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1 Preamble: Longitudinal Fixed Models for Two Multinomial Response Variables Ignoring Correlations

1 Preamble: Longitudinal Fixed Models for Two Multinomial Response Variables Ignoring Correlations

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340

6 Multinomial Models for Longitudinal Bivariate Categorical Data

t = 1, . . . , T. Using individual specific covariate (i.e., common covariate for both
responses), a separate marginal and conditional probabilities model for zit , similar
to (6.1)–(6.2) may be written as
Marginal probability at t = 1 for z variable:
(r)

P[zi1 = zi1 ] = π(i1)r (z) =

exp(w∗i1 αr∗ )

R−1
1+∑g=1
exp(w∗i2 αg∗ )
1
R−1
1+∑g=1
exp(w∗i1 αg∗ )

for r = 1, . . . , R − 1
for r = R,

(6.3)

(r)

where αr∗ = (αr0 , αr ) is the effect of w∗i1 influencing zi1 to be zi1 , where r =
1, . . . , R − 1. Here αr = [αr1 , . . . , αru , . . . , αrp ] .
Lag 1 based conditional probabilities at t = 2, . . . ,T, for z variable:

(r)
ηit|t−1 (h|z) = P

(r)
Zit = zit

(h)
Zi,t−1 = zi,t−1

=

(h)

exp w∗it αr∗ +λr zi,t−1
(h)

R−1 exp w∗ α ∗ +λ z
1 + ∑v=1
v i,t−1
it v
1
(h)
R−1

1 + ∑v=1 exp wit αv +λv zi,t−1

, for r = 1, . . . , R − 1
, for r = R,

(6.4)

where h = 1, . . . , R is a given (known) category, and λr = (λr1 , . . . , λrv , . . . , λr,R−1 )
denotes the dynamic dependence parameters.

6.2 Correlation Model for Two Longitudinal Multinomial
Response Variables
Notice that yit = (yit1 , . . . , yit j , . . . , yit,J−1 ) and zit = (zit1 , . . . , zitr , . . . , zit,R−1 ) are
two multinomial responses recorded at time t for the ith individual. Thus, two
multinomial responses for the ith individual may be expressesd as
yi = [yi1 , . . . , yit , . . . , yiT ] : (J − 1)T × 1,
zi = [zi1 , . . . , zit , . . . , ziT ] : (R − 1)T × 1.

(6.5)

Similar to Sect. 5.4 (specifically see (5.109)–(5.110)), it is quite reasonable to
assume that these two responses are influenced by a common random or latent
effect ξi∗ . Conditional on this random effect ξi∗ , one may then modify (6.1)–(6.2),
and develop the marginal and conditional probabilities for the repeated responses
yi1 , . . . , yit , . . . , yiT as

P[yi1 =

( j)
yi1 |ξi∗ ]

= π(i1)
j· =

exp(w∗i1 β j∗ +ξi∗ )
∗ ∗

1+∑J−1
g=1 exp(wi1 βg +ξi )
1
∗ ∗

1+∑J−1
g=1 exp(wi1 βg +ξi )

for j = 1, . . . , J − 1
for j = J,

(6.6)

6.2 Correlation Model for Two Longitudinal Multinomial Response Variables

341

and
∗( j·)

( j)

(g)

ηit|t−1 (g) = P Yit = yit Yi,t−1 = yi,t−1 , ξi∗

(g)
exp w∗it β j∗ +γ j yi,t−1 +ξi∗

, for j = 1, . . . , J − 1
(g)
J−1
∗ ∗

= 1 + ∑v=1 exp wit βv +γv yi,t−1 +ξi

1

⎩ 1 + J−1 exp w∗ β ∗ +γ y(g) +ξ ∗ , for j = J,
∑v=1

it

v

v i,t−1

(6.7)

i

for t = 2, . . . , T. We will use

β = [β1∗ , . . . , β j∗ , . . . , βJ−1
] : (J − 1)(p + 1) × 1

to denote regression parameters under all categories, and

γ = [γ1∗ , . . . , γ ∗j , . . . , γJ−1
] : (J − 1)(J − 1) × 1

to denote all dynamic dependence parameters.
Similarly, by modifying (6.3)–(6.4), one may develop the marginal and conditional probabilities for zi1 , . . . , zit , . . . , ziT as

P[zi1 =

(r)
zi1 |ξi∗ ]

= π(i1)·r

=

exp(w∗i1 αr∗ +ξi∗ )

R−1
1+∑g=1
exp(w∗i2 αg∗ +ξi∗ )
1
R−1
1+∑g=1
exp(w∗i1 αg∗ +ξi∗ )

for r = 1, . . . , R − 1
for r = R,

(6.8)

and
∗(·r)

(r)

(h)

ηit|t−1 (h) = P Zit = zit Zi,t−1 = zi,t−1 , ξi∗

(h)
exp w∗it αr∗ +λr zi,t−1 +ξi∗

, for r = 1, . . . , R − 1
(h)
R−1
∗ ∗

= 1 + ∑v=1 exp wit αv +λv zi,t−1 +ξi
1

⎩ 1 + R−1 exp w∗ α ∗ +λ z(h) +ξ ∗ , for r = R,
∑v=1

it

v

v i,t−1

(6.9)

i

for t = 2, . . . , T. All regression and dynamic dependence parameters corresponding
to the z response variable will be denoted by

α = [α1∗ , . . . , αr∗ , . . . , αR−1
] : (R − 1)(q + 1) × 1,

and

λ = [λ1∗ , . . . , λr∗ , . . . , λR−1
] : (R − 1)(R − 1) × 1,

respectively.

342

6 Multinomial Models for Longitudinal Bivariate Categorical Data

6.2.1 Correlation Properties For Repeated Bivariate Responses
6.2.1.1

(a) Marginal Expectation Vector and Covariance Matrix for y
Response Variable at Time t
ξ∗

Conditional on ξi∗ or equivalently ξi = σi (see (5.1)), these expectation vector
ξ
and covariance matrix may be written following Chap. 4, specifically following
Sect. 4.4.1. Thus these properties will be constructed as follows by combining
the longitudinal properties of a variable from Sect. 4.4.1 and its possible common
correlation property with another variable as discussed in Sect. 5.4.1. That is, in
notation of (6.6)–(6.7), we write
⎧ ∗

[π(i1)1· , . . . , π(i1)
for t = 1,

j· , . . . , π(i1)(J−1)· ] = π(i1)∗·

∗(∗·)

π
(β , γ , σξ |ξi ) = π(i1)∗·
for t = 1,

⎨ (i1)
∗(∗·)
∗(∗·)
(6.10)
E[Yit |ξi ] = π(it) (β , γ , σξ |ξi ) = η(it|t−1) (J)

∗(∗·)
∗(∗·)
∗(∗·)

⎪ + η(it|t−1),M − η(it|t−1) (J)1J−1 π(i,t−1) (β , γ , σξ |ξi )

∗(1·)
∗( j·)
∗((J−1)·)
= [π(it) , . . . , π(it) , . . . , π(it)
]
for t = 2, . . . , T,
and
var[Yit |ξi ] =

, . . . , π(i1)
diag[π(i1)1·
j· , . . . , π(i1)(J−1)· ] − π(i1)∗· π (i1)∗· for t = 1,
∗(1·)

∗( j·)

∗((J−1)·)
∗(∗·) ∗(∗·)

diag[π(it) , . . . , π(it) , . . . , π(it)− π(it)] π(it)

for t = 2, . . . , T.

∗(∗·)

= Σ(i,tt) (β , γ , σξ |ξi ) : (J − 1) × (J − 1), (say), for t = 1, . . . , T.

(6.11)
(6.12)

In (6.10)–(6.11),
∗(∗·)

∗(1·)

∗( j·)

∗((J−1)·)

η(it|t−1) (J) = [η(it|t−1) (J), . . . , η(it|t−1) (J), . . . , η(it|t−1) (J)]
∗(∗·)

= π(i1) (β , γ = 0, σξ |ξi )
∗(∗·)

(6.13)

η(it|t−1),M

⎛ ∗(1·)
∗(1·)
∗(1·)
η(it|t−1) (1) · · · η(it|t−1) (g) · · · η(it|t−1) (J − 1)

..
..
..
..
..

.
.
.
.
.

⎜ ∗( j·)
∗( j·)
∗( j·)
⎟ : (J−1)×(J−1), (6.14)
=⎜
η
(1)
·
·
·
η
(g)
·
·
·
η
(J

1)

⎜ (it|t−1)
(it|t−1)
(it|t−1)

.
.
.
.
.

..
..
..
..
..

∗((J−1)·)
∗((J−1)·)
∗((J−1)·)
η(it|t−1) (1) · · · η(it|t−1) (g) · · · η(it|t−1) (J − 1)
∗( j·)

with η(it|t−1) (g) as in (6.7), for j, g = 1, . . . , J − 1.

6.2 Correlation Model for Two Longitudinal Multinomial Response Variables

343

Now to obtain the unconditional mean vector and covariance matrix for yit ,
similar to (5.1), we assume that the random effects are independent and they follow
ξ ∗ iid
the standard normal distribution, i.e., ξi = σi ∼ N(0, 1). More specifically, after
ξ

taking the average over the distribution of ξi , these unconditional moment properties
are obtained from (6.10)–(6.11), as follows:
Unconditional mean vector at time t = 1, . . . , T :

−∞ π(i1)∗· (β , σξ |ξi ) f N (ξi )d ξi
∗(∗·)

−∞ π(it) (β , γ , σξ |ξi ) f N (ξi )d ξi

E[Yit ] = Eξi E[Yit |ξi ] =

for t = 1,
for t = 2, . . . , T,

(6.15)

(∗·)

= π(it) (β , γ , σξ ) : (J − 1) × 1, (say), for t = 1, . . . , T, (6.16)
where fN (ξi ) = √12π exp[− 12 ξi2 ]. Note that all integrations over the desired functions
in normal random effects in this chapter including (6.16) may be computed by using
the binomial approximation, for example, introduced in the last chapter, specifically
in Sect. 5.3.
Unconditional covariance matrix at time t = 1, . . . , T :
var[Yit ] = Eξi var[Yit |ξi ] + varξi E[Yit |ξi ]
=
=

−∞

−∞

∗(∗·)

∗(∗·)

Σ(i,tt) (β , γ , σξ |ξi ) fN (ξi )d ξi + varξi [π(it) (β , γ , σξ |ξi )]

∗(∗·)

Σ(i,tt) (β , γ , σξ |ξi ) fN (ξi )d ξi +

−∞

∗(∗·)

[{π(it) (β , γ , σξ |ξi )}

∗(∗·)

{π(it) (β , γ , σξ |ξi ) }] fN (ξi )d ξi
(∗·)

(∗·)

− π(it) (β , γ , σξ )π(it) (β , γ , σξ )
(∗·)

= Σ (i,tt) (β , γ , σξ ),

(6.17)

(∗·)

where π(it) (β , γ , σξ ) is given in (6.16).
6.2.1.1

(b) Auto-Covariances Between Repeated Responses for y Variable
Recorded at Times u < t

Conditional on ξi , following (4.71), the covariance matrix between the response
vectors yiu and yit has the form
∗(∗·)

∗(∗·)

t
η(is|s−1),M − η(is|s−1) (J)1J−1 var[Yiu |ξi ], for u < t
cov[{Yiu ,Yit }|ξi ] = Πs=u+1

344

6 Multinomial Models for Longitudinal Bivariate Categorical Data
∗(∗·)

= (cov(Yiu j ,Yitk )) = (σ(i,ut) jk ), j, k = 1, . . . , J − 1
∗(∗·)

= Σ(i,ut) (β , γ , σξ |ξi ).

(6.18)

One then obtains the unconditional covariance matrix as follows by using the
conditioning un-conditioning principle.
Unconditional covariance matrix between yiu and yit for u < t :
cov[Yiu ,Yit ] = Eξi cov[{Yiu ,Yit }|ξi ] + covξi [E[Yiu |ξi ], E[Yit |ξi ]]
=
=

−∞

−∞

∗(∗·)

∗(∗·)

∗(∗·)

Σ(i,ut) (β , γ , σξ |ξi ) f N (ξi )d ξi + covξi [{π(iu) (β , γ , σξ |ξi )}, {π(it) (β , γ , σξ |ξi )}]
∗(∗·)

Σ(i,ut) (β , γ , σξ |ξi ) f N (ξi )d ξi +

(∗·)

−∞

∗(∗·)

∗(∗·)

[{π(iu) (β , γ , σξ |ξi )}{π(it) (β , γ , σξ |ξi ) }] fN (ξi )d ξi

(∗·)

− π(iu) (β , γ , σξ )π(it) (β , γ , σξ )
(∗·)

= Σ(i,ut) (β , γ , σξ ),
(∗·)

(6.19)
∗(∗·)

where π(it) (β , γ , σξ ) is given in (6.16), and Σ(i,ut) (β , γ , σξ |ξi ) for u < t is given by
(6.18).

6.2.1.2

(a) Marginal Expectation Vector and Covariance Matrix for z
Response Variable at Time t

The computation for these moments is similar to those for the moments of y. More
∗(∗·)
∗( j·)

specifically, replacing the notations π(i1)
j· , π(it) (β , γ , σξ |ξi ), and η(it|t−1) (g) in the
∗(·∗)

formulas for the moments of y variable, with π(i1)·r
(6.8), π(it) (α , λ , σξ |ξi ), and
∗(·r)

η(it|t−1) (h) (6.9), respectively, one may write the formulas for the moments for z
variable. To be brief, we write the formulas only as follows, without giving any
further explanation on computation.
⎧ ∗

[π(i1)·1 , . . . , π(i1)·r
, . . . , π(i1)·(R−1)
] = π(i1)·∗
for t = 1,

∗(·∗)

π
(α , λ , σξ |ξi ) = π(i1)·∗
for t = 1,

⎨ (i1)
∗(·∗)
∗(·∗)
(6.20)
E[Zit |ξi ] = π(it) (α , λ , σξ |ξi ) = η(it|t−1) (R)

∗(·∗)
∗(·∗)
∗(·∗)

+ η(it|t−1),M − η(it|t−1) (R)1R−1 π(i,t−1) (α , λ , σξ )

∗(·1)
∗(·r)
∗(·(R−1))
]
for t = 2, . . . , T,
= [π(it) , . . . , π(it) , . . . , π(it)

6.2 Correlation Model for Two Longitudinal Multinomial Response Variables

345

∗(·∗)

using π(i1) (α , λ = 0, σξ ) = π(i1)·∗
, and

var[Zit |ξi ] =

diag[π(i1)·1
, . . . , π(i1)·r
, . . . , π(i1)·(R−1)
] − π(i1)·∗
π ∗ (i1)·∗ for t = 1,

⎩ diag[π ∗(·1) , . . . , π ∗(·r) , . . . , π ∗(·(R−1)) ] − π ∗(·∗) π ∗(·∗)
(it)

(it)

(it)

(it)

(it)

(6.21)
for t = 2, . . . , T.

∗(·∗)

= Σ (i,tt) (α , λ , σξ |ξi ) : (R − 1) × (R − 1), (say), for t = 1, . . . , T.

(6.22)

In (6.20)–(6.21),
∗(·∗)

∗(·1)

∗(·r)

∗(·(R−1))

η(it|t−1) (R) = [η(it|t−1) (R), . . . , η(it|t−1) (R), . . . , η(it|t−1) (R)]
∗(·∗)

= π(i1) (α , λ , σξ |ξi )

(6.23)

∗(·∗)

η(it|t−1),M

⎛ ∗(·1)
∗(·1)
∗(·1)
η(it|t−1) (1) · · · η(it|t−1) (g) · · · η(it|t−1) (R − 1)

..
..
..
..
..

.
.
.
.
.

⎜ ∗(·r)
∗(·r)
∗(·r)

= ⎜ η(it|t−1) (1) · · · η(it|t−1) (g) · · · η(it|t−1) (R − 1) ⎟
⎟ : (R−1)×(R−1),(6.24)

.
.
.
.
.

..
..
..
..
..

∗(·(R−1))
∗(·(R−1))
∗(·(R−1))
η(it|t−1) (1) · · · η(it|t−1) (g) · · · η(it|t−1) (R − 1)
∗(·r)

with η(it|t−1) (g) as in (6.9), for r, g = 1, . . . , R − 1.
Next the unconditional moments are obtained by taking the average of the desired
quantities over the distribution of the random effects xii .:
Unconditional mean vector at time t = 1, . . . , T :
E[Zit ] = Eξi E[Zit |ξi ] =

−∞ π(i1)·∗ (α , σξ |ξi ) f N (ξi )d ξi
∗(·∗)

−∞ π(it) (α , λ , σξ |ξi ) f N (ξi )d ξi

for t = 1,
for t = 2, . . . , T,

(6.25)

(·∗)

= π(it) (α , λ , σξ ) : (J − 1) × 1, (say), for t = 1, . . . , T, (6.26)
where fN (ξi ) =

√1

exp[− 12 ξi2 ].

Unconditional covariance matrix at time t = 1, . . . , T :
var[Zit ] = Eξi var[Zit |ξi ] + varξi E[Zit |ξi ]
=
=

−∞

−∞

∗(∗·)

∗(·∗)

Σ(i,tt) (α , λ , σξ |ξi ) fN (ξi )d ξi + varξi [π(it) (α , λ , σξ |ξi )]
∗(·∗)

Σ(i,tt) (α , λ , σξ |ξi ) fN (ξi )d ξi +

−∞

∗(·∗)

∗(·∗)

[{π(it) (α , λ , σξ |ξi )}{π(it) (α , λ , σξ |ξi ) }] fN (ξi )d ξi

346

6 Multinomial Models for Longitudinal Bivariate Categorical Data
(·∗)

(·∗)

− π(it) (α , λ , σξ )π(it) (α , λ , σξ )
(·∗)

= Σ(i,tt) (α , λ , σξ ),

(6.27)

(·∗)

where π(it) (α , λ , σξ ) is given in (6.26).
6.2.1.2

(b) Auto-Covariances Between Repeated Responses for z Variable
Recorded at Times u < t

Similar to (6.18), the covariance matrix between the response vectors ziu and zit has
the form
∗(·∗)

∗(·∗)

t
cov[{Ziu , Zit }|ξi ] = Πs=u+1
η(is|s−1),M − η(is|s−1) (R)1R−1 var[Ziu |ξi ], for u < t
∗(·∗)

= (cov(Ziu j , Zitk )) = (σ(i,ut)r ), r, = 1, . . . , R − 1
∗(·∗)

= Σ(i,ut) (α , λ , σξ |ξi ).

(6.28)

One then obtains the unconditional covariance matrix as follows by using the
conditioning un-conditioning principle.
Unconditional covariance matrix between ziu and zit for u < t :]
cov[Ziu , Zit ] = Eξi cov[{Ziu , Zit }|ξi ] + covξi [E[Ziu |ξi ], E[Zit |ξi ]]
=
=

−∞

−∞

∗(·∗)

∗(·∗)

∗(·∗)

Σ(i,ut) (α , λ , σξ |ξi ) fN (ξi )d ξi + covξi [{π(iu) (α , λ , σξ |ξi )}, {π(it) (α , λ , σξ |ξi )}]
∗(·∗)

Σ(i,ut) (α , λ , σξ |ξi ) fN (ξi )d ξi +

(·∗)

−∞

∗(·∗)

∗(·∗)

[{π(iu) (α , λ , σξ |ξi )}{π(it) (α , λ , σξ |ξi ) }] fN (ξi )d ξi

(·∗)

− π(iu) (α , λ , σξ )π(it) (α , λ , σξ )
(·∗)

(6.29)

= Σ (i,ut) (α , λ , σξ ),

(·∗)

∗(·∗)

where π(it) (α , λ , σξ ) is given in (6.26), and Σ(i,ut) (α , λ , σξ |ξi ) for u < t is given by
(6.28).

6.2.1.3

(a) Covariance Matrix Between yit and zit of Dimension
(J − 1) × (R − 1)

Because y and z are uncorrelated conditional on ξi , the covariance matrix between
these two multinomial response variables at a given time point may be computed as:
cov[Yit , Zit ] = Eξi cov[{Yit , Zit }|ξi ] + covξi [E[Yit |ξi ], E[Zit |ξi ]]

6.2 Correlation Model for Two Longitudinal Multinomial Response Variables

347

= covξi [E[Yit |ξi ], E[Zit |ξi ]]

⎨ cov π ∗(∗·) (β , γ = 0, σ |ξi ), π ∗(·∗) (α , λ = 0, σ |ξi ) for t = 1
ξi
ξ
ξ
(i1)
(i1)
=
⎩ cov π ∗(∗·) (β , γ , σ |ξ ), π ∗(·∗) (α , λ , σ |ξ )
for t = 2, . . . , T.
ξi
ξ i
ξ i
(it)
(it)

∗(∗·)
∗(·∗)

−∞ π(i1) (β , γ = 0, σξ |ξi )π(i1) (α , λ = 0, σξ |ξi ) f N (ξi )d ξi

⎨ −π (∗·) (β , γ = 0, σ )π (·∗) (α , λ = 0, σ )
for t = 1
ξ (i1)
ξ
(i1)
=
∗(∗·)
∗(·∗)

−∞ π(it) (β , γ , σξ |ξi )π(it) (α , λ , σξ |ξi ) f N (ξi )d ξi

⎩ −π (∗·) (β , γ , σ )π (·∗) (α , λ , σ )
for t = 2, . . . , T.
ξ (it)
ξ
(it)
(∗∗)

= Σ(i,tt) (β , γ , α , λ , σξ ).

6.2.1.3

(6.30)

(b) Covariance Matrix Between yiu and zit of Dimension
(J − 1) × (R − 1)

For all u,t, this covariance matrix may be obtained as follows. Thus, the following
formulas accommodate the u = t case provided in Sect. 6.2.1.3(a). The general
formula is given by
cov[Yiu , Zit ] = Eξi cov[{Yiu , Zit }|ξi ] + covξi [E[Yiu |ξi ], E[Zit |ξi ]]
= covξi [E[Yiu |ξi ], E[Zit |ξi ]]

∗(∗·)
∗(·∗)

covξi π(i1) (β , γ = 0, σξ |ξi ), π(i1) (α , λ = 0, σξ |ξi )

⎨ covξ π ∗(∗·) (β , γ = 0, σξ |ξi ), π ∗(·∗) (α , λ , σξ |ξi )
(i1)
(it)
i
=
∗(∗·)
∗(·∗)

covξi π(iu) (β , γ , σξ |ξi ), π(i1) (α , λ = 0, σξ |ξi )

⎩ cov π ∗(∗·) (β , γ , σ |ξ ), π ∗(·∗) (α , λ , σ |ξ )
ξi

ξ

(iu)

i

(it)

ξ

i

for u = 1,t = 1
for u = 1,t = 2, . . . , T
for u = 2, . . . , T ;t = 1
for u,t = 2, . . . , T.

∗(∗·)
∗(·∗)

−∞ π(i1) (β , γ = 0, σξ |ξi )π(i1) (α , λ = 0, σξ |ξi ) fN (ξi )d ξi

(∗·)
(·∗)

−π(i1) (β , γ = 0, σξ )π(i1) (α , λ = 0, σξ )

⎪ ∞
∗(∗·)
∗(·∗)

−∞ π(i1) (β , γ = 0, σξ |ξi )π(it) (α , λ , σξ |ξi ) fN (ξi )d ξi

⎨ −π (∗·) (β , γ = 0, σ )π (·∗) (α , λ , σ )
ξ (it)
ξ
(i1)
=
⎪ ∞ π ∗(∗·) (β , γ , σ |ξi )π ∗(·∗) (α , λ = 0, σ |ξi ) fN (ξi )d ξi

ξ
ξ

−∞
(iu)
(i1)

(∗·)
(·∗)

−π(iu) (β , γ , σξ )π(i1) (α , λ = 0, σξ )

∗(∗·)
∗(·∗)

−∞ π(iu) (β , γ , σξ |ξi )π(it) (α , λ , σξ |ξi ) f N (ξi )d ξi

⎩ −π (∗·) (β , γ , σ )π (·∗) (α , λ , σ )
(iu)

(∗∗)

ξ

(it)

= Σ(i,ut) (β , γ , α , λ , σξ ).

ξ

for u = t = 1
for u = 1;t = 2, . . . , T
for u = 2, . . . , T ;t = 1
for u,t = 2, . . . , T.

(6.31)

348

6 Multinomial Models for Longitudinal Bivariate Categorical Data

6.3 Estimation of Parameters
6.3.1 MGQL Estimation for Regression Parameters
Recall that the regression parameters involved in the marginal (6.6) and conditional
(6.7) probabilities for the y variable with J categories are denoted by

β j∗ = (β j0 , β j ) , j = 1, . . . , J − 1
= (β j0 , β j1 , . . . , β jp ) , j = 1, . . . , J − 1,
and similarly the regression parameters involved in the marginal (6.8) and conditional (6.9) probabilities for the z variable with R categories are denoted by

αr∗ = (αr0 , αr ) , r = 1, . . . , R − 1
= (αr0 , αr1 , . . . , αrq ) , r = 1, . . . , R − 1.
Also recall that

β = (β1∗ , . . . , β j∗ , . . . , βJ−1
) : (J − 1)(p + 1) × 1

α = (α1∗ , . . . , αr∗ , . . . , αR−1
) : (R − 1)(q + 1) × 1,

and by further stacking we write

μ = [β , α ] : {(J − 1)(p + 1) + (R − 1)(q + 1)} × 1.

(6.32)

Next, for dynamic dependence parameters we use

θ = (γ , λ ) : {(J − 1)2 + (R − 1)2 } × 1,

(6.33)

where

γ = (γ1 , . . . , γ j , . . . , γJ−1 ) : (J − 1)2 × 1
λ = (λ1 , . . . , λr , . . . , λR−1 ) : (R − 1)2 × 1.
In this section, it is of interest to estimate the regression (μ ) parameters by
11) approach. Note that this GQL approach was used in Chap. 5, specifically
in Sect. 5.4, for the estimation of both regression and random effects variance
parameters involved in the cross-sectional bivariate multinomial models, whereas
in this chapter, more specifically in this section, the GQL approach is used only
for the estimation of regression parameters involved in the longitudinal bivariate
multinomial model. The application of the GQL approach for the estimation of
dynamic dependence parameters as well as σξ2 would be complex for the present

6.3 Estimation of Parameters

349

bivariate longitudinal model. For simplicity we estimate these latter parameters by
using the traditional method of moments (MM) in the next two sections. Note that,
similar to the GQL approach, the MM approach also produces consistent estimators
but they will be less efficient than the corresponding GQL estimators.
We now turn back to the GQL estimation of the regression parameter μ =
(β , α ) . This is also referred to as the marginal GQL (MGQL) approach as it is
constructed for a marginal set of parameters. For known γ , λ , the MGQL estimating
equation for μ = (β , α ) is given by
f (μ ) =

(∗·)

K

(·∗)

∂ (π(i) (β , γ , σξ ), π(i) (α , λ , σξ ))
∂μ

i=1

(∗·)

yi − π(i) (β , γ , σξ )

−1
× Σ(i)
(μ , γ , λ , σξ )

(·∗)

zi − π(i) (α , λ , σξ )

= 0,

(6.34)

where, for
yi = [yi1 , . . . , yit , . . . , yiT ] : (J − 1)T × 1, and
zi = [zi1 , . . . , zit , . . . , ziT ] : (R − 1)T × 1,
(∗·)

E[Yi ] = π(i) (β , γ , σξ )
(∗·)

(∗·)

(∗·)

(·∗)

(·∗)

= [π(i1) (·), . . . , π(it) (·), . . . , π(iT ) (·)] ,

(6.35)

by (6.16). Similarly by (6.26), one writes
(·∗)

E[Zi ] = π(i) (α , λ , σξ )
(·∗)

= [π(i1) (·), . . . , π(it) (·), . . . , π(iT ) (·)] .

(6.36)

Note that in (6.35),
(∗·)

(∗·)

π(it) (·) ≡ π(it) (β , γ , σξ )
(1·)

( j·)

((J−1)·)

= [π(it) (β , γ , σξ ), . . . , π(it) (β , γ , σξ ), . . . , π(it)

(β , γ , σξ )] , (6.37)

with
( j·)

π(it) (β , γ , σξ ) =

−∞

∗( j·)

π(it) (β , γ , σξ |ξi ) fN (ξi )d ξi ,

350

6 Multinomial Models for Longitudinal Bivariate Categorical Data

as in (6.10) and (6.16). Similarly, in (6.36),
(·∗)

(·∗)

π(it) (·) ≡ π(it) (α , λ , σξ )
(·1)

(·r)

(·(R−1))

= [π(it) (α , λ , σξ ), . . . , π(it) (α , λ , σξ ), . . . , π(it)

(α , λ , σξ )] ,

(6.38)

with
(·r)

π(it) (α , λ , σξ ) =

−∞

∗(·r)

π(it) (α , λ , σξ |ξi ) fN (ξi )d ξi ,

as in (6.20) and (6.26).

6.3.1.1

Construction of the Covariance Matrix Σ i ( μ , γ , λ , σξ )

In (6.34), the covariance matrix has the formula

Σi (μ , γ , λ , σξ ) = cov
=

Yi
Zi

: {(J − 1) + (R − 1)}T × {(J − 1) + (R − 1)}T

var(Yi ) : (J − 1)T × (J − 1)T cov(Yi , Zi ) : (J − 1)T × (R − 1)T
, (6.39)
cov(Zi ,Yi ) : (R − 1)T × (J − 1)T var(Zi ) : (R − 1)T × (R − 1)T

where

Yi1
⎜ .. ⎟
⎜ . ⎟
⎜ ⎟

var(Yi ) = var ⎜
⎜ Yit ⎟ : (J − 1)T × (J − 1)T
⎜ . ⎟
⎝ .. ⎠

YiT

var[Yi1 ] · · · cov[Yi1 ,Yit ] · · · cov[Yi1 ,YiT ]
..
..
..

.
.
.

=⎜
,Y
]
·
·
·
var[Y
]
·
·
·
cov[Y
,Y
]
cov[Y
it i1
it
it iT ⎟

.
.
.
..
..
..

cov[YiT ,Yi1 ] · · · cov[YiT ,Yit ] · · · var[YiT ]

⎛ (∗·)
(∗·)
(∗·)
Σ(i,11) (β , γ , σξ ) · · · Σ(i,1t) (β , γ , σξ ) · · · Σ(i,1T ) (β , γ , σξ )

..
..
..

.
.
.

⎜ (∗·)
(∗·)
(∗·)

=⎜
Σ
(
β
,
γ
,
σ
)
·
·
·
Σ
(
β
,
γ
,
σ
)
·
·
·
Σ
(
β
,
γ
,
σ
)
ξ
ξ
ξ ⎟ , (6.40)
⎜ (i,t1)
(i,tt)
(i,tT )

..
..
..

.
.
.

(∗·)
(∗·)
(∗·)
Σ(i,T 1) (β , γ , σξ ) · · · Σ(i,Tt) (β , γ , σξ ) · · · Σ(i,T T ) (β , γ , σξ )