Table 2.
Summary statistics for the estimation of Model (B) λ(t | X1, X2) = λ0(t) exp(β1X1 + β2X2 + β3X1X2) when ρ = 1.
| Proportion censored | β1 | β2 | β3 | ρ | Λ(0.3) | Λ(0.7) | |||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| n | Coef | Bias | ES | RE | Bias | ES | RE | Bias | ES | RE | Bias | ES | Bias | ES | RE | Bias | ES | RE | |
| 0% | 100 | PL | -1 | 17 | – | 2 | 24 | – | -2 | 24 | – | – | – | 0 | 3 | – | 0 | 10 | – |
| DEL | 0 | 6 | 9.45 | 0 | 18 | 1.80 | -1 | 17 | 1.84 | – | – | 0 | 2 | 1.84 | 0 | 6 | 2.54 | ||
| DELρ | 0 | 6 | 9.38 | 2 | 23 | 1.02 | -2 | 18 | 1.72 | 6 | 26 | 0 | 3 | 1.01 | 0 | 10 | 1.04 | ||
| 400 | PL | -1 | 8 | – | 1 | 12 | – | 0 | 11 | – | – | – | 0 | 1 | – | 0 | 5 | – | |
| DEL | 0 | 3 | 9.44 | 1 | 9 | 1.77 | 0 | 8 | 1.64 | – | – | 0 | 1 | 2.01 | 0 | 3 | 2.35 | ||
| DELρ | 0 | 3 | 9.41 | 1 | 12 | 1.01 | 0 | 9 | 1.50 | 1 | 12 | 0 | 1 | 1.03 | 0 | 5 | 1.03 | ||
| 30% | 100 | PL | -2 | 21 | – | 3 | 29 | – | -2 | 28 | – | – | – | 0 | 3 | – | 0 | 12 | – |
| DEL | -1 | 6 | 13.8 | 0 | 21 | 1.92 | -1 | 20 | 1.93 | – | – | 0 | 2 | 1.96 | 0 | 8 | 2.44 | ||
| DELρ | -1 | 6 | 13.7 | 2 | 28 | 1.04 | -2 | 21 | 1.84 | 8 | 31 | 0 | 3 | 1.03 | 0 | 12 | 1.05 | ||
| 400 | PL | -1 | 9 | – | 1 | 14 | – | 0 | 13 | – | – | – | 0 | 2 | – | 0 | 6 | – | |
| DEL | 0 | 3 | 13.7 | 0 | 10 | 1.88 | -1 | 10 | 1.78 | – | – | 0 | 1 | 2.04 | 0 | 4 | 2.42 | ||
| DELρ | 0 | 3 | 13.6 | 1 | 13 | 1.04 | -1 | 10 | 1.70 | 2 | 14 | 0 | 1 | 1.04 | 0 | 5 | 1.06 | ||
| 50% | 100 | PL | -2 | 27 | – | 4 | 35 | – | -2 | 36 | – | – | – | 0 | 3 | – | 0 | 14 | – |
| DEL | 0 | 6 | 23.2 | -1 | 24 | 2.12 | -4 | 25 | 2.13 | – | – | 0 | 2 | 2.05 | 1 | 9 | 2.45 | ||
| DELρ | 0 | 6 | 23.0 | 2 | 33 | 1.09 | -4 | 25 | 2.11 | 9 | 35 | 0 | 3 | 1.05 | 0 | 14 | 1.08 | ||
| 400 | PL | -1 | 12 | – | 1 | 16 | – | 0 | 16 | – | – | – | 0 | 2 | – | 0 | 7 | – | |
| DEL | 1 | 3 | 21.0 | 0 | 12 | 1.94 | -2 | 11 | 1.95 | – | – | 0 | 1 | 2.04 | 0 | 4 | 2.35 | ||
| DELρ | 1 | 3 | 21.0 | 0 | 16 | 1.03 | -2 | 11 | 1.91 | 2 | 15 | 0 | 2 | 1.04 | 0 | 6 | 1.06 | ||
NOTE: β1, β2, and, β3 are the regression coefficients, where the true parameter values are (−0.5, 1, −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.