Empirical relative efficiencies (ERE) and empirical relative interquartile ranges (ERI) compared with the asymptotically-optimal design based on true (unknown) parameters and empirical coverage probabilities (ECP) of estimators for βx based on 2000 simulated datasets with N = 800 and n = 200.
| Binary X | Continuous X | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Parametric | Empirical | Parametric | Empirical | |||||||||
| 100n(b)/n | ERE | ERI | ECP | ERE | ERI | ECP | ERE | ERI | ECP | ERE | ERI | ECP |
| Two-stage adaptive – proportional sampling in phase-IIa | ||||||||||||
| 0* | 76.6 | 88.7 | 95.5 | 76.6 | 88.7 | 95.5 | 88.9 | 94.4 | 95.2 | 88.9 | 94.4 | 95.2 |
| 10 | 85.0 | 93.4 | 95.3 | 84.3 | 90.2 | 95.2 | 94.7 | 97.7 | 94.8 | 89.4 | 97.9 | 94.8 |
| 20 | 86.3 | 89.9 | 94.7 | 84.6 | 90.9 | 94.8 | 94.0 | 96.9 | 94.7 | 87.8 | 92.2 | 94.0 |
| 30 | 94.0 | 97.8 | 95.2 | 89.3 | 97.9 | 93.8 | 102.6 | 99.8 | 95.2 | 93.2 | 97.1 | 94.6 |
| 40 | 96.3 | 96.3 | 95.0 | 103.1 | 100.8 | 95.4 | 100.7 | 104.3 | 94.8 | 86.0 | 91.1 | 93.2 |
| 50 | 98.2 | 95.0 | 94.8 | 92.6 | 96.3 | 94.2 | 103.2 | 102.4 | 95.1 | 88.5 | 94.7 | 93.3 |
| 60 | 103.1 | 100.0 | 95.2 | 95.2 | 96.5 | 94.8 | 105.0 | 102.0 | 95.5 | 88.2 | 97.0 | 93.7 |
| 70 | 97.2 | 98.2 | 95.0 | 95.4 | 98.8 | 94.4 | 103.0 | 103.3 | 94.6 | 93.9 | 96.9 | 94.8 |
| 80 | 100.7 | 103.1 | 95.1 | 100.8 | 101.3 | 95.0 | 107.9 | 105.2 | 95.6 | 85.6 | 96.6 | 94.5 |
| 90 | 98.6 | 102.2 | 95.4 | 97.7 | 97.5 | 94.5 | 103.7 | 103.8 | 95.0 | 82.8 | 94.0 | 94.7 |
| Two-stage adaptive – balanced sampling in phase-IIa | ||||||||||||
| 0* | 95.2 | 99.6 | 95.2 | 95.2 | 99.6 | 95.2 | 90.6 | 90.6 | 95.0 | 90.6 | 90.6 | 95.0 |
| 10 | 101.6 | 102.3 | 95.3 | 102.9 | 102.5 | 95.5 | 98.3 | 98.6 | 94.6 | 95.2 | 99.2 | 95.0 |
| 20 | 96.8 | 101.5 | 93.8 | 98.4 | 98.9 | 95.0 | 102.0 | 99.4 | 94.7 | 95.9 | 99.8 | 94.6 |
| 30 | 105.2 | 103.7 | 95.3 | 99.9 | 100.7 | 95.0 | 107.5 | 103.1 | 95.9 | 88.2 | 92.2 | 93.8 |
| 40 | 100.1 | 103.2 | 95.0 | 102.5 | 98.4 | 94.9 | 109.7 | 105.4 | 95.5 | 93.9 | 95.3 | 94.0 |
| 50 | 103.8 | 99.3 | 95.3 | 100.0 | 100.1 | 94.8 | 101.0 | 101.8 | 95.0 | 96.7 | 95.5 | 95.5 |
| 60 | 99.8 | 97.6 | 95.0 | 98.9 | 96.4 | 95.2 | 100.8 | 98.3 | 94.8 | 86.5 | 90.8 | 94.2 |
| 70 | 103.7 | 98.9 | 95.5 | 96.6 | 98.5 | 94.3 | 103.2 | 103.6 | 95.0 | 85.0 | 90.7 | 93.8 |
| 80 | 100.2 | 101.1 | 94.7 | 97.7 | 98.0 | 94.7 | 103.9 | 104.0 | 95.2 | 86.6 | 91.5 | 94.8 |
| 90 | 95.8 | 98.5 | 95.2 | 96.0 | 96.2 | 94.8 | 102.3 | 100.6 | 95.2 | 84.0 | 95.9 | 94.8 |
| Fully adaptive | ||||||||||||
| 102.4 | 107.0 | 95.3 | 99.0 | 96.7 | 94.5 | 102.4 | 101.9 | 95.3 | 86.8 | 89.1 | 94.3 | |
The parameters were set to (β0,βx,βv) = (−1.95,1.00,0.90), (α0,αv) = (1.05,−0.41), and γ0=−0.04 for the case with binary X and to (β0,βx,βv) = (−2.18,0.03,.84), (α0,α1,αv) = (1.40,10,5), and γ0=−0.04 for the setting with continuous X. Two-stage adaptive designs select n − n(b) individuals using proportional or balanced sampling and use these individuals to estimate the design components either through parametric estimation (parametric) or through empirical estimation (empirical) to approximate optimal selection of the remaining n(b) individuals. Nonadaptive designs are a special case of the two-stage sampling where all individuals are selected in phase-IIa. Fully adaptive designs involved selecting an initial balanced sample of size 40 (corresponding to 20% of n) and then selecting the remaining individuals one at a time while updating estimates of the design component after each individual is selected.
*The nonadaptive design does not require estimation of the design components, so there is no distinction between design components being estimated ‘empirically’ or ‘parametrically’.