Table 1.
Simulation study. Point estimates for β1, with true value 1, together with the coefficients of determination (R2) and the 95% confidence intervals based on the empirical likelihood (EL) and normal approximation (Norm) methods. Also displayed are the averages of the lower and upper endpoints of the confidence intervals, their average midpoints, their average lengths, and the associated coverage probabilities for the simulated data (a) when X and Z are independent, and (b) when X and Z are strongly correlated.
| Estimate | R2 | Confidence Interval | Midpoint | Length | Coverage | ||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| n | Case | Naive | ME | Naive | X observed | EL | Norm | EL | Norm | EL | Norm | EL | Norm |
| (a) X and Z are independent | |||||||||||||
| 100 | 1 | 0.649 | 1.027 | 0.464 | 0.525 | (0.511, 1.624) | (0.438, 1.615) | 1.072 | 1.027 | 1.114 | 1.177 | 93.7 | 94.3 |
| 2 | 0.643 | 1.021 | 0.467 | 0.525 | (0.500, 1.627) | (0.438, 1.605) | 0.969 | 1.021 | 1.126 | 1.167 | 94.0 | 94.7 | |
| 3 | 0.655 | 1.042 | 0.861 | 0.915 | (0.806, 1.318) | (0.740, 1.324) | 1.082 | 1.042 | 0.522 | 0.544 | 95.7 | 92.7 | |
| 4 | 0.657 | 1.048 | 0.473 | 0.532 | (0.505, 1.676) | (0.364, 1.733) | 1.090 | 1.048 | 1.171 | 1.369 | 96.3 | 94.3 | |
| 500 | 1 | 0.665 | 1.007 | 0.446 | 0.502 | (0.793, 1.221) | (0.793, 1.221) | 0.981 | 1.007 | 0.427 | 0.428 | 94.7 | 94.8 |
| 2 | 0.658 | 0.997 | 0.445 | 0.501 | (0.786, 1.217) | (0.777, 1.218) | 1.006 | 0.997 | 0.432 | 0.441 | 95.8 | 94.5 | |
| 3 | 0.660 | 1.000 | 0.858 | 0.917 | (0.913, 1.071) | (0.916, 1.083) | 1.007 | 1.000 | 0.158 | 0.167 | 93.7 | 95.8 | |
| 4 | 0.663 | 1.003 | 0.446 | 0.502 | (0.788, 1.144) | (0.785, 1.221) | 0.990 | 1.003 | 0.363 | 0.436 | 94.8 | 94.9 | |
| (b) X and Z are strongly correlated | |||||||||||||
| 100 | 1 | 0.569 | 1.082 | 0.65 | 0.688 | (0.306, 2.016) | (-0.001, 2.165) | 1.061 | 1.082 | 1.71 | 2.166 | 92.7 | 97.0 |
| 2 | 0.589 | 1.094 | 0.641 | 0.677 | (0.357, 1.916) | (0.256, 1.931) | 0.971 | 1.094 | 1.569 | 1.675 | 96.3 | 95.0 | |
| 3 | 0.568 | 1.088 | 0.879 | 0.922 | (0.530, 1.907) | (0.044, 2.132) | 1.019 | 1.088 | 1.377 | 2.088 | 94.6 | 96.7 | |
| 4 | 0.564 | 0.983 | 0.645 | 0.674 | (0.261, 2.082) | (-0.296, 2.445) | 1.172 | 0.983 | 1.821 | 2.741 | 94.7 | 97.7 | |
| 500 | 1 | 0.586 | 1.013 | 0.631 | 0.665 | (0.725, 1.227) | (0.720, 1.306) | 1.026 | 1.013 | 0.522 | 0.586 | 94.8 | 94.2 |
| 2 | 0.582 | 1.005 | 0.631 | 0.664 | (0.716, 1.221) | (0.715, 1.296) | 0.981 | 1.005 | 0.544 | 0.581 | 95.6 | 94.5 | |
| 3 | 0.582 | 0.991 | 0.912 | 0.942 | (0.897, 1.062) | (0.892, 1.125) | 0.993 | 0.991 | 0.168 | 0.233 | 98.6 | 95.3 | |
| 4 | 0.586 | 1.011 | 0.630 | 0.664 | (0.728, 1.228) | (0.727, 1.295) | 1.023 | 1.011 | 0.510 | 0.568 | 96.4 | 94.6 | |