In this first GE interaction analysis of mathematics ability, we found significant GE interactions, after Bonferroni correction, between a mathematics 10-SNP set and two measures of the home environment: Home Chaos and Parent Negativity. The association between the SNP set and mathematics was greater when children’s homes were disorganized and when parents were negative. Moreover, for these two environmental measures there was an opportunity to compare results across ages. For both measures, GE interaction was in the same direction at ages 9 and 12 (Fig. ), although the GE interactions only reached significance at age 12 (Table ). In addition, composite measures of Home Chaos and Parent Negativity across ages 9 and 12 showed greater interaction effects. Power is likely to have played a role in this result, as the creation of a composite measure, in which even individuals possessing data at only one age were included, increased sample size. Another possibility involves the fact that subjects with missing data on environmental measures had slightly, but significantly, lower mathematics scores than those missing no environmental data. By including those individuals with missing data at some time points, our composite measures thus permitted GE interaction analyses across a greater range of mathematics performance. It may also be the case that the composite scores captured the cumulative effects of Home Chaos and Parent Negativity over time. As continued exposure to environmental pathogens is likely to be important in shaping a phenotype, it has been suggested that GE interaction studies may profit from the use of repeated measures rather than one-time snap-shots (Moffitt et al. 2005
). Our results support the use of repeated-measure GE interaction study designs.
The significant GE interactions we have reported allow us to conclude that the association between the SNP set and mathematics ability in our sample differed as a function of the environment. It is more difficult to interpret the form of the interaction (Manuck 2009
). Because the environmental measures are only weakly correlated with mathematics performance, low and high scores could not strictly be construed as good and bad environments, at least specifically in relation to mathematics ability. Nonetheless, the significant GE interactions could be viewed as supporting the diathesis-stress model, in which individuals at genetic risk (diathesis) have worse-than-expected outcomes when subjected to environmental risk (stress) (Asbury et al. 2005
). For Home Chaos at age 12, the GE interaction suggests that the genetic effect of a low SNP-set score on mathematics performance was greatest in high-Chaos environments. Similarly, for Parent Negativity at age 12––as well as at age 7 (graph not shown)––the genetic effect of a low SNP-set score on mathematics performance was greatest when parents were negative. These significant GE interactions were in the poor-get-poorer direction of the diathesis-stress model. Diathesis-stress GE interactions suggest a ‘dark side’ to development: Bad environments make bad genotypes worse. However, a more positive way to frame the diathesis-stress model is to say that good environments are good for everyone whereas bad environments are especially bad for children with certain genotypes.
One limitation of our use of SNPs that show ‘main effect’ associations with an outcome is that main effects and interactions are theoretically independent. One would expect ordinal associations (in which associations are stronger in some environments than in others), such as those fitting the diathesis-stress model, to dilute main-effect associations, making them harder to find. Furthermore, disordinal (or cross-over) associations are likely to completely mask any main-effect associations. In the most extreme example, gene-phenotype associations could be in opposite directions in good and bad environments, which would conceal a main-effect association across environments. A new hypothesis about GE interaction suggests that such interactions may be common (Belsky et al. 2009
; Belsky and Pluess 2009
). This theory posits that individuals are differentially susceptible to environmental influences, both good and bad. Though our study was more likely to identify ordinal GE interactions, like those of the SNP-set with Home Chaos and Parent Negativity at 12, because the main-effect association of the SNP set was so modest it is possible that true disordinal interactions could be found. Although it did not survive Bonferroni correction, Teacher Negativity appeared to show a disordinal (cross-over) interaction that could be interpreted as support for the ‘plasticity’ hypothesis that genes affect sensitivity to both ‘good’ and ‘bad’ environments (Belsky et al. 2009
; Belsky and Pluess 2009
). Because Teacher Negativity was not significantly correlated with mathematics ability, it was especially difficult to interpret high or low scores as good or bad.
The effect sizes of the two GE interactions surviving Bonferroni correction were very small: 10-SNP-set interactions with Home Chaos at age 12 and Parent Negativity at age 12 each explained only 0.49% of the variance in mathematics performance. The joint effect of the two interactions was larger––but still very small––explaining only 0.71% of the variance in mathematics in our sample. This figure suggests that these interactions did not act completely independently and additively, which is not surprising because the environmental variables were correlated. In statistical interaction analyses based on the analysis-of-variance model, variance attributed to interaction is independent of variance attributed to main effects. Thus, in our analyses, variance attributed to GE interaction was independent of the main effects of G (SNP set) and E (environmental measures). Although parsimony favors the statistical model of no interaction in that main effects are more parsimonious than interactions, it could be argued that some of the variance attributed to main effects could be attributed logically to GE interaction (Rutter 2007
). The fact that our GE interaction term was independent of the main effects of G and E also indicates that our findings of GE interaction were not caused by GE correlation. Moreover, the results in Table indicate that our mathematics SNP set was uncorrelated with our measures of the environment. Nonetheless, GE correlation is likely to form an important part of GE interplay in the etiology of complex traits such as mathematical ability, and it is possible that in much larger samples significant correlations between the 10-SNP set and the environment may be detected.
The overlap between the samples used in our GE interaction analyses was incomplete, and these sample differences could explain why the significant GE interactions we reported at age 12 did not reach significance at age 9. For both Parental Negativity and Home Chaos the individuals unique to ages 9 and 12 did not differ significantly from one another in their mathematics scores. Furthermore, when analysed alone in a linear regression, the 10-SNP set’s association with mathematical ability was not affected by these differences in sample composition. However, it is worth noting that the effect of the SNP set on mathematics does vary in the results presented in Table . For example, the association of the SNP-set with mathematics was no longer significant when Home Chaos at 12 and a GE interaction term were controlled for in the model. It could follow that much of the observed main effect of the 10-SNP-set was tied up in this GE interaction with Home Chaos at 12. However, as the 10-SNP-set score would be correlated with the GE interaction terms, and as the 10-SNP-set’s effect was also non-significant in the multiple regression analysis involving Harsh Parental Discipline at 9––in which neither the environment or the interaction term was significantly associated with mathematics––caution is advised in attempting to interpret these differences.
For the two significant GE interactions based on the 10-SNP set, we explored GE interactions with each of the 10 SNPs individually. The significant GE interactions were not due to just one SNP––most of the SNPs in the SNP set showed effects in similar directions but we found no systematic patterns of results. However, this exploratory analysis was greatly underpowered: As compared to the SNP set, individual SNPs’ associations with mathematics ability had about one-tenth the effect size but increased multiple testing tenfold. Moreover, the reason for analyzing the individual SNPs would be to look for differential patterns of GE interaction across the SNPs, but the power needed to detect significant differences in GE interaction between SNPs was about four times greater than the power needed to detect significant GE interaction for one SNP without considering the added multiple-testing of comparing 10 SNPs two at a time.
Nonetheless, analysis of individual SNPs is relevant to the issue of what it means to find a GE interaction with a SNP set when each of the constituent SNPs is likely to have very different mechanisms. Our view is that the SNP set for mathematics was meant as an index of the heritability of mathematics ability even though our SNP set only indexed a small proportion of the total heritability. In quantitative genetics, GE interaction involves finding that heritability differs as a function of the environment. In our analysis, we examined the extent to which the SNP set’s association with mathematics ability differs as a function of the environment. This hypothesis-free GE interaction approach limited our ability to interpret our results. As we do not know the mechanisms by which any of the 10 SNPs in the SNP set affects mathematics ability, we have no idea how the SNP set for mathematics ability might interact with the environment. Given the general rules of pleiotropy (each gene affects many traits) and polygenicity (each trait is affected by many genes), it is safe to predict that the answer will be complicated (Kovas and Plomin 2006
Although the strengths of our study included its use of a SNP set and its composite measure of mathematical ability, the study was limited in terms of its sample, its measures of mathematics-relevant environments and its power. Concerning the sample, although the study’s representative sample could be considered as a strength, it might also be a weakness for identifying GE interaction if, as some have suggested, GE interaction is most likely to be found at the extremes of the environment (Caspi et al. 2010
). Nonetheless, a counterargument is that in addition to attempting to demonstrate the existence of GE interaction at the extremes of environment, it is also useful to know the extent of GE interaction in the population.
Concerning the measures of the environment, we were limited to measures obtained in TEDS which was itself limited by the fact that there are few measures of mathematics-relevant environments. Nonetheless, the 10 measures included in our study seemed a reasonable starting point in the search for GE interaction in mathematics. Furthermore, we believe our use of proximal rather than distal measures of the environment to be a strength of the study. Though previous quantitative genetic GE interaction studies of cognition have focused on environmental measures such as parental education (Friend et al. 2008
; Kremen et al. 2005
), parental employment (Guo and Stearns 2002
) and socioeconomic status (SES) (Fischbein 1980
; Scarr-Salapatek 1971
; Harden et al. 2007
), it has been suggested that proximal environments afford more power in GE interaction analyses (Moffitt et al. 2005
). Moreover, proximal environments such as Home Chaos may be easier to adjust than distal environments such as SES. The results of GE investigations involving proximal measures should therefore be more readily transferable into practical interventions. Nevertheless, though they were not as all-encompassing as SES, the measures used here were still limited in being fairly general. It has been suggested that, at least in the case of stress, specific stressors yield more replicable GE interaction results than do more general ratings of stressful life events (Caspi et al. 2010
). If this suggestion is correct for mathematics-relevant environments, then it is a conservative bias in our study in that it would have made it more difficult to for us to show GE interaction given that our environmental measures were general ratings of the home, parents and teachers.
Another possible weakness of the environmental measures is that they were not all obtained at the same age as the mathematics measures. As we analysed a phenotypic outcome at age 10, our use of environmental measures at age 12 is a disadvantage. It is certainly difficult to conclude that significant GE interactions of the 10-SNP-set with Home Chaos and Parent Negativity at age 12 have a causal effect over mathematics at age 10. However, as Chaos and Parent Negativity are both moderately stable over time (shown in Table ), we considered both age 9 and age 12 data to be a proxy for the environment at age 10. Though age 9 GE interactions of Home Chaos and Parent Negativity did not reach significance, composite measures across ages 9 and 12 did. Furthermore, we have reported a significant interaction between the 10-SNP-set and Parent Negativity at age 7. The effect of this interaction can more easily be interpreted as causal in the influence of mathematics performance at age 10. For Home Chaos on the other hand, one can not rule out the possible influence of mathematics performance at age 10 over the significant GE interaction reported at age 12.
The major limitation of the present study is power and the need for replication. The original genome-wide association study (Docherty et al. 2010
) that identified the 10 SNPs used in the present study was underpowered to detect association effect sizes of the magnitude found for the individual SNPs (i.e., less than 0.6% of the variance), and this set of SNPs has yet to be tested for replication in an independent sample. Even greater power is needed to detect GE interaction, and the small effects reported here were at the limit of our study’s power to detect them. These results must therefore be viewed as preliminary until they are replicated in independent samples. However, there is some weak evidence for replication in the present study in that the environmental measures that were assessed across ages suggested consistent results in terms of direction of effect, as noted above.
A possible limitation of GE interaction studies that is not often appreciated is that the power to detect GE interaction depends on the distribution of the genotypes, environments, and outcomes (Caspi et al. 2010
). Our study is less limited by these issues because our measures of environments and outcomes were continuous and representative of the population. SNP sets are also normally distributed, unlike most individual SNPs, and so for this reason, our use of a SNP set was a strength (Plomin et al. 2009
). However, the aggregation of 10 SNPs limited our ability to interpret our findings on a biological level. Furthermore, our SNP set was created under the assumption that the 10 SNPs interacted additively, which means epistatic or multiplicative interactions between SNPs would not have been well represented. We adopted this approach because our sample was underpowered to detect epistasis, and because quantitative genetic studies suggest additive genetic influence over mathematics.
Our theory-free genome-wide association approach is complementary to the alternative theory-guided candidate-gene approach that investigates a nomological network of convergent evidence (Caspi et al. 2010
). Because there are no candidate-gene studies relevant to mathematical ability we could not focus on candidates but we hope that our nominated GE interactions, after replication, become candidate GE interaction targets in studies that extend the construct validity using a more theory-guided approach.
Research into GE interplay in the etiology of mathematics is only just beginning. Many more GE interactions involving many more genetic and environmental factors are likely to influence this complex trait. Indeed, though we did not find significant GE interaction between the 10-SNP-set and environments such as Classroom Chaos and Harsh Parental Discipline in our sample, we would predict that significant GE interactions with measures such as these will emerge in the future. As neither genetic nor environmental factors act in isolation, GE interplay studies are likely to have important practical implications. Understanding why some individuals suffer or benefit under certain environmental conditions, while others do not, will assist in the development of tailored environmental interventions aimed at improving mathematics and other skills.