1 Derivation of Expected Covariances

(47) | |||

(48) |

Or, in matrices:

which in turn we can write more economically as

Following the rules for matrix algebra set out in Chapters 4 and , we can rearrange this equation, as before:

(49) | |||

(50) | |||

(51) |

and then, multiplying both sides of this equation by the inverse of (

In this case, the matrix (

which has determinant , so is

The symbol is used to represent the Kronecker product, which in this case simply means that each element in the matrix is to be multiplied by the constant . We have a vector of phenotypes on the left hand side of equation 8.8. In the chapter on matrix algebra (p. ) we showed how the covariance matrix could be computed from the raw data matrix by expressing the observed data as deviations from the mean to form matrix , and computing the matrix product . The same principle is applied here to the vector of phenotypes, which has an expected mean of

Now in the middle of this equation we have the matrix product . This is the covariance matrix of the

We now have all the pieces required to compute the covariance matrix, recalling that for this case,

The reader may wish to show as an exercise that by substituting the right hand sides of equations 8.11 to 8.13 into equation 8.10, and carrying out the multiplication, we obtain:

We can use this result to derive the effects of sibling interaction on the variance and covariance due to a variety of sources of individual differences. For example, when considering:

- additive genetic influences, and , where is 1.0 for MZ twins and 0.5 for DZ twins;
- shared environment influences, and ;
- non-shared environmental influences, and ;
- genetic dominance, and , where for MZ twins and for DZ twins.

Source | Variance | Covariance |

Additive genetic | ||

Dominance genetic | ||

Shared environment | ||

Non-shared environment | ||

represents the scalar obtained from equation 8.14. |