However, this is strictly forbidden since the full scale IQ score is disqualified based on the significant difference between the verbal and performance contributing scales. If the verbal IQ score of 80 were used to predict calculation achievement level according to a regression formula, the same steps would be applied with new reliability figures. A reliability of .95 for verbal IQ, multiplied with the .92 reliability for calculation would yield a contrived correlation between intelligence and achievement of .87. Now the predicted calculation achievement score is 82.6. Or, if the performance IQ is used, this time with a reliability of .91, contrived correlation of .83, predicted calculation achievement is 116.6. Which is it? The same child’s word attack (reliability = .91) score might be predicted using estimated correlations as well. The predictions would be essentially the same. The slight change in word attack reliability would predict reading scores that would be moderately different but effectively equivalent to those in math. Given the verbal IQ score of 80, word attack achievement would be predicted at 82.8. Basing word attack achievement on performance IQ would yield a predicted score of 116.4. Common sense says that this cannot be so. Low verbal ability cannot predict a math score virtually the same as a reading score. Likewise, high performance ability would not have implications for reading and math equivalent to those associated with high verbal ability. This can only happen according to a statistical model that places measures of central tendency above well documented real, on the ground, well known relationships. A Would be Solution that Accentuates the Problem While regression formulas laudably try to address the not-so-simple relationships among intelligence and achievement, they do so by making the problem of oversimplification even worse. Traditional discrepancy subtraction calculations, where achievement is subtracted from intelligence, continue to hold the advantage on several bases: They are transparent. They do not bias the findings based on assumptions that conflict with the neurological and conceptual dynamics fundamental to learning disabilities. They are more responsive to the idiosyncratic dynamics of individual differences among specific achievement and processing areas. To the extent that regression formulas are meant to ease the problem of over identification by making it harder to find a discrepancy, this can be done in far more forthright ways. “True” cases of learning disability might be found best by requiring more stringent and less diluted assessment of those processes known to be contributory to learning in specific skill and content areas. These are reasonably well known but too important to be given short shrift here. 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