Proceeding of International Conference On Research, Implementation And Education Of Mathematics And Sciences 2014, Yogyakarta State University, 18-20 May 2014
M-51
n i
i i
y it
T t
y it
it it
g g
L
1 1
1
1 ,
,
12
Then the log-likelihood function is
i
T t
i y
it y
it n
i
it it
g g
L LL
1 1
1
1 log
, ,
log 13
There are some methods of iteration to get the solution of equatian 13 : Newton-Raphson methods, BFGS methods, etc.
4. GENERATING DATA SIMULATION AND RESULT
We will generate simulation data with T=3, n=1000. Then, the equations of utility are U
i0t
=
i0
+ 0
0t
+
0t
X
i
+
t
Z
i0t
+
i01
and U
i1t
=
i1
+ 0
1t
+
1t
X
i
+
t
Z
i1t
+
i11
14 for i=1,...,N; j=0,1 and t=1,...,3;
ijt
~Extreme Value Type I,
i
~ N0,
2
. Equation 14 can be presented in difference of utility U
it
= U
i1t
–U
i0t
. On Logit Model, equations of utility difference are
U
it
= 0
t
+
t
X
i
+
t
Z
it
+
i
+
it
15 or
U
i1
= V
i1
+
i
+
i1
; U
i2
= V
i2
+
i
+
i2
;U
i3
= V
i3
+
i
+
i3
with V
it
=
V
i1t
–
V
i0t
=
it t
i t
t
Z X
. Z
it
= Z
i1t
– Z
i0t
; 0
t
= 0
0t
- 0
1t
;
t
=
0t
-
1t
;
i
=
i1
-
i0
. We generate data on
0
t
=-1;
t
= 0.5 ,
t
=0.3 and some of variance
2
=0; 0.5; 1; 2; 4, 6 using program R.2.8.1. From the data simulation, we built the Logit Model using three
approximations. First, we assume independence each other for all i, j and t independent logit model and estimate parameters using MLE
MLECS
. Second, we estimate parameter using GEE
GEE
. The last approximation is Random Effect Model using MLE
MLERE
. Results of the simulations presented in Figure 1 to Figure 10.
Figure 1. Estimator of
2
Figure β. Estimator of 0
1
Based on simulation result representing at Figure 1., estimator of
2
using Random effect Model
MLERE
more accurate than GEE
GEE
.
MLERE
have value closer to real value of parameter than
GEE
.
Jaka Nugraha Random Effect Model... ISBN.978-979-99314-8-1
M-52 Overall, estimator of 0
t
,
t
and
t
in MLE as same as GEE see Figure 2 to Figure 10.
Figure γ. Estimator of 0
2
Figure 4. Estimator of 0
3
Figure 5. Estimator of
1
Figure 6. Estimator of
2
Figure 7. Estimator of
3
Figure 8. Estimator of
1
Figure 9. Estimator of
2
Figure 10. Estimator of
3
We can see that on
2
= 0 no effect individual estimators by three approximations are same.
Proceeding of International Conference On Research, Implementation And Education Of Mathematics And Sciences 2014, Yogyakarta State University, 18-20 May 2014
M-53
On general, estimator of independent Logit Model
MLECS
ˆ
and estimator of GEE
GEE
ˆ
are
not different but increasing of
2
impact to increasing bias of estimator. On data having individual effect see Figure 2 to Figure 10, the random effect model
MLERE
ˆ
better than
MLECS
ˆ
and
GEE
ˆ
.
By GEE, we estimate coefficient regressions and correlation among alternative. Using effect random
model we
can estimate
coefficient regressions,
individual effect
and covarianscorrelations among alternative. On value of individual effect
2
about 1, random effect model can estimate covarians of utility more appropriate than the others value of
2
.
5. CONCLUSION
