For simplicity, let’s first consider Nsib sibpairs. For the quantity of interest xi (i = 1, 2,⋯2Nsib, the 2Nsib × 2Nsib correlation matrix for xi is: (1)R=(1r00.r100.001r.00r1......) Each 2-by-2 sub-matrix in Eq.(1) represents a sibpair with off-diagonal element r being the correlation coefficient Cor(xi, xi+1) between two sibs i and i + 1. The variance of the extensive variable X = ∑i xi is then equal to the weighted sum: VarX=∑ijσiσjRij where σi and σj is the standard deviation of x for person i and j, and the variance of the intensive quantity x = ∑i xi/(2Nsib) is Varx=∑ijσiσjRij/(2Nsib)2 Since here we are dealing with sibpairs of the same affection status, σi = σj = σ, which simplifies the variance for the correlation matrix in Eq.(1): VarX=Nsib⋅σ2⋅2(1+r) and Varx=σ2(1+r)2Nsib The equivalent number independent samples that lead to the same variance for x can be derived by equating σ2 · 2(1 + r)/(2Nsib) = σ2/NE, or, the ESS for sibpairs is: (2)NE=2Nsib1+r.
