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 not particularly 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 additive. This is in part because the
trait with the greater mean value 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).
of variance is a crucial assumption of many
biological experiments and analyses, especially Analysis of Variance (ANOVA) and Heritability studies.