On the sources of the height–intelligence correlation: New insights from a bivariate ACE model with assortative mating
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DOI: 10.1007/s1051901093767
 Cite this article as:
 Beauchamp, J.P., Cesarini, D., Johannesson, M. et al. Behav Genet (2011) 41: 242. doi:10.1007/s1051901093767
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Abstract
A robust positive correlation between height and intelligence, as measured by IQ tests, has been established in the literature. This paper makes several contributions toward establishing the causes of this association. First, we extend the standard bivariate ACE model to account for assortative mating. The more general theoretical framework provides several key insights, including formulas to decompose a crosstrait genetic correlation into components attributable to assortative mating and pleiotropy and to decompose a crosstrait withinfamily correlation. Second, we use a large dataset of male twins drawn from Swedish conscription records and examine how well genetic and environmental factors explain the association between (i) height and intelligence and (ii) height and military aptitude, a professional psychogologist’s assessment of a conscript’s ability to deal with wartime stress. For both traits, we find suggestive evidence of a shared genetic architecture with height, but we demonstrate that point estimates are very sensitive to assumed degrees of assortative mating. Third, we report a significant withinfamily correlation between height and intelligence \((\hat{\rho}=0.10),\) suggesting that pleiotropy might be at play.
Keywords
Assortative mating Bivariate ACE model Genetic correlation Height Intelligence IQ Withinfamily correlationIntroduction
A robust positive correlation between height and intelligence, as measured by IQ tests,^{1} has been established in the literature with little consensus regarding its cause (Humphreys et al. 1985; Johnson 1991; Kanazawa and Reyniers 2009; Tanner 1979; Teasdale et al. 1989; Wheeler et al. 2004). Both environmental and genetic explanations have been advanced. For instance, previous research has found that markers of prenatal quality and nutritional status during childhood are associated with height (Eide et al. 2005; Steckel 1995) and cognition in adulthood (See GomezPinilla 2008; Martyn et al. 1996; Seidman et al. 1992; Sørensen et al. 1997), suggesting that early environmental factors may be responsible for the correlation between height and intelligence (Abbott et al. 1998). Other evidence in support of environmental channels is the decline in the correlation between height and intelligence over time observed in the Scandinavian countries (Teasdale et al. 1989; Sundet et al. 2005; Tuvemo et al. 1999). On the other hand, reported heritability estimates in adults for measures of intelligence (e.g. 75%) (Neisser et al. 1996) and height (e.g. 87–93%) (Silventoinen et al. 2003) are high, suggesting that the relationship may instead be genetically mediated. For example, growth hormone deficiency, which is sometimes caused by genetic mutations, is characterized by both short stature and cognitive impairments (van Dam et al. 2005). Evidence of a shared genetic architecture between brain volume, intracranial space, and height (Posthuma et al. 2000) may also be interpreted as evidence of genetic mediation of the height–intelligence correlation.
Two studies, using twin data, have attempted to estimate the components of the height–intelligence correlation that are due to genetic and environmental effects. The results were mixed. Silventoinen et al. (2006) found in several samples of Dutch twins that the association between height and intelligence is primarily genetic in origin. Sundet et al. (2005), using conscription data from a considerably larger and more representative sample of Norwegian twins, found that the association between height and intelligence is primarily explained by common environmental factors. Both papers use a standard bivariate ACE model to arrive at these conclusions.
The present study makes several contributions toward elucidating the sources of covariation between height and intelligence. First, we extend the bivariate ACE model to allow for assortative mating and derive a general formula for decomposing a crosstrait genetic correlation into components attributable to assortative mating and pleiotropy. This provides a clear theoretical framework for studying the correlation between two traits and is particularly useful when examining phenotypes for which there is high assortative mating, height and intelligence being prime examples. The model demonstrates how sensitive estimates from the bivariate ACE model are to assumptions about assortative mating. Second, we use a sample of male Swedish twins matched to conscription records to examine the relative importance of genetic and environmental factors in explaining the association between height and intelligence. The sample used is the largest to date for such a study. It includes an additional measure of cognitive function other than intelligence, measured during enlistment through interviews by a military psychologist. This measure, which we label military aptitude, has a strong predictive validity for labor market outcomes independent of intelligence, such as wages, earnings and unemployment (Lindqvist and Vestman forthcoming). We apply the standard bivariate ACE model to decompose the height–intelligence and the height–military aptitude correlations, but we caution that the resulting estimates are very sensitive to assumptions about assortative mating and illustrate the sensitivity of these estimates by reporting results for different assumed levels of assortative mating. Lastly, we report a significant withinfamily height–intelligence correlation \(\left(\hat{\rho} = 0{\text{.}}10\right),\) consistent with the hypothesis that pleiotropy (or linkage) accounts for part of the genetic correlation.
Method
Sample
Our data comes from two main sources: the Swedish Twin Registry (STR) and the Swedish National Service Administration (SNSA). The STR contains information on nearly all twin births in Sweden since 1886, and has been described in further detail elsewhere (Lichtenstein et al. 2006). The sample includes those individuals who have participated in at least one of the Twin Registry’s surveys. The primary datasource is SALT (Screening Across the Lifespan Twin study). This was a survey administered to all Swedish twins born between 1926 and 1958 and attained a response rate of 74%. Fifty percent of the subjects in the dataset are from the SALT cohort. The secondary source is the webbased survey STAGE (The Study of Twin Adults: Genes and Environment). This was a webbased survey administered between November 2005 and March 2006 to all twins born in Sweden between 1959 and 1985. It attained a response rate of 61%. Approximately 30% of our subjects are drawn from STAGE. Our final datasource comes from a survey sent out in 1973 to the same cohort as SALT (Lichtenstein et al. 2006).
We matched the Swedish twins to the conscription data provided by SNSA. All Swedish men are required by law to participate in a nationwide military conscription at the age of 18. Before 1990, exemptions were very rare. The actual drafting procedure can take several days during which recruits undergo medical and psychological examination. The basic structure of the administered intelligence test has remained unchanged during our study period, though minor changes took place in 1980 and 1994. Recruits take four subtests (logical, verbal, spatial and technical) which, for most of the study period, are graded on a scale from 0 to 40. These raw scores are converted to a ordinal variable ranging from 1 to 9. Carlstedt (2000) discusses the history of psychometric testing in the Swedish military and provides evidence that this test of intelligence is a good measure of general intelligence. Thus, this test differs from the AFQT, which focuses more on “crystallized” intelligence.
All conscripts also see a psychologist for a structured interview. The psychologist has access to background information on the interviewee, such as school grades, medical background, cognitive ability and answers to a battery of questions on friends, family, and life. In conducting the interview, the psychologist is required to follow a manual and, ultimately, to make an assessment of the prospective recruit’s capacity to handle stress in a war situation. In making the assessment, the psychologist considers an individual’s ability to function in a group, adapt to new environments, as well as his persistance and emotional stability. Motivation for doing the military service is not among the set of characteristics that is considered beneficial for succeeding in the military. The psychologist assigns each interviewee an ordinal score from 1 to 9, but again these are constructed from four raw scores, this time ranging from 1 to 5. Like the intelligence test score, the military aptitude score is subject to measurement error because of random influences on conscript performance and because conscripts may differ in their motivation for the military service.^{2} Lindqvist and Vestman (forthcoming) provide a more detailed description of the personality measure used by SNSA.
Background variables
 MZ  DZ  Population 

Income (in SEK)  342,631  335,987  325,245 
SD  241,060  340,271  258,867 
Education (years)  12.50  12.14  12.54 
SD  2.68  2.63  2.35 
1 if married  0.50  0.51  0.51 
SD  –  –  – 
Age in 2005  48.85  51.69  45.43 
SD  7.66  6.37  3.63 
Intelligence  5.12  4.94  5.13 
SD  1.88  1.94  1.94 
Military aptitude  5.32  5.17  5.08 
SD  1.69  1.78  1.78 
Height (in cm)  178.45  178.60  178.94 
SD  6.60  6.46  6.56 
The bivariate ACE model, assortative mating and the crosstrait genetic correlation
The bivariate ACE model
Assumed values of E[A^{i}A^{j}′], E[C^{i}C^{j}′] and E[E^{i}E^{j}′] for different relationships between individuals I and J
 Relationship between individuals i and j  

i = j  MZ twins  DZ twins  Unrelated  
E[A^{i}A^{j}′]  \(\Upgamma_{A}\)  \(\Upgamma_{A}\)  \(\frac{1}{2}\left(\Upgamma_{A}+\Upgamma_{M}\right)\)  0 
E[C^{i}C^{j}′]  \(\Upgamma_{C}\)  \(\Upgamma_{C}\)  \(\Upgamma_{C}\)  0 
E[E^{i}E^{j}′]  \(\Upgamma_{E}\)  0  0  0 
Behavior geneticists have previously studied models which allow for assortative mating, but they were mostly concerned with the case of a single phenotype (see, for instance, Eaves et al. 1978; Eaves and Heath 1981; Martin et al. 1986; Keller et al. 2009). Eaves et al. (1984) consider the multivariate case and develop a model in which there is assortative mating on a latent phenotype. We add to this literature by deriving formulas that describe, among others, the effects of assortative mating on the genetic correlation between two traits and by augmenting the bivariate ACE model to account for these effects.
The resulting model has 19 free parameters: ρ_{A}, ρ_{C}, ρ_{E}, m_{11}, m_{12}, m_{21}, m_{22} and 4 free parameters for each of the matrices a, c and e. However, only nine moments can be computed from the data: the crosstrait covariance between the MZ twins and, for each trait, the covariance between the MZ twins; the corresponding three moments for the DZ twins; and the population variances of the two traits as well as the population crosstrait covariance. Since the number of parameters exceeds the number of independently informative equations, at least ten identifying assumptions need to be made. In the standard decomposition, it is assumed that a, c, and e are diagonal (i.e. that a_{12} = a_{21} = c_{12} = c_{21} = e_{12} = e_{21} = 0) and that \(\Upgamma_{M}=0.\) Under these assumptions, the remaining parameters are identified. The substantive implication of the diagonality assumption is that, while the latent factors underlying the two traits may be correlated, each latent factor may only influence its respective trait.^{3} The restriction that \(\Upgamma_{M}=0\) means that there is no assortative mating (including crosstrait assortative mating) for phenotypes Y_{1} and Y_{2}.^{4} Below, we also discuss the consequence of assuming different, more realistic values of \(\Upgamma_{M}.\)
Assortative mating in the bivariate ACE model
So far, we have only considered assortative mating at the additive genetic level. The parameters of the matrix of genetic assortative mating correlations \(\Upgamma_{M}\) are not directly observable, so it would be useful to have a mapping relating \(\Upgamma_{M}\) to observable parameters. Unfortunately, to our knowledge, no one has yet derived such a mapping for the general case of unconstrained multivariate assortative mating.^{9} Gianola (1982)^{10} considers two special cases of interest to the livestock industry. The first case is when assortative mating is actively practiced on one phenotype only and a second phenotype is genetically correlated with the first due to pleiotropy (or linkage)—such as when, for instance, large bulls are mated with large cows, and the second trait is genetically correlated with cattle size. The second case is when mating pairs are assorted to have a certain correlation between phenotype X in males and phenotype Y in females—such as when, to use Gianola’s example, high milk production females are mated to fast growing males. Assortative mating for height and intelligence in humans is more complex and unlikely to fit either of these cases. We leave the derivation of a general mapping to future research. However, the above discussion should make it clear that it is important to investigate how sensitive the results from the standard bivariate decomposition are to the assumption that \(\Upgamma_{M}\) is equal to zero for traits with high assortative mating.
We now turn our attention to three additional features of the model which merit further exploration. First, we show that in the augmented model (in which the elements of the \(\Upgamma_{M}\) matrix are not constrained to be zero), the correlation between two traits can be decomposed into parts attributable to assortative mating, pleiotropy, common environment, and individual environment. Second, we investigate the bias which arises if a standard bivariate ACE model is estimated in the presence assortative mating. Finally, we consider how withinfamily correlations can be used to shed light on the sources of a phenotypic correlation.
Decomposition of ρ_{A}
The correlation between phenotypes Y_{1} and Y_{2} can thus be decomposed into parts attributable to assortative mating, pleiotropy, common environment, and individual environment.
A lower bound for the share of the genetic correlation
Interestingly, when there is no crosstrait assortative mating at the genetic level \(\left(\bar{m}_{12}=0\right),\) the share of the crosstrait phenotypic correlation attributable to additive genetic factors \(\left(\frac{ a_{11}a_{22}\rho_{A}}{corr(Y_{1},Y_{2})}\right) \) does not depend on the sametrait assortative mating genetic correlations (m_{11} and m_{22}). Thus, under the maintained assumptions of the model, if crosstrait assortative mating is nonnegative at the genetic level \(\left(\bar{m}_{12}\geq 0\right),\) estimates from the standard bivariate ACE model still provide a lower bound for the share of the crosstrait phenotypic correlation attributable to additive genetic factors. The crosstrait genetic correlation itself (ρ_{A}) is however a function of both the crosstrait and sametrait assortative mating genetic correlations, since the coefficients a_{11} and a_{22} depend on the latter.
Withinfamily analysis
Therefore, crosstrait assortative mating does not affect the withinfamily correlation. Unless ρ_{S} and ρ_{E} have different signs, the empirical observation that corr_{WF}(Y_{1}, Y_{2}) ≅ 0 would rule out pleiotropy (or linkage) as a source of genetic correlation, thereby suggesting that assortative mating is responsible. Lastly, observe that expression Eq. 8 collapses nicely to Corr_{WF}(Y_{1}, Y_{2}) ≅ ρ_{S} = ρ_{A,Pleiotropy} when e_{11} ≅ e_{22} ≅ 0.
Results
Results from the estimation of the bivariate ACE model for height–intelligence and for height–military aptitude

 Height–intelligence  Height–military aptitude  

Estimate  SE  Estimate  SE  
Height  \(a_{1}^{2}\)  0.77  0.03  0.77  0.03 
\(c_{1}^{2}\)  0.16  0.04  0.16  0.04  
\(e_{1}^{2}\)  0.07  0.003  0.07  0.003  
Intelligence  \(a_{2}^{2}\)  0.63  0.04  
\(c_{2}^{2}\)  0.20  0.04  
\(e_{2}^{2}\)  0.18  0.01  
Military  \(a_{2}^{2}\)  0.55  0.05  
Aptitude  \(c_{2}^{2}\)  0.11  0.04  
\(e_{2}^{2}\)  0.35  0.01  
ρ_{A}  0.08  0.04  0.09  0.04  
ρ_{C}  0.59  0.15  0.23  0.21  
ρ_{E}  0.15  0.03  0.09  0.03  
ln(L)  −13332.32  −13765.34 
Results suggest that 77%, 16% and 7% of the observed variance in height is attributable to additive genetic, common environmental, and unique environmental factors, respectively. The corresponding figures are 62%, 20%, and 18% for intelligence, and 55%, 11%, and 35% for military aptitude. Our point estimates also suggest that the correlation between the latent genetic factors underlying the height and intelligence phenotypes (ρ_{A}) is a rather modest 0.08, and that the corresponding correlation from the height–military aptitude decomposition is 0.09. We obtain much larger estimates for the common environmental correlations (ρ_{C}): 0.59 for height–intelligence and 0.23 for height–military aptitude, but we note the fairly low precision of these estimates. Finally, we obtain modest but very significant estimates for the unique environment correlations (ρ_{E}) for both variance decompositions.
Shares of the height–intelligence and height–military aptitude correlations accounted for by the three latent factors
Share of the phenotypic correlations accounted for by...  Height–intelligence  Height–military aptitude  

Estimate  SE  Estimate  SE  
A share  0.31  0.13  0.56  0.26 
C share  0.59  0.13  0.30  0.26 
E share  0.09  0.02  0.14  0.05 
These results suggest that 59% of the height–intelligence correlation is mediated by common environment, that 31% is due to additive genetic factors, and that 9% is due to unique environment. The results of the variance decomposition for height and military aptitude suggest that most of the correlation is accounted for by additive genetic factors (56%); the estimated share due to common environment appears quite large at 30%, but is not significantly different from zero.
Parameter estimates for height–intelligence under different assumed values of the \(\Upgamma_{M}\) matrix
\(\Upgamma_{M}=\)  \(\left[\begin{array}{cc} 0.10 & 0.02 \\ 0.02 & 0.10 \end{array}\right]\)  \(\left[\begin{array}{cc} 0.15 & 0.04 \\ 0.04 & 0.15 \end{array}\right]\)  \(\left[\begin{array}{cc} 0.20 & 0.06 \\ 0.06 & 020 \end{array}\right]\)  \(\left[\begin{array}{cc} 0.25 & 0.08 \\ 0.08 & 0.25 \end{array}\right]\) 

\(a_{1}^{2}\)  0.85  0.89  0.91  0.94 
0.04  0.03  0.03  –  
\(c_{1}^{2}\)  0.07  0.04  0.02  0.00 
0.04  0.03  0.02  –  
\(e_{1}^{2}\)  0.07  0.07  0.07  0.07 
0.003  0.003  0.003  –  
\(a_{2}^{2}\)  0.70  0.73  0.77  0.83 
0.04  0.04  0.04  –  
\(c_{2}^{2}\)  0.13  0.10  0.06  0.00 
0.04  0.04  0.04  –  
\(e_{2}^{2}\)  0.18  0.18  0.18  0.18 
0.01  0.01  0.01  –  
A Share  0.40  0.54  0.70  0.91 
0.14  0.12  0.12  –  
C Share  0.51  0.36  0.20  0.00 
0.13  0.12  0.121  –  
E Share  0.09  0.09  0.09  0.10 
0.02  0.02  0.02  – 
These results demonstrate that the point estimates from the standard bivariate ACE model are sensitive to assumptions about the spousal crosstrait genetic correlation. As the values of the offdiagonal parameters and the values of the diagonal parameters of \(\Upgamma_{M}\) matrix are progressively increased from 0 to 0.08 and from 0.10 to 0.25, respectively, the value of the genetic share goes from 0.314 to 0.905 and the value of the common environmental share declines from 0.592 to zero. These changes are quite dramatic and illustrate the sensitivity of the model to even small departures from the assumptions on which it is based. Nonetheless, as we demonstrated analytically above, in the presence of positive crosstrait assortative mating, the standard bivariate ACE model still provides a lower bound for the share of the crosstrait phenotypic correlation attributable to additive genetic factors. Table 5 also illustrates another wellknown result, namely that as the diagonal elements of the \(\Upgamma_{M}\) matrix are progressively increased, the estimated heritabilities rise at the expense of the estimated shared environmental variance components.
Our framework also showed how the withinDZ correlation can be used to help shed light on the sources of the phenotypic correlation between height and intelligence. In our sample, the withinfamily height–intelligence correlation is 0.10 (SE 0.017). The withinfamily height–military aptitude correlation is 0.09 (SE 0.022).
Discussion
Our point estimates from the standard bivariate ACE model with zero assortative mating suggest that both environmental and genetic factors are responsible for the height–intelligence and the height–military aptitude correlations. Specifically, they suggest that common environment and additive genetic effects account for 59% and 31% of the height–intelligence correlation, respectively; the corresponding numbers for the height–military aptitude correlation are 30% and 56%. Our height–intelligence results are strikingly similar to those of Sundet et al. (2005): applying the standard bivariate ACE model to a sample of Norwegian conscripts, they found that the correlation between the latent common environmental factors which underlie height and intelligence was 0.56, only marginally lower than our point estimate of 0.59. They also estimated the heritability of height at 0.76 and the heritability of intelligence at 0.64, while our own estimates are 0.77 and 0.63, respectively.^{12}
This finding contrasts with those reported by Silventoinen et al. (2006) who found that for all cohorts in which there was an association between height and intelligence, the association was explained entirely by additive genetic correlation. One possible reason for the discrepancy between our results and those reported by Silventoinen et al. (2006) is the difference between the samples. Silventoinen et al. (2006) perform independent analyses on four cohorts of Dutch twins ranging in age from children to middleage adults and in size from 156 to 567 twin pairs.
However, a more likely explanation is that Silventoinen et al. (2006) use a different model selection procedure. Procedurally, they do the following: (1) for each cohort, they begin by fitting univariate models to the data for both height and intelligence; (2) they compare the χ^{2}goodnessoffit statistics of the models and select the bestfitting model for each trait; and (3) they use those bestfitting models for all subsequent analyses. They find that the univariate AE model offers adequate fit in every cohort for height and in almost every cohort for intelligence. Thus, in their bivariate model, they only include latent factors for additive genetic effects and for unique environment for height for every cohort, and they do likewise for intelligence in almost every cohort. As a result, there is no ρ_{C} parameter in any of their bivariate models. Moreover, it appears that they drop the ρ_{E} parameter from their model whenever this improves the fit. In other words, they decompose the height–intelligence correlation either into additive genetic and individual environmental effects, or uniquely into additive genetic effects. Dropping ρ_{C}—the most important component of the height–intelligence correlation in both in Sundet et al.’s (2005) data and the results reported here—from their model, and sometimes also dropping ρ_{E}, quite mechanically leads to the conclusion that ρ_{A} is an important source of the correlation.Therefore, the confidence intervals presented by Silventoinen et al. (2006) do not reflect the uncertainty that stems from the modelselection procedure that is employed. It is not clear to us why the fact that a parameter is not statistically significant should justify the restriction that the parameter is equal to zero. This point is further elaborated on by Goldberger (2002).
We also derived the bivariate ACE model for the more general and realistic case where there is assortative mating at the genetic level. Sundet et al. (2005, p. 310) and Silventoinen et al. (2006, p. 587) note that their estimates may be biased if there is assortative mating for the studied traits. Our analysis demonstrates the sensitivity of the estimates from the bivariate ACE models to one of several problematic assumptions. Specifically, changing the assumed values of the crosstrait and sametrait spousal genetic assortative mating correlations from 0 to 0.08 and from 0.10 to 0.25, respectively, raises the estimated share of the height–intelligence correlation mediated by genes from 0.314 to 0.905. Given that there is ample uncertainty about the relationship between the elements of \(\Upgamma_{M}\) and the spousal trait and crosstrait correlations, this further reinforces our conclusion that for traits with assortative mating, bivariate ACE estimate must be approached with caution.
It is important to emphasize that we have only explored one dimension of the model selection uncertainty in multivariate behavior genetic models. The interpretation would be further complicated by attempts to allow for the latent factors to be correlated or for gene action to be nonadditive, questions which we have not investigated here. As is well known, information on MZ and DZ twins alone is not sufficient to estimate models with latent factors for additive genetic effects, nonadditive genetic effects, common environment, and individual environment. Since the twin model suffers from parameter indeterminacy when both dominance and common environmental effects are present (Keller and Coventry 2005), we suggest that future work use extended twin family samples with data on twins’ relatives, thus allowing richer models to be identified. Also, rapid advances in molecular genetics may soon make it possible to study the sources of covariation between traits from a new angle.
Despite our reservations, we believe that our framework provides valuable insights into what could plausibly be learned from studies of this kind. For example, Jensen (1980) notes that the observation that the withinfamily height–intelligence correlation is small and only marginally significant, in contrast to the betweenfamily correlation, is consistent with the hypothesis that the height–intelligence correlation stems from assortative mating. This reasoning is consistent with expression Eq. 8. Johnson (1991), on the other hand, in a review of the literature, concludes that there is a marginally significant positive withinfamily height–intelligence correlation, and that this suggests that pleiotropy plays some role in the height–intelligence correlation. We also report a significant positive withinfamily height–intelligence correlation, adding further weight to the hypothesis that pleiotropy is at play. Though expression Eq. 8 shows that this withinfamily correlation could in part be the result of a correlation between the latent unique environment factors underlying height and intelligence rather than of pleiotropy, the conclusion that pleiotropy cannot account for part of the height–intelligence correlation appears to be premature.
Previous research provides little insight into the genetic mechanisms responsible for the association between height and various aspects of cognition. The difficulty of pinpointing specific mechanisms is partly due to the fact that both height and intelligence are polygenic traits. While a number of promising markers for height have been identified, they only explain a small share of the heritable variation (Weedon and Frayling 2008). As for decoding the molecular genetic structure underlying intelligence, an extraordinarily complex trait, the problems appear to be even less tractable (Butcher et al. 2007; Deary et al. 2009). However, with the advent of genomewide association technology, we are hopeful that key genetic markers will be identified resulting in a greater understanding of the genetics of the association between height and various aspects of cognition.
Concluding remark
We developed a general theoretical framework to study the correlation between two traits. We used this framework to extend the standard bivariate ACE model to account for assortative mating and to derive formulas to decompose a crosstrait genetic correlation into components attributable to assortative mating and pleiotropy and to decompose a crosstrait withinfamily correlation. Our results from the standard bivariate ACE model without assortative mating suggest that common environment explains most of the height–intelligence correlation. However, we caution that our estimates from the standard bivariate ACE model are sensitive to assumptions about assortative mating. In fact, we show that assuming more plausible values for the matrix of assortative mating correlations \(\Upgamma_{M}\) dramatically changes the results of the height–intelligence decomposition, implying that the correlation is primarily mediated by genetic factors rather than by common environment. Also, although we use the largest sample to date for this kind of analysis, and although our data comes from standardized measurements of male conscripts in a tight age range, our estimates are not very precise, as reflected by the size of the standard errors. This leads us to emphasize the difficulty of disentangling the sources of covariation between two traits with samples consisting only of MZ and DZ twins. Future work could benefit from the use of large extended twin family samples. Alternatively, given the rapid pace at which knowledge about the genome is increasing, it may be that such questions will soon be more easily tackled through an approach that seeks to understand how specific genes directly affect the phenotypes of interest.
However, the military aptitude score is subject to an additional form of measurement error since psychologists will vary in their judgement of identical conscripts. Lilieblad and Ståhlberg (1977) estimated the correlation between the SNSA psychologists’ assessment to be .85 after letting thirty SNSA psychologists listen to tape recordings of thirty enlistment interviews.
An alternative way to proceed is to assume that the matrices a, c, and e are lower triangular (i.e. that a_{12} = c_{12} = e_{12} = 0) and that ρ_{A} = ρ_{C} = ρ_{E} = 0, while maintaining the assumption that \(\Upgamma_{M}=0.\) The substantive implication of the first assumption is that the latent factors A_{1}, C_{1}, and E_{1} of the first trait Y_{1} can affect the second trait Y_{2}, whereas the latent factors A_{2}, C_{2}, and E_{2} of the second trait Y_{2} cannot affect the first trait Y_{1}. The second assumption implies that the latent factors of the two traits are not correlated, even within individuals. The model resulting from this alternative set of assumptions is sometimes referred to as a Cholesky decomposition. That model spans the same space as our preferred model, and it is thus possible to transform the parameters from either model into the parameters of the other (see Loehlin 1996, for a more thorough discussion).
Alternatively, that restriction implies that if there is assortative mating, it does not have consequences at the genetic level. This might be the case, for example, if social homogamy fully explained the spousal phenotypic resemblance.
The term “share” in this context can be misleading, as correlations can be negative and these “shares” can therefore be negative.
In a more realistic model, the distribution of \(S^{i}=\left(S_{1}^{i},S_{2}^{i}\right)^{\prime}\) would vary as a function of \(A^{F_{i}}\) and \(A^{M_{i}}.\) For simplicity and tractability, we do not consider such a model.
\(S_{k}^{i}\) and \(S_{k}^{j}\) are uncorrelated with each other because, conditional on the parents’ genomes, the precise genetic draw from one DZ twin does not affect that of the other.
Thus, ρ_{A,kl} = 1 if k = l and ρ_{A,kl} = ρ_{A} if k = 1 and l = 2 as in the previous section. Also, we use \(\bar{m}_{kl}\) to denote m_{kl} when k = l.
Observe that it is not correct to simply assume that \(m_{kl}=\sqrt{a_{kk}^{2} \, a_{ll}^{2}} \, r_{kl},\) where r_{kl} is the phenotypic spousal correlation for traits k and l (k, l\(\epsilon\) {1, 2}). To see, consider Gianola’s first case below. There, \(\bar{m}_{12}>r_{12}\) because the phenotypic crosstrait correlation follows from the genetic crosstrait correlation, and not the other way around.
Acknowledgements
We are grateful to Bill Dickens, James Lee, Joe Rodgers, Peter Visscher and two anonymous reviewers for helpful comments. We are also grateful to The Jan Wallander and Tom Hedelis Foundation, the Swedish Research Council, and the Swedish Council for Working Life and Social Research for financial support. Jonathan Beauchamp also thanks the Trudeau Foundation and the Social Sciences and Humanities Research Council of Canada for financial support.
Open Access
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