Although the above approach can reduce the effect of shared environment, it may exclude many samples and substantially reduce statistical power. To maximize the use of the collected samples, we considered a second approach suggested by Do et al [4], in which a common variance component is used to account for the impact of shared environment. Specifically, we considered the following linear mixed model (5)y=Xβ+Wu+Cv+e,u~N(0,σu2I),v~N(0,συ2I),e~N(0,σe2I),where v is the random effect accounting for the shared environment, C = [cij] ∈ ℝn×F is its design matrix, and F is the number of families. The design matrix C is an incidence matrix, i.e., cij = 1 if the i-th individual belongs to the j-th family, otherwise cij = 0. Under this model setting, it is assumed that, the family members within the j-th family share the same environmental factor captured by the random effect vj and these random effects share a common variance component συ2. The advantage of this model is that it makes full use of the collected samples, but its estimation may be an approximation because the individuals within a family may not have the same environmental impact.
