3 Analyzing Genetic and Environmental Correlations

We illustrate the estimation of the genetic and environmental covariance matrices for a simple case of skinfold measures made on 11 year-old male twins from the Medical College of Virginia Twin Study (Schieken et al., 1989). Our skinfold assessments include four different measures which were obtained using standard anthropometric techniques. The measures were obtained for biceps (BIC), subscapular (SSC), suprailiac (SUP), and triceps (TRI)skinfolds. The raw data were averaged for the left and right sides and subjected to a logarithmic transformation prior to analysis in order to remove the correlation between error variance and skinfold measure. The $8 \times 8$ covariance matrices for the male MZ and DZ twins are given in Table 10.7.

Table 10.7: Covariance matrices for skinfold measures in adolescent Virginian male twins.

	Dizygotic Male Pairs (N=33)
	BIC1	SSC1	SUP1	TRI1	BIC2	SSC2	SUP2	TRI2
BIC1	.1538
SSC1	.1999	.3007
SUP1	.2266	.3298	.3795
TRI1	.1285	.1739	.2007	.1271
BIC2	.0435	.0336	.0354	.0376	.1782
SSC2	.0646	.0817	.0741	.0543	.2095	.3081
SUP2	.0812	.0901	.0972	.0666	.2334	.3241	.3899
TRI2	.0431	.0388	.0376	.0373	.1437	.1842	.2108	.1415
	Monozygotic Male Pairs (N=84)
	BIC1	SSC1	SUP1	TRI1	BIC2	SSC2	SUP2	TRI2
BIC1	.1285
SSC1	.1270	.1759
SUP1	.1704	.2156	.3031
TRI1	.1035	.1101	.1469	.1041
BIC2	.0982	.1069	.1491	.0824	.1233
SSC2	.0999	.1411	.1848	.0880	.1295	.1894
SUP2	.1256	.1654	.2417	.1095	.1616	.2185	.2842
TRI2	.0836	.0907	.1341	.0836	.1010	.1134	.1436	.1068
Variable Labels: BIC=Biceps; SSC=Subscapular; SUP=Suprailiac;
TRI=Triceps. ``1'' and ``2'' refer to measures on first and second twins

An example Mx program for estimating the Cholesky factors of the additive genetic and within-family environmental covariance matrices is given in Appendix . The matrices X and Z are now declared as free lower triangular matrices. When this program is run with the data from male twins, we obtain a goodness-of-fit chi-squared of 68.92 for 52 d.f. ($p=.058$) suggesting that the AE model gives a reasonable fit to these data. Setting the off-diagonal elements of the genetic factors to zero yields a chi-squared that may be compared using the difference test to see whether the measures can be regarded as genetically independent. This chi-squared turns out to be 110.96 for 6 d.f. which is highly significant. Therefore, the genetic correlations between these skinfold measures cannot be ignored. Similarly, setting the environmental covariances to zero yields a significant increase in chi-squared of 356.98, also for 6 d.f. Clearly, there are also highly significant environmental covariances among the four variables. Table 10.8 gives the estimates of the Cholesky factors of the genetic and environmental covariance matrices produced by Mx.

Table 10.8: Parameter estimates of the cholesky factors in the genetic and environmental covariance matrices.

	Genetic Factor					Environmental Factor
Variable	1	2	3	4		1	2	3	4
BIC	0.340	0.000	0.000	0.000		0.170	0.000	0.000	0.000
SSC	0.396	0.182	0.000	0.000		0.160	0.138	0.000	0.000
SUP	0.487	0.159	0.148	0.000		0.180	0.117	0.093	0.000
TRI	0.288	0.016	0.036	0.110		0.117	0.039	-0.004	0.085

Carrying out the pre- and post-multiplication of the Cholesky factors (see equations 10.8 and 10.9) gives the maximum-likelihood estimates of the genetic and environmental covariance matrices, which we present in the upper part of Table 10.9. The lower part of Table 10.9 gives the matrices of genetic and environmental correlations derived from these covariances (see 10.5 and 10.6).

Table 10.9: Maximum-likelihood estimates of genetic and environmental covariance (above the diagnoals) and correlation (below the diagonals) matrices for skinfold measures.

	Genetic					Environmental
Variable	BIC	SSC	SUP	TRI		BIC	SSC	SUP	TRI
BIC	0.116	0.135	0.166	0.098		0.029	0.027	0.030	0.020
SSC	0.909	0.190	0.222	0.117		0.759	0.044	0.045	0.024
SUP	0.914	0.955	0.284	0.148		0.769	0.908	0.054	0.025
TRI	0.927	0.863	0.894	0.097		0.778	0.757	0.716	0.023
Note: The variances are given on the diagonals of the two matrices

We see that the genetic correlations between the four skinfold measures are indeed very large, suggesting that the amount of fat at different sites of the body is almost entirely under the control of the same genetic factors. However, in this example, the environmental correlations also are quite large, suggesting that environmental factors which affect the amount of fat at one site also have a generalized effect over all sites.