GENERALIZED ESTIMATING EQUATIONS (GEE) FOR MIXED LOGISTIC MODELS Mohand-Larbi Feddag (1)
(1,2)
, Ion Grama
(1)
and Mounir Mesbah
(1)
Dept of Applied Statistics (SABRES)
University of South Brittany, Rue Yves Mainguy, Tohannic F56000 Vannes, France (2)
University of Sciences and Technology of Algiers (USTHB)
BP 32, El Alia 16111 Algiers, Algeria E-Mail :
[email protected] Key Words: Generalized linear mixed model; Logistic model; Generalized Estimating Equations; Fisher-scoring algorithm; Rasch model. ABSTRACT In this paper, the problem of estimating the xed eects parameters and the variance components in logistic mixed models is considered. It is well known that estimating these parameters by the method of maximum likelihood faces computational di culties. As an alternative, we propose the Generalized Estimating Equations approach (denoted, GEE) de ned by Liang and Zeger. The estimators obtained are consistent and asymptotically normal. We illustrate this method with simulations and with an analysis of real data in quality of life. 1. INTRODUCTION Repeated binary data in which each subject is observed repeatedly, occur frequently in public health and medicine. Some examples of these studies include cross-over trials, clustered data and longitudinal data. The mixed logistic model is usually used to model the heterogeneity between the subjects and the correlation among repeated observations. It is well known that using maximum likelihood to estimate the xed eect parameters and variance components is computationally di cult. As a result, the EM algorithm is frequently used to estimate these parameters. An alternative to the maximum likelihood approach is 1
the method of generalized estimating equation (GEE) developed by Liang and Zeger (1). It is de ned as an extension of Quasi-likelihood approach (see Wedderburn (2)) to the repeated measures. This approach does not require the complete speci cation of the joint distribution of the repeated responses but rather only the rst two moments. A major advantage of GEE is that it provides a consistent estimate for the regression parameter even when the correlation matrix is misspeci ed. The simulation results (see Liang and Zeger (1) and Feddag (3)) indicate that the e ciency loss relative to the maximum likelihood is small. Several modi cations and extensions of the GEE methodology have been studied. Prentice (4) extends this approach for correlated binary data by specifying supplementary generalized estimating equations, based on the empirical pairwise covariances that permit the estimation of the correlation parameters. Zhao and Prentice (5), and Prentice and Zhao (6) have generalized this work to the correlated binary regression using a quadratic exponential model. The application of the GEE approach for xed eects parameters requires the rst and second order marginal moments of the response. Under the mixed model, the exact form of these two rst moments is not available and approximations are often necessary. This approach has been used by Zeger et al. (7) to estimate the regression parameters through the use of logit approximations by the probit function ( Kotz et al. (8) and Johnson et al. (9) ), while the variance components are estimated empirically. The estimation of the variance component of the random eects in the generalized linear mixed models is however more complex as compared to the xed eects parameter estimation. This is because the construction of the estimating equations for the variance components requires the joint moments up to order four of the response. Sutradhar and Rao (10) has used approximations for the joint moments to estimate the regression parameters and the small variance components separately. In this article we propose the estimating equations based on the empirical pairwise covariances to estimate the variance components. These equations are used by Prentice (4) to estimate the correlation parameters. We use the approximations proposed by Sutradhar and
2
Rao (10) to estimate by the GEE approach, the xed eects parameters and the univariate variance component of a random eect simultaneously. The estimators obtained are shown to be consistent and asymptotically normal. The paper is organized as follows. The mixed logistic model is given in Section 2. Thereafter in Section 3, we give approximations for the joint moments up to order 4 and use them for approximating the mean and the covariance matrix of the data. Next, we propose the equations for simultaneous estimation of the xed eects parameters and of the variance of the random eects and establish the asymptotic properties of the estimators. In Section 4, we present some simulation results and give an application using real data from a quality of life experiment. A summary discussion is presented in Section 5. 2. LOGISTIC MIXED MODELS Consider a sample of K independent random multivariate binary observations yi =
(yi1 , . . . , yini )t , i = 1, . . . , K, where yij = 1 if the i-th individual has a positive response (say success) on the j-th response and yij = 0 otherwise. We shall assume that yij depends on a p × 1 vector of xed covariates xij associated with a xed eect β = (β1 , . . . , βp )t and on a xed covariate zij associated with the univariate non observed random eect bi . The logistic mixed model belongs to the family of generalized linear mixed models (see McCullagh et al. (11)), satisfying the following conditions:
• Given bi , the variables yi1 , . . . , yini are mutually independent with density function given by
© ¡ ¡ ¢¢ª f (yij | bi ) = exp yij (xtij β + zij bi ) − ln 1 + exp xtij β + zij bi ,
(1)
where β = (β1 , . . . , βp )t is the parameter vector of xed eects.
• The random eects, b1 , . . . , bK , are mutually independent with a common underlying distribution G. For the model introduced above, the conditional mean and the conditional variance are 3
given by
E(yij | bi ) = h−1 (xtij β + zij bi ) ¢¢ª © ¡ ¡ , = exp xtij β + zij bi − ln 1 + exp xtij β + zij bi V ar (yij | bi ) = v(E(yij | bi )) n ¡ ¡ t ¢¢2 o t = exp xij β + zij bi − ln 1 + exp xij β + zij bi , where h and v are the link and the variance function de ned respectively by h(x) = logit (x) and v(x) = x(1 − x). >From now on, we will assume that G is the cumulative distribution function of the normal distribution with mean 0 and variance σ 2 . We are interested in estimating (β, σ 2 ). Usually, one uses the maximum likelihood approach. The marginal likelihood of (y1 , ..., yK ) is given by
L(y; β, σ 2 ) = ³√
1 2πσ 2
(Z K Y ´K i=1
¡ ¡ ¢¢ µ 2¶ ) exp yij xtij β + zij bi −bi ¡ t ¢ exp dbi . 2 2σ 1 + exp x β + z b ij i ij j=1
ni +∞ Y −∞
(2)
The maximization of the above function is computationally di cult and requires iterative techniques. Most researchers now use the EM algorithm. The integrals at two steps of the algorithm, the E step and the M step, are often approximated by the Gauss-Hermite procedure (Rigdon et al. (12)), a method with a slow rate of convergence. To avoid these computational di culties, we propose the GEE approach, which turns out to be computationally less intensive. For the generalized linear mixed model introduced above, the mean, µi = (µij )j=1,...,ni and the variance, Vi,11 = (σi,jl )j,l=1,...,ni are respectively de ned as follows: ¡ ¢ µ 2¶ Z +∞ exp xtij β + zij bi −bi 1 2 ¡ t ¢ exp dbi , µij = µij (β, σ ) = √ 2σ 2 2πσ 2 −∞ 1 + exp xij β + zij bi
σijl = σijl (β, σ 2 ) ¢ ¡ Z +∞ exp (xtij + xtil )β + (zij + zil )bi ¢ ¡ ϕσ2 (bi )dbi = 1 + exp(xtij β + zij bi ) (1 + exp(xtil β + zil bi )) −∞ −µij µil , 4
(3)
(4)
where ϕσ2 (θ) =
√ 1 2πσ 2 2
³ exp
−θ2 2σ 2
´
is the density function of the normal distribution with mean
0 and variance σ .
3. GEE FOR THE LOGISTIC MIXED MODELS For a generalized linear logistic model, Prentice (4) proposed estimating the correlation parameters by supplementary estimating equations, based on the empirical pairwise covariances. We propose using this idea, along with basic equations, to estimate simultaneously the regression parameter β and the parameter of the random eect σ 2 for the model formulated in Section 2. This method requires the computation of the joint moments of the variables up to order four. 3.1. THE APPROXIMATION OF MARGINAL LIKELIHOOD AND JOINT MOMENTS The aim of this section is to give the approximations of the joint moments up to order four of the observed variable yi , which we shall use later on in the GEE. Their computation requires an approximation of the marginal likelihood of yi , i = 1, . . . , K given by formula (2). This approximation is obtained by using a Taylor series expansion about the random eects
bi = 0. For all i = 1, . . . , K, j = 1, . . . , ni , let fij denote the density fij (yij ; bi , β) = exp (yij αij − aij ) , ¡ ¡ ¢¢ where αij = xtij β + zij bi and aij = ln 1 + exp xtij β + zij bi . Conditional on bi , the likelihood of yi = (yi1 , . . . , yini )t is given by
Lci (β | bi , yi ) =
ni Y
fij (yij ; bi , β).
j=1
and the marginal likelihood of yi is
1 Li (yi ; β, σ ) = √ 2πσ 2
Z
+∞ −∞
½ ¾ b2i c exp ln (Li (β | bi , yi )) − 2 dbi . 2σ
(5)
2r It is well known that E (b2r i ) = O(σ ), for all r ≥ 1. Up to now we assume that the following
assumption is satis ed
¡ ¢ E b2r = 0(σ 5 ), for all r ≥ 3. i
The following theorem gives approximations for the joint distributions of yi . 5
(6)
Theorem 1 Up to a term of order O (σ 6 ), we have the following: 1. The likelihood L(y; β, σ 2 ) de ned in (2), is given by ∗
K Y
2
L (y; β, σ ) =
L∗i (yi ; β, σ 2 ),
i=1
with
L∗i (yi ; β, σ 2 )
=
ni Y
fij∗ (yij ; β)
j=1
µ ¶ σ2 2 σ4 1 + (Ai − Bi ) + Qi . 2 8
(7)
2. For all subsets Jm = {j1 , . . . , jm }, 1 ≤ m ≤ ni , i = 1, . . . , K , the joint density of
(yij1 , . . . , yijm ) is given by L∗i,j1 ...jm (yij1 , . . . , yijm ; β, σ 2 ) = µ ¶ m Y σ2 2 σ4 ∗ fijl (yijl ; β) × 1 + (Ai,j1 ...jm − Bi,j1 ...jm ) + Qi,j1 ...jm . 2 8 l=1 3. For all (i, j), i = 1, . . . , K, j = 1, . . . , ni , the density of yij is given by ¶ µ σ4 σ2 2 ∗ 2 ∗ Li,j (yij ; β, σ ) = fij (yij ; β) 1 + (Ai,j − Bi,j ) + Qi,j . 2 8 The quantities involved in the points 1., 2. and 3. are de ned below:
n o t fij∗ (yij ; β) = exp yij xtij β − ln(1 + exij β ) , (1)
Ai,j = zij (yij − aij ),
(2)
Bi,j = zij2 aij ,
(3)
(4)
Ci,j = zij3 aij , Fi,j = zij4 aij , ni ni X X Ai = Ai,j , Bi = Bi,j , Ci = Qi =
j=1 ni X
Ci,j ,
j=1 A4i −
Fi =
j=1 ni X
Fi,j ,
j=1
6A2i Bi − 4Ai Ci + 3Bi2 − Fi ,
6
(8)
(9)
Ai,j1 ...jm = Ci,j1 ...jm =
m X
Ai,jl ,
Bi,j1 ...jm =
m X
l=1
l=1
m X
m X
Ci,jl ,
Fi,j1 ...jm =
Bi,jl , Fi,jl ,
l=1
l=1
Qi,j1 ...jm = A4i,j1 ...jm − 6A2i,j1 ...jm Bi,j1 ...jm 2 −4Ai,j1 ...jm Ci,j1 ...jm + 3Bi,j − Fi,j1 ...jm , 1 ...jm 2 − Fi,j , Qi,j = A4i,j − 6A2i,j Bi,j − 4Ai,j Ci,j + 3Bi,j t
(1) aij
=
t
exij β t
(1 + exij β ) t
(3) aij
=
(2) aij
,
=
exij β t
(1 + exij β )2 t
t
exij β (1 − exij β ) t
(1 + exij β )3
,
(4) aij
=
, t
t
exij β (e2xij β − 4exij β + 1) t
(1 + exij β )4
.
Proof: The proof of this theorem requires the following lemma. Lemma 1 Let Y be a random variable with density function de ned by fθ (y) = exp {yθ − a(θ) + c(y)} , where a(.) and c(.) are functions respectively of θ and y . We have the following results
¡ ¢ E(Y ) = a(1) (θ), V (Y ) = a(2) (θ), E (Y − a(1) )3 = a(3) (θ), ¡ ¢ ¡ ¢ E (Y − a(1) )4 = a(4) (θ) + 3(a(2) (θ))2 , E (Y − a(1) )5 = a(5) (θ) + 10a(3) (θ)a(2) (θ) ¡ ¢ ¡ ¢ ¡ ¢ E Y (Y − a(1) )r = E (Y − a(1) )r+1 + a(1) E (Y − a(1) )r , (r = 2, 3, 4). Based on the assumption made in (6) and on expanding the function fij in a Taylor series about bi = 0 up to order four, we obtain the expression µ ¶ b2i b3i b4i ∗ 5 fij (yij ; β, bi ) = fij (yij ; β) 1 + Ai,j bi + Ri,j + Pi,j + Qi,j + O(bi ) , 2 6 24 where
Ri,j = A2i,j − Bi,j ,
Pi,j = A3i,j − 3Ai,j Bi,j − Ci,j .
7
Then we can approximate the expression (5) by
ˆ ci (β L
| b i , yi ) =
ni Y
µ fij∗ (yij ; β)
j=1
= T (yi ; β, bi )
ni Y
b3 b4 b2 1 + Ai,j θi + Ri,j i + Pi,j i + Qi,j i 2 6 24
¶
fij∗ (yij ; β)
j=1
with
T (yi ; β, bi ) = 1 + Ai bi + Ri Ri = A2i − Bi ,
b3 b4 b2i + P i i + Qi i , 2 6 24
Pi = A3i − 3Ai Bi − Ci .
ˆ ci (β/bi , yi ) over bi and using Taking the expectation of L E(bi ) = E(b3i ) = 0, E(b2i ) = σ 2 , E(b4i ) = 3σ 4 , we obtain the expression (7). The joint density of (yij1 , . . . , yijm ) is given by
X
L∗i,j1 ...jm (yij1 , . . . , yijm ; β, σ 2 ) =
L∗i (yi ; β, σ 2 )
yil = 0, 1 1 ≤ l 6∈ {j1 , . . . , jm } ≤ ni
By using the previous lemma for the density function fij∗ , we deduce that
E(Ai,j ) = E(Ri,j ) = E(Pi,j ) = E(Qi,j ) = 0. These results yield the expression (8). We obtain the marginal density of yij easily and it is given by :
L∗i,j (yij ; β, σ 2 ) = L∗i,jk (yij , yik = 0; β, σ 2 ) + L∗i,jk (yij , yik = 1; β, σ 2 ). 2 We now use the above theorem to derive the joint moments of the variables up to order four.
Theorem 2 For all i, j, k, l, h satisfying 1 ≤ i ≤ K, 1 ≤ j 6= k 6= l 6= h ≤ ni , the following holds: 8
1. E (yij ) = µij + O (σ 6 ) , where t
µij =
t
exij β t
1 + exij β
t
σ 2 exij β (1 − exij β ) + zij2 t 2 (1 + exij β )3 t
t
t
t
σ 4 exij β (−e3xij β + 11e2xij β − 11exij β + 1) + zij4 . t 8 (1 + exij β )5
(10)
2. E (yij yik ) = ζijk + O (σ 6 ) , where t
t
exij β exik β
ζijk =
t
t
(1 + exij β )(1 + exik β ) i σ 2 h 2 (3) (1) (2) (2) (1) (3) + zij aij aik + 2zij zik aij aik + zil2 aij aik 2 σ 4 h 4 (5) (1) (4) (2) 2 (3) (3) + aij aik z a a + 4zij3 zik aij aik + 6zij2 zik 8 ij ij ik i 3 (2) (4) 4 (1) (5) +4zij1 zik aij aik + zik aij aik ,
(11)
3. E (yij yik yil ) = ζijkl + O (σ 6 ) , where t
t
t
exij β exik β exil β
ζijkl =
t
t
t
(1 + exij β )(1 + exik β )(1 + exil β ) · ¸ ´ σ4 σ 2 ³ ˆ2 ˆ × 1+ Ai,jkl − Bi,jkl + Qi,jkl , 2 8
(12)
4. E (yij yik yil yih ) = ζijklh + O (σ 6 ) , where t
t
t
t
exij β exik β exil β exih β
ζijklh = t t t t (1 + exij β )(1 + exik β )(1 + exil β )(1 + exih β ) · ¸ ´ σ4 σ 2 ³ ˆ2 ˆ × 1+ Ai,jklh − Bi,jklh + Qi,jklh . 2 8
(13)
The quantities involved in the points 1., 2., 3. and 4. are de ned in Theorem 1 and below: t
(5) aij
=
t
t
t
exij β (−e3xij β + 11e2xij β − 11exij β + 1) t
(1 + exij β )5
(1) (1) (1) Aˆi,jkl = zij (1 − aij ) + zik (1 − aik ) + zil (1 − ail ), 2 ˆ i,jkl = Aˆ4i,jkl − 6Aˆ2i,jkl Bi,jkl − 4Aˆi,jkl Ci,jkl + 3Bi,jkl − Di,jkl , Q (1) (1) (1) (1) Aˆi,jklh = zij (1 − aij ) + zik (1 − aik ) + zil (1 − ail ) + zih (1 − aih ), 2 ˆ i,jklh = Aˆ4i,jklh − 6Aˆ2i,jklh Bi,jklh − 4Aˆi,jklh Ci,jklh + 3Bi,jklh − Di,jklh . Q
9
Proof: We obtain the assertions of this theorem by substituting the derivatives a(t) ij , t = 1, . . . , 4, by their explicit expressions given by Theorem 1 in the following equations E(yij ) =
1 X
yij L∗i,j (yij ; β, σ 2 ) = L∗i,j (1; β, σ 2 ),
yij =0
E(yij yik ) =
1 1 X X
yij yik L∗i,jk (yij , yik ; β, σ 2 ) = L∗i,jk (1, 1; β, σ 2 ),
yij =0 yik =0
E(yij yik yil ) = L∗i,jkl (1, 1, 1; β, σ 2 ), E(yij yik yil yih ) = L∗i,jklh (1, 1, 1, 1; β, σ 2 ). 2 This theorem is used to compute the covariance matrix Vi , i = 1, . . . , K , given in Section 3.3. 3.2. ESTIMATION OF THE PARAMETERS Our approach for estimating parameters β and σ 2 is as follows. Along with the basic estimating equations for the mean of yi , which gives estimators for the regression parameter
β , we shall use supplementary equations to estimate the parameter of the random eect σ 2 . These equations are based on the empirical covariances which are unbiased estimators of the true covariances of the vector yi . The idea goes back to Prentice (4) and Prentice and Zhao (6), who used these equations for estimating the correlation parameter in a generalized logistic model with xed eects only. Consider yi = (yi1 , . . . , yini )t , i = 1, . . . , K, to be the outcomes of the logistic mixed model de ned in Section 2. Let si be a
ni (ni −1) 2
× 1 vector of empirical pairwise covariances
de ned by
si = (si,jl )t1≤j²
√
3 ) Tr(BK
o
k z − E (Zi ) k2 dFi (z),
and Fi is the cumulative density function of Zi , i = 1, ..., K. 3. For i = 1, . . . , K , the function Dit V−1 i (Zi − E (Zi )) is twice dierentiable with respect to (β, σ 2 ) and the rst derivative is integrable. 11
4. The covariance matrix Cov(Zi ) of Zi , i = 1, . . . , K, satis es K 1 X lim diag {Cov(Zi )} = 0. K→∞ K 2 i=1
Theorem 3 Under assumption 1-4, ³
´ 2 ˆ β, σ ˆ is consistent for (β, σ 2 ) and is asymptotically normal K
1/2
µ³ ¶ ´t ¡ ¢ t L 2 2 βˆ − β , σ ˆ −σ −→ N (0, W )
where
as
K → ∞,
¡ ¢ −1 W = lim K A−1 1 A2 A1 K7−→∞
with
A1 =
K X
Dit V−1 i Di ,
A2 =
K X
(16)
−1 Dit V−1 i Cov(ξi )Vi Di .
i=1
i=1
The proof of this theorem follows the same lines as in Liang and Zeger (1) and Prentice (4) and therefore will not be detailed here. For details on a similar proof, we refer to Feddag (3). The covariance matrix W is consistently estimated by
´ ³ ˆ = lim K Aˆ−1 Aˆ2 Aˆ−1 , W 1 1 K7−→∞
(17)
ˆ σ where Aˆ1 , Aˆ2 are the values of A1 and A2 respectively at (β, ˆ 2 ). ˆ σ The computation of (β, ˆ 2 ) is obtained by the Fisher-scoring algorithm. The iterative procedure at step (j + 1) is given by ÃK !−1 Ã K ! (j+1) (j) ˆ ˆ X t X t β β t 1 ˆ −1 D ˆ −1 ξˆi , ˆi V ˆi ˆi V = + D D i i 2(j+1) 2(j) K σ ˆ σ ˆ i=1 i=1
ˆ i , ξˆi are respectively the values of Di , Vi and ξi at (βˆ(j) , σ ˆi , V where D ˆ 2(j) ). 3.3. THE COMPUTATION OF THE COVARIANCE MATRIX Vi 12
The approximations Vi of the matrices Vi are constructed using the approximations of the joint moments of yi up to order 4 given in Section 3.1. We shall consider three variants for Vi in the simulations presented in Section 4.1 latter on. In any case the working matrix
Vi will be of the form
Vi =
Vi,11 Vi,12 Vi,21 Vi,22
.
Here the elements of the matrix Vi,11 = (σijl )j,l=1,...,ni , with σijl = ζijl − µij µil , are approximations of the rst two moments of yi , and can be computed easily from Theorem 2, where the approximations of the mean µi = (µij )j=1,...,ni and of the joint moments of order two
(ζijl )j,l=1,...,ni of yi , i = 1, . . . , K , are given by (10) and (11). The remaining entries are speci ed as follows. The case of completely speci ed matrix Vi corresponds to a ni ×
ni (ni −1) 2
matrix Vi,12 = VTi,21 de ned by Cov(yij , yi,jl ) = (1 − µij )σijl ,
Cov(yij , si,kj ) = (1 − µij )σijk ,
Cov(yij , si,kl ) = ζijkl − µij σikl − µik σijl − µil σijk − µij µik µil , and a
ni (ni −1) 2
×
ni (ni −1) 2
(18)
matrix Vi,22 de ned by
2 Cov(si,jl , si,jl ) = (1 − 2µij )(1 − 2µil )σijl + σijj σill − σijl ,
Cov(si,jl , si,jm ) = (1 − 2µij ) (ζijml − µim σijl − µil σijm − µij µim µil ) + µ2ij σilm − σijl σijm , (19) Cov(si,jl , si,kj ) = (1 − 2µij ) (ζijkl − µik σijl − µil σijk − µij µik µil ) + µ2ij σilk − σijl σikj ,
(20)
Cov(si,jl , si,lm ) = (1 − 2µil ) (ζijlm − µim σijl − µij σilm − µij µim µil ) + µ2il σijm − σijl σilm , (21) Cov(si,jl , si,kl ) = (1 − 2µil ) (ζijlk − µij σilk − µik σijl − µij µik µil ) + µ2il σijk − σijl σikl ,
(22)
Cov(si,jl , si,km ) = ζijlkm − µij ζilkm − µil ζijkm − µik ζijlm − µim ζijlk + µij µik ζilm
+µij µim ζilk + µij µil ζikm + µil µik ζijm + µil µim ζijk +µik µim ζijl − 3µij µil µik µim − σijl σikm .
13
(23)
The second case consider that all the covariances given by formula (23) are taken to be 0. The last one assumes in addition that the matrix Vi,22 is diagonal (i.e. the covariances given by formulas (19 - 22) are 0) and all elements given by formula (18) are 0. 4. ILLUSTRATION BY RASCH MODEL IRT (Item Response Theory) models rst appeared in the eld of psychometry and educational sciences to quantify human behavior. They are now increasingly used in medicine to study psychological traits in psychiatry and more recently, to assess quality of life in clinical trials or epidemiology. Generally, the quality of life of the patients is evaluated using questionnaires with dichotomous items. One of the most popular IRT model is the Rasch model (Fisher and al. (13), Hamon (14)). In our application, we consider the mixed Rasch model as a particular case of the logistic mixed model de ned in section 2, where the covariate
xij associated with the xed eects is (0, . . . , 0, −1, 0, . . . , 0) and the covariate zij associated with the random eect is equal to 1 for all i, j . This example will illustrate how the Rasch model explains the occurrence of a data matrix containing the dichotomously scored answers of a sample of K people to J items xed in advance (ni = J, i = 1, . . . , K), that measure the same latent trait b. It is assumed that each subject i has a real value person parameter bi , denoting his/her position on the latent trait. Each item j , has a real valued item parameter
βj denoting the di culty of this item. Let yi = (yij )j=1,...,J be the binary response vector for the i-th individual for all J items i = 1, . . . , K, where yij = 1 if the i-th individual has a positive response (correct, agree) for item j and yij = 0 (false, disagree) otherwise. The probability pij of the response of the i-th individual to the j-th item is given by
pij = fij (yij ; bi , βj ) =
exp ((bi − βj )yij ) , 1 + exp (bi − βj )
(24)
where bi , i = 1, . . . , K , are independent realizations of random variable b called latent variable and βj the di culty parameter of item j. Since all variables yi , i = 1, . . . , K, have the same mean and the same covariance matrix V, then the GEE de ned by (15) reduces to 2
t
U (β, σ ) = D V
−1
K X i=1
14
ξi = 0,
(25)
where
D=
D11 D12 D21 D22
,
V =
V11 V12 V21 V22
,
ξi =
yi − µ si − η
.
In this case, the mean µ = (µj )j=1,...,J and the covariance matrix V11 = (σjl )j,l=1,...,J of
yi , i = 1, . . . , K are given by 1 σ 2 eβj (eβj − 1) σ 4 eβj (e3βj − 11e2βj + 11eβj − 1) µj (βj , σ) = + + , 1 + eβj 2 (1 + eβj )3 8 (1 + eβj )5 σjj (βj , σ) =
eβj σ 2 eβj (eβj − 1)2 + (1 + eβj )2 2 (1 + eβj )4 σ 4 eβj (e4βj − 14e3βj + 26e2βj − 14eβj + 1) , + 8 (1 + eβj )6
· eβj eβl σ 4 (e3βj − 4e2βj + eβj )eβl σjl (β, σ) = σ + (1 + eβj )2 (1 + eβl )2 2 (1 + eβj )4 (1 + eβl )2 ¸ eβj (eβj − 1)eβl (eβl − 1) eβj (e3βl − 4e2βl + eβl ) + + , (1 + eβj )3 (1 + eβl )3 (1 + eβj )2 (1 + eβl )4
(26)
(27)
2
(28)
and the matrices D11 , D12 , D22 , D21 are respectively given by µ µ µ ¶ ¶ ¶ ∂σjl ∂σjl ∂µj ∂µj , D22 = , D12 = , )j=1,...,J , D12 = D11 = diag( ∂βj ∂σ 2 j=1,...,J ∂σ 2 1≤j
