Table 2 The bias and RMSE for \(\tau _{\textrm{SATE}}\) under quasi-rerandomization (QReR) and rerandomization (ReR) under different combinations of the acceptance probability \(p_a\), covariate scenarios and response surfaces when \(r=N_0/N_1=1\). The average Monte Carlo standard errors (MCSEs) of bias are 0.03, 0.04 and 0.31 and those of RMSE are 0.02, 0.03 and 0.31 for Linear, NonLinear (Interaction) and NonLinear (Polynomial) models, respectively
Response | \(p_a\) | Method | Scenario 1 | Scenario 2 | Scenario 3 | |||
|---|---|---|---|---|---|---|---|---|
| Â | Â | Â | Bias | RMSE | Bias | RMSE | Bias | RMSE |
Linear | 0.1 | \(\textrm{QReR}_{\textrm{S}}\) | -0.002 | 0.35 | 0.005 | 0.44 | 0.020 | 0.42 |
| Â | Â | \(\textrm{QReR}_{\textrm{M}}\) | -0.022 | 0.12 | -0.032 | 0.12 | -0.028 | 0.12 |
| Â | Â | ReR | 0.004 | 0.31 | 0.001 | 0.39 | 0.003 | 0.41 |
| Â | 0.5 | \(\textrm{QReR}_{\textrm{S}}\) | -0.053 | 0.47 | -0.059 | 0.60 | -0.067 | 0.66 |
| Â | Â | \(\textrm{QReR}_{\textrm{M}}\) | -0.037 | 0.12 | -0.059 | 0.13 | -0.058 | 0.13 |
| Â | Â | ReR | 0.029 | 0.45 | 0.038 | 0.53 | -0.001 | 0.56 |
| Â | 1 | \(\textrm{QReR}_{\textrm{S}}\) | -0.035 | 0.60 | -0.017 | 0.67 | -0.062 | 0.76 |
| Â | Â | \(\textrm{QReR}_{\textrm{M}}\) | -0.054 | 0.12 | -0.064 | 0.14 | -0.060 | 0.14 |
| Â | Â | ReR | -0.023 | 0.57 | -0.038 | 0.68 | 0.072 | 0.72 |
NonLinear | 0.1 | \(\textrm{QReR}_{\textrm{S}}\) | -0.017 | 0.49 | -0.060 | 0.64 | -0.071 | 0.62 |
(Interaction) | Â | \(\textrm{QReR}_{\textrm{M}}\) | -0.033 | 0.19 | -0.123 | 0.24 | -0.132 | 0.25 |
| Â | Â | ReR | 0.014 | 0.46 | 0.011 | 0.54 | 0.032 | 0.62 |
| Â | 0.5 | \(\textrm{QReR}_{\textrm{S}}\) | -0.079 | 0.65 | -0.162 | 0.83 | -0.199 | 0.93 |
| Â | Â | \(\textrm{QReR}_{\textrm{M}}\) | -0.055 | 0.19 | -0.148 | 0.25 | -0.165 | 0.27 |
| Â | Â | ReR | 0.041 | 0.63 | 0.053 | 0.74 | -0.020 | 0.76 |
| Â | 1 | \(\textrm{QReR}_{\textrm{S}}\) | -0.061 | 0.84 | -0.095 | 0.94 | -0.166 | 1.10 |
| Â | Â | \(\textrm{QReR}_{\textrm{M}}\) | -0.075 | 0.20 | -0.148 | 0.25 | -0.152 | 0.27 |
| Â | Â | ReR | -0.037 | 0.80 | -0.058 | 0.95 | 0.090 | 1.02 |
NonLinear | 0.1 | \(\textrm{QReR}_{\textrm{S}}\) | 0.078 | 2.31 | 5.001 | 7.46 | -5.931 | 7.69 |
(Polynomial) | Â | \(\textrm{QReR}_{\textrm{M}}\) | 0.025 | 2.04 | 4.869 | 7.02 | -6.118 | 7.58 |
| Â | Â | ReR | 0.030 | 2.03 | -0.571 | 5.75 | 0.368 | 5.77 |
| Â | 0.5 | \(\textrm{QReR}_{\textrm{S}}\) | -0.051 | 2.54 | 4.958 | 7.75 | -6.869 | 8.77 |
| Â | Â | \(\textrm{QReR}_{\textrm{M}}\) | -0.063 | 2.12 | 4.861 | 7.23 | -6.551 | 7.98 |
| Â | Â | ReR | -0.007 | 1.94 | -0.289 | 5.60 | 0.119 | 6.16 |
| Â | 1 | \(\textrm{QReR}_{\textrm{S}}\) | -0.051 | 2.65 | 4.774 | 7.54 | -7.008 | 8.77 |
| Â | Â | \(\textrm{QReR}_{\textrm{M}}\) | -0.071 | 2.15 | 4.703 | 7.05 | -7.229 | 8.61 |
| Â | Â | ReR | -0.048 | 2.07 | -0.029 | 6.03 | -0.020 | 6.48 |