Familial Aggregation Patterns in Mathema

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genetic contributions at both ends of the age spectrum (McClearn et al.,1997; Petrill et al., 1997, 1998). In addition, a genetic linkage study obtained suggestive evidence for quantitative trait loci (QTL) related to gen-eral cognitive ability (Fisher et al.,1999), although this was not confirmed in another study by the same group (Plominet al.,2001).

There are strong intercorrelations between mea-sures of IQ or general cognitive ability and meamea-sures of individual cognitive processes or abilities. One study reported high correlations between a measure of both mathematical and verbal reasoning and general intelli-gence,g (Carroll, 1993). Another study reports high correlations between mathematics achievement and general cognitive ability in twins (Alarcón et al.,2000) and estimates that 90% of the variance in a latent factor for mathematical performance can be accounted for by

1_{Division of Medical Genetics, Department of Medicine.} 2_{Department of Biostatistics.}

3_{Department of Genome Sciences.}

4_{Department of Psychiatry and Behavioral Sciences.}

5_{Regional Primate Research Center, University of Washington,}

Seattle.

6_{To whom all correspondence should be addressed at Division of}

Medical Genetics, Box 357720, University of Washington, Seattle, Washington 98195-7720. Tel: 206-543-8987. Fax: 206-616-1973. e-mail: wijsman@u.washington.edu

Familial Aggregation Patterns in Mathematical Ability

Ellen M. Wijsman,

1,2,3,6

_{Nancy M. Robinson,}

_{Kathryn H. Ainsworth,}

Elisabeth A. Rosenthal,

_{Ted Holzman,}

_{and Wendy H. Raskind}

Received 15 Mar. 2002—Final 6 May 2003

Mathematical talent is an asset in modern society both at an individual and a societal level. En-vironmental factors such as quality of mathematics education undoubtedly affect an individual’s performance, and there is some evidence that genetic factors also may play a role. The current study was performed to investigate the feasibility of undertaking genetics studies on mathe-matical ability. Because the etiology of low ability in mathematics is likely to be multifactorial and heterogeneous, we evaluated families ascertained through a proband with high mathemati-cal performance in grade 7 on the SAT to eliminate, to some degree, adverse environmental fac-tors. Families of sex-matched probands, selected for high verbal performance on the SAT, served as the comparison group. We evaluated a number of proxy measures for their usefulness in the study of clustering of mathematical talent. Given the difficulty of testing mathematics perfor-mance across developmental ages, especially with the added complexity of decreasing exposure to formal mathematics concepts post schooling, we also devised a semiquantitative scale that incorporated educational, occupational, and avocational information as a surrogate for an aca-demic mathematics measure. Whereas several proxy measures showed no evidence of a genetic basis, we found that the semiquantitative scale of mathematical talent showed strong evidence of a genetic basis, with a differential response as a function of the performance measure used to select the proband. This observation suggests that there may be a genetic basis to specific mathematical talent, and that specific, as opposed to proxy, investigative measures that are de-signed to measure such talent in family members could be of benefit for this purpose.

KEY WORDS: Cognitive ability; aggregation analysis; familial correlation; polygenic; complex genetic model.

INTRODUCTION

There is increasing recognition that cognitive function is strongly influenced by genetic factors. Twin, family, and adoption studies support a substantial genetic con-tribution to IQ (e.g., Bouchard and McGue, 1981; Bouchardet al.,1990; Chorney et al.,1998), and mea-sures of overall cognitive abilities show evidence of

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genetic effects. Finally, mathematics disability is often comorbid with reading disability in studies with ascer-tainment through individuals with learning disabilities (Gilliset al.,1992; Light and DeFries, 1995; McCleod and Armstrong, 1982). These correlations between measures of mathematics and verbal ability in both the normal and disability ends of the spectrum, coupled with evidence that deficits in verbal measures associ-ated with learning disabilities may have a genetic basis (Raskind, 2001), suggest that there may also be a genetic basis to mathematical ability.

There have been a few investigations into genetic contributions to mathematical cognitive ability. Most have studied the possible genetic etiology of mathe-matics disability, at least partly because of its comor-bidity with reading disability (Gillis et al.,1992; Light and DeFries, 1995; McCleod and Armstrong, 1982). Results from a twin study were consistent with a ge-netic basis for mathematics disability whether com-bined with reading disability or not (Alarcón et al.,

1997), and estimates of high heritability of mathemat-ical ability were obtained in a sample of twins with nor-mal intelligence ascertained for reading disability (Gillis et al.,1992). This study and a subsequent one by the same group implicated shared genetic factors for both disabilities (Knopik et al.,1997). In addition to mathematics disability, mathematics performance along the normal continuum also provides evidence of a strong heritable component (Alarcón et al.,2000; Gillis

et al.,1992; Thompson et al.,1991; Wilson, 1975). There are only a few published reports bearing specifically on family patterns of high mathematics achievement. Both the early emergence of distinctive mathematical talent (Robinson et al.,1996, 1997) and sex differences at the highest level of mathematical competence (Benbow, 1988; Benbow and Stanley, 1981) suggest that this ability may have a biological basis. Mathematical talent was well represented in the families of students who had placed high in a mathe-matics contest in East Germany (Weiss, 1982, 1994). A study of adolescents with high mathematical ability and their parents (Benbow et al.,1983a, b) found high general ability in the parents and considerable assorta-tive mating, but did not specifically examine genetic correlations in mathematical competence.

A study of the extremes of mathematical cognitive ability represents a potentially effective strategy to study the genetic basis of such ability. Within this ap-proach, study of subjects with high ability may be more effective than would be study of individuals with low ability. The high end of the distribution represents the normal to optimal interface of environmental and genetic factors, whereas the study of the etiology of

low ability in mathematics is likely to be confounded by many deleterious influences (Knopik et al.,1997). A similar approach has recently been taken in the study of the genetic basis of perfect pitch (Baharloo et al.,

2000) and of resistance to AIDS (Martin et al., 1998; Smithet al.,1997). A further, pragmatic, advantage of studying high-end talent is the fact that it is readily identifiable. Many thousands of seventh-graders each year participate in regional talent searches in the United States by taking either the Scholastic Aptitude Test (SAT) given nationally by the Educational Testing Ser-vice, or the American College Test (ACT) given by the American College Testing Program.

However, the study of the etiology of mathemati-cal talent may present its own difficulties. Environ-mental factors demonstrate substantial correlations with both verbal and numerical abilities (Thompson et al.,

1991; Walberg and Marjoribanks, 1973) and are also likely to influence the level of mathematical attainment as well as talent, and secular trends in educational prac-tices and social factors may confound the analysis. Although mathematical talent can be assessed by stan-dardized tests directly within an age cohort of young people, measurement of such ability in older family members may be much more problematic. Adding to the complexity of research in this area is the fact that mathematical ability is strongly correlated with general intelligence. Therefore any study of families ascer-tained for high mathematical ability must attempt to control for general level of intelligence if the goal is to determine whether there are components of mathemat-ical ability that are unique and are not simply a side ef-fect of overall general ability. Most of the studies of the familial nature of mathematical ability mentioned above did not control for this overall general ability.

Here we present the results of a preliminary study designed to determine whether there is evidence for a genetic basis, which is distinct from overall talent, for several measures of mathematical talent. We evaluated the prevalence of several measures of mathematical ability within families of young people with demon-strated advanced mathematical reasoning ability. To control for general level of intelligence within the fam-ilies, we also assessed mathematical ability in compa-rable families chosen for high verbal ability of the proband.

METHODS Overall Design

We used a case-control/family design to control for the effects of general cognitive ability. By comparing

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various measures of mathematical attainment in rela-tives of mathematically precocious probands (cases) with relatives of equivalently verbally precocious probands (controls), we could expect that overall cog-nitive abilities of cases and controls and their relatives were likely to be similar. Evidence for familial factors contributing specifically to high mathematical ability could then be evaluated by determining whether or not there were differences in measures of high mathema-tical ability in relatives of case vs. control probands, as well as whether there was a relationship between the measures and the degree of kinship with the proband. In this design, the cases were the probands with high SAT-M scores, and the controls were the probands with equivalently high SAT-V scores. We use the broad definition of a proband here: the individual who brings the family to attention (Lynch and Walsh, 1998). Here the ascertainment is through the higher percentile of the two SAT scores, so that both groups of preco-cious individuals are, by definition, probands. Cases and controls were approximately frequency matched, both on sex and on the number of each type of proband (verbally or mathematically talented) recruited into the study. Regression analysis methods were used to pro-vide comparisons of measures in relatives of case vs. control proband families.

Sample

Families of Washington State seventh-graders who, from 1988 through 1998 had participated in the Johns Hopkins Talent Search (Barnett and Juhasz, 2002; Stanley and Brody, 2002), constituted the origi-nal pool from which students were selected. All of these students had taken the SAT during seventh grade, gen-erally at age 12, having qualified to do so by virtue of having previously scored at the 97th percentile or higher on an age-normed comprehensive test of verbal skills, mathematical skills, or general aptitude. From records maintained by the Center for Talented Youth of Johns Hopkins, scores in the highest percentiles were adjusted for year-effects due to fluctuations in test dif-ficulty and recentering of the SAT. Lists were con-structed separately for males and females who scored in at least the 97th percentile of Talent Search partici-pants on either the mathematical reasoning scale (SAT-M) or on the verbal reasoning scale (SAT-V), or on both scales. These students represented the top 0.1% in academic ability of students their age.

Within each sex, individuals were assigned to either the high mathematics or the high verbal list, with assignment based on which of the two score percentiles was in the highest sex-specific percentile. Students with

apparently Asian surnames were excluded from recruitment, to avoid the need to match on ethnicity in addition to sex because of the marked difference in the sex discrepancy among individuals with the highest SAT-M scores in Asian-American compared to Caucasian samples (Lubinski and Benbow, 1992). No attempt was made to contact these individuals or their family members. Parents from this list of potential probands were contacted, starting with the highest per-centile scores on each list, until 22 of each type of proband had agreed to participate (44 male and 44 fe-male probands, respectively, split evenly, in each sex, between math and verbal probands). Analyses were per-formed on the subset of these (66, total) who had re-turned questionnaires by a specific date. Three subjects whose siblings with higher scores were among the probands were also excluded as probands, although they were eligible as family members.

Questionnaire

Information on cases and controls and their fam-ily members was collected by questionnaire. A tele-phone call to a parent of each proband described the study, determined whether English was the language of the home (no students were dropped because commu-nication with the parents would have been impaired), and ascertained interest in continuing. During this or a second call, a pedigree was drawn, after which a visual record of the pedigree was mailed to that parent, along with copies of a consent form to be distributed to po-tential informants, up to two on the maternal and two on the paternal side of the family. Information from only one informant was used for any given relative, based on the quantity of completed questions about that relative. In most cases, at least one of the parents was one of the informants, but where another relative (e.g., a grandparent) had more information on family mem-bers, such a relative was used as an informant. After consent forms were returned, together with any cor-rections in the pedigree, the same interviewer tele-phoned the informants who had consented to participate and conducted a detailed interview lasting 45–75 min about each member of the family, including grandpar-ents, pargrandpar-ents, siblings, avuncular relatives (aunts and uncles), and cousins. Questionnaires included date of birth, ethnicity, educational information such as major areas of study and honors attained, occupations, and other interests and skills as children and as adults. Ad-ditional information on these measures is described below.

To process the interview data, a list of educational domains, occupations, and avocational interests was

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created, each of them characterized by up to three of the eight ability domains (or “intelligences”) described by Gardner (1983, 1999). Coders were blind to the group to which the proband belonged. Exact agreement by two independent coders exceeded 95% in sampled cases. In addition, using a 1–5 summary rating scale reflecting lifetime level of ability and interest based on all available information, coders rated the mathemati-cal domain for each individual. This smathemati-cale is referred to as MRATE (see variables below) and was a subjec-tive lifetime measure designed to maximally stratify the relatives of the probands with regard to overall evidence of mathematical talent. Also, because it was based on all available data, it had a lower missing data rate than other measures. A rating of 1 indicated below-average ability; 2 indicated below-average educational, voca-tional, and avocational mathematical interests; and 3–5 indicated increasingly higher above-average mathe-matics performance. A rating of 3 indicated, for ex-ample, long-term employment as a shopkeeper or bookkeeper or an interest in playing mathematical games; 4 was used for juveniles when there was evi-dence of high mathematical ability through outstand-ing performance in math competitions, or for adults, when mathematics played a significant role in a pro-fession, such as engineering, computer science, or physics; and a rating of 5 indicated that the individual was truly remarkable or eminent, including demon-stration through named mathematics professorships, prestigious academic prizes, etc. Juveniles (including the probands) could not obtain a score of 5 because of the need for demonstration of eminence. Individuals with at least some historical occupational and educa-tional data (beyond high school) but no distinctive in-formation in the domain were rated a “2,” on the assumption that a reputation for high or low ability in an area would have been recognized within the family. A few individuals had no occupational data and no ed-ucation beyond the high school level; such individuals were coded as unknown.

Variables

Seven measures were used as proxy measures of mathematical talent in the statistical analyses. Each of these was used, individually, as the response variable in a regression analysis. Five binary-outcome measures were used: whether or not the undergraduate or gradu-ate major field of study was mathematics (UGMATH, GMATH) and whether or not the individual partici-pated in mathematics activities during school (MACT), or, as adults, had hobbies (MHOB), or a long-term

occupation that is indicative of mathematics talent (MOCC). Only pure mathematics as a field of study was considered for the variables UGMATH and GMATH because of the difficulty of distinguishing between applied and theoretical mathematics in other quantitative fields such as physics, engineering, ac-counting, and computer science. Mathematics hobbies and occupations were based on involvement of logical/ mathematical “intelligence” as defined by Gardner (1983, 1999). A seven-level, ordered categorical vari-able (HIMATH) indicated the highest-level mathemat-ics class taken. A 5-point mathematmathemat-ics rating scale, as described above, was used to summarize the overall evidence for mathematical talent and interest across the life span (MRATE).

Variables that were not direct measures of math-ematical talent, but that might be predictive of, corre-lated with, or confounded with measures of such talent, were included as covariates in some or all regression analyses. These variables were the kinship coefficient, extracted from the pedigree structure data with the pro-gram RELATE (S.A.G.E., 1997); education level; sex; generation number; and proband status. For most re-gression analyses, the proband variable corresponding to the relatives’ response variable was also included as a covariate. The exception to this was final regression models in which the probands’ measure was not a sig-nificant predictor of the relatives’ measure. The kin-ship coefficient between relatives of probands was used as a covariate in all regression analyses designed to de-termine whether there was a potential genetic basis, as described below. Other variables were included as co-variates in some regression analyses: education level was a nine-level, ordered categorical variable, with highest completed level of education as the outcome variable (HI-ED); sex (with female as the baseline); generation number to control for cohort effects (with the grandparents in generation 1 and the probands in generation 3); and proband status (with the control as baseline), were all also used in some analyses. Some of these variables (e.g., sex) could only be used in analyses that focused on individual relatives rather than average scores across a class of relatives (described under statistical analyses, below). These variables, as well as others, which were not restricted to analyses of individual relatives (proband status, cohort effects), were explored as possible confounders and only re-tained in the model if they provided significant im-provement of fit to the model.

Variables that are proxy measures of mathemati-cal talent were only occasionally used as covariates in the analysis of other response variables. A few models

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were fit to one such measure as the response variable, with others proxy measures of mathematical talent as covariates, in the absence of terms for proband status and kinship. These models were constructed to obtain a better picture of the relationship among certain vari-ables, particularly MRATE vs. the remaining, more specific measures, such as MOCC. The models fit in this fashion were always based on the expectation that there might be a relationship among the covariates in the model and the response variable, such as the ex-pectation that there might be a relationship between the score for MRATE and MOCC, or knowledge that there may be secular changes in education pursuits and level, and different trends in males and females.

Statistical Analyses

Regression Models

If genetic factors contribute specifically to math-ematical talent, certain expectations for regression coefficients should be observed when relatives’ scores are regressed on probands’ scores. First, a regression toward the mean for measures of mathematical talent would be expected in relatives of probands that were selected on extreme scores. Second, there should be a statistically significant relationship between the kin-ship coefficient and the measure in question in the probands vs. in their relatives. In addition, if the cases are more extreme on the measures in question than are the controls, we would expect the slope of the regres-sion to be more pronounced in relatives of cases than controls. In the absence of specific genetic contribu-tions to mathematical talent, we would expect the re-gression coefficients for the kinship coefficient to be similar for relatives of the math and verbal probands, even if, as might be expected in families selected through talented probands, there is a significant over-all regression to the mean as a function of the kinship coefficient. Use of regression to the mean as a measure of genetic effect is the basis behind the design proposed by DeFries and Fulker (1985), originally in the context of a twin design, but easily adaptable to other pedigree-based designs. Under this framework, model 1 was the basis for all regression analyses that were used for ini-tial investigation of the genetic basis of measures of mathematical talent:

yj f =␮+␣xj f +␤␾j f +ej f, (1)

whereyj f is the observed phenotypic measure in rela-tive j in family f, xj f is the observed measure in the proband in family f, ␾j f is the kinship coefficient of

individualjand the proband in family f,ej f is the error term for individual j in family f, and the remaining terms are regression coefficients. The coefficient ␣ is the partial regression of the relatives’ scores on the probands’ scores and accounts for resemblance that is independent of genetic factors. ␤is the partial regres-sion of the relatives’ scores on the kinship coefficient, with a significant value indicative of genetic factors, thus providing a test of significance for genetic etiology.

Initial models based on Eq. (1) were fit for each individual measure as the response variable, using the kinship coefficient and equivalent proband measure as the independent variable. The purpose of constructing these initial models was to determine which, if any, measures showed evidence for a relationship between relatives and probands that could be indicative of a ge-netic basis, for example, a value for ␤that was statis-tically significantly different from zero. For measures that showed some evidence for a relationship between the phenotype and the kinship coefficient, additional models were fit to determine whether the slope of the regression differed between proband types. This was achieved by adding indicator variables for proband type (␲j =1, 0 if individual jis a relative of a math or verbal

proband, respectively) to include the possibility of proband-specific intercepts and slopes, to give model 2:

yj f =(␮+␥ ␲j f)+␣xj f +(␤+␦␲j f)␾j f +ej f, (2)

where ␥ and ␦give the shift, relative to the control probands, in the intercept and slope.

For those measures for which ␤and/or␦was sig-nificantly different from zero, additional models were fit to include other measures as covariates in linear models. The purpose of building these additional mod-els was to explain more of the variance than did the ini-tial models and to further explore the relationship among the available variables. The only additional vari-ables that were considered as covariates in these analy-ses were those that were not themselves candidates for measures of mathematical talent. The basic structure behind these models (model 3) can be represented as:

yj f =_(␮+_{␥ ␲}_{j f}₎+_␣xj f +_(␤+_␦␲_{j f}_)␾_{j f} +

␯izi j f +ej f, (3)

wherezi j f is the value of covariate iin individual jin familyf. Interactions between some of these covariates were also considered.

Linear regression was used to fit the coefficients in the models when the dependent variable was con-tinuous or multilevel integer, and logistic regression

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was used when the dependent variable was a binary trait. Covariates were added first as main effects, and then as interaction terms, for any covariates that were significant as main effects. Stepwise regression, using both forward and backward searching with inclusion of coefficients at the 5% significance level or better, was used to identify the most parsimonious models. In all cases, main effects were left in the model when inter-actions were also included, even if then the regression coefficients for the main effects were statistically in-significant. The coefficient ␣, however, was not forced to remain in the final regression model. The rationale for this was that this coefficient simply reflects the shared covariance of relatives with the probands re-sulting from, for example, shared environment, after allowing for the relationship and other covariates. It is possible, in the absence of shared environment, that this coefficient is not significant. Adequacy of final mod-els was evaluated through plots of residuals, by pre-dicted values and normal quantiles, and by investigation of the effect of the most influential data points identi-fied through Cook’s distance. Because of the ex-ploratory nature of the current study, no corrections were made for multiple testing.

Pedigree Data

Unlike twin studies, in which the data come in nat-ural pairs, families come in a variety of sizes. Analy-ses were done in two ways to accommodate this size variation. First, all observations were given equal weight (individual-based regressions), such that yj f

refers to the response value for individual jin family

f. This makes maximal use of the information, but is subject to undue influence of large sibships. Second, the average phenotypic value for all relatives of a par-ticular type within a family was used (family-based re-gressions), where the types of relatives in the study were siblings, first cousins, parents, grandparents, and avuncular relatives. For this approach, yj f refers to the average value of a measure among all the relatives of a particular type (e.g., siblings). This gives equal weight to families, and was the type of regression given the most weight in interpreting the results, because it is closest to the original framework described by DeFries and Fulker (1985). In addition, by averaging over rel-atives of a particular type, the measures become con-tinuous rather than categorical, which is advantageous in the regression framework. All analyses are based on the joint analysis of all relative pairs, using one of these two weighting schemes to combine relative-pair types in the regression analyses. Some analyses were also

done separately for different relative pair types, but be-cause results were consistent with those for the com-bined relative pair types, these other analyses are not discussed here. For all analyses, pedigrees consisting of parents, siblings, grandparents, avuncular relatives, and first cousins were used in analysis, by extracting these individuals from larger pedigree with the program

RELATEfrom the S.A.G.E. package.

RESULTS

Participant Statistics

Of subjects who took the SAT in seventh grade in the state of Washington between 1988 and 1998, 342 with non-Asian surnames were eligible as probands on the basis of either their high SAT-M or SAT-V scores (the higher score being at least 650). Of these, information about proband sex was available on 304 eligible probands. Of the 98 eligible female probands, 35 qualified as math probands (higher SAT-M than SAT-V percentiles), 62 qualified as ver-bal probands (higher SAT-V than SAT-M percentiles), and 1 had identical scores on both tests. For purposes of the discussion that follows, we refer to the lower of the two SAT scores (measured as a percentile score) as the nonqualifying score. Of the 206 eligible male probands, 144 qualified as math probands, 58 quali-fied as verbal probands, and 4 had identical scores on both tests. Starting with eligible probands in each of the four possible proband groups (male math, female math, male verbal, female verbal) with the highest qualifying scores, 92 of these qualifying probands ini-tially were contacted. Four contacted probands were subsequently eliminated based on ineligible scores (e.g., because of initial recording errors) or because they had a sibling who was already a proband, leaving 23 female verbal, 21 female math, 22 male verbal, and 22 male math probands. Ultimately, parents of 26 probands chose not to participate. The final data set of probands consisted of 66 probands: 34 females and 32 males. Of the females, 15 were verbal probands (female controls) and 19 were math probands (female cases). Of the males, 18 were verbal probands (male controls) and 14 were math probands (male cases). The SAT scores and ages of the contacted nonpartici-pating probands were not markedly different from those of the participating probands, although the total SAT scores and the SAT-M scores for the female non-participating probands were slightly lower than the totals for the other participating and nonparticipating proband groups (results not shown).

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Fig. 1. Distribution of SAT scores by proband category. Dark bars: males; light bars: females. White lines: median score. Boxes: in-terquartile range. Whiskers extend to 1.5 times inin-terquartile range. Individual black lines: extreme scores.

Descriptive Statistics

Correlations between SAT-V and SAT-M scores in the participating probands were generally high (Table I). Correlations between the two SAT scores were higher for the verbal probands than for the math probands for both sexes combined (p=₀.05) and were higher for male probands than for female probands for both types of probands. In addition, the correlation be-tween the two scores in the female math probands was considerably lower than the same correlation in the other proband groups, although the results within sex are not statistically significant, given the small sample sizes. In addition, differences between the minimum and maximum scores were similar among the four groups for both the qualifying (100–140 points) and nonqualifying SAT scores (180–220 points). The most discrepant of the probands in this regard was the fe-male verbal group, with a spread of 140 points in the qualifying score, and 180 points in the nonqualifying score. The other proband groups all had a spread of 100 points in the qualifying score, and a spread of 190–220 points in the nonqualifying score.

The distribution of qualifying vs. nonqualifying SAT scores was similar for the math and verbal par-ticipating proband groups, when sexes were combined. For the verbal probands, the SAT-V scores averaged 698.82 (SD=₃₁.2), while for the math probands the SAT-M scores averaged 706.18 (SD=₃₆.9). For the nonqualifying SAT, the SAT-M scores for the ver-bal probands averaged 583.53 (SD=₆₀.8), while the SAT-V scores for the math probands averaged 571.18 (SD=₆₀.8). The medians were also similar: on the qualifying test, the median SAT was 700 for both groups, while for the nonqualifying test it was 580 for the SAT-M among the verbal probands and 570 for the SAT-V among the math probands.

When participating proband groups are separated by sex, differences between male and female probands become apparent (Fig. 1). These differences are quite large for the SAT-M scores, but considerably lower for the SAT-V scores. For the SAT-M scores, the male math probands averaged 47.7 points more than the fe-male math probands, while this difference between

SAT-M scores was 23.4 points for the male vs. female verbal probands. In contrast, whereas females had slightly higher SAT-V scores, on average, the differ-ence between average female and male SAT-V scores for the verbal probands was a negligible 3.41, and for the math probands the difference between the SAT-V scores for female and male probands was 8.04. The large difference between qualifying and nonqualifying SAT scores was observed not only in the mean scores within each proband type but also in the mean dif-ference between the two scores for each proband type. For the math probands, the mean difference between the math and verbal scores was 129.7 (SD=62.4) overall, and 157.1 (SD=58.6) and 109.5 (SD=58.4) for the male and female math probands, respectively. For the verbal probands, the mean difference between verbal and math scores was 115.2 (SD=52.1) over-all and 102.8 (SD=51.4) and 130.0 (SD=50.6) for the male and female probands, respectively. Only three probands (1 female math, 1 female verbal, and 1 male verbal proband) had a difference between qualifying and nonqualifying scores of less than 30 points.

Total SAT scores (SAT-V +_{SAT-M) were}

simi-lar in the four participating proband groups. Among fe-male probands, the mean (SD) total scores were 1260.0 (69.6) and 1271.9 (84.16) for the math and verbal probands, respectively. Among male probands, mean scores were 1299.0 (78.2) and 1291.7 (82.6) for the math and verbal probands, respectively, for slightly higher total scores for the male than the female probands. However, all differences were within 1 SD and are statistically nonsignificant.

Table I. Correlations Between SAT-V and SAT-M Scores in Participating Proband Groups

Proband group Female Male Total

Verbal 0.58 (15) 0.64 (18) 0.57 (33)

Math 0.23 (19) 0.37 (14) 0.22 (33)

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The female math participating probands were younger at the time of this study than were the other three groups of probands (Table II). Although the me-dian age of the female math probands was the same as that of the other probands, the distribution of ages was different, as reflected in the lower mean age of the fe-male math probands. Only one fefe-male math proband was older than 18 years of age, but seven of the male math probands were over 18 years of age at the time of the study. The age difference was the result of a dif-ferent distribution of birth years, not a difference in the age at which the SAT was taken. Over the period dur-ing which probands had taken the SAT, sex differences in extreme SAT-M scores were decreasing from proximately 13:1 (Benbow and Stanley, 1981) to ap-proximately 5 or 6:1 (Brody et al.,1992), where they appear to have stabilized (personal communication, Brody, 2002). The females and males of the verbal probands had a similar age distribution, although the females were younger than the males by 1 year, on average.

The amount of missing data varied widely among measures. Of the 1082 total family members of probands (parents, siblings, grandparents, cousins, and avuncular relatives), three were no data other than re-lationship information available for 129, so these were not included in the estimation of missing data rates. In analyses based on individual family members, of the 953 relatives with data on at least one measure, the missing data rates were 4.1%, 11.6%, 44%, 45%, 60.7%, 61%, 78.6%, and 83.1% for MRATE, HI-ED, UGMATH, MHOB, MACT, MOCC, GMATH, and HIMATH, respectively. When family members were grouped into age/kinship classes and data averaged within class, the total sample size was 309, and the missing data rates in the classes of family members were: 0.6%, 10.7%, 24.9%, 20.4%, 34%, 38.5%, 60.8%, and 70.2% for MRATE, HI-ED, UGMATH, MHOB, MACT, MOCC, GMATH, and HIMATH, re-spectively. Among the probands, the missing data rates were: 0%, 30.3%, 65.2%, 39.4%, 36.4%, 74.2%, 90.9%, and 84.8% for MRATE, HI-ED, UGMATH, MHOB, MACT, MOCC, GMATH, and HIMATH,

respectively. The number of relatives with each of the different MRATE categories, stratified by proband type, is shown in Table III, and demonstrates signifi-cant differences (p=0.013) between the relatives of the case vs. control probands.

Regression Analyses

Three measures (MRATE, MHOB, and MACT) produced regression coefficients for ␤ and/or ␦that were suggestively or significantly different from zero in one or more of the regression analyses for model 1 (Table IV). When the regression model was expanded to include separate coefficients that are a function of proband status (model 2), two of these measures, MRATE and MHOB, continued to provide evidence of a significant regression on kinship coefficient, which was different in the families of cases vs. controls (Table IV). For all three measures, the sign of ␦was positive, as would be expected if the regression to the mean in the relatives of case probands was more pro-nounced than in relatives of control probands. How-ever, the only measure for which there was a strong significant relationship with kinship (coefficient ␤) was MRATE; for this measure, both individual and family-based regressions gave statistically significant results

(p< .05) for both ␥ and␦. MRATE was therefore

pur-sued further, with additional analyses based on model 3. Results for the remaining measures were not signif-icant in this early exploratory analysis (results not shown).

For MRATE and the full model 3, there was a sig-nificant effect of the kinship coefficient in the regres-sion model (Table V). The regresregres-sion coefficient for kinship-by-proband interaction was statistically signif-icant (p=0.024) for both individual-based and family-based regressions. In addition, for the models family-based on family-based regressions, the estimate for␤ was also significant (p=0.014). This parameter was insignifi-cant for the individual-based regressions, suggesting that there was no effect of kinship on MRATE in the families of the control probands, after accounting for the other coefficients in the model (including the

Table II. Distribution of Age at Time of Study Minimum Median Mean Maximum

Proband group age age age age

Female verbal 14 17 17.4 24

Male verbal 15 18.5 18.44 25

Female math 15 17 16.59 19

Male math 14 17 17.93 24

Table III. Distribution of MRATE Scores in Families of Math and Verbal Probands

MRATE score in relative

Proband type 1 2 3 4 5

Math 2 298 129 81 12

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Table V. Best Regression Model 3 for MRATE Weights

Familya _Individualb

Model coefﬁcient Estimate (SD) pvalue Estimate (SD) pvalue

␮: Intercept .818 (.46) .075 .968 (.22) <_.0001

␤: Kinship 1.622 (.65) .014 .407 (.60) .5

␥: Proband −.244 (.16) .133 −.256 (.12) .039

␦: Kinship ×proband 2.114 (.93) .024 2.307 (.82) .005

HI-ED .654 (.17) .0001 .626 (.07) <_.0001

HI-ED2 ₋_{.055 (.01)} _.0002 ₋_{.045 (.006)} <_.0001

Generation .327 (.15) .026 .319 (.07) <_.0001

Generation×HI-ED −.087 (.044) .052 −.08 (.02) .0002

Sex ND −.262 (.09) .005

Sex×HI-ED ND .228 (.048) <_.0001

Sex×HI-ED2 _ND ₋_{.027 (.006)} <_.0001

a_{266 df, R}2₌_.175;b_{797 df, R}2₌_.272.

proband-specific coefficient, ␥). This difference in whether the value for␤ is statistically significant ap-pears to be due to interactions between sex and higher education, since models for family-based regressions that do not include terms for sex and higher education do produce statistically significant estimates for ␤ (results not shown). Higher education and generation effects were significant for both family-based and indi-vidual-based regressions, as were interaction terms be-tween these effects. Under individual-based regression, sex effects were highly significant (p<0.0027), as were sex-by-higher education interactions (p< .0001 for HI-ED-by-sex,p=.03 for female-by-HI-ED2, and

p< .0001 for male-by-HI-ED2).

The regression coefficients for the kinship coeffi-cient remained significant in almost all separate analy-ses of the families of case and control probands. In all cases of analysis of the case probands’ families the re-gression coefficient remained significant at p< .0001. For a model based on individual weights with a sex effect included, the significance of the coefficient in the control-family regression weakened to p<0.1. In addition, the coefficient in the models involving fam-ilies of the case probands was always considerably larger than the equivalent coefficient in the models in-volving the control probands (Table IV), consistent with the faster regression to the mean seen in case vs. control families.

Table IV. Regression Coefﬁcients (SD) for Regression Models 1 and 2 for Measures of Mathematics Talent

Measure Wts Model ␮(SE) ␣(SE) ␤(SE) ␥(SE) ␦(SE)

MRATE Fam 1 2.05 (.15)*** .08 (.04)* 2.48 (.44)*** ND ND

2 2.22 (.18)*** .08 (.05) 1.44 (.62)** −.33 (.16)** 2.06 (.87)**

Indiv 1 2.13 (.13)*** .04 (.04) 2.59 (.4)*** ND ND

2 2.33 (.15)*** .05 (.05) 1.2 (.61)* −.35 (.13)*** 2.46 (.82)***

MHOB Fam 1 .15 (.06)** −.09 (.05)* .12 (.31) ND ND

2 .23 (.09)** −.09 (.05)* −.59 (.53) −.12 (.11) 1.08 (.65)*

Indiv 1 1.11 (.05)*** −.02 (.02) .1 (.28) ND ND

2 −2.04 (.8)*** −.19 (.19) −1.74 (5.2) −.02 (.9) 3.24 (5.84)

MACT Fam 1 .42 (.1)*** −.02 (.08) .54 (.51) ND ND

2 .49 (.15)*** −.003 (.08) .24 (.83) −.13 (.19) .49 (1.06)

Indiv 1 −.39 (.27) .04 (.13) 2.35 (1.76)* ND ND

2 .39 (.5) .1 (.14) −.89 (3.19) −1.1 (.6) 4.49 (3.84)

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For the regression models for MRATE, the signs of the regression coefficients were consistent with ex-pectations if there is a specific heritable component to mathematical talent. For models 1 and 2 (Table IV), the sign of ␣was positive, as would be expected if there is a relationship between MRATE in probands and their relatives. For all three models (Tables IV and V), the sign of ␤ was positive, as would be expected if there is a genetic basis to the phenotype. The sign of ␣was positive, as would be expected if the relatives of cases had higher values of MRATE than relatives of controls, and the interaction term ␦was positive, as would be expected if there is a stronger relationship for kinship in the relatives of cases than in the relatives of con-trols. Finally, most of the other variables that might be expected to be predictive of MRATE gave results con-sistent with this expectation: increased levels of higher education were associated with higher values for MRATE; and individuals in more recent generations were associated with higher values for MRATE. The only possible exception to a difference in expected sign for a regression coefficient was that for male sex, which was associated with lower values for MRATE. However, the sex by higher education interaction term indicated that males had more years of education, on average, than did females after adjusting for other covariates.

DISCUSSION

By examining reports of talent and interests in three-generation kinships, we have found evidence of a gradient of association between high mathematical talent in probands and life-span mathematics attainment in members of their families. This gradient is stronger in the families of probands selected through outstand-ing performance on the mathematics SAT than it is in families of controls whose overall performance on the SAT was equally outstanding but whose scores were higher in the verbal than in the mathematical domain. The regression of relatives’ scores on probands’ scores through the kinship coefficient supports an inter-pretation of heritability of mathematical talent. In ad-dition, the increased slope of the regression coefficient in families of case probands compared to control probands is consistent with an effect that is specific to mathematical talent, as opposed to simply reflecting overall cognitive ability. Such a familial association is, of course, consistent with shared familial environmen-tal effects as well as genetic effects, but had there been no evidence for such an association, our expectation of the likelihood of genetic factors in mathematical talent

would have been diminished. A more elaborate segre-gation analysis, with a larger sample, would be needed to provide further support for genetic vs. environmen-tal sources of the familial correlations.

We explored a number of proxy measures of such mathematics talent. Because so few family members had majored in pure mathematics as undergraduates or grad-uate students (UMATH and GMATH), these variables did not prove useful. MOCC (math-relatedness of occupation) and MHOB (math-relatedness of avoca-tional interests), both based on the Gardner (1983, 1999) scheme, predicted MRATE (the overall rating) more effectively than did MACT, a variable reflecting math-relatedness of activities pursued concurrent with the school years. The most useful and promising vari-able was MRATE, a summary, semiquantitative com-posite measure of mathematical talent and attainment over the life span. This broad variable enabled the coders to take into account the consistency and strength of mathematical behavioral patterns in ways that discrete measures did not; furthermore, MRATE could be rated even when some componential data were missing. Thus the missing data rate was far lower for MRATE than for most other measures, which may also have influenced which proxy measures gave useful results in the current study. It is notable, also, that one other study that re-ported evidence for familial transmission of mathe-matical giftedness also used a (different) composite measure, based also on a combination of education, occupation, and performance (Weiss, 1994).

Although evidence for a genetic basis often de-rives from estimates of heritabilities, this is not possi-ble here. Estimation of heritabilities in the context of the Fulker-DeFries approach, which we used, is possi-ble, in principle, from the estimated regression coeffi-cient, ␤. However, estimation of heritability in this framework requires data on the phenotype distributions in unselected normal populations. We do not have this information for the variables used here and thus can only determine the sign of such regression coefficients and whether there is evidence for an interaction of this effect with proband status, indicative of a genetic basis for specific mathematics talent.

The younger age of female math probands may derive from secular trends toward diminished sex dif-ferences on the SAT-M, particularly at the extreme upper reaches of the distribution (Brody et al.,1992). With such trends, more girls would have qualified later than earlier in the decade during which the students took the SAT, 1988–1998. Just why these changes have occurred—for example, whether because of greater encouragement given to girls, improvements in early

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mathematics education, or some other factor—is out-side the scope of this study. The significant sex effects found in the individual-based regressions for MRATE also may reflect persistent differential social effects and opportunities for men and women in the mathematical domain, although biological variables are not ruled out. The differences in correlations between the two SAT scores in the female math probands vs. the other probands may also reflect such social trends, possibly by resulting in more stringent ascertainment of mathe-matically talented girls from the pool of all seventh-grade girls, relative to ascertainment of the other three groups of probands. However, the numbers are small, and additional studies would be needed to replicate this observation.

This study admittedly has its shortcomings. The sample sizes are too small to estimate the size of some parameters of interest. The data were derived from the report of one or two informants about members of their kinship who were sometimes not well known to them. In individuals in prior generations—some now deceased—youthful educational and vocational attain-ments had sometimes occurred before the birth of the informant. The educational, vocational, and avocational variables we measured were, at best, proxy stand-ins for underlying mathematical ability. As mentioned pre-viously, however, had we attempted direct measures of mathematical ability in the fewer individuals available for direct testing, their varying educational experience and the unknown results of disuse and of social, gen-der, and age-specific effects on current procedural knowledge and mathematical reasoning would have complicated the picture at least as seriously as the mea-sures we used. It might be possible to develop process-based rather than content-process-based measures, but none exists at this time with the very wide range needed in a study of the mathematically talented across the age span. An alternative strategy—making use of data such as archived SAT scores across generations— would present still further complications in that only family members who actually had taken the SAT or the equivalent ACT could be included, thereby excluding those who had not spent their high-school years in the United States, those who had not applied to college, and older individuals who had completed their educa-tions before the widespread use of the SAT or ACT. CONCLUSION

This preliminary study provides encouragement for the development of methods to measure and study the genetic basis of mathematical talent. Despite the

limitations of the proxy measures used here and the small sample size, we were able to obtain some indi-cation that there may be a genetic basis that is specific to mathematical talent. The case-control/family design used here has the potential to control for some of the confounding variables that make study of specific tal-ent difficult. A larger and more complete data set stud-ied with more direct measures may shed further light on the etiology of mathematical talent.

ACKNOWLEDGMENTS

This work was partially supported by funding from the University of Washington Royalty Research Fund and NIH HD33812. Some of the results of this paper were obtained by using the program package S.A.G.E., which is supported by a U.S. Public Health Service Resource Grant (1 P41 RR03655) from the National Center for Research Resources.

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Descriptive Statistics

higher for male probands than for female probands for both types of probands. In addition, the correlation be-tween the two scores in the female math probands was considerably lower than the same correlation in the other proband groups, although the results within sex are not statistically significant, given the small sample sizes. In addition, differences between the minimum and maximum scores were similar among the four groups for both the qualifying (100–140 points) and nonqualifying SAT scores (180–220 points). The most discrepant of the probands in this regard was the fe-male verbal group, with a spread of 140 points in the qualifying score, and 180 points in the nonqualifying score. The other proband groups all had a spread of 100 points in the qualifying score, and a spread of 190–220 points in the nonqualifying score.

SAT-M scores averaged 706.18 (SD=₃₆.9). For the

nonqualifying SAT, the SAT-M scores for the ver-bal probands averaged 583.53 (SD=₆₀.8), while the

SAT-V scores for the math probands averaged 571.18 (SD=₆₀.8). The medians were also similar: on the

qualifying test, the median SAT was 700 for both groups, while for the nonqualifying test it was 580 for the SAT-M among the verbal probands and 570 for the SAT-V among the math probands.

overall, and 157.1 (SD=58.6) and 109.5 (SD=58.4)

for the male and female math probands, respectively. For the verbal probands, the mean difference between verbal and math scores was 115.2 (SD=52.1)

over-all and 102.8 (SD=51.4) and 130.0 (SD=50.6)

for the male and female probands, respectively. Only three probands (1 female math, 1 female verbal, and 1 male verbal proband) had a difference between qualifying and nonqualifying scores of less than 30 points.

Total SAT scores (SAT-V +_{SAT-M) were} simi-lar in the four participating proband groups. Among fe-male probands, the mean (SD) total scores were 1260.0 (69.6) and 1271.9 (84.16) for the math and verbal probands, respectively. Among male probands, mean scores were 1299.0 (78.2) and 1291.7 (82.6) for the math and verbal probands, respectively, for slightly higher total scores for the male than the female probands. However, all differences were within 1 SD and are statistically nonsignificant.

Table I. Correlations Between SAT-V and SAT-M Scores in Participating Proband Groups

Proband group Female Male Total

Verbal 0.58 (15) 0.64 (18) 0.57 (33)

Math 0.23 (19) 0.37 (14) 0.22 (33)

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the case vs. control probands. Regression Analyses

pur-sued further, with additional analyses based on model 3. Results for the remaining measures were not signif-icant in this early exploratory analysis (results not shown).

family-based regressions. In addition, for the models family-based on family-based regressions, the estimate for␤ was also significant (p=0.014). This parameter was

insignifi-cant for the individual-based regressions, suggesting that there was no effect of kinship on MRATE in the families of the control probands, after accounting for the other coefficients in the model (including the

Table II. Distribution of Age at Time of Study

Minimum Median Mean Maximum

Proband group age age age age

Female verbal 14 17 17.4 24

Male verbal 15 18.5 18.44 25

Female math 15 17 16.59 19

Male math 14 17 17.93 24

Table III. Distribution of MRATE Scores in Families of Math and Verbal Probands

MRATE score in relative

Proband type 1 2 3 4 5

Math 2 298 129 81 12

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Table V. Best Regression Model 3 for MRATE Weights

Familya _Individualb

Model coefﬁcient Estimate (SD) pvalue Estimate (SD) pvalue

␮: Intercept .818 (.46) .075 .968 (.22) <_.0001

␤: Kinship 1.622 (.65) .014 .407 (.60) .5

␥: Proband −.244 (.16) .133 −.256 (.12) .039

␦: Kinship ×proband 2.114 (.93) .024 2.307 (.82) .005

HI-ED .654 (.17) .0001 .626 (.07) <_.0001

HI-ED2 ₋_{.055 (.01)} _.0002 ₋_{.045 (.006)} <_.0001

Generation .327 (.15) .026 .319 (.07) <_.0001

Generation×HI-ED −.087 (.044) .052 −.08 (.02) .0002

Sex ND −.262 (.09) .005

Sex×HI-ED ND .228 (.048) <_.0001

Sex×HI-ED2 _ND ₋_{.027 (.006)} <_.0001

a_{266 df, R}2₌_.175;b_{797 df, R}2₌_.272.

were sex-by-higher education interactions (p< .0001

for HI-ED-by-sex,p=.03 for female-by-HI-ED2, and p< .0001 for male-by-HI-ED2).

For a model based on individual weights with a sex effect included, the significance of the coefficient in the control-family regression weakened to p<0.1. In

addition, the coefficient in the models involving fam-ilies of the case probands was always considerably larger than the equivalent coefficient in the models in-volving the control probands (Table IV), consistent with the faster regression to the mean seen in case vs. control families.

Table IV. Regression Coefﬁcients (SD) for Regression Models 1 and 2 for Measures of Mathematics Talent

Measure Wts Model ␮(SE) ␣(SE) ␤(SE) ␥(SE) ␦(SE)

MRATE Fam 1 2.05 (.15)*** .08 (.04)* 2.48 (.44)*** ND ND

2 2.22 (.18)*** .08 (.05) 1.44 (.62)** −.33 (.16)** 2.06 (.87)**

Indiv 1 2.13 (.13)*** .04 (.04) 2.59 (.4)*** ND ND

2 2.33 (.15)*** .05 (.05) 1.2 (.61)* −.35 (.13)*** 2.46 (.82)***

MHOB Fam 1 .15 (.06)** −.09 (.05)* .12 (.31) ND ND

2 .23 (.09)** −.09 (.05)* −.59 (.53) −.12 (.11) 1.08 (.65)*

Indiv 1 1.11 (.05)*** −.02 (.02) .1 (.28) ND ND

2 −2.04 (.8)*** −.19 (.19) −1.74 (5.2) −.02 (.9) 3.24 (5.84)

MACT Fam 1 .42 (.1)*** −.02 (.08) .54 (.51) ND ND

2 .49 (.15)*** −.003 (.08) .24 (.83) −.13 (.19) .49 (1.06)

Indiv 1 −.39 (.27) .04 (.13) 2.35 (1.76)* ND ND

2 .39 (.5) .1 (.14) −.89 (3.19) −1.1 (.6) 4.49 (3.84)

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DISCUSSION

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This preliminary study provides encouragement for the development of methods to measure and study the genetic basis of mathematical talent. Despite the

ACKNOWLEDGMENTS

REFERENCES

Alarcón, M., DeFries, J. C., Light, J. C., and Pennington, B. F. (1997). A twin study of mathematics disability. J. Learn. Disabil.

30:617–623.

Alarcón, M., Knopik, V., and DeFries, J. (2000). Covariation of math-ematics achievement and general cognitive ability in twins.

J. School Psychol.38:63–77.

Baharloo, S., Service, S., Risch, N., Gitschier, J., and Freimer, N. (2000). Familial aggregation of absolute pitch. Am. J. Hum. Genet.67:755–758.

Barnett, L., and Juhasz, S. (2002). The Johns Hopkins University Talent Searches today. Gift. Talent. Int.16:96–99.

Benbow, C. (1988). Sex differences in mathematical reasoning ability in intellectually talented preadolescents: Their nature, effects, and possible causes. Behav. Brain Sci.11:169–232.

Benbow, C., and Stanley, J. (1981). Mathematical ability: Is sex a factor?Science212:118–119.

Benbow, C., Stanley, J., Kirk, M., and Zonderman, A. (1983a). Struc-ture of intelligence in intellectually precocious children and in their parents. Intelligence7:129–152.

Benbow, C., Zonderman, A., and Stanley, J. (1983b). Assortative marriage and the familiarity of cognitive abilities in families of extremely gifted students. Intelligence7:153–161.

Bouchard, T. J., and McGue, M. (1981). Familial studies of intelli-gence: A review. Science212:1055–1059.

Bouchard, T. J., Lykken, D., McGue, M., Segal, N., and Tellegen, A. (1990). Sources of human psychological differences: The Minnesota Study of Twins Reared Apart. Science

250:223–228.

(pp. 204–210). Munich, Germany: Hogrefe & Huber. Carroll, J. (1993). Human Cognitive Abilities: A Survey of

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trait locus associated with cognitive ability inh children. Psychol. Sci.9:159–166.

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Familial Aggregation Patterns in Mathema

Familial Aggregation Patterns in Mathematical Ability

Ellen M. Wijsman,

Nancy M. Robinson,

Kathryn H. Ainsworth,

Elisabeth A. Rosenthal,

Ted Holzman,

and Wendy H. Raskind

Parts

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Steady State Spatial Patterns in a Cell

Carrier Aggregation Technique to Improve Capacity in LTE-Advanced Network

METAPHORICAL PATTERNS IN BARACK OBAMA’S SPEECHES.

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Familial congenital heart disease in Bandung, Indonesia.

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_{Nancy M. Robinson,}

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