Additivity of Variances

In the **first **block, two traits A and B contribute to a combined
trait A+B. The
contributions of the two loci are (1) approximately equal
and (2) independent
(A & B are not correlated).
Then, the phenotypic value
of the combined trait is simply the sum of the two
contributing loci, and the variance of (A+B) is the sum of the
variances of A & B separately. That is,
variance is **additive**.

In the**second **block, the result is
moderately sensitive to moderate differences in the relative contribution of
traits. In the second data set, B contributes only 33% of the value of A to the combined trait, but the variances
remain essentially additive (expected 1048 vs observed 1056).
This is in part because the trait with the greater mean
contribution will also tend to contribute most of the
variance.

In the**third **block, the result is
more sensitive to non-independence
of the contribution of traits. Here, the second trait
is perfectly correlated with A
by the addition of 10 units. (Note that the variances of A and A' are identical).
However, the variance of the combined trait is much greater
than the sum of the contributing traits. (This is an extreme
example: with partial correlation the effect is much less
exaggerated).

**Additivity of variance**** is a
crucial assumption** of many biological experiments and
analyses, especially Analysis
of Variance (ANOVA)
and Heritability studies.

In the

In the

All text material ©2016 by Steven M. Carr