Journal of Probability and Statistics / 2012 / Article / Tab 1 / Research Article
Mixed-Effects Tobit Joint Models for Longitudinal Data with Skewness, Detection Limits, and Measurement Errors Table 1 A summary of the estimated posterior mean (PM) of population (fixed-effects) and scale parameters, the corresponding standard deviation (SD) and lower limit (
𝐿
𝐶
𝐼
) and upper limit (
𝑈
𝐶
𝐼
) of 95% equal-tail credible interval (CI) as well as DIC values based on the joint modeling (JM), the naive (NV), and the two-step (TS) methods.
Method Model
𝛼
0
𝛼
1
𝛼
2
𝛽
1
𝛽
2
𝛽
3
𝛽
4
𝛽
5
𝜏
2
𝜎
2
DIC JM SN PM
−0.95
0.15
−
0
.
2
3
5
.
6
2
−
1
4
.
6
−
2
.
3
4
1
1
.
7
−
4
.
7
6
0
.
0
7
0
.
1
4
863.0
𝐿
C
I
−
1
.
5
8
0.06
−
1
5
.
2
4
.
1
7
−
2
2
.
1
−
5
.
1
4
4
.
5
2
−
9
.
9
2
0
.
0
4
0
.
0
1
𝑈
C
I
−
0
.
0
1
0
.
9
0
1
4
.
8
7
.
5
9
−
8
.
1
4
1
.
4
4
2
1
.
7
−
0
.
6
2
0
.
1
1
0
.
6
4
SD
0
.
4
7
0
.
3
7
7
.
6
3
0
.
9
6
3
.
9
8
1
.
6
5
5
.
2
5
2
.
3
4
0
.
0
2
0
.
1
8
ST PM
−
0
.
9
4
0
.
3
4
−
0
.
3
1
5
.
8
4
−
1
2
.
0
−
1
.
2
0
8
.
1
2
−
6
.
3
1
0
.
0
4
0
.
2
1
9
8
5
.
6
𝐿
C
I
−
1
.
4
1
0
.
1
8
−
1
4
.
1
4
.
1
5
−
1
6
.
5
−
5
.
7
2
2
.
2
0
−
1
2
.
6
0
.
0
2
0
.
0
1
𝑈
C
I
−
0
.
0
6
0
.
8
8
1
3
.
4
8
.
0
2
−
7
.
7
2
2
.
7
2
1
9
.
2
−
1
.
4
1
0
.
0
5
0
.
8
6
SD
0
.
3
5
0.26
7
.
0
9
1
.
1
0
2
.
2
2
2
.
2
2
4
.
1
4
2
.
7
7
0
.
0
1
0
.
2
6
𝑁
PM
−
0
.
2
1
0
.
4
5
−
2
.
8
7
7
.
7
4
−
1
5
.
4
−
0
.
8
0
1
3
.
6
−
6
.
2
6
0
.
1
3
2
.
6
3
1242.3
𝐿
C
I
−
0
.
4
6
0
.
2
2
−
1
5
.
9
7
.
2
0
−
1
8
.
3
−
4
.
1
6
9
.
9
7
−
1
1
.
7
0
.
1
1
2
.
0
6
𝑈
C
I
0
.
0
4
0
.
6
8
9
.
9
0
8
.
2
9
−
1
2
.
6
2
.
5
3
1
7
.
2
−
1
.
4
3
0
.
1
6
3
.
3
5
SD
0
.
1
3
0
.
1
2
6
.
5
4
0
.
2
8
1
.
4
8
1
.
7
3
1
.
8
5
2
.
6
1
0
.
0
1
0
.
3
3
NV SN PM — — —
5
.
0
3
−
1
1
.
1
0
.
5
8
6
.
8
3
−2.10
—
0
.
1
0
1083.5
𝐿
C
I
— — —
3
.
8
2
−
1
3
.
6
−
0
.
9
4
4
.
5
2
−4.18
—
0
.
0
1
𝑈
C
I
— — —
6
.
5
9
−
8
.
7
3
2
.
0
8
9
.
1
8
0
.
0
7
—
0
.
3
5
SD — — —
0
.
7
5
1
.
2
3
0
.
7
7
1
.
1
9
1
.
0
4
—
0
.
0
9
TS SN PM
−
0
.
9
9
0
.
1
9
2
.
7
1
5
.
9
1
−
1
4
.
4
−
1
.
2
4
8
.
4
7
−
5
.
9
0
0
.
0
9
0
.
1
4
1
0
2
3
.
8
𝐿
C
I
−
1
.
5
8
−
0
.
4
3
−
1
2
.
1
4
.
1
2
−
2
2
.
1
−
5
.
0
1
1
.
8
3
−
1
0
.
6
0
.
0
5
0
.
0
1
𝑈
C
I
0
.
0
7
0
.
9
0
1
7
.
1
7
.
7
2
−
8
.
5
0
2
.
1
6
2
1
.
2
−
0
.
8
0
0
.
1
4
0
.
6
5
SD
0
.
4
2
0
.
3
6
7
.
5
4
1
.
0
5
3
.
8
8
1
.
7
9
5
.
1
4
2
.
5
2
0
.
0
2
0
.
1
8
