Page 160 - Applied Probability

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8. The Polygenic Model
∂
Note that
µ and
∂µ i
Since E(Y )= Aµ, the expected information matrix has entries
2
t
∂
∂
∂

−1
t
µ
A
=
µ A Ω
E −
∂µ i
∂µ j
∂µ i ∂µ j L ∂σ ∂ i 2 Ω=Γ i are treated as constants in these derivations.
∂ ∂ t −1 −1
2 t
E − 2 L = µ A Ω Γ i Ω [E(Y ) − Aµ]
∂σ ∂µ j ∂µ j
i
= 0 (8.3)
∂ 1 −1 −1 −1 −1 −1
2
E − 2 2 L = − tr(Ω Γ i Ω Γ j ) + tr(Ω Γ i Ω Γ j Ω Ω)
∂σ ∂σ 2
i j
1 −1 −1
= tr(Ω Γ i Ω Γ j ).
2
Some simplification in the above formulas can be achieved by defining
the partial score d µ L =( ∂ L, ... , ∂ L) and the corresponding partial
∂µ 1 ∂µ p
∂
2
∂
observed information matrix −d L. Since ( ∂µ 1 µ,..., ∂µ p µ) is the identity
µ
matrix I,
t
d µ L t = A Ω −1 (y − Aµ)
2
t
−d L = A Ω −1 A.
µ
The expected information matrix J evidently has the block diagonal form
2
E[−d L] 0
J = µ 2 ,
0 E[−d 2 L]
σ
2
2
2
2 t
where −d 2 L is the observed information matrix on σ =(σ ,...,σ ) .
σ 1 r
Since the block form of J is retained under matrix inversion, the current
µ is updated in the scoring algorithm by
t
t
t
t
µ +(A Ω −1 A) −1 A Ω −1 (y − Aµ) = (A Ω −1 A) −1 A Ω −1 y. (8.4)
t
t
If there is more than one pedigree, the quantities A Ω −1 A and A Ω −1 y must
be summed over all pedigrees before matrix inversion and multiplication.
2
2
The scoring increment ∆σ to σ is similarly expressed as
t
2
∆σ 2 =E(−d 2L) −1 d σ L .
2
σ
2
t
It is convenient to initialize σ at (0,... , 0, 1) . Since Ω = I in this case,
the first iteration of the scoring algorithm (8.4) produces the standard lin-
ear regression estimate of µ. Of course, this estimate does not take into
account the correlational architecture of the pedigree. In those unlikely cir-
2
cumstances when µ is known exactly, the initial value σ =(0,... , 0, 1) t
2
leads to a least-squares estimate of σ after a single iteration of scoring.
(See Problem 4.) Computation of the score dL and expected information