Two-component mixtures of generalized linear mixed effects models ...

26 DB Hall and L Wang
adaptive Gaussian quadrature (AGQ) (Liu and Pierce, 1994; Pinheiro and Bates, 1995).
Let ^b 1 i and ^b 2 i denote the modes of the integrands in the numerator and denominator,
respectively, of Equation (3.2), and let g 1 (b i ) P n i
j¼1 ‘c (d; y ij ; u (h)
ij
(b i )j
b i )f (y i jb i ; d (h) )f q (b i ) and g 2 (b i ) f (y i jb i ; d (h) )f q (b i ) from equation (3.2). In addition,
let ^G 1i and ^G 2i be the Hessian matrices of log g 1 (b i ) and log g 2 (b i ) evaluated at ^b 1 i and
^b 2 i , and let p ‘1 ;...;‘ q
¼ (p ‘1
; ...; p ‘q
) T and z ‘1 ;...;‘ q
¼ (z ‘1
; ...; z ‘q
) T , where p 1 ; ...; p m and
z 1 ; ...; z m are m-point ordinary Gaussian quadrature (OGQ) weights and abscissas,
respectively. Then the quadrature points under AGQ are shifted and rescaled versions
of z ‘1 ;...;‘ q
as follows: b 1
i‘ 1 ;...;‘ q
¼ (b 1
i‘ 1
; ...; b 1
i‘ q
) T ¼ ^b 1 i þ 2 q=2 1=2 ^G
1i
z ‘1 ;...;‘ q
and
b 2
i‘ 1 ;...;‘ q
¼ (b 2
i‘ 1
; ...; b 2
i‘ q
) T ¼ ^b 2 i þ 2 q=2 1=2 ^G
2i
z ‘1 ;...;‘ q
for g 1 (b i ) and g 2 (b i ), respectively.
The corresponding AGQ weights are (p ‘ 1
; ...; p ‘ q
) T , where p i ¼ p i exp (z 2 i ).
Hence, at the E step, Q(djd (h) ) is approximated by
X
X m
w (h)
i‘ 1 ;...;‘ q
[u (h)
ij
(b 1
i‘ 1 ;...;‘ q
) log p ij (c) þ {1 u (h)
ij
(b 1
i‘ 1 ;...;‘ q
)} log {1 p ij (c)}]
i;j ‘ 1 ;...;‘ q
n
o
i‘ 1 ;...;‘ q
) log f 1 (y ij jb 1
i‘ 1 ;...;‘ q
; ~a; s 1 )
where
þ Xm
w (h)
i‘ 1 ;...;‘ q
u (h)
ij
(b 1
‘ 1 ;...;‘ q
þ Xm
‘ 1 ;...;‘ q
w (h)
i‘ 1 ;...;‘ q
{1 u (h)
w (h)
i;‘ 1 ;...;‘ q
¼
ij
(b 1
are weights that do not involve d.
3.1.2 M step
n
i‘ 1 ;...;‘ q
)} log f 2 (y ij jb 1
i‘ 1 ;...;‘ q
; b; ~ o
s 2 )
(3:3)
j ^G 1i j 1=2 f (y i jb 1
i‘ 1 ;...;‘ q
; d (h) )f q (b 1
i‘ 1 ;...;‘ q
) Q q
n¼1 p ‘ n
j ^G 2i j 1=2 P h
m
‘ 1 ;...;‘ q
f (y i jb 2
i‘ 1 ;...;‘ q
; d (h) )f q (b 2
i‘ 1 ;...;‘ q
) Q q
n¼1 p ‘ n
In the (h þ 1)th iteration of the algorithm, the M step maximizes the approximation to
Q(djd (h) ) given by Equation (3.3) with respect to d. Notice that Q(djd (h) ) has a relatively
simple form that allows it to be maximized in a straightforward way. From Equation
(3.3), the approximation can be seen to be a sum of three terms: first, a weighted
binomial loglikelihood involving c only; secondly, a weighted exponential dispersion
family loglikelihood involving only a, h 1 and s 1 ; and thirdly, a weighted exponential
dispersion family loglikelihood involving only b, h 2 , h 3 and s 2 . Therefore, the M step
for d can be done by separately maximizing the three terms in Q(djd (h) ). For each term,
this can be done by fitting a weighted version of a standard GLM.
M Step for c. Maximization of Q(djd (h) ) with respect to c can be accomplished by
fitting a weighted binomial regression of the u (h)
ij
(b 1
i‘ 1 ;...;‘ q
)’s on W i 1 m
q with weights
w (h)
i‘ 1 ;...;‘ q
. Here 1 k is the k 1 vector of ones. For instance, for g p taken to be the logit
i