Dependent Observations
(I) Quasi-likelihood estimating equations
Suppose now that
positive-definite
V (Y ) = σ 2V (µ ) , where V (µ )
n×n
is a symmetric
matrix of known functions
Vij (µ ) ,
no
longer diagonal. The quasi-score function is
U
where
(β ) =
D = [Dir ]n× p
and
1
σ
Dir =
following properties,
E [U (β )] = 0, Cov[U (β )] =
−1
D tV
2
∂µ i
∂β r
(y
− µ
. The score function has the
⎡ ∂U (β ) ⎤
t −1
=
=
−
D
V
D
i
E
β
⎢ ∂β ⎥ .
σ2
⎣
⎦
1
Under suitable limiting conditions, the root βˆ
equation
( )
U βˆ = Dˆ t Vˆ
−1
is approximately unbiased for β
(y
− µˆ
(
)=
of the estimating
0 ,
and asymptotically normally
distributed with limiting variance
()
),
Cov βˆ ≈ σ 2 D tV −1 D
)
−1
= iβ−1 .
Note:
Block-diagonal covariance matrices arise most commonly in
longitudinal studies, in which repeat measurements made on the
same subject are usually positively correlated. A well-known
application is the generalized estimating equation proposed by Liang
and Zeger (1986).
1
(II) Quasi-likelihood function
The quasi-score function for dependent observations is different from
the one for independent observations in that
∂ U r ( β ) ∂U s ( β )
(for dependent observatio ns )
≠
∂β s
∂β r
∂ 2Q (µ ) ∂U r (β ) ∂U s (β ) ∂ 2Q (µ )
(for indpendent observatio ns )
=
=
=
∂β r ∂β s
∂β s
∂β r
∂β s ∂β r
The line integral
Q (µ , y , t (s )) =
t ( s 1 )= µ
t
−1
(
)
(
)
y
−
t
[
V
t
]
n × n dt n × 1 (s ) ,
1
n
×
σ ∫t ( s )= y
1
2
0
along a smooth path t (s
t (s 0
)= µ
)
in
R
from t (s 0
n
y
is path independent if and only if
∂ 2Q(µ , y, t (s )) ∂ 2Q(µ , y, t (s ))
=
∂β r ∂β s
∂β s ∂β r
where y =
)=
[y 1
y2
L
yn
]t
,
.
Note:
Let
⎡ P ( x , y )⎤
F = ⎢
⎥ , t (s ) =
(
)
Q
x
y
,
⎣
⎦
{
t (b )
b
t a
a
∫( )
⎡ x (s )⎤
⎢ y (s )⎥ . Then,
⎣
⎦
}
Fdt (s ) = ∫ P [x (s ), y (s )]x ' (s ) + Q [x (s ), y (s )]y ' (s ) ds .
For example,
⎡ 2 xy ⎤
, t (s ) =
F = ⎢ 2
2⎥
+
x
y
⎣
⎦
⎡ x (s )⎤ ⎡ s ⎤
⎢ y (s )⎥ = ⎢ s 2 ⎥ , 1 ≤ s ≤ 3 .
⎣
⎦ ⎣ ⎦
2
to
Then,
∫ ( ) F ⋅ dt (s ) = ∫ [2 s
t (3 )
3
t 1
1
=
3
(
) ]
∫ (4 s
3
⋅ 1 + s + s ⋅ 2 s ds =
2
4
1
3
)
+ 2 s 5 ds
968
3
◆
If V
(µ ) =
b
''
(θ ), µ
is some function, then
( ) (µ ) , b(θ )
= b ' (θ ), θ = b
'
−1
Q (µ , y , t (s )) is path-independent. Thus, we
can construct
V
−1
k
(µ ) = ∑
j =1
A tj W
j
(A
which will ensure the path-independent of
construct
the
V − 1 (µ )
,
we
can
j
µ )A j
,
Q (µ , y , t (s )) . After we
further
construct
the
quasi-likelihood function. Consider the straight-line path
t (s
so that
)=
y +
(µ
− y )s , 0 ≤ s ≤ 1 ,
t (0 ) = y and t (1) = µ . Provided the straight-line path
exists, the quasi-likelihood function is given by
t ⎡ 1
Q (µ , y ) = − ( y − µ ) ⎢ 2
⎣σ
If
V − 1 (µ )
⎤
−1
(
(
)
)
[
]
s
V
t
s
ds
∫0
⎥(y − µ )
⎦
1
is approximately linear in t over the straight-line path,
the above integral may be approximated by
Q (µ , y ) ≈
−1
t
(
)
y
−
µ
V − 1 (µ )( y − µ )
2
3σ
1
t
(
)
−
y
−
µ
V − 1 ( y )( y − µ )
2
6σ
3