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 (means 53 & 59) 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,t he 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 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,t he 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.

All text material ©2014 by Steven M. Carr