Specifically, we assumed that refractive error was a normally distributed random variable, such that the
i-th observation,
yi, can be modeled as
Display Formula
where μ is a constant common to all animals and
xi is a set of recorded nongenetic contributors to refractive error (i.e., sex, coat color, and right or left eye measurement). We did not include litter size in this analysis, as some of the offspring information was incomplete and exact litter size was not known in all cases. We defined
βi as the unknown effects associated with these nongenetic contributors,
ai the additive genetic contribution to refractive error,
pei the unique environmental contribution to refractive error, and
ei the unknown residual variances. In addition
ai was assumed to be sampled from a multivariate normal density, with mean zero and covariance
Display Formula
, with
Display Formula
being the numerator relationship matrix, the
pei to be sampled from a multivariate normal density, independent of
ai, with mean zero and variance of the unique environmental contribution
Display Formula
and the residuals to follow from a normal density, with mean zero and variance of the temporary or unknown environmental contribution
Display Formula
. The objective was to estimate the unknown vector
βi, along with the unknown variances
σ2a,
σ2pe, and
σ2e. With that, the narrow sense heritability of refractive error also can be estimated (
h2 =
σ2a/[
σ2a +
σ2pe +
σ2e]; genetic variance as a proportion of total variance) and the phenotypic changes that we observed were affected by genes or genetic factors (single allele or multiple alleles) and/or by nongenetic factors (residual, environment, and so forth). The correlation of refractive error between the two eyes of the same animal also can be estimated (
r = [
σ2a +
σ2pe]/[
σ2a +
σ2pe +
σ2e]).