Alternatively, the bivariate twin model in Fig. 28.5 can be used to decompose a phenotypic correlation between two traits into its three possible sources: overlapping genetic, overlapping shared environmental, or overlapping unique environmental factors. The word "overlapping" can be defined more precisely as the correlation between the latent genetic (Rg), shared environmental (Rc), or unique environmental (Re) factors influencing SBP at rest (A, C, E) and SBP under stress (As, Cs, Es). More generally Rg between two traits is derived as the genetic covariance between the traits divided by the square root of the product of their genetic variances:

If the genetic correlation is close to 1, the two genetic factors (A and As) overlap completely. In this case a22 will be zero and there are no genes that emerge specifically for the second trait. If the genetic correlation is significantly less than 1, there may be overlap in the genetic factors (A and As) that influence both traits but the overlap is imperfect and a22 will be non-zero (this was the case in our SBP example above). If the genetic correlation is close to 0, there is no overlap in the genetic factors that influence both traits and the heritability of the second trait is determined completely by a22.

Analogously, the environmental correlations Rg and Rc between two traits are derived as the environmental covariance divided by the square root of the product of the environmental variances of the two traits:

The actual observed or phenotypic (Rp) correlation between two traits is a function of Rg, Rc, Re and the square roots of the standardized genetic (a2), shared environmental (c2), and unique environmental (e2) variances of these two traits. In a general notation for traits x and y, the phenotypic correlation is

In the adolescent sample used above, the observed phenotypic correlation between SBP at rest and during stress in our study was Rp = 0.81. Only additive genetic and unique environmental factors contributed to the variance in SBP at rest (a2est = 0.59 e2est = 0.41) and under stress (a2tress = 0.72 e2tress = 0.28) with Rg = 0.874, Rc = 0, and Re = 0.707. Hence, Eq. (28.18) correctly estimates the phenotypic correlation: ^0.59 x 0.874 x ^0.72 + ^0 x 0 x ^0 + V0.41 x 0.707 x ^0.28 = 0.81. The extent to which the phenotypic correlation is explained by the correlation at the genetic level (Va^st x Rg x VaLess) is often reported as a percentage and represents the "heritability of the covariance." For the phenotypic correlation between SBP at rest and SBP under stress, the percentage explained by correlation at the genetic level is (V0.59 x 0.874 x ^0.72)/0.81 = 70%.

The bivariate model in Fig. 28.5 can be further expanded to a multivariate design when more than two measurements are available, e.g., during ambulatory blood pressure recording (Kupper et al, 2005b). More importantly, these multivariate models can also be used to compute genetic and environmental correlations between different traits, rather than the repeated measures of the same trait. A typical example is given in the left-hand side of Fig. 28.6, where three components of the metabolic syndrome, body mass index (BMI), SBP, and oral glucose tolerance test-derived insulin sensitivity (INS) have been assessed in Danish MZ and DZ twins. The study set out to detect the source of the clustering of a group of symptoms related to insulin resistance (obesity, glucose intolerance, hypertension, dys-lipidemia) (Benyamin et al, 2007).5 This cluster may be a better predictor of diabetes and cardiovascular disease than each of these risk factors separately.

5 The original study computed genetic and environmental correlations between nine phenotypes, of which three were selected for the example figure used here.

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Fig. 28.6 A multivariate AE model for SBP, body mass index (BMI), and insulin sensitivity (Insul_Sens)

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When multivariate models are depicted in a path diagram, the amount of arrows (and path loadings) can be become overwhelming so Fig. 28.6 differs from Fig. 28.5 in that only the part of one twin is depicted (the other twin and non-twin sibling is still used in the analyses with their latent factors correlated as before; they are just no longer drawn). The multivariate model drawn on the left is a so-called Cholesky decomposition, where n traits load on exactly n latent A, C, E factors with all traits loading on the first factor, n-1 loading on the second factor, and so on. The right-hand side of Fig. 28.6 describes exactly the same model but it uses a different notation. In a correlated factors model the pathways from the latent factor of a trait to the observed value of the other traits are now replaced by correlations between the latent factors. The Cholesky model is the preferred model when there is a clear theoretical causal or temporal ordering of the traits (as in repeated measurements), but when the ordering of the traits is arbitrary a correlated factors model may be more appropriate.

Using the model on the right in Fig. 28.6 as the null model we can test the relative fit of various nested models. For instance, we can test a model that freely estimates Rg but fixes Rc and Re to zero. This would test the hypothesis that the associations between SBP and BMI and between SBP and insulin sensitivity derive entirely from a common set of genetic factors. Empirical data showed this hypothesis to be false (Benyamin et al, 2007). The phenotypic correlations between SBP and INS (r = -0.31) as well as between SBP and BMI (r = 0.26) were due to not only genetic (Rg = -0.23 and 0.26) but also unique environmental (Re = -0.26 and 0.27) factors. Furthermore, the genetic correlations were rather modest which argues against the idea that all variables in the metabolic syndrome have a common (genetic) etiology.

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