Journal of Probability and Statistics

Research Article

Secondary Analysis under Cohort Sampling Designs Using Conditional Likelihood

Table 3

Maximum likelihood estimates of parameters 𝛽 of the outcome models 𝑌 𝑖 ∣ 𝑍 𝑖 , 𝛽 ∼ 𝑁 ( 𝛽 0 + 𝛽 1 𝑍 𝑖 , 𝛽 2 2 ) and 𝑌 𝑖 ∣ 𝑍 𝑖 , 𝛽 ∼ N P Q M ( 𝛽 0 + 𝛽 1 𝑍 𝑖 , 𝛽 2 , 𝛽 3 , 𝛽 4 ) and the population frequency 𝛾 of the binary covariate 𝑍 𝑖 (means over 1000 replications). Outcome variable was simulated from 𝑌 𝑖 ∣ 𝑍 𝑖 ∼ 𝑍 𝑖 + 𝑁 ( 4 , 4 ) and 𝑌 𝑖 ∣ 𝑍 𝑖 ∼ 𝑍 𝑖 + G a m m a ( 4 , 1 ) .
Normal data 𝑛 Normal model NPQM model
̂ 𝛽 0 ̂ 𝛽 1 ̂ 𝛽 S E ( 1 ) ̂ 𝛽 2 ̂ 𝛾 ̂ 𝛽 0 ̂ 𝛽 1 ̂ 𝛽 S E ( 1 ) ̂ 𝛽 2 ̂ 𝛽 3 ̂ 𝛽 S E ( 3 ) ̂ 𝛽 4 ̂ 𝛾
Full lik.1000 4.00 1.00 0.16 2.00 0.20 4.00 1.00 0.16 2.00 0.00 0.01 0.14 0.20
500 4.00 1.01 0.22 1.99 0.20 4.00 1.01 0.22 1.99 0.00 0.02 0.14 0.20
200 4.00 1.00 0.34 1.99 0.20 4.00 1.00 0.34 1.99 0.00 0.02 0.14 0.20
100 4.00 1.00 0.47 1.99 0.20 4.00 1.01 0.48 1.99 0.00 0.02 0.14 0.20
Cond. lik.500 4.00 1.01 0.22 1.99 0.20 4.00 1.01 0.22 1.99 0.00 0.02 0.14 0.20
200 4.01 1.00 0.35 1.99 0.20 4.01 1.00 0.35 1.99 0.00 0.03 0.13 0.20
100 4.00 1.01 0.50 1.98 0.20 4.00 1.01 0.51 1.98 0.00 0.05 0.13 0.20
Gamma data 𝑛 Normal model NPQM model
̂ 𝛽 0 ̂ 𝛽 1 ̂ 𝛽 S E ( 1 ) ̂ 𝛽 2 ̂ 𝛾 ̂ 𝛽 0 ̂ 𝛽 1 ̂ 𝛽 S E ( 1 ) ̂ 𝛽 2 ̂ 𝛽 3 ̂ 𝛽 S E ( 3 ) ̂ 𝛽 4 ̂ 𝛾
Full lik.1000 4.00 1.01 0.16 1.99 0.20 3.99 1.00 0.13 2.01 0.15 0.01 0.14 0.20
500 3.97 1.15 0.24 1.98 0.20 3.97 1.08 0.18 2.00 0.15 0.02 0.14 0.20
200 3.83 2.01 0.36 1.86 0.19 3.92 1.28 0.23 1.97 0.16 0.02 0.14 0.21
100 3.72 2.95 0.26 1.70 0.16 3.87 1.44 0.26 1.94 0.16 0.02 0.14 0.23
Cond. lik.500 4.00 1.00 0.22 1.99 0.20 4.00 1.00 0.19 2.01 0.15 0.02 0.14 0.20
200 4.00 0.99 0.35 1.98 0.20 4.01 1.00 0.29 2.00 0.15 0.03 0.14 0.20
100 4.00 0.99 0.50 1.97 0.20 4.01 1.01 0.41 2.00 0.15 0.05 0.13 0.20