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##

2 Variance Components Model: Example of Unstandardized Tracing Rules

Following the unstandardized tracing rules, the expected covariances
of twin pairs in the variance components
model of Figure 5.3b, are also easily derived. For the case
of monozygotic twin pairs reared together (MZT), we can trace the following routes:

so that the expected covariance between MZ twin pairs reared together
will be

Only the latter two chains contribute to the expected covariance of MZ
twin pairs reared apart, as they do not share their environment. The
expected covariance of MZ twin pairs reared apart (MZA)is thus

In the case of dizygotic twin pairs reared together (DZT), we can trace the following routes:

yielding an expected covariance between DZ twin reared together of

Similarly, the expected covariance of DZ twin pairs reared
apart (DZA) is

In deriving expected variances of
unstandardized variables, any chain from a dependent variable to an
independent variable will include the double-headed arrow from the
independent variable to itself (unless it also includes a
double-headed arrow connecting that variable to another independent
variable) and each path from an dependent variable to an independent
variable and back to itself is only counted once. In this example the
expected phenotypic variance, for all groups of relatives, is easily
derived by tracing all the paths from to itself:

yielding the predicted variance for P1 or P2 in Figure 5.3b
of

The equivalence between Figures 5.3a and 5.3b comes
from the biometrical principles outlined in Chapter 3:
, , , and are defined as
,
,
, and
,
respectively. Since correlations are calculated as covariances
divided by the product of the square roots of the variances (see
Chapter 2), the twin correlations in Figure 5.3a
may be derived using the covariances and variances in
Figure 5.3b. Thus, in Figure 5.3b, the correlation
for MZ pairs reared together is

Similarly, the correlations for MZ twins reared apart, and for DZ
twins together and apart are

as in the case of Figure 5.3a.

** Next:** 7 Identification of Models
** Up:** 6 Path Models for
** Previous:** 1 Path Coefficients Model:
** Index**
Jeff Lessem
2002-03-21