As an alternative to scoring in the polygenic model, one can implement the EM algorithm [7]. In the notation of the text, consider a multivariate normal random vector Y with mean ν = Aµ and covariance Ω = r k=1 σ2 kΓk, where A is a fixed design matrix, the σ2 k > 0, the Γk are positive definite covariance matrices, and Γr = I. Let the complete data consist of independent, multivariate normal random vectors X1,...,Xr such that Y = r k=1 Xk and such that Xk has mean 1{k=r}Aµ and covariance σ2 kΓk. If γ = (µ1,...,µp, σ2 1,...,σ2 r )t

, and the observed data are amalgamated into a single pedigree with m people, then prove the following assertions:

(a) The complete data loglikelihood is ln f(X | γ) = −1 2

r k=1

{ln det Γk + m ln σ2 k

+

1

σ2 k

[Xk − E(Xk)]t

Γ−1 k [Xk − E(Xk)]}.

(b) Omitting irrelevant constants, the Q(γ | γn) function of the EM algorithm is Q(γ | γn)
= − 1 2 r−1 k=1 {m ln σ2 k +
1 σ2 k [tr(Γ−1 k Υnk) + νt nkΓ−1 k νnk]}
− m 2 ln σ2 r − 1 2σ2 r [tr(Υnr)+(νnr − Aµ)
t (νnr − Aµ)], where νnk is the conditional mean vector νk = 1{k=r}Aµ + σ2 kΓkΩ−1[y − Aµ]
and Υnk is the conditional covariance matrix Υk = σ2 kΓk − σ2 kΓkΩ−1σ2 kΓk of Xk given Y = y evaluated at the current iterate γn. (Hint:
Consider the concatenated random normal vector
Xk Y and use fact

(e) proved in the text.)

(c) The solution of the M step is µn+1 = (At A)
−1At νnr σ2 n+1,k = 1 m[tr(Γ−1 k Υnk) + νt nkΓ−1 k νnk], 1 ≤ k ≤ r − 1 σ2 n+1,r = 1 m[tr(Υnr)+(νnr − Aµn+1)
t (νnr − Aµn+1)].
In the above update, µn+1 is the next iterate of the mean vector µ and not a component of µ.