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 web-based survey STAGE (The Study of Twin Adults: Genes and Environment). This was a web-based 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.Footnote 2 Lindqvist and Vestman (forthcoming) provide a more detailed description of the personality measure used by SNSA.
For both intelligence and military aptitude, the raw scores underlying an individual’s ordinal score are available to us. The raw scores are percentile rank-transformed and then converted by taking the inverse of the standard normal distribution to produce normally distributed test scores. The transformation is done separately for each year, but when less than 100 pairs are available for a particular year, two adjoint years are pooled. Since the sample of women who enlist in the military comprises a small and highly self-selected group, this paper focuses exclusively on men. We restrict the sample to all male twin pairs for whom complete data on cognitive ability and military aptitude is available for both twins. This leaves 1246 pairs of monozygotic (MZ) twins and 1568 pairs of dizygotic (DZ) twins for analysis, all of which are born between 1950 and 1976. Descriptive statistics for the twins in our sample are presented in Table 1, disaggregated by zygosity. For expositional convenience, we report the ordinal instead of the normalized scores in the Table, even though the latter are used in the analyses that follow.
Table 1 Background variables
The bivariate ACE model, assortative mating and the cross-trait genetic correlation
The bivariate ACE model
We follow biometrical genetic theory (Falconer and Mackay 1981) and previous papers on this subject (Silventoinen et al. 2006; Sundet et al. 2005) to decompose the variance of a phenotype into shares attributable to additive genetic factors, common environment, and individual environment. Our empirical analysis uses the standard bivariate ACE model (Neale and Cardon 1992). Let \(Y^{i}=\left( Y_{1}^{i},Y_{2}^{i}\right)^{\prime}\) be a vector of two observable phenotypes of individual i, and suppose that
$$ Y^{i}={\bf a}A^{i}+{\bf c}C^{i}+{\bf e}E^{i}, $$
where a, c, and e are 2 × 2 matrices of coefficients and where \(A^{i}=\left( A_{1}^{i},A_{2}^{i}\right)^{\prime},\, C^{i}=\left( C_{1}^{i},C_{2}^{i}\right)^{\prime}\) and \(E^{i}=\left( E_{1}^{i},E_{2}^{i}\right)^{\prime}\) are, respectively, the latent additive genetic, common environmental, and individual environmental factors underlying traits \(Y_{1}^{i}\) and \(Y_{2}^{i}.\) Throughout, we make the standard assumption that A, C, and E are mutually independent. The model assumes that all genetic variance is additive, thereby ruling out dominance and epistasis. Suppose further, without loss of generality, that all the variables have been standardized to have mean zero and unit variance. The correlation between the traits of two individuals i and j will then be equal to
$$ {\bf E} \left[ Y^{i}Y^{j \prime}\right] ={\bf aE}\left[ A^{i}A^{j\prime}\right] {\bf a}^{\prime}+{\bf cE}\left[ C^{i}C^{j\prime}\right] {\bf c}^{\prime}+{\bf eE}\left[ E^{i}E^{j\prime}\right] {\bf e}^{\prime}, $$
(1)
where E denotes the expectations operator. It follows from the standardization that the expectation of the product of two variables is simply their correlation. Table 2 summarizes a set of assumptions about E[A
i
A
j′], E[C
i
C
j′] and E[E
i
E
j′]. These assumptions are customary in the behavior genetics literature, except for the presence of the \({\Upgamma}_{M}\) matrix which, as we will show, accounts for assortative mating at the additive genetic level. The usual bivariate ACE model does not account for assortative mating and thus implicitly assumes that \(\Upgamma_{M}\) is a zero matrix. In the next section of the paper, we show that assortative mating enters the model this way, through \(\Upgamma_{M}.\)
Table 2 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
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 \({ \Upgamma }\) matrices in Table 2 have the following elements:
$$ \begin{aligned} \Upgamma_{A} & = \left(\begin{array}{cc} 1 & \rho_{A} \\ \rho_{A} & 1 \end{array} \right), \Upgamma_{C}=\left(\begin{array}{cc} 1 & \rho_{C} \\ \rho_{C} & 1 \end{array}\right), \Upgamma_{E}=\left(\begin{array}{cc} 1 & \rho_{E} \\ \rho_{E} & 1 \end{array}\right) \\ \hbox{and }\Upgamma_{M} &= \left(\begin{array}{cc} m_{11} & \bar{m}_{12} \\ \bar{m}_{12} & m_{22} \end{array} \right). \end{aligned} $$
As we show in the next section, \(\bar{m}_{12}=\frac{m_{12}+m_{21}}{2},\) where m
kl
is the correlation between the latent additive genetic factors underlying phenotype Y
k
in fathers and phenotype Y
l
in mothers (k,l
\(\epsilon\){1, 2}). Thus, there is positive assortative mating at the additive genetic level for phenotypes k and l if m
kl
> 0 and/or m
lk
> 0.
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 cross-trait 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 cross-trait 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.Footnote 3 The restriction that \(\Upgamma_{M}=0\) means that there is no assortative mating (including cross-trait assortative mating) for phenotypes Y
1 and Y
2.Footnote 4 Below, we also discuss the consequence of assuming different, more realistic values of \(\Upgamma_{M}.\)
Estimates of the shares of the observed variance in either trait attributable to additive genetic factors (a
2), common environment (c
2), and unique environment (e
2) can be obtained from the above parameters. Examination of Eq. 1 and some algebra reveals that
$$ Var(Y_{k})={\bf E}\left[\left( Y_{k}^{i}\right)^{2}\right]=1=(a_{kk})^{2}+(c_{kk})^{2}+(e_{kk})^{2}. $$
We thus see that \(a_{k}^{2}=a_{kk}^{2},\, c_{k}^{2}=c_{kk}^{2},\) and \(e_{k}^{2}=e_{kk}^{2}\, (k=1,2).\) It also follows from Eq. 1 that the within-individual cross-trait correlation is given by
$$ corr(Y_{1},Y_{2})={\bf E}\left[ Y_{1}^{i}Y_{2}^{i}\right] =a_{11}a_{22}\rho_{A}+c_{11}c_{22}\rho_{C}+e_{11}e_{22}\rho_{E}. $$
(2)
The shares of the cross-trait phenotypic correlation attributable to additive genetic factors, common environment, and unique environment are thus \(\frac{a_{11}a_{22}\rho_{A}}{corr(Y_{1},Y_{2})}, \, \frac{c_{11}c_{22}\rho_{C}}{corr(Y_{1},Y_{2})},\) and \(\frac{e_{11}e_{22}\rho_{E}}{corr(Y_{1},Y_{2})},\) respectively.Footnote 5 These shares depend on the correlations between the latent factors underlying both traits (ρ
A
, ρ
C
, and ρ
E
) and on the shares of observed phenotypic variance explained by each of the latent factors (a
2, c
2, and e
2).
Assortative mating in the bivariate ACE model
We now show that assortative mating enters the bivariate ACE model as described in the previous section. For this, it will be useful to augment the model to include individual i’s father and mother. We assume that we are in genetic equilibrium and that all the parameters of the model are fixed across generations. We can write
$$ {\bf E}\left[A_{k}^{i}|A_{k}^{F_{i}},A_{k}^{M_{i}}\right]=\frac{1}{2} \left(A_{k}^{F_{i}}+A_{k}^{M_{i}}\right)\Longrightarrow A_{k}^{i}=\frac{1}{2} \left(A_{k}^{F_{i}}+A_{k}^{M_{i}}\right)+\epsilon_{k}^{i}, $$
(3)
where, as above, \(A_{k}^{i}\) is the latent additive genetic factor underlying phenotype \(Y_{k}^{i}\,(k=1,2)\) for individual i; F
i
and M
i
refer to i’s father and mother, respectively; and \(\epsilon_{k}^{i}\) is an error term independent of the parental genotypes. We rewrite the error term as
$$ \epsilon_{k}^{i}=\theta_{k}S_{k}^{i}, $$
(4)
where θ
k
is a normalizing constant and \({\bf E}\left[\left( S_{k}^{i}\right)^{2}\right]=1,\) thus adhering to the convention of working with standardized variables. The expression \(\theta_{k}S_{k}^{i}\) is the deviation due to Mendelian segregation (Otto et al. 1994). Finally, we let \({\bf E}\left[S_{1}^{i}S_{2}^{i}\right]=\rho_{S}\).Footnote 6 The first equality in Eq. 3 implies that \({\bf E} \left[S_{k}^{i}|A_{k}^{F_{i}}\right]={\bf E}\left[S_{k}^{i}|A_{k}^{M_{i}}\right]={\bf E} \left[S_{k}^{i}\right]=0,\) and thus \({\bf E}\left[S_{k}^{i}A_{k}^{F_{i}}\right]={\bf E} \left[S_{k}^{i}A_{k}^{M_{i}}\right]=0.\) For any pair of full siblings (including DZ twins) i and j, it follows that
$$ \begin{aligned} {\bf E}\left(A_{k}^{i}A_{l}^{j}\right) & = {\bf E}\left[ \left( \frac{1}{2} \left(A_{k}^{F_{i}}+A_{k}^{M_{i}}\right)+\theta_{k}S_{k}^{i}\right) \left( \frac{1}{2} \left(A_{l}^{Fj}+A_{l}^{M_{j}}\right)+\theta_{l}S_{l}^{j}\right) \right] \\ & = \frac{1}{4}\left( {\bf E}\left[ A_{k}^{F_{i}}A_{l}^{F_{j}}\right] + {\bf E}\left[ A_{k}^{M_{i}}A_{l}^{M_{j}}\right] + {\bf E}\left[ A_{k}^{F_{i}}A_{l}^{M_{j}}\right] + {\bf E}\left[ A_{k}^{M_{i}}A_{l}^{F_{j}}\right] \right) \\ & = \frac{1}{4} \left( \rho_{A,kl}+\rho_{A,kl}+m_{kl}+m_{lk}\right) ={{1 }\over {2}}\left( \rho_{A,kl}+\bar{m}_{kl}\right), \end{aligned} $$
where the second equality holds because \(S_{k}^{i}\) and \(S_{l}^{j}\) are uncorrelated with each other and with the other variables.Footnote 7 The third equality follows from the fact that F
i
= F
j
and M
i
= M
j
and from our assumption of genetic equilibrium, which implies that \({\bf E}\left[\left(A_{k}^{i}\right)^{2}\right]=1\) is constant across generations and thus that \({\bf E} \left[A_{k}^{i}A_{l}^{i}\right]=Cov\left(A_{k}^{i},A_{l}^{i}\right)=\rho_{A,kl}.\) Here, ρA,kl is the correlation between the latent additive genetic factors underlying phenotypes k and l.Footnote 8 Therefore, \({\bf E}\left[A^{i}A^{j\prime}\right]=\frac{1}{2} \left(\Upgamma_{A}+\Upgamma_{M}\right) .\)
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.Footnote 9 Gianola (1982)Footnote 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 within-family correlations can be used to shed light on the sources of a phenotypic correlation.
Decomposition of ρ
A
Observe that
$$ \begin{aligned} 1 &= {\bf E}\left[\left(A_{k}^{i}\right)^{2}\right] ={\bf E}\left[ \left( \frac{1}{2}\left( A_{k}^{Fi}+A_{k}^{M_{i}}\right) +\theta_{k}S_{k}\right)^{2} \right] \\ &=\frac{1}{2}\left(1+m_{kk}\right) +\left( \theta_{k}\right)^{2}\Longrightarrow \theta_{k}=\sqrt{\frac{1-m_{kk}}{2}},\, (k=1,\,2), \end{aligned} $$
and that
$$ \begin{aligned} \rho_{A} &={\bf E}\left[ A_{1}^{i}A_{2}^{i}\right] = {\bf E}\left[\left( \frac{1}{2}\left(A_{1}^{Fi}+A_{1}^{M_{i}}\right) +\theta_{1}S_{1}\right) \cdot \left( \frac{1}{2}\left( A_{2}^{Fi}+A_{2}^{M_{i}}\right) +\theta_{2}S_{2}\right) \right] ,\\ &=\frac{1}{2}\left( \rho_{A}+\bar{m}_{12}\right) +\theta_{1}\theta_{2}\rho_{S}. \end{aligned} $$
It follows that
$$ \rho_{A}=\bar{m}_{12}+\sqrt{( 1-m_{11}) ( 1-m_{22})} \rho_{S}. $$
(5)
Therefore, in equilibrium, the correlation between the latent additive genetic factors underlying phenotypes Y
1 and Y
2 is equal to the sum of a term accounting for cross-trait assortative mating and a term accounting for the genetic correlation arising from pleiotropy (or linkage). In the limiting case where there is no assortative mating, \(\rho_{A|m_{11}= \bar{m}_{12}=m_{22}=0}=\rho_{S}.\) Without assortative mating, the genetic correlation must be entirely attributable to pleiotropy (or linkage), and thus ρ
s
is the genetic correlation that is attributable to pleiotropy (or linkage). We can thus rewrite Eq. 5 as
$$ \rho_{A}=\bar{m}_{12}+\sqrt{(1-m_{11}) (1-m_{22})} \rho_{A,Pleiotropy} $$
(6)
In the limiting case where there is no pleiotropy, \(\rho_{A|\rho_{S}=0}= \bar{m}_{12}\) and the genetic correlation is entirely due to cross-trait assortative mating. Substituting Eq. 6 into Eq. 2 yields
$$ \begin{aligned} corr(Y_{1},Y_{2}) =& a_{11}a_{22}\bar{m}_{12}+a_{11}a_{22}\sqrt{( 1-m_{11}) ( 1-m_{22}) }\rho_{A,Pleiotropy}\\ &+c_{11}c_{22}\rho_{C}+e_{11}e_{22}\rho_{E}. \end{aligned} $$
(7)
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
Observe that for a pair of MZ twins,
$$ {\bf E}^{MZ}\left[Y_{i}Y_{j}^{\prime}\right]={\bf a}\Upgamma_{A}{\bf a}^{\prime } +{\bf c}\Upgamma_{C}{\bf c}^{\prime}, $$
and that for a pair of DZ twins,
$$ {\bf E}^{DZ}\left[Y_{i}Y_{j}^{\prime}\right]=\frac{1}{2}{\bf a}\left( \Upgamma_{A}+\Upgamma_{M}\right) {\bf a}^{\prime } +{\bf c}\Upgamma_{C}{\bf c}^{\prime}, $$
implying that
$$ {\bf a}\Upgamma_{A}{\bf a}^{\prime }=2\left( {\bf E}^{MZ}\left[Y^{i}Y^{j\prime}\right]-{\bf E}^{DZ}\left[Y^{i}Y^{j\prime}\right]\right) +{\bf a }\Upgamma_{M}{\bf a}^{\prime}. $$
Computing the off-diagonal elements of the symmetric matrices on both sides of the above equation gives
$$ a_{11}a_{22}\rho_{A}=2\left( {\bf E}^{MZ}\left[Y_{1}^{i}Y_{2}^{j}\right]-{\bf E}^{DZ}\left[Y_{1}^{i}Y_{2}^{j}\right]\right) +a_{11}a_{22}\bar{m}_{12}. $$
Interestingly, when there is no cross-trait assortative mating at the genetic level \(\left(\bar{m}_{12}=0\right),\) the share of the cross-trait 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 same-trait assortative mating genetic correlations (m
11 and m
22). Thus, under the maintained assumptions of the model, if cross-trait 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 cross-trait phenotypic correlation attributable to additive genetic factors. The cross-trait genetic correlation itself (ρ
A
) is however a function of both the cross-trait and same-trait assortative mating genetic correlations, since the coefficients a
11 and a
22 depend on the latter.
Within-family analysis
The within-family correlation between traits Y
1 and Y
2 is defined as \(corr_{WF}(Y_{1},Y_{2})=corr\left(Y_{1}^{i}-Y_{1}^{j},Y_{2}^{i}-Y_{2}^{j}\right),\) where individuals i and j are either non-twin siblings or DZ twins. Thus,
$$ \begin{aligned} corr_{WF}(Y_{1},Y_{2})&= \frac{{\bf E}\left[ \left( Y_{1}^{i}-Y_{1}^{j}\right) \left( Y_{2}^{i}-Y_{2}^{j}\right) \right] }{\sqrt{ {\bf E}\left[Y_{1}^{i}-Y_{1}^{j}\right]^{2}{\bf E}\left[Y_{2}^{i}-Y_{2}^{j}\right]^{2}}}\\ &=\frac{{\bf E}\left[ \left( a_{11}A_{1}^{i}+e_{11}E_{1}^{i}-a_{11}A_{1}^{j}-e_{11}E_{1}^{j}\right) \left(a_{22}A_{2}^{i}+e_{22}E_{2}^{i}-a_{22}A_{2}^{j}-e_{22}E_{2}^{j}\right) \right]}{\sqrt{{\bf E} \left[a_{11}A_{1}^{i}+e_{11}E_{1}^{i}-a_{11}A_{1}^{j}-e_{11}E_{1}^{j}\right]^{2}{\bf E}\left[a_{22}A_{2}^{i}+e_{22}E_{2}^{i}-a_{22}A_{2}^{j}-e_{22}E_{2}^{j}\right]^{2}}}\\ &= \frac{{\bf E}\left[ a_{11}\theta_{1}\left( S_{1}^{i}-S_{1}^{j}\right) a_{22}\theta_{2}\left( S_{2}^{i}-S_{2}^{j}\right) +e_{11}\left( E_{1}^{i}-E_{1}^{j}\right) e_{22}\left( E_{2}^{i}-E_{2}^{j}\right) \right] }{\sqrt{{\bf E}\left[ a_{11}\theta_{1}\left( S_{1}^{i}-S_{1}^{j}\right) +e_{11}\left( E_{1}^{i}-E_{1}^{j}\right) \right]^{2}{\bf E}\left[ a_{22}\theta_{2}\left( S_{2}^{i}-S_{2}^{j}\right) +e_{22}\left( E_{2}^{i}-E_{2}^{j}\right) \right]^{2}}}\\ &=\frac{a_{11}a_{22}\theta_{1}\theta_{2}\rho_{S}+e_{11}e_{22}\rho_{E}}{\sqrt{\left(\left( a_{11}\theta_{1}\right)^{2}+\left( e_{11}\right)^{2}\right)\cdot \left(\left( a_{22}\theta_{2}\right)^{2}+\left( e_{22}\right)^{2}\right)}}. \end{aligned} $$
(8)
Therefore, cross-trait assortative mating does not affect the within-family 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
= ρA,Pleiotropy when e
11 ≅ e
22 ≅ 0.