Table 1.
Summary statistics for the estimation of Model (A) λ(t | X1, X2) = λ0(t)exp(β1X1 + β2X2) when ρ = 1.
| Proportion censored | β1 | β2 | ρ | Λ(0.3) | Λ(0.7) | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| n | Coef | Bias | ES | RE | Bias | ES | RE | Bias | ES | Bias | ES | RE | Bias | ES | RE | |
| 0% | 100 | PL | -2 | 12 | – | 3 | 22 | – | – | – | 0 | 3 | – | 0 | 10 | – |
| DEL | -1 | 5 | 4.96 | 0 | 18 | 1.48 | – | – | 0 | 2 | 1.67 | 0 | 6 | 2.59 | ||
| DELρ | -1 | 5 | 4.86 | 2 | 22 | 1.03 | 7 | 26 | 0 | 3 | 1.03 | 0 | 10 | 1.05 | ||
| 400 | PL | 0 | 6 | – | 1 | 11 | – | – | – | 0 | 1 | 0 | 5 | – | ||
| DEL | 0 | 2 | 5.62 | 0 | 9 | 1.47 | – | – | 0 | 1 | 1.86 | 0 | 3 | 2.35 | ||
| DELρ | 0 | 2 | 5.52 | 0 | 11 | 1.02 | 1 | 11 | 0 | 1 | 1.05 | 0 | 5 | 1.05 | ||
| 30% | 100 | PL | -1 | 14 | – | 3 | 26 | – | – | – | 0 | 3 | – | 0 | 11 | – |
| DEL | -1 | 5 | 6.84 | 1 | 20 | 1.67 | – | – | 0 | 2 | 1.76 | 0 | 7 | 2.50 | ||
| DELρ | -1 | 5 | 6.68 | 2 | 25 | 1.04 | 7 | 31 | 0 | 3 | 1.05 | 0 | 11 | 1.05 | ||
| 400 | PL | -1 | 7 | – | 1 | 13 | – | – | – | 0 | 2 | – | 0 | 5 | – | |
| DEL | 0 | 2 | 7.10 | 0 | 10 | 1.60 | – | – | 0 | 1 | 1.87 | 0 | 4 | 2.36 | ||
| DELρ | 0 | 2 | 6.98 | 0 | 13 | 1.02 | 1 | 13 | 0 | 1 | 1.05 | 0 | 5 | 1.04 | ||
| 50% | 100 | PL | -2 | 17 | – | 0 | 31 | – | – | – | 0 | 3 | – | 1 | 13 | – |
| DEL | -1 | 5 | 9.64 | -2 | 23 | 1.84 | – | – | 0 | 2 | 1.81 | 1 | 8 | 2.54 | ||
| DELρ | -1 | 5 | 9.61 | -1 | 30 | 1.04 | 6 | 32 | 0 | 3 | 1.01 | 2 | 13 | 1.03 | ||
| 400 | PL | -1 | 8 | – | 1 | 15 | – | – | – | 0 | 2 | – | 0 | 6 | – | |
| DEL | 1 | 2 | 9.66 | 0 | 12 | 1.72 | – | – | 0 | 1 | 1.87 | 0 | 4 | 2.50 | ||
| DELρ | 1 | 2 | 9.56 | 0 | 15 | 1.02 | 1 | 14 | 0 | 2 | 1.05 | 0 | 6 | 1.04 | ||
NOTE: β1 and β2 are the regression coefficients, where the true parameter values are (−0.5, 0.5); Λ(t) = t2 is the baseline cumulative hazard function evaluated at t; PL, the maximum partial likelihood estimator β̂PL; DEL, the double empirical likelihood estimator β̂; DELρ, the extended double empirical likelihood estimator β̂ρ that allows for a different baseline hazard function for the aggregate data; Bias and ES, empirical bias (×100) and empirical standard deviation (×100) of 1,000 regression parameter estimates; RE, the empirical variance of the maximum partial likelihood estimator divided by that of the double empirical likelihood estimators.