Original Article
Evidence That Dyslexia May Represent the Lower Tail of a Normal Distribution of Reading Ability
N Engl J Med 1992; 326:145-150January 16, 1992
- Abstract
- Abstract
Background
Dyslexia is now widely believed to be a biologically based disorder that is distinct from other, less specific reading problems. According to this view, reading ability is considered to follow a bimodal distribution, with dyslexia as the lower mode. We hypothesized that, instead, reading ability follows a normal distribution, with dyslexia at the lower end of the continuum.
Methods and Results
We used data from the Connecticut Longitudinal Study, a sample survey of 414 Connecticut children who entered kindergarten in 1983 and were followed as a longitudinal cohort. Dyslexia was defined in terms of a discrepancy score, which represents the difference between actual reading achievement and achievement predicted on the basis of measures of intelligence. Data were available from intelligence tests administered in grades 1, 3, and 5 and achievement tests administered yearly in grades 1 through 6. For each child there were 108 possible discrepancy scores ([3 × 3 years] × [2 × 6 years]) based on combinations of the ability scores (full-scale, verbal, and performance IQ) in each of three years and two achievement scores (reading and mathematics) in each of six years. We demonstrated that each of the discrepancy scores followed a univariate normal distribution and that the interrelation of two different discrepancy scores followed a bivariate normal distribution. At most, only 9 of 108 discrepancy scores (8.3 percent) and 171 of 3402 pairs of discrepancy scores (5.0 percent) were significantly different (at the 5 percent level) from the expected scores — well within the expected values for data with univariate and bivariate normal distributions, respectively. We also examined the stability of dyslexia over time. The normal-distribution model predicted (and the data indicated) that only 7 of the 25 children (28 percent) classified as having dyslexia in grade 1 would also be classified as having dyslexia in grade 3.
Conclusions
Reading difficulties, including dyslexia, occur as part of a continuum that also includes normal reading ability. Dyslexia is not an all-or-none phenomenon, but like hypertension, occurs in degrees. The variability inherent in the diagnosis of dyslexia can be both quantified and predicted with use of the normal-distribution model. (N Engl J Med 1992;326:145–50.)
Media in This Article
Figure 1
The Relation between the Age-Standardized Reading Score and Full-Scale IQ of Children in Grade 3 (Panel A), an Example of a Bivariate Normal Distribution with a Correlation of 0.6 for a Sample Size of 414 (Panel B), and the Relation between Discrepancy Scores for Reading-Disability Classifications in Two Different Grades (Panel C).Article Activity
- Article
DYSLEXIA is a neurologically based disorder in which there is an unexpected failure to read. As defined by the World Federation of Neurology, the disorder is "manifested by difficulty in learning to read despite conventional instruction, adequate intelligence, and sociocultural opportunity."1 Dyslexia is also referred to as "specific reading disability" or "specific reading retardation." It is generally assumed that the failure to learn to read represents a specific syndrome that is distinct from the normal distribution of poor readers. Rather than representing the lower end of a continuum of reading disability and reading ability, dyslexia (or specific reading disability) is viewed as a biologically coherent disorder that is distinct from other, less specific reading problems. Support for this point of view comes from the work of Rutter and Yule,2 who found that "children with [specific reading disability] form a `hump' at the bottom of the normal curve."3 They used these findings to argue that reading ability is bimodally distributed, with specific reading disability appearing as the extreme lower tail. This notion of reading disability as a specific, discrete entity serves as the basis both for investigations into the neurobiology of dyslexia and for the diagnosis of dyslexia and the provision of services to persons with the disorder.
Rather than following the bimodal-distribution model posited by Rutter and Yule,2 , 3 a model that continues to dominate thinking in the field, we hypothesized that dyslexia occurs along a continuum and is best conceptualized as the tail of a normal distribution of reading ability. To examine this issue we investigated both the distribution and the temporal stability of reading disability, using data from the Connecticut Longitudinal Study, a sample survey of Connecticut children who entered kindergarten in 1983. The results of our study indicate that a normal-distribution model of dyslexia fits the data extremely well and that this model can be used to predict the stability of dyslexia over time. These findings provide support for a fundamental revision in the concept of dyslexia; rather than existing as a discrete entity, dyslexia, like hypertension and obesity, occurs along a continuum and varies in severity.
Methods
Sample Selection
The target population for the Connecticut Longitudinal Study was made up of the children attending Connecticut public kindergartens during the 1983–1984 school year. The children included in the study were selected by means of a two-stage probability-sample survey of children who entered public kindergarten in Connecticut in 1983; this sample has been described in detail elsewhere.4 Briefly, two kindergarten classes within each of 12 selected communities were chosen randomly, so that each class within a town had an equal probability of selection. All the children in the 24 selected classes were invited to participate in the study. Exclusion criteria were limited to substantial sensory impairment and serious psychiatric problems. Four hundred forty-five children, 96.5 percent of the selected sample, participated; there were 235 girls (52.8 percent) and 210 boys (47.2 percent). Of these children, 375 were non-Hispanic whites (84.3 percent), 50 were black (11.2 percent), 9 were Hispanic (2.0 percent), 4 were Asian (0.9 percent), and 7 were children whose race or ethnic group was unknown (1.6 percent). A complete set of data was available for 414 of these children. The children's ability was assessed with the revised version of the Wechsler Intelligence Scale for Children (WISC-R),5 administered in grades 1, 3, and 5, and their achievement with the reading and mathematics subtests of the Woodcock—Johnson Psychoeducational Battery6 administered yearly in grades 1 through 6. To ensure the quality of the results, the WISC-R was administered only by trained school psychologists and the Woodcock—Johnson tests by experienced educators, each of whose adherence to administrative and scoring standards was closely monitored.
Statistical Analysis
Specific reading disability, which is defined as an unexpected failure to read, is expressed as a discrepancy between the level of reading achievement predicted on the basis of intelligence (ability) and the actual level of reading achievement.2 3 4 , 7 8 9 Predicted achievement is determined by the regression of actual achievement on ability (measured in terms of IQ).4 , 8 , 9 In this study, following the definition of specific reading retardation proposed by Rutter and Yule2 and the U.S. Office of Education's definition of reading disability,10 we defined the discrepancy score as the difference between the observed achievement level and the predicted achievement level based on the IQ score. The discrepancy score therefore depended on the choice of achievement and IQ tests. In practice, any of the three components of IQ (verbal, performance, or full-scale IQ) may be used to measure ability. We reasoned that if dyslexia follows a normal-distribution model, each discrepancy score should follow a univariate normal distribution regardless of which particular component of IQ, which particular reading-achievement score, or which year of administration was used to determine the discrepancy score. By systematically varying the component of ability (the full-scale, verbal, and performance IQ), the area of achievement (reading and mathematics scores), and the year the ability and achievement tests were administered, we obtained 108 discrepancy scores ([3 ability scores × 3 years] × [2 achievement scores × 6 years] = 108). We proposed that two distinct discrepancy scores would be interrelated by a bivariate normal distribution. This distribution describes two quantities, each of which has a univariate normal distribution and whose correlation with one another follows an explicitly defined equation. Assuming such a distribution allowed us to calculate the predicted agreement of the diagnosis of dyslexia over time on the basis of two discrepancy scores obtained when a child was in two different grades. Since a test of achievement in mathematics was also administered yearly, we were able to test whether achievement in mathematics also followed a normal distribution.
We tested our normal-distribution model of dyslexia and confirmed its validity by demonstrating that each of the discrepancy scores followed a univariate normal distribution and that the interrelation between two different discrepancy scores followed a bivariate normal model. Both empirical, graphic methods and more formal statistical tests were used (a more detailed description of our statistical methods appears in the Appendix).
Results
We first examined whether the normal-distribution model adequately fit the IQ and achievement data by examining all the discrepancy scores for normality. Figure 1AFigure 1
The Relation between the Age-Standardized Reading Score and Full-Scale IQ of Children in Grade 3 (Panel A), an Example of a Bivariate Normal Distribution with a Correlation of 0.6 for a Sample Size of 414 (Panel B), and the Relation between Discrepancy Scores for Reading-Disability Classifications in Two Different Grades (Panel C). shows a typical plot of the age-standardized reading score as compared with the full-scale IQ for our sample; the degree of correlation between these two variables is 0.68. Using a sample with the same size (414), we obtained a typical plot of a computer-generated bivariate normal distribution with a correlation coefficient of 0.6 (Fig. 1B). The similarity of these figures is evident, giving empirical support to the appropriateness of the normal-distribution model. More formal tests of normality (described in the Appendix) also indicated that the data conformed to a normal distribution (P>0.20 for all comparisons).The distribution of the discrepancy scores was examined with the strategy proposed by Rutter and Yule2; these investigators reported an overrepresentation of subjects in the extreme lower end of the distribution (i.e., a "hump"). Assumptions of normality were examined by comparing the number of reading and mathematics classifications that were significantly (at the 5 percent level) different from the expected results at cutoff levels of -2 SD, -1.5 SD, +1.5 SD, and +2 SD. There were 108 different discrepancy scores, defined as the difference between the observed achievement level and that predicted on the basis of the IQ score. At each of the four cutoff points, we found that for 1 of 108, 1 of 108, 1 of 108, and 9 of 108 of the pairs, respectively, a significantly different number of children were classified as having a disability in reading or mathematics than was predicted by the normal model. These data were within expected values and were thus consistent with the hypothesis that a continuum exists with respect to the discrepancy between ability and achievement.
Next, we used the model to investigate the longitudinal stability of dyslexia — that is, to examine the relations of the same reading-disability classifications employed in different school years. Children were first classified according to a particular set of ability and achievement measures at a specific grade level and then reclassified according to the same set of measures at a different grade level. Figure 1C, which shows discrepancy scores for grades 3 and 5, illustrates the nature of this relation. Some children were classified as having dyslexia in both grades (those in the lower left quadrant), whereas others were not considered dyslexic in either grade (upper right quadrant). Other children (in the upper left and lower right quadrants) were classified as having dyslexia in one grade but not in the other.
According to the normal model, the percentage of children who are classified as having dyslexia in both grades is a direct function of the correlation of the discrepancy scores. Figure 2Figure 2
Probability of Classification as Dyslexic According to a Second Criterion (DS2) after Classification as Dyslexic According to the First Criterion (DS1) in Relation to the Correlation between DS1 and DS2. shows the correlation of two distinct criteria for classifying a child as having dyslexia (DS1 and DS2) and the probability that a child classified as dyslexic according to the first criterion (DS1) would also be so classified according to the second (DS2). Thus, if one has knowledge only of the correlation of the discrepancy scores, this figure can be used to calculate the predicted stability of the diagnosis of dyslexia over time. Specifically, we were able to predict accurately the stability of the diagnosis for grades 1 through 6. For example, using age-standardized reading scores and full-scale IQ scores, we found that 25 children were classified as dyslexic in grade 1 and 31 in grade 3; 7 of them were identified as dyslexic in both grades. On the basis of the correlation of 0.53 between the discrepancy scores for these two years, our model predicted that there would be 28.3 children in each group, with 8.38 identified as having dyslexia in both grades. Thus, the observed instability of dyslexia across grades is also reflected in the degree of instability predicted by our model.Similarly, when we examined the stability of the diagnosis of reading disability later on, from grade 3 to grade 5, we found 30 children classified as dyslexic in grade 3 and 24 in grade 5, with 14 so classified in both grades. The correlation between the discrepancy scores for these two grades is 0.67. Our model predicted that 10.8 children would be identified as having dyslexia in both grades. Thus, the adequacy of the normal model was confirmed, since the predictions based on the model were similar to the observed values (Poisson's P>0.40 for both comparisons).
To evaluate the model further, we systematically varied each component of the definition of disability in reading or mathematics, including the measures used to test academic achievement and intelligence, the cutoff points for disability, and the year or grade. Specifically, we calculated the expected degree of overlap between each of the pairs of disability classifications at each of four cutoff points (-2 SD, -1.5 SD, +1.5 SD, and +2 SD), using each of the three IQ scores (verbal, performance, and full-scale) for grades 1, 3, and 5 and two achievement scores (reading and mathematics) for grades 1 through 6. A complete enumeration of all possible bivariate combinations of discrepancy scores yielded a total of 3402 comparisons. Our model fit the data well for the 3402 pairs; only rarely was there a lack of fit at the 5 percent level (two-sided). There were 26 failures of the model to fit the data at -2 SD, 49 at -1.5 SD, 171 at +1.5 SD, and 51 at +2 SD. Because of the use of multiple tests, we would expect to find a lack of fit for 5 percent of the pairs at each cutoff value. The observed values were consistent with what would be predicted by chance, further supporting the appropriateness of the model.
Discussion
This study allowed us to investigate the commonly held belief that dyslexia is a discrete diagnostic entity. Our data do not support this notion. Rather, they suggest that dyslexia occurs along a continuum that blends imperceptibly with normal reading ability. These results indicate that no distinct cutoff point exists to distinguish children with dyslexia clearly from children with normal reading ability; rather, the dyslexic children simply represent the lower portion of a continuum of reading capabilities. These findings support the need for a fundamental revision in the theoretical framework underlying the study of dyslexia; such a conceptual shift has both investigational and clinical implications.
Our results differ from those of Rutter and Yule,2 whose finding of an overrepresentation of subjects in the lower tail of the distribution provided the basis for the widely held belief that dyslexia is a discrete entity. Careful review of the study design and measures used in their Isle of Wight studies11 , 12 raises serious methodologic concerns, however. The results were based on group tests in which a measure of reading selected to identify the poorest readers was used; this measure imposed a ceiling on reading ability and therefore skewed the reading scores and caused the artifactual appearance of a "hump" or lower mode. Thus, the hump observed may reflect not so much an overrepresentation of poor readers but an underrepresentation of readers at the upper limits of the distribution due to the ceiling imposed by the test itself. According to van der Wissel and Zegers,13 the imposition of this ceiling could easily skew the distribution and thus produce the appearance of a hump at the lower end of the distribution. Our results, based on an individually administered reading test that did not impose a ceiling on scores, are consistent with those of other investigators14 , 15 who, in the 20 years since the publication of the Isle of Wight findings, have also been unable to replicate Rutter and Yule's results.
Although every effort was made to minimize sources of variation, the possibility still exists that characteristics associated with our sampled population, with the measures we used, or with the procedures for administering the tests could have influenced our results. For example, tails of distributions are especially sensitive to small effects, since the number of observations is low in relation to the number of observations in the central portion of the distribution. Thus, the absence (or presence) of a small mode, or hump, could simply reflect certain characteristics of the study population. Another possible explanation for the absence of a hump in our study and others involves statistical power. If the true rate of dyslexia were very low (for example, 0.5 to 1 percent), our study would have low statistical power (<20 percent) to detect a hump. Although theoretically possible, such a low rate is inconsistent with the prevalence of dyslexia reported in the literature. Yule et al.12 based their findings on an estimated prevalence of 2.3 percent, reflecting a cutoff at 2 SD. At this cutoff, they reported an observed prevalence ranging from 3.1 percent to 9.3 percent, with an average of 5 percent. At 5 percent prevalence, our study has great statistical power (89 percent) to detect a low-end hump. In addition, the absence of a small mode could reflect the test-standardization procedures themselves, although the WISC-R and the Woodcock—Johnson tests are the current gold standard for measuring IQ and reading, respectively.16 , 17 Thus, although our data are consistent with the hypothesis that dyslexia follows a normal distribution, it is still possible that a small second mode may have gone unnoticed. Many or even most children who are labeled dyslexic may come from the lower end of the normal distribution; however, we do not wish to rule out the possibility that some may, in fact, have a reading disability of qualitatively different origin or a unique biologic deficit.
The demonstration that dyslexia conforms to a normal-distribution model has important implications for common clinical practices related to dyslexia. Current public policy for kindergarten screening and the early identification of dyslexic children is based on the premise that dyslexia is a discrete entity that is stable over time. Our findings indicate that the diagnosis of dyslexia is not constant over time but will show a predictable year-to-year variability. For example, as predicted by the model, we found that only 28 percent of the children classified as dyslexic in grade 1 were also classified as dyslexic in grade 3. These findings come at a time when there are increasing pressures to diagnose learning disabilities as early as possible in children's school careers. The possible benefits of early diagnosis must now be tempered by the knowledge that as many as two thirds of the children given this diagnosis early (in the first grade) will not meet the criteria for reading disability two years later. These findings also have important implications for intervention studies designed to treat dyslexia. Investigators and clinicians should be cautious in attributing the apparent cure of a child's reading disability to a particular intervention, since without any intervention two thirds of first-grade children with dyslexia would no longer be considered dyslexic in the third grade.
These new findings also explain the puzzling results of many efforts to develop a screening battery that, when administered in kindergarten, can reliably predict reading disability in later grades. In contrast to the initial sanguine reports based on the analysis of short-term outcomes (from kindergarten to grade 1 or 2),18 , 19 the results of more recent, longer-term follow-up studies of these children to the sixth grade have been almost uniformly disappointing.20 , 21 Our data not only are consistent with such findings but also go one step further in providing an explanation for these results. Our findings suggest that the early measures that serve as predictors have not failed, but rather the outcome itself, the diagnosis of dyslexia, is unstable over time. As a result, children who meet the criteria for the diagnosis one year will not necessarily meet the criteria another year. Thus, the outcome measure, the classification as having dyslexia, varies from year to year. For example, our data indicate that only 17 percent of the children classified as dyslexic in grade 1 will be so classified in grade 6. Almost every school system administers a variation of a kindergarten screening battery to each child who enters school, and placement or tracking in school is frequently dictated by the results of such screening. Parents often turn to their pediatrician or family physician when children "fail" such screenings,22 and physicians need to be aware of the absence of long-term validity of these predictive measures.
Finally, the notion that dyslexia is a discrete entity has provided the basis for a special-education policy that provides services only to those who satisfy what are seen as specific, unvarying criteria for dyslexia. In contrast, our findings indicate that dyslexia is not an all-or-none phenomenon but, like hypertension and obesity, occurs in varying degrees of severity. Although limitations on resources may necessitate the imposition of cutoff points for the provision of services, physicians must recognize that such cutoffs may have no biologic validity. Instead, children who do not meet these arbitrarily imposed criteria may still require and profit from special help.
Supported by grants (PO1 HD21888 and P50 HD25802) from the National Institute of Child Health and Human Development, by a grant (GOO-8535118) from the U.S. Office of Education, by a contract with the Connecticut Department of Education, by a National Research Service Award (MH15758) from the National Institute of Mental Health, and by a grant (CA50287) from the National Cancer Institute.
We are indebted to Ralph I. Horwitz, M.D., and Joseph B. Warshaw, M.D., for their thoughtful review of the manuscript; to the children, their parents, and the educators whose participation in the Connecticut Longitudinal Study made this research possible; to the staff of the Yale Center for the Study of Learning and Attention Disorders, particularly Abigail Sadler for her efforts in maintaining contact with the children's families; to John M. Holahan, Ph.D., for technical assistance; and to Carmel Lepore for assistance in the preparation of the manuscript.
Source Information
From the Departments of Pediatrics (S.E.S., B.A.S.) and Neurology (B.A.S.), the Child Study Center (S.E.S., B.A.S.), and the Division of Biostatistics (R.M.), Yale University School of Medicine, New Haven, Conn.; the Department of Statistics, Carnegie—Mellon University, Pittsburgh (M.D.E.); and the Department of Pediatrics, University of Texas at Houston (J.M.F.). Address reprint requests to Dr. Sally E. Shaywitz at the Department of Pediatrics, P.O. Box 3333, Yale University School of Medicine, New Haven, CT 06510–8064.
Appendix
To characterize the relation between the type of achievement test and the type of IQ score, we let A denote a random variable for a type of achievement test and I a random variable for a type of IQ test. For example, A might represent the age-standardized reading-achievement score in grade 1 and I the verbal IQ score in grade 1. We assumed that the random variables follow a multivariate normal distribution with a mean and standard deviation of MA and SDA for A and a mean and standard deviation of MI and SDI for I. The correlation of A and I is R. The value of the discrepancy score (DS) is defined as the residual of a least-squares fit of A on I. Therefore, DS is defined as follows: DS = (A - MA) - R · SDA/SDI · (I - MI), (1) where SD denotes standard deviation. DS designates a continuous variable that represents the amount of reading and mathematics disability (termed "learning disability" [LD]), defined by achievement test A and IQ test I. Although an expectation of normality is embedded in the IQ and achievement-test scores, it does not necessarily follow that DS is normally distributed.
In terms of notation, we defined LD as follows: 1 if DS < C · SDA · √1 - R2, and 0 otherwise, (2) where C is defined as the cutoff point used in a particular discrepancy analysis. To study the effects of two different definitions of DS1 and DS2, we note that DS1 and DS2 are functions of the sets of random variables A1 and I1 and A2 and I2 according to equation 1. If the distribution of A1, A2, I1, and I2 is multivariate normal, then DS1 and DS2 are jointly bivariate normal variables with a mean of 0 and covariance (COV(DS1,DS2)) defined as follows: COV(DS1,DS2) = SDA1 · SDA2 · [Corr(A1,A2) + Corr(A1,I1) · Corr(A2,I2) · Corr(I1,I2) - Corr(A1,I2) · Corr(A2,I2) - Corr(A1,I1) · Corr(A2,I1)], (3) where Corr denotes correlation. The correlation of DS1 and DS2 is as follows:
Since DS1 and DS2 are bivariate normal variables with the correlation defined in equation 3, functions of DS1 and DS2 arc defined according to standard normal theory. Let F(a,b;r) denote the standard normal bivariate cumulative distribution function with correlation r, where (a,b) is the point defining the lower left quadrant from which the probability F(a,b;r) is calculated. Then the probability (P) that a child will be classified as learning-disabled according to two different definitions LD(A1,I1,C1) and LD(A2,I2,C2) is
To examine the goodness of fit of the observed data to the normal-distribution model, several steps were taken. Empirical, graphic methods were used in addition to more formal tests involving skewness (symmetry) and kurtosis (peakedness). In addition, we used the total number of individual tests that exhibited lack of fit at the 5 percent level to measure the goodness of fit of the observed data to the normal model. The exact distribution of this test statistic is difficult to ascertain because of the correlated nature of the tests. Thus, the binomial distribution was selected as a guide in evaluating the discrepancy between the observed and predicted values. This nonconservative test tends to reject the hypothesis of multivariate normality too frequently. This feature is desirable, since the failure to reject the hypothesis (despite nonconservatism) indicates that the observed distribution is quite consistent with the assumption of multivariate normality.23 24 25
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