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 𝐢 𝐼 ) and upper limit ( π‘ˆ 𝐢 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.

MethodModel 𝛼 0 𝛼 1 𝛼 2 𝛽 1 𝛽 2 𝛽 3 𝛽 4 𝛽 5 𝜏 2 𝜎 2 DIC

JM SNPM βˆ’0.95 0.15 βˆ’ 0 . 2 3 5 . 6 2 βˆ’ 1 4 . 6 βˆ’ 2 . 3 4 1 1 . 7 βˆ’ 4 . 7 6 0 . 0 7 0 . 1 4 863.0
𝐿 C I βˆ’ 1 . 5 8 0.06 βˆ’ 1 5 . 2 4 . 1 7 βˆ’ 2 2 . 1 βˆ’ 5 . 1 4 4 . 5 2 βˆ’ 9 . 9 2 0 . 0 4 0 . 0 1
π‘ˆ C I βˆ’ 0 . 0 1 0 . 9 0 1 4 . 8 7 . 5 9 βˆ’ 8 . 1 4 1 . 4 4 2 1 . 7 βˆ’ 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
STPM βˆ’ 0 . 9 4 0 . 3 4 βˆ’ 0 . 3 1 5 . 8 4 βˆ’ 1 2 . 0 βˆ’ 1 . 2 0 8 . 1 2 βˆ’ 6 . 3 1 0 . 0 4 0 . 2 1 9 8 5 . 6
𝐿 C I βˆ’ 1 . 4 1 0 . 1 8 βˆ’ 1 4 . 1 4 . 1 5 βˆ’ 1 6 . 5 βˆ’ 5 . 7 2 2 . 2 0 βˆ’ 1 2 . 6 0 . 0 2 0 . 0 1
π‘ˆ C I βˆ’ 0 . 0 6 0 . 8 8 1 3 . 4 8 . 0 2 βˆ’ 7 . 7 2 2 . 7 2 1 9 . 2 βˆ’ 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 βˆ’ 0 . 2 1 0 . 4 5 βˆ’ 2 . 8 7 7 . 7 4 βˆ’ 1 5 . 4 βˆ’ 0 . 8 0 1 3 . 6 βˆ’ 6 . 2 6 0 . 1 3 2 . 6 3 1242.3
𝐿 C I βˆ’ 0 . 4 6 0 . 2 2 βˆ’ 1 5 . 9 7 . 2 0 βˆ’ 1 8 . 3 βˆ’ 4 . 1 6 9 . 9 7 βˆ’ 1 1 . 7 0 . 1 1 2 . 0 6
π‘ˆ C I 0 . 0 4 0 . 6 8 9 . 9 0 8 . 2 9 βˆ’ 1 2 . 6 2 . 5 3 1 7 . 2 βˆ’ 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

NVSNPMβ€”β€”β€” 5 . 0 3 βˆ’ 1 1 . 1 0 . 5 8 6 . 8 3 βˆ’2.10 β€” 0 . 1 0 1083.5
𝐿 C I β€”β€”β€” 3 . 8 2 βˆ’ 1 3 . 6 βˆ’ 0 . 9 4 4 . 5 2 βˆ’4.18 β€” 0 . 0 1
π‘ˆ C I β€”β€”β€” 6 . 5 9 βˆ’ 8 . 7 3 2 . 0 8 9 . 1 8 0 . 0 7 β€” 0 . 3 5
SDβ€”β€”β€” 0 . 7 5 1 . 2 3 0 . 7 7 1 . 1 9 1 . 0 4 β€” 0 . 0 9
TS SN PM βˆ’ 0 . 9 9 0 . 1 9 2 . 7 1 5 . 9 1 βˆ’ 1 4 . 4 βˆ’ 1 . 2 4 8 . 4 7 βˆ’ 5 . 9 0 0 . 0 9 0 . 1 4 1 0 2 3 . 8
𝐿 C I βˆ’ 1 . 5 8 βˆ’ 0 . 4 3 βˆ’ 1 2 . 1 4 . 1 2 βˆ’ 2 2 . 1 βˆ’ 5 . 0 1 1 . 8 3 βˆ’ 1 0 . 6 0 . 0 5 0 . 0 1
π‘ˆ C I 0 . 0 7 0 . 9 0 1 7 . 1 7 . 7 2 βˆ’ 8 . 5 0 2 . 1 6 2 1 . 2 βˆ’ 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

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