Statistical Power to Detect Genetic Loci Affecting Environmental Sensitivity

Abstract

There is evidence in different species of genetic control of environmental variation, independent of scale effects. The statistical power to detect genetic control of environmental or phenotypic variability for a quantitative trait was investigated analytically using a monozygotic (MZ) twin difference design and a design using unrelated individuals. The model assumed multiplicative or additive effects of alleles on trait variance at a bi-allelic locus and an additive (regression) model for statistical analysis. If genetic control acts on phenotypic variance then the design using unrelated individuals is more efficient but 10,000s of observations are needed to detect loci explaining at most 3.5% of the variance of the variance at genome-wide significance. If genetic control acts purely on environmental variation then an MZ twin difference design is more efficient when the MZ trait correlation is larger than ~0.3. For a locus that explains a given proportion of the variation in variance, twice the number of observations is needed for detection when compared to a locus that explains the same proportion of variation in phenotypes.

Appendix: Additive model

The model is summarised in Table 2. Phenotypic variances (= E(y − μ)²) of the 3 genotypes are (1 + xδ), with x = 0, 1, 2. Expectations of the fourth moments are 3(1 + xδ)².

Table 2 Additive model for phenotypic variances

The phenotypic variance in the population is 1 + 2pδ and the covariance between x and (y − μ)² is hδ. The variance of (y − μ)² is,

$$ {\text{var}}\left( {{\text{y}} - \mu } \right)^{ 2} = 2\left[ { 1+ 4 {\text{p}}\delta + {\text{p}}\delta^{ 2} \left( { 3+ {\text{p}}} \right)} \right] \approx 2\left[ { 1+ 4 {\text{p}}\delta } \right], $$

when δ is small. The regression R² is therefore,

$$ {\text{R}}^{ 2} = \raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} {\text{h}}\delta^{ 2} /\left[ { 1+ 4 {\text{p}}\delta + {\text{p}}\delta^{ 2} \left( { 3+ {\text{p}}} \right)} \right] \approx \raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} {\text{h}}\delta^{ 2} \left[ { 1- 4 {\text{p}}\delta } \right] \approx \raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} {\text{h}}\delta^{ 2} . $$

As expected, the additive and multiplicative model are similar when δ is small because 1 + 2δ ≈ (1 + δ)² = λ².

See Table 2.