Saran KESIMPULAN DAN SARAN
DAFTAR PUSTAKA
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Unpublish manuscript.
LAMPIRAN
lampiran Lampiran 1. Output SAS untuk analisis data longitudinal
Covariance Parameter Estimates Standard Z
Cov Parm Subject Estimate Error Value Pr Z UN1,1 patient 15.9111 1.1702 13.60 .0001
UN2,1 patient -0.1300 0.06169 -2.11 0.0350 UN2,2 patient 0.02854 0.005968 4.78 .0001
Residual 3.0716 0.1713 17.93 .0001 Fit Statistics
-2 Log Likelihood 7006.3 AIC smaller is better 7026.3
AICC smaller is better 7026.4 BIC smaller is better 7067.7
Null Model Likelihood Ratio Test DF Chi-Square Pr ChiSq
3 1181.63 .0001
Solution for Fixed Effects Standard
Effect Estimate Error DF t Value Pr |t| Intercept 8.0129 0.3511 463 22.82 .0001
obstime -0.1668 0.02038 404 -8.19 .0001 obstimerandgrp1 0.02998 0.02891 535 1.04 0.3003
gender1 -0.1582 0.3249 535 -0.49 0.6265 prevoi1 -2.3152 0.2382 535 -9.72 .0001
stratum1 -0.1309 0.2352 535 -0.56 0.5780 Type 3 Tests of Fixed Effects
Num Den Effect DF DF F Value Pr F
obstime 1 404 67.00 .0001 obstimerandgrp1 1 535 1.08 0.3003
gender1 1 535 0.24 0.6265 prevoi1 1 535 94.51 .0001
stratum1 1 535 0.31 0.5780
Lampiran 2. Boxplot nilai ARB untuk pengaruh acak menyebar normal ganda dan bivariate-t diasumsikan normal pada frekuensi
pengamatan longitudinal sering dan jarang
Parameter=a12
nn4 nn9
tn34 tn39
tn44 tn49
tn54 tn59
-1000 1000
2000 3000
4000 5000
R B
kode
Parameter=a11
nn4 nn9
tn34 tn39
tn44 tn49
tn54 tn59
-500 500
1000 1500
2000 2500
R B
kode
Parameter=b10
nn4 nn9
tn34 tn39
tn44 tn49
tn54 tn59
-150 -100
-50 50
100
R B
kode
Parameter=a22
nn4 nn9
tn34 tn39
tn44 tn49
tn54 tn59
-1000 1000
2000 3000
4000
R B
kode
Parameter=b20
nn4 nn9
tn34 tn39
tn44 tn49
tn54 tn59
-100 -50
50
R B
kode
Parameter=b11
nn4 nn9
tn34 tn39
tn44 tn49
tn54 tn59
-100 -50
50 100
R B
kode
Parameter=b22
nn4 nn9
tn34 tn39
tn44 tn49
tn54 tn59
-400 -300
-200 -100
100
R B
kode
Parameter=b21
nn4 nn9
tn34 tn39
tn44 tn49
tn54 tn59
-250 250
500 750
1000
R B
kode
Parameter=s22
nn4 nn9
tn34 tn39
tn44 tn49
tn54 tn59
-100 100
200 300
R B
kode
Parameter=s21
nn4 nn9
tn34 tn39
tn44 tn49
tn54 tn59
-40 -20
20 40
60
R B
kode
Lampiran 3. Boxplot nilai MSE untuk pengaruh acak menyebar normal ganda dan bivariate-t diasumsikan normal pada frekuensi pengamatan
longitudinal sering dan jarang
Par am et er =a11
nn4 nn9
t n34 t n39
t n44 t n49
t n54 t n59
500 1000
1500 2000
2500
S E
kode
Parameter=a12
nn4 nn9
tn34 tn39
tn44 tn49
tn54 tn59
1000 2000
3000 4000
S E
kode
Parameter=b10
nn4 nn9
tn34 tn39
tn44 tn49
tn54 tn59
0.25 0.50
0.75 1.00
1.25
S E
kode
Parameter=a22
nn4 nn9
tn34 tn39
tn44 tn49
tn54 tn59
1000 2000
3000 4000
5000 6000
S E
kode
Parameter=b20
nn4 nn9
tn34 tn39
tn44 tn49
tn54 tn59
10 20
30 40
50
S E
kode
Parameter=b11
nn4 nn9
tn34 tn39
tn44 tn49
tn54 tn59
0.25 0.50
0.75 1.00
S E
kode
Parameter=b22
nn4 nn9
tn34 tn39
tn44 tn49
tn54 tn59
100 200
300 400
S E
kode
Parameter=b21
nn4 nn9
tn34 tn39
tn44 tn49
tn54 tn59
100 200
300 400
S E
kode
Parameter=s22
nn4 nn9
tn34 tn39
tn44 tn49
tn54 tn59
10 20
30 40
S E
kode
Parameter=s21
nn4 nn9
tn34 tn39
tn44 tn49
tn54 tn59
0.25 0.50
0.75 1.00
S E
kode
Lampiran 4. Nilai ARB penduga parameter submodel-1
Parameter b10
Sebaran pengaruh acak frekuensi pengamatan longitudinal
2 3
4 6
12 bivariate t, db3
-8.42 -7.45
-5.19 -2.37
0.67 bivariate t, db4
-7.99 -4.68
-4.16 -1.98
-0.82 bivariate t, db5
-7.79 -3.38
-2.84 -1.12
1.37 normal ganda
-2.45 -4.48
-4.48 0.12
0.38
Parameter b11
Sebaran pengaruh acak frekuensi pengamatan longitudinal
2 3
4 6
12 bivariate t, db3
-9.98 -7.36
-8.32 -2.65
-0.01 bivariate t, db4
-7.74 -4.24
-5.47 -2.33
0.28 bivariate t, db5
-6.52 -3.97
-3.70 -0.68
1.19 normal ganda
-3.57 -4.86
-2.83 -1.47
0.72
Parameter a11
Sebaran pengaruh acak frekuensi pengamatan longitudinal
2 3
4 6
12 bivariate t, db3
22.18 18.44
7.18 -3.14
-3.57 bivariate t, db4
9.15 13.48
9.03 4.84
-2.40 bivariate t, db5
6.82 14.63
2.56 0.06
-4.85 normal ganda
4.86 6.61
4.82 0.93
-0.68
Parameter a12
Sebaran pengaruh acak frekuensi pengamatan longitudinal
2 3
4 6
12 bivariate t, db3
35.53 20.72
16.51 2.10
-1.51 bivariate t, db4
23.12 15.77
18.46 5.85
-0.86 bivariate t, db5
18.83 14.87
12.67 2.00
-2.41 normal ganda
16.50 11.55
8.63 3.10
-1.13
Parameter a22
Sebaran pengaruh acak frekuensi pengamatan longitudinal
2 3
4 6
12 bivariate t, db3
10.67 3.65
2.64 -0.36
-1.87 bivariate t, db4
4.39 5.38
6.88 0.33
-1.93 bivariate t, db5
4.96 4.61
4.23 1.12
-1.82 normal ganda
3.55 3.93
2.60 0.74
-1.31
Parameter s21
Sebaran pengaruh acak frekuensi pengamatan longitudinal
2 3
4 6
12 bivariate t, db3
4.46 -0.15
2.92 0.40
-0.05 bivariate t, db4
3.19 -0.08
1.10 -0.29
-0.13 bivariate t, db5
1.45 -0.85
1.01 -0.15
0.03 normal ganda
3.27 0.46
-0.18 0.27
-0.22
Lampiran 5. Nilai MSE penduga parameter submodel-1
Parameter b10
Sebaran pengaruh acak frekuensi pengamatan longitudinal
2 3
4 6
12 bivariate t, db3
0.20 0.14
0.15 0.04
0.03 bivariate t, db4
0.19 0.10
0.08 0.05
0.03 bivariate t, db5
0.18 0.11
0.07 0.05
0.03 normal ganda
0.15 0.12
0.07 0.04
0.03
Parameter b11
Sebaran pengaruh acak frekuensi pengamatan longitudinal
2 3
4 6
12 bivariate t, db3
0.08 0.06
0.07 0.04
0.02 bivariate t, db4
0.08 0.05
0.06 0.04
0.02 bivariate t, db5
0.06 0.04
0.05 0.03
0.02 normal ganda
0.05 0.05
0.04 0.03
0.02
Parameter a11
Sebaran pengaruh acak frekuensi pengamatan longitudinal
2 3
4 6
12 bivariate t, db3
2.49 2.09
1.20 1.34
3.15 bivariate t, db4
1.01 1.11
0.73 3.06
0.65 bivariate t, db5
1.15 1.01
0.52 0.48
0.29 normal ganda
0.97 0.67
0.47 0.26
0.15
Parameter a12
Sebaran pengaruh acak frekuensi pengamatan longitudinal
2 3
4 6
12 bivariate t, db3
1.65 0.92
0.67 1.32
3.74 bivariate t, db4
0.61 0.53
0.43 1.22
0.39 bivariate t, db5
0.54 0.37
0.31 0.19
0.18 normal ganda
0.43 0.30
0.22 0.11
0.08
Parameter a22
Sebaran pengaruh acak frekuensi pengamatan longitudinal
2 3
4 6
12 bivariate t, db3
3.21 0.97
1.24 2.21
5.79 bivariate t, db4
0.31 0.63
0.43 0.81
0.49 bivariate t, db5
0.26 0.21
0.24 0.24
0.27 normal ganda
0.11 0.12
0.12 0.10
0.08
Parameter s21
Sebaran pengaruh acak frekuensi pengamatan longitudinal
2 3
4 6
12 bivariate t, db3
0.14 0.07
1.28 0.02
0.01 bivariate t, db4
0.12 0.06
0.04 0.02
0.01 bivariate t, db5
0.12 0.06
0.04 0.02
0.01 normal ganda
0.12 0.06
0.04 0.02
0.01
Lampiran 6. Nilai ARB penduga parameter submodel-2
Parameter b20
Sebaran pengaruh acak frekuensi pengamatan longitudinal
2 3
4 6
12 bivariate t, db3
-9.98 -7.36
-8.32 -2.65
-0.01 bivariate t, db4
-7.40 -4.12
-5.44 -2.30
0.10 bivariate t, db5
-6.63 -3.81
-3.56 -0.69
1.22 normal ganda
-3.31 -5.04
-3.26 -1.28
0.79
Parameter b21
Sebaran pengaruh acak frekuensi pengamatan longitudinal
2 3
4 6
12 bivariate t, db3
54.78 43.02
45.55 28.33
5.81 bivariate t, db4
57.90 35.86
38.61 19.59
5.41 bivariate t, db5
60.36 36.13
42.32 14.85
6.05 normal ganda
55.75 37.92
29.98 14.74
2.45
Parameter b22
Sebaran pengaruh acak frekuensi pengamatan longitudinal
2 3
4 6
12 bivariate t, db3
-23.32 -18.38
-18.85 -10.71
-1.89 bivariate t, db4
-22.32 -13.53
-15.14 -6.60
-1.35 bivariate t, db5
-20.97 -13.55
-15.28 -4.47
-1.70 normal ganda
-19.34 -13.61
-10.63 -4.97
-0.60
Parameter s22
Sebaran pengaruh acak frekuensi pengamatan longitudinal
2 3
4 6
12 bivariate t, db3
8.65 7.70
4.87 -9.93
-10.16 bivariate t, db4
-2.59 -7.50
-7.32 -10.07
-9.39 bivariate t, db5
-14.74 -7.41
-16.6 -8.24
-9.51 normal ganda
-23.86 -7.48
-11.53 -13.23
-5.27
Lampiran 7. Nilai MSE penduga parameter submodel-2
Parameter b20
Sebaran pengaruh acak frekuensi pengamatan longitudinal
2 3
4 6
12 bivariate t, db3
3.86 3.03
3.13 1.74
0.94 bivariate t, db4
3.25 2.26
2.67 1.63
0.88 bivariate t, db5
2.86 2.07
2.17 1.34
0.96 normal ganda
2.09 2.41
1.74 1.18
0.86
Parameter b21
Sebaran pengaruh acak frekuensi pengamatan longitudinal
2 3
4 6
12 bivariate t, db3
4.30 2.37
10.89 2.16
0.27 bivariate t, db4
3.19 1.96
2.42 1.48
0.18 bivariate t, db5
3.83 2.27
2.84 0.76
0.18 normal ganda
2.87 1.67
1.58 0.73
0.11
Parameter b22
Sebaran pengaruh acak frekuensi pengamatan longitudinal
2 3
4 6
12 bivariate t, db3
3.48 2.46
7.56 2.06
0.22 bivariate t, db4
2.71 1.63
2.33 1.17
0.12 bivariate t, db5
2.51 2.18
2.64 0.50
0.12 normal ganda
2.14 1.43
1.16 0.53
0.08
Parameter s22
Sebaran pengaruh acak frekuensi pengamatan longitudinal
2 3
4 6
12 bivariate t, db3
3.72 3.17
3.29 1.77
0.36 bivariate t, db4
3.40 2.72
2.95 1.41
0.32 bivariate t, db5
3.30 2.57
2.53 1.16
0.33 normal ganda
2.82 2.67
2.12 1.19
0.30
ABSTRACT
INDAHWATI. A Robust Approach for Joint Models Based on t Distribution. Supervised by AUNUDDIN, KHAIRIL ANWAR NOTODIPUTRO and I GUSTI
PUTU PURNABA.
Existing methods for joint modeling are usually based on normality assumption of random effects and intra-subject errors. We propose a joint model
based on t distribution of the intra subject errors to improve robustness of the estimation. In addition, study is also performed to evaluate the effects of number
of longitudinal data series on normality assumption. Our model consists of two submodels: a mixed linear mixed effects model for the longitudinal data, and a
generalized linear model for continuousbinary primary response. The proposed method is evaluated by means of simulation studies as well as application to HIV
data. Results of simulation study show that the effects of random effect distribution on bias and MSE of parameter estimates will be small if large number
of longitudinal data series are used. Otherwise, the number of longitudinal data series give little effects when intra-subject error is not normal. But long tail intra-
subject error distribution will give large bias and MSE if modeled as normal. For small number of longitudinal data series, robust approach based on t distribution
give smaller bias and MSE, mainly for parameters that joint longitudinal covariate with the the primary response variable.
Keywords: longitudinal data, joint model, mixed model, generalized linear model,
robust, t-distribution