. Author manuscript; available in PMC: 2020 Apr 1.

Published in final edited form as: Stat Biosci. 2017 Dec 6;11(1):47–90. doi: 10.1007/s12561-017-9208-x

Algorithm 1.

AIM procedure

input: matrices Z, X, positive semi-definite matrix Q, λ₂ > 0

1: define Q̃ := Q+λ₂I_p;

2: decompose Q̃ as Q̃ = L^⊤L;

3: find REML estimates σ̂² and

{\hat{σ}}_{b}^{2}

, of σ² and

σ_{b}^{2}

, in LMM with X as matrix of fixed effects, ZL⁻¹ as matrix of random effects,

b ~ N (0, σ_{b}^{2} I_{p})

and ε ~ 𝒩(0,σ²I_n);

4: define

λ : = {\hat{σ}}^{2} / {\hat{σ}}_{b}^{2}

and find estimates β̂^AIM and b̂^AIM by applying formula (2.4) with matrix Q̃.