The 9-year means and standard deviations for the three mathematics scores and for the composite are presented in . Two-by-two (sex BY zygosity) ANOVA analyses on the mathematics measures yielded a significant main effect of sex on two of the four scores (using and applying: p = .069, η2 = 0.001; numbers and algebra: p = .001, η2 = 0.002; shapes, space and measures: p = .408, η2 < 0.001; composite: p = .04, η2 = 0.001), with boys performing significantly better than girls. However, the significance of these effects is attributable to the large sample size because the effect size was very small, accounting for less than 1% of the variance. Similarly, there were significant main effects of zygosity on all of the mathematics measures (using and applying: p = .004, η2 = 0.002; numbers and algebra: p = .012, η2 = 0.001; shapes, space and measures: p = .002, η2 = 0.002; composite: p = .003, η2 = 0 .002), with DZ twins performing significantly better than MZ twins. Again, the effect sizes of these significant effects were very small, accounting for less than 1% of the variance. Whether the twins had the same or different teacher did not significantly affect their mathematics scores. All analyses were repeated separating the sample into twins assessed by the same or different teacher; however, the findings remained unchanged and there was minimal difference between the two subsamples. Therefore, these results are not presented here.
Means (and Standard Deviations) for 9-Year Teacher Assessments of Mathematics (Adjusted for Age), by Zygosity and Sex; and ANOVA Results Showing Significance and Effect Size, by Sex and Zygosity
Genetic Analysis of Individual Differences for the Entire Sample
The intraclass twin correlations for the four scores are shown in . For the entire sample, doubling the difference between these correlations indicates that genetics substantially influences all mathematics scores: .66 for using and applying; .64 for numbers and algebra; .62 for shapes, space and measures; and .68 for the composite score. Estimates of the shared environment are consistently modest (average .09).
Intraclass Correlations and Estimated A, C and E Parameters for Three Mathematics Measures and the Composite for Twins by Zygosity at 9 Years
Univariate model-fitting analyses were carried out for each of the three mathematics measures and the composite. The results of these univariate analyses for the entire sample are shown in .
Individual Differences Univariate Model Fitting for all Measures for the Entire Sample: Model Fit and Parameter Estimates
For the full ACE models, shared environmental (C) estimates range from .08 to .11, and heritabilities range from .62 to .68. For all measures, the parameter estimates from the model fitting are highly similar to the estimates made from the intra-class correlations (). Furthermore, in line with our estimates from the correlations, the best-fitting and most parsimonious model for every measure is the AE model. For this best-fitting AE model, heritability estimates are greater than for the full ACE model because the A in the AE model tends to subsume the small amount of variance due to shared environment in the full ACE model. The remainder of the variance is attributed to nonshared environment plus error of measurement.
Genetic Analysis of Low Mathematics Performance
Probandwise concordances are shown in . As with the intraclass correlations, these concordance rates suggest genetic influence on the risk of being a proband because, in every case, concordance rates are substantially higher for MZ twins than those for DZ twins. Average MZ and DZ concordances across the four scores are .69 and .38, respectively, suggesting substantial genetic influence.
Low Mathematics Performance: MZ and DZ Probandwise Concordances and Results of DF Extremes Analysis (Twin Group Correlations and h2g and c2g Parameter Estimates) Using a 15% Cutoff
also presents the results from the DF extremes analysis, which gives estimates of group heritability and group environmental influences. When calculating these results, group heritability cannot exceed MZ group correlation, in such cases group heritability was constrained. The results of the DF extremes analyses are highly similar to the results of the individual differences analyses for the entire sample ( and ). For example, for the mathematics composite, MZ and DZ twin group correlations () are .75 and .33, respectively. The group heritability estimate for the mathematics composite () is .75 and the influence of shared environment is .00. Results for the three mathematics scales are similar to those for the composite measure, yielding substantial estimates of group heritability (.69 to .75 in ) and modest estimates of group shared environment (.00 to .04).
Longitudinal Analysis of Individual Differences for the Entire Sample
Longitudinal data were analyzed for the composite mathematics scores at 7 and 9 years. The phenotypic correlation between the mathematics composites at 7 and 9 years is .60. presents the results from the longitudinal Cholesky model, which decomposes the variance and covariance of the mathematics composite scores at 7 and 9 years into common and independent additive genetic, shared environmental and nonshared environmental influences. displays the Cholesky results for the genetic influences that are common and independent between the mathematics composite at 7 and 9.
Table 5 Standardized Cholesky Squared Path Estimates (95% Confidence Intervals) for Mathematics at 7 and Mathematics at 9 Indicating Proportions of Genetic (A), Shared Environmental (C) and Nonshared Environmental (E) Influences on Each Trait That Are Shared (more ...)
Genetic results from bivariate Cholesky decomposition. Results for additive genetic effects that are common and independent for mathematics performance (composite score) at 7 and 9 years-of-age.
The genetic contribution to the phenotypic correlation can be estimated from this longitudinal model as the product of the paths to the latent variable A1, which represents genetic influences in common across the two ages. Thus, the genetic contribution to the phenotypic correlation of .60 is .48 (i.e.,√.62 * √.37). In other words, 80% of the phenotypic correlation is mediated genetically (i.e., .48/.60 = .80), which is bivariate (longitudinal) heritability. As shown in , the model-fitting estimate of bivariate heritability is .81 (.67–.95 95% confidence interval [CI]). Although most of the phenotypic continuity between 7 and 9 years is mediated genetically, there is genetic variance at each age that is unique to that age. For example, shows that the model-fitting estimate of heritability at age 7 is .62; the path estimate of .37 from A1 to 9 years indicates that roughly half of this genetic influence at 7 years also affects scores at 9 years. Similarly, heritability at 9 years is estimated as .71, the sum of the two paths to 9 years (.37 + .34). Again, about half of the genetic influence at 9 years is in common with 7 years and about half is independent.
Genetic, Shared Environment, and Nonshared Environment Correlations for Mathematics Composite at 7 years and Mathematics Composite at 9 years; and Proportion of Phenotypic Correlation Between These Variables Mediated by A, C, and E
As noted earlier, this longitudinal genetic analysis also provides an estimate of the genetic correlation from 7 to 9 years. The genetic correlation is independent of heritabilities at each age: The genetic correlation can be low even if the heritabilities are high and vice versa. It indicates the extent to which the same genes are operating at each age regardless of the magnitude of their effect on the phenotype. As shown in , the genetic correlation is estimated as .72 (.64–.82 CI), indicating substantial genetic overlap between mathematics performance at 7 and 9 years. The genetic correlation can be gleaned from : The genetic contribution to the phenotypic correlation (.48) mentioned above is the product of the square roots of the two heritabilities and the genetic correlation (Plomin & DeFries, 1979
). Because we know the heritabilities at 7 years (.62) and 9 years (.71), we can solve for the genetic correlation, 48/(√.62 * √.71) = .72.
Of the phenotypic correlation of .60 from 7 to 9 not explained by genetics, .13 (22%) can be attributed to shared environment and .06 (10%) to nonshared environment. Although shared environment does not account for much variance at either age, what shared environment exists is largely in common between 7 and 9 years. In contrast, nonshared environmental influences are largely independent at the two ages.
Longitudinal Analysis of Low Mathematics Performance
DF extremes longitudinal analyses based on the lowest 15% of each sample also indicated substantial genetic continuity from 7 to 9 years for low mathematics performance. Prospective group heritability, with probands defined at 7 years of age and compared to co-twins’ 9-year scores, was .81 (SE
= .14). Retrospective group heritability, with probands selected at 9 years and compared to co-twins’ 7-year scores, was .66 (SE
= .15). From these two longitudinal group heritabilities it is possible to calculate a genetic correlation between low mathematics performance at 7 and 9 years (Knopik et al., 1997
). The genetic correlation was high and in fact exceeded 1; although confidence intervals for this group genetic correlation have not been worked out, they would obviously be very large. As found in the individual differences longitudinal analyses, nonshared environmental influences largely contributed to change in low mathematics performance from 7 to 9 years. Group shared environmental influences were not significant in this longitudinal analysis.