| ASE | 0.1824 | 0.2930 | 0.1837 | 0.1825 | 0.1825 | 0.1804 | 0.1822 | 0.1819 | 0.1862
| MSE | 0.0382 | 0.1083 | 0.0429 | 0.0385 | 0.0386 | 0.0385 | 0.0386 | 0.0371 | 0.0401
| CP | 0.9310 | 0.9350 | 0.9170 | 0.9310 | 0.9310 | 0.9270 | 0.9310 | 0.9330 | 0.9390
| $n=1,500$
$\beta_{0}$ | Bias | 0.0031 | 0.0734 | 0.0102 | 0.0066 | 0.0053 | 0.0066 | 0.0053 | -0.0530 | 0.0014
| SD | 0.1238 | 0.1834 | 0.1312 | 0.1285 | 0.1282 | 0.1285 | 0.1282 | 0.1145 | 0.1409
| ASE | 0.1203 | 0.1798 | 0.1256 | 0.1217 | 0.1218 | 0.1217 | 0.1231 | 0.1299 | 0.1354
| MSE | 0.0153 | 0.0390 | 0.0173 | 0.0166 | 0.0165 | 0.0166 | 0.0165 | 0.0159 | 0.0199
| CP | 0.9510 | 0.9450 | 0.9430 | 0.9410 | 0.9430 | 0.9400 | 0.9460 | 0.9480 | 0.9460
$\beta_{1}$ | Bias | 0.0078 | 0.0356 | 0.0143 | 0.0108 | 0.0105 | 0.0108 | 0.0105 | -0.2013 | 0.0390
| SD | 0.1759 | 0.2794 | 0.1928 | 0.1858 | 0.1843 | 0.1858 | 0.1843 | 0.1772 | 0.2279
| ASE | 0.1745 | 0.2668 | 0.1859 | 0.1774 | 0.1776 | 0.1796 | 0.1813 | 0.2122 | 0.2216
| MSE | 0.0310 | 0.0793 | 0.0374 | 0.0346 | 0.0341 | 0.0346 | 0.0341 | 0.0719 | 0.0534
| CP | 0.9470 | 0.9370 | 0.9400 | 0.9360 | 0.9350 | 0.9390 | 0.9410 | 0.8640 | 0.9450
$\beta_{2}$ | Bias | 0.0080 | 0.0209 | 0.0261 | 0.0166 | 0.0138 | 0.0166 | 0.0138 | -0.2568 | 0.0095
| SD | 0.1670 | 0.2544 | 0.1871 | 0.1783 | 0.1773 | 0.1783 | 0.1773 | 0.1482 | 0.2150
| ASE | 0.1598 | 0.2442 | 0.1737 | 0.1633 | 0.1638 | 0.1636 | 0.1677 | 0.1919 | 0.2013
| MSE | 0.0280 | 0.0652 | 0.0357 | 0.0321 | 0.0316 | 0.0321 | 0.0316 | 0.0879 | 0.0463
| CP | 0.9450 | 0.9390 | 0.9370 | 0.9300 | 0.9320 | 0.9310 | 0.9370 | 0.7590 | 0.9440
$\beta_{3}$ | Bias | 0.0096 | 0.1226 | 0.0177 | 0.0106 | 0.0105 | 0.0106 | 0.0105 | -0.0158 | 0.0196
| SD | 0.1458 | 0.2339 | 0.1512 | 0.1464 | 0.1463 | 0.1464 | 0.1463 | 0.1445 | 0.1496
| ASE | 0.1484 | 0.2363 | 0.1493 | 0.1486 | 0.1486 | 0.1478 | 0.1487 | 0.1479 | 0.1513
| MSE | 0.0213 | 0.0698 | 0.0232 | 0.0215 | 0.0215 | 0.0215 | 0.0215 | 0.0211 | 0.0228
| CP | 0.9530 | 0.9340 | 0.9490 | 0.9510 | 0.9520 | 0.9510 | 0.9520 | 0.9600 | 0.9560
$\widehat{\bm{\beta}}_{M1}$: MI1 estimator whose value is the solution of
$\bm{U}_{M1}(\bm{\beta})=\bm{0}$ in Equation (18) with Rubin’s type
estimated variance in Equation (26)
$\widehat{\bm{\beta}}_{M2}$: MI2 estimator whose value is the solution of
$\bm{U}_{M2}(\bm{\beta})=\bm{0}$ in Equation (19) with Rubin’s type
estimated variance in Equation (27)
$\widehat{\bm{\beta}}_{M1n}$: MI1 estimator whose value is the solution of
$\bm{U}_{M1}(\bm{\beta})=\bm{0}$ in Equation (18) with proposed
estimated variance in Equation (40)
$\widehat{\bm{\beta}}_{M2n}$: MI2 estimator whose value is the solution of
$\bm{U}_{M2}(\bm{\beta})=\bm{0}$ in Equation (19) with proposed
estimated variance in Equation (41)

Table 4: Relative efficiencies in Study 2 ($M=15$; $n=500,1,000,1,500$);
$\bm{\beta}=(1.2,1,1,1)^{\texttt{T}}$;
$\bm{\alpha}=(1.4,0.6,0.6)^{\texttt{T}}$;
$\bm{\gamma}=(0.7,-0.2,0.1,-1.2)^{\texttt{T}}$; observed selection
probabilities: $(0.45,0.20,0.20,0.15)$ Relative efficiency
---
$\bm{\beta}$ | $C1$ | $W1$ | $M11$ | $M21$ | $RF1$ | $EM1$ | $C2$ | $W2$ | $M12$ | $M22$ | $RF2$ | $EM2$ | $M12n$
$n=500$
$\beta_{0}$ | 1.541 | 1.078 | 1.026 | 1.028 | 1.088 | 1.162 | 1.489 | 1.042 | 0.992 | 0.993 | 1.051 | 1.123 | 0.966
$\beta_{1}$ | 1.542 | 1.065 | 1.005 | 1.009 | 1.187 | 1.280 | 1.505 | 1.040 | 0.981 | 0.985 | 1.159 | 1.250 | 0.976
$\beta_{2}$ | 1.591 | 1.114 | 1.032 | 1.038 | 1.198 | 1.314 | 1.517 | 1.062 | 0.984 | 0.989 | 1.142 | 1.252 | 0.953
$\beta_{3}$ | 1.673 | 1.058 | 1.036 | 1.036 | 1.034 | 1.064 | 1.635 | 1.034 | 1.012 | 1.012 | 1.010 | 1.039 | 0.977
$n=1,000$
$\beta_{0}$ | 1.498 | 1.041 | 1.006 | 1.007 | 1.067 | 1.127 | 1.472 | 1.022 | 0.988 | 0.989 | 1.048 | 1.107 | 0.982
$\beta_{1}$ | 1.501 | 1.040 | 0.991 | 0.993 | 1.176 | 1.244 | 1.482 | 1.027 | 0.979 | 0.981 | 1.161 | 1.229 | 0.987
$\beta_{2}$ | 1.522 | 1.073 | 1.007 | 1.011 | 1.172 | 1.255 | 1.473 | 1.038 | 0.974 | 0.978 | 1.134 | 1.215 | 0.968
$\beta_{3}$ | 1.624 | 1.019 | 1.012 | 1.012 | 1.008 | 1.032 | 1.608 | 1.008 | 1.001 | 1.002 | 0.998 | 1.022 | 0.990
$n=1,500$
$\beta_{0}$ | 1.477 | 1.032 | 1.000 | 1.001 | 1.067 | 1.113 | 1.460 | 1.020 | 0.988 | 0.989 | 1.055 | 1.100 | 0.988
$\beta_{1}$ | 1.486 | 1.035 | 0.988 | 0.989 | 1.182 | 1.234 | 1.472 | 1.026 | 0.978 | 0.980 | 1.170 | 1.223 | 0.991
$\beta_{2}$ | 1.492 | 1.062 | 0.998 | 1.001 | 1.173 | 1.230 | 1.456 | 1.036 | 0.974 | 0.977 | 1.145 | 1.201 | 0.976
$\beta_{3}$ | 1.599 | 1.010 | 1.005 | 1.005 | 1.001 | 1.024 | 1.589 | 1.004 | 0.999 | 0.999 | 0.995 | 1.018 | 0.994
$Ar$, $A\in\\{C,W,M1,M2,RF,EM\\}$, $r=1,2$, are the relative efficiencies of
CC, SIPW, MI1, MI2,
RFMI, and SAEM estimators to MI$r$n estimators for each $\beta_{i}$,
$i=0,1,2,3$.
$Ar$ is the ratio of ASE of $A$ estimator to ASE of MI$r$n estimator for each
$\beta_{i}$.
$Ar=ASE_{A}/ASE_{MIrn}$, $M12n=ASE_{MI1n}/ASE_{MI2n}$
MI1: MI1 estimation method with Rubin’s type estimated variance in Equation
(26)
MI2: MI2 estimation method with Rubin’s type estimated variance in Equation
(27)
MI1n: MI1 estimation method with proposed estimated variance in Equation (40)
MI2n: MI2 estimation method with proposed estimated variance in Equation (41)

Table 5: Simulation results of Study 3 $(n=1,000)$; $\bm{\beta}=(1.2,1,1,1)^{\texttt{T}}$; $\bm{\alpha}=(1.4,0.6,0.6)^{\texttt{T}}$; $\bm{\gamma}=(0.7,-0.2,0.1,-1.2)^{\texttt{T}}$; observed selection probabilities: $(0.45,0.20,0.20,0.15)$; $M=(10,20,30)$ | $M=10$ |  | $M=20$ |  | $M=30$ |
---|---|---|---|---|---|---
|  | $\widehat{\bm{\beta}}_{F}$ | $\widehat{\bm{\beta}}_{C}$ | $\widehat{\bm{\beta}}_{W}$ | $\widehat{\bm{\beta}}_{EM}$ |  | $\widehat{\bm{\beta}}_{M1}$ | $\widehat{\bm{\beta}}_{M2}$ | $\widehat{\bm{\beta}}_{M1n}$ | $\widehat{\bm{\beta}}_{M2n}$ | $\widehat{\bm{\beta}}_{RF}$ |  | $\widehat{\bm{\beta}}_{M1}$ | $\widehat{\bm{\beta}}_{M2}$ | $\widehat{\bm{\beta}}_{M1n}$ | $\widehat{\bm{\beta}}_{M2n}$ | $\widehat{\bm{\beta}}_{RF}$ |  | $\widehat{\bm{\beta}}_{M1}$ | $\widehat{\bm{\beta}}_{M2}$ | $\widehat{\bm{\beta}}_{M1n}$ | $\widehat{\bm{\beta}}_{M2n}$ | $\widehat{\bm{\beta}}_{RF}$
$\beta_{0}$ | Bias | 0.0109 | 0.0827 | 0.0189 | 0.0134 |  | 0.0145 | 0.0130 | 0.0145 | 0.0130 | -0.0466 |  | 0.0143 | 0.0126 | 0.0143 | 0.0126 | -0.0465 |  | 0.0142 | 0.0125 | 0.0142 | 0.0125 | -0.0466
| SD | 0.1521 | 0.2257 | 0.1661 | 0.1712 |  | 0.1585 | 0.1567 | 0.1585 | 0.1567 | 0.1398 |  | 0.1583 | 0.1565 | 0.1583 | 0.1565 | 0.1394 |  | 0.1584 | 0.1565 | 0.1584 | 0.1565 | 0.1384
| ASE | 0.1479 | 0.2223 | 0.1544 | 0.1673 |  | 0.1494 | 0.1495 | 0.1484 | 0.1511 | 0.1586 |  | 0.1493 | 0.1494 | 0.1484 | 0.1511 | 0.1585 |  | 0.1492 | 0.1494 | 0.1483 | 0.1510 | 0.1581
| MSE | 0.0233 | 0.0578 | 0.0279 | 0.0295 |  | 0.0253 | 0.0247 | 0.0253 | 0.0247 | 0.0217 |  | 0.0253 | 0.0246 | 0.0253 | 0.0246 | 0.0216 |  | 0.0253 | 0.0246 | 0.0253 | 0.0246 | 0.0213
| CP | 0.9490 | 0.9430 | 0.9350 | 0.9530 |  | 0.9400 | 0.9380 | 0.9360 | 0.9410 | 0.9580 |  | 0.9400 | 0.9370 | 0.9360 | 0.9400 | 0.9590 |  | 0.9390 | 0.9400 | 0.9350 | 0.9410 | 0.9590
$\beta_{1}$ | Bias | 0.0016 | 0.0310 | 0.0112 | 0.0263 |  | 0.0056 | 0.0059 | 0.0056 | 0.0059 | -0.2149 |  | 0.0058 | 0.0058 | 0.0058 | 0.0058 | -0.2103 |  | 0.0057 | 0.0054 | 0.0057 | 0.0054 | -0.2118
| SD | 0.2131 | 0.3394 | 0.2331 | 0.2753 |  | 0.2232 | 0.2211 | 0.2232 | 0.2211 | 0.2132 |  | 0.2232 | 0.2213 | 0.2232 | 0.2213 | 0.2122 |  | 0.2232 | 0.2212 | 0.2232 | 0.2212 | 0.2100
| ASE | 0.2143 | 0.3291 | 0.2280 | 0.2728 |  | 0.2174 | 0.2178 | 0.2192 | 0.2220 | 0.2588 |  | 0.2173 | 0.2177 | 0.2192 | 0.2220 | 0.2569 |  | 0.2172 | 0.2176 | 0.2192 | 0.2219 | 0.2566
| MSE | 0.0454 | 0.1161 | 0.0544 | 0.0765 |  | 0.0499 | 0.0489 | 0.0499 | 0.0489 | 0.0916 |  | 0.0498 | 0.0490 | 0.0498 | 0.0490 | 0.0893 |  | 0.0498 | 0.0490 | 0.0498 | 0.0490 | 0.0889
| CP | 0.9530 | 0.9500 | 0.9510 | 0.9480 |  | 0.9460 | 0.9460 | 0.9470 | 0.9510 | 0.8960 |  | 0.9470 | 0.9490 | 0.9470 | 0.9530 | 0.8950 |  | 0.9440 | 0.9480 | 0.9470 | 0.9510 | 0.9030
$\beta_{2}$ | Bias | 0.0090 | 0.0206 | 0.0281 | 0.0170 |  | 0.0176 | 0.0149 | 0.0176 | 0.0149 | -0.2615 |  | 0.0172 | 0.0141 | 0.0172 | 0.0141 | -0.2587 |  | 0.0170 | 0.0138 | 0.0170 | 0.0138 | -0.2594
| SD | 0.2053 | 0.3019 | 0.2406 | 0.2591 |  | 0.2241 | 0.2211 | 0.2241 | 0.2211 | 0.1759 |  | 0.2236 | 0.2208 | 0.2236 | 0.2208 | 0.1782 |  | 0.2236 | 0.2209 | 0.2236 | 0.2209 | 0.1747
| ASE | 0.1964 | 0.3015 | 0.2125 | 0.2486 |  | 0.1996 | 0.2003 | 0.1982 | 0.2048 | 0.2316 |  | 0.1993 | 0.2001 | 0.1981 | 0.2047 | 0.2316 |  | 0.1993 | 0.2001 | 0.1981 | 0.2046 | 0.2307
| MSE | 0.0422 | 0.0916 | 0.0587 | 0.0674 |  | 0.0505 | 0.0491 | 0.0505 | 0.0491 | 0.0993 |  | 0.0503 | 0.0489 | 0.0503 | 0.0489 | 0.0987 |  | 0.0503 | 0.0490 | 0.0503 | 0.0490 | 0.0978
| CP | 0.9460 | 0.9530 | 0.9200 | 0.9480 |  | 0.9200 | 0.9240 | 0.9170 | 0.9290 | 0.8180 |  | 0.9210 | 0.9260 | 0.9190 | 0.9310 | 0.8180 |  | 0.9200 | 0.9250 | 0.9210 | 0.9300 | 0.8280
$\beta_{3}$ | Bias | 0.0112 | 0.1394 | 0.0287 | 0.0229 |  | 0.0126 | 0.0125 | 0.0126 | 0.0125 | -0.0130 |  | 0.0127 | 0.0125 | 0.0127 | 0.0125 | -0.0127 |  | 0.0127 | 0.0125 | 0.0127 | 0.0125 | -0.0130
| SD | 0.1952 | 0.2982 | 0.2052 | 0.1988 |  | 0.1959 | 0.1960 | 0.1959 | 0.1960 | 0.1923 |  | 0.1959 | 0.1960 | 0.1959 | 0.1960 | 0.1921 |  | 0.1960 | 0.1960 | 0.1960 | 0.1960 | 0.1923
| ASE | 0.1824 | 0.2930 | 0.1837 | 0.1862 |  | 0.1825 | 0.1825 | 0.1804 | 0.1822 | 0.1819 |  | 0.1825 | 0.1825 | 0.1804 | 0.1822 | 0.1819 |  | 0.1825 | 0.1825 | 0.1804 | 0.1822 | 0.1819
| MSE | 0.0382 | 0.1083 | 0.0429 | 0.0401 |  | 0.0386 | 0.0386 | 0.0386 | 0.0386 | 0.0371 |  | 0.0385 | 0.0386 | 0.0385 | 0.0386 | 0.0371 |  | 0.0386 | 0.0386 | 0.0386 | 0.0386 | 0.0372
| CP | 0.9310 | 0.9350 | 0.9170 | 0.9390 |  | 0.9310 | 0.9310 | 0.9280 | 0.9310 | 0.9340 |  | 0.9310 | 0.9320 | 0.9300 | 0.9310 | 0.9310 |  | 0.9310 | 0.9320 | 0.9300 | 0.9300 | 0.9310
$\widehat{\bm{\beta}}_{M1}$: MI1 estimator whose value is the solution of
$\bm{U}_{M1}(\bm{\beta})=\bm{0}$ in Equation (18) with Rubin’s type estimated
variance in Equation (26)
$\widehat{\bm{\beta}}_{M2}$: MI2 estimator whose value is the solution of
$\bm{U}_{M2}(\bm{\beta})=\bm{0}$ in Equation (19) with Rubin’s type estimated
variance in Equation (27)
$\widehat{\bm{\beta}}_{M1n}$: MI1 estimator whose value is the solution of
$\bm{U}_{M1}(\bm{\beta})=\bm{0}$ in Equation (18) with proposed estimated
variance in Equation (40)
$\widehat{\bm{\beta}}_{M2n}$: MI2 estimator whose value is the solution of
$\bm{U}_{M2}(\bm{\beta})=\bm{0}$ in Equation (19) with proposed estimated
variance in Equation (41)

Table 6: Relative efficiencies in Study 3 $(n=1,000)$;
$\bm{\beta}=(1.2,1,1,1)^{\texttt{T}}$;
$\bm{\alpha}=(1.4,0.6,0.6)^{\texttt{T}}$;
$\bm{\gamma}=(0.7,-0.2,0.1,-1.2)^{\texttt{T}}$; observed selection
probabilities: $(0.45,0.20,0.20,0.15)$; $M=10,20,30$ Relative efficiency
---
$\bm{\beta}$ | $C1$ | $W1$ | $M11$ | $M21$ | $RF1$ | $EM1$ | $C2$ | $W2$ | $M12$ | $M22$ | $RF2$ | $EM2$ | $M12n$
$M=10$
$\beta_{0}$ | 1.498 | 1.040 | 1.006 | 1.008 | 1.069 | 1.127 | 1.471 | 1.022 | 0.988 | 0.989 | 1.050 | 1.107 | 0.982
$\beta_{1}$ | 1.501 | 1.040 | 0.992 | 0.994 | 1.180 | 1.244 | 1.482 | 1.027 | 0.979 | 0.981 | 1.166 | 1.229 | 0.987
$\beta_{2}$ | 1.522 | 1.072 | 1.007 | 1.011 | 1.169 | 1.255 | 1.472 | 1.038 | 0.975 | 0.978 | 1.131 | 1.214 | 0.968
$\beta_{3}$ | 1.624 | 1.019 | 1.012 | 1.012 | 1.008 | 1.032 | 1.608 | 1.008 | 1.001 | 1.002 | 0.998 | 1.022 | 0.990
$M=20$
$\beta_{0}$ | 1.498 | 1.041 | 1.006 | 1.007 | 1.068 | 1.127 | 1.472 | 1.022 | 0.988 | 0.989 | 1.049 | 1.107 | 0.982
$\beta_{1}$ | 1.501 | 1.040 | 0.991 | 0.993 | 1.172 | 1.244 | 1.482 | 1.027 | 0.979 | 0.981 | 1.157 | 1.229 | 0.987
$\beta_{2}$ | 1.522 | 1.073 | 1.006 | 1.010 | 1.169 | 1.255 | 1.473 | 1.038 | 0.974 | 0.978 | 1.132 | 1.215 | 0.968
$\beta_{3}$ | 1.624 | 1.019 | 1.012 | 1.012 | 1.008 | 1.032 | 1.608 | 1.008 | 1.002 | 1.002 | 0.998 | 1.022 | 0.990
$M=30$
$\beta_{0}$ | 1.499 | 1.041 | 1.006 | 1.007 | 1.066 | 1.127 | 1.472 | 1.022 | 0.988 | 0.989 | 1.047 | 1.107 | 0.983
$\beta_{1}$ | 1.501 | 1.040 | 0.991 | 0.993 | 1.171 | 1.245 | 1.483 | 1.027 | 0.979 | 0.980 | 1.156 | 1.229 | 0.988
$\beta_{2}$ | 1.522 | 1.073 | 1.006 | 1.010 | 1.165 | 1.255 | 1.473 | 1.039 | 0.974 | 0.978 | 1.127 | 1.215 | 0.968
$\beta_{3}$ | 1.624 | 1.019 | 1.012 | 1.012 | 1.008 | 1.032 | 1.608 | 1.008 | 1.002 | 1.002 | 0.998 | 1.022 | 0.990
$Ar$, $A\in\\{C,W,M1,M2,RF,EM\\}$, $r=1,2$, are the relative efficiencies of
CC, SIPW, MI1, MI2,
RFMI, and SAEM estimators to MI$r$n estimators for each $\beta_{i}$,
$i=0,1,2,3$.
$Ar$ is the ratio of ASE of $A$ estimator to ASE of MI$r$n estimator for each $\beta_{i}$. |  |
$Ar=ASE_{A}/ASE_{MIrn}$, $Mn12=ASE_{MI1n}/ASE_{MI2n}$
MI1: MI1 estimation method with Rubin’s type estimated variance in Equation
(26)
MI2: MI2 estimation method with Rubin’s type estimated variance in Equation
(27)
MI1n: MI1 estimation method with proposed estimated variance in Equation (40)
MI2n: MI2 estimation method with proposed estimated variance in Equation (41)

Table 7: Simulation results of Study 4 ($M=15$; $n=1,500$); $\bm{\beta}=(1.2,1,1,1)^{\texttt{T}}$; $\bm{\gamma}=(0.7,-0.2,0.1,-1.2)^{\texttt{T}}$; three sets of $\bm{\alpha}$ are $(2.4,0.6,0.6)^{\texttt{T}}$, $(1.6,0.6,0.6)^{\texttt{T}}$, and $(1,0.6,0.6)^{\texttt{T}}$ to create the observed selection probabilities: $(0.70,0.12,0.12,0.06)$, $(0.50,0.19,0.19,0.12)$, and $(0.36,0.24,0.24,0.16)$, respectively |  | $\widehat{\bm{\beta}}_{F}$ | $\widehat{\bm{\beta}}_{C}$ | $\widehat{\bm{\beta}}_{W}$ | $\widehat{\bm{\beta}}_{M1}$ | $\widehat{\bm{\beta}}_{M2}$ | $\widehat{\bm{\beta}}_{M1n}$ | $\widehat{\bm{\beta}}_{M2n}$ | $\widehat{\bm{\beta}}_{RF}$ | $\widehat{\bm{\beta}}_{EM}$
---|---|---|---|---|---|---|---|---|---|---
Observed selection probabilities: $(0.70,0.12,0.12,0.06)$;
$\bm{\alpha}=(2.4,0.6,0.6)^{\texttt{T}}$
$\beta_{0}$ | Bias | 0.0104 | 0.0508 | 0.0144 | 0.0130 | 0.0164 | 0.0130 | 0.0164 | 0.1364 | 0.0130
| SD | 0.1532 | 0.1839 | 0.1720 | 0.1658 | 0.1619 | 0.1658 | 0.1619 | 0.1532 | 0.1612
| ASE | 0.1521 | 0.1833 | 0.1668 | 0.1583 | 0.1585 | 0.1604 | 0.1631 | 0.1645 | 0.1613
| MSE | 0.0236 | 0.0364 | 0.0298 | 0.0277 | 0.0265 | 0.0277 | 0.0265 | 0.0421 | 0.0261
| CP | 0.9520 | 0.9430 | 0.9440 | 0.9370 | 0.9490 | 0.9420 | 0.9540 | 0.9080 | 0.9550
$\beta_{1}$ | Bias | 0.0197 | 0.0239 | 0.0231 | 0.0243 | 0.0196 | 0.0243 | 0.0196 | -0.1695 | 0.0251
| SD | 0.2187 | 0.2622 | 0.2677 | 0.2567 | 0.2526 | 0.2567 | 0.2526 | 0.1995 | 0.2502
| ASE | 0.2108 | 0.2579 | 0.2561 | 0.2298 | 0.2311 | 0.2364 | 0.2473 | 0.2302 | 0.2382
| MSE | 0.0482 | 0.0693 | 0.0722 | 0.0665 | 0.0642 | 0.0665 | 0.0642 | 0.0685 | 0.0632
| CP | 0.9490 | 0.9530 | 0.9450 | 0.9310 | 0.9360 | 0.9420 | 0.9530 | 0.9050 | 0.9480
$\beta_{2}$ | Bias | -0.0041 | -0.0044 | -0.0072 | -0.0076 | -0.0108 | -0.0076 | -0.0108 | -0.1948 | -0.0026
| SD | 0.2012 | 0.2518 | 0.2596 | 0.2393 | 0.2353 | 0.2393 | 0.2353 | 0.1824 | 0.2324
| ASE | 0.2099 | 0.2567 | 0.2547 | 0.2287 | 0.2302 | 0.2353 | 0.2459 | 0.2293 | 0.2373
| MSE | 0.0405 | 0.0634 | 0.0674 | 0.0573 | 0.0555 | 0.0573 | 0.0555 | 0.0712 | 0.0540
| CP | 0.9560 | 0.9680 | 0.9510 | 0.9340 | 0.9450 | 0.9470 | 0.9610 | 0.9000 | 0.9560
$\beta_{3}$ | Bias | 0.0072 | 0.0839 | 0.0108 | 0.0071 | 0.0062 | 0.0071 | 0.0062 | -0.0054 | 0.0102
| SD | 0.2085 | 0.2675 | 0.2145 | 0.2096 | 0.2090 | 0.2096 | 0.2090 | 0.2078 | 0.2095
| ASE | 0.2104 | 0.2656 | 0.2122 | 0.2111 | 0.2110 | 0.2099 | 0.2114 | 0.2098 | 0.2118
| MSE | 0.0435 | 0.0786 | 0.0461 | 0.0440 | 0.0437 | 0.0440 | 0.0437 | 0.0432 | 0.0440
| CP | 0.9600 | 0.9520 | 0.9560 | 0.9580 | 0.9610 | 0.9580 | 0.9610 | 0.9540 | 0.9600
Observed selection probabilities: $(0.50,0.19,0.19,0.12)$;
$\bm{\alpha}=(1.6,0.6,0.6)^{\texttt{T}}$
$\beta_{0}$ | Bias | 0.0104 | 0.0760 | 0.0201 | 0.0188 | 0.0236 | 0.0188 | 0.0236 | 0.2109 | 0.0142
| SD | 0.1532 | 0.2131 | 0.1880 | 0.1777 | 0.1713 | 0.1777 | 0.1713 | 0.1562 | 0.1697
| ASE | 0.1521 | 0.2151 | 0.1813 | 0.1611 | 0.1618 | 0.1673 | 0.1698 | 0.1725 | 0.1688
| MSE | 0.0236 | 0.0512 | 0.0357 | 0.0319 | 0.0299 | 0.0319 | 0.0299 | 0.0689 | 0.0290
| CP | 0.9520 | 0.9490 | 0.9460 | 0.9290 | 0.9350 | 0.9340 | 0.9480 | 0.8100 | 0.9530
$\beta_{1}$ | Bias | 0.0197 | 0.0240 | 0.0192 | 0.0174 | 0.0104 | 0.0174 | 0.0104 | -0.2777 | 0.0295
| SD | 0.2187 | 0.2991 | 0.3098 | 0.2775 | 0.2716 | 0.2775 | 0.2716 | 0.1820 | 0.2677
| ASE | 0.2108 | 0.3053 | 0.2979 | 0.2365 | 0.2402 | 0.2551 | 0.2677 | 0.2399 | 0.2600
| MSE | 0.0482 | 0.0900 | 0.0964 | 0.0773 | 0.0739 | 0.0773 | 0.0739 | 0.1102 | 0.0725
| CP | 0.9490 | 0.9630 | 0.9440 | 0.9150 | 0.9220 | 0.9400 | 0.9520 | 0.8230 | 0.9570
$\beta_{2}$ | Bias | -0.0041 | 0.0033 | -0.0005 | -0.0106 | -0.0152 | -0.0106 | -0.0152 | -0.2948 | -0.0003
| SD | 0.2012 | 0.3009 | 0.3152 | 0.2723 | 0.2615 | 0.2723 | 0.2615 | 0.1786 | 0.2598
| ASE | 0.2099 | 0.3044 | 0.2965 | 0.2355 | 0.2388 | 0.2531 | 0.2658 | 0.2398 | 0.2586
| MSE | 0.0405 | 0.0905 | 0.0994 | 0.0742 | 0.0686 | 0.0742 | 0.0686 | 0.1188 | 0.0675
| CP | 0.9560 | 0.9630 | 0.9360 | 0.9090 | 0.9230 | 0.9310 | 0.9540 | 0.8060 | 0.9570
$\beta_{3}$ | Bias | 0.0072 | 0.1173 | 0.0302 | 0.0074 | 0.0057 | 0.0074 | 0.0057 | -0.0103 | 0.0128
| SD | 0.2085 | 0.3233 | 0.2323 | 0.2092 | 0.2084 | 0.2092 | 0.2084 | 0.2067 | 0.2094
| ASE | 0.2104 | 0.3207 | 0.2157 | 0.2116 | 0.2115 | 0.2077 | 0.2117 | 0.2095 | 0.2127
| MSE | 0.0435 | 0.1183 | 0.0549 | 0.0438 | 0.0435 | 0.0438 | 0.0435 | 0.0428 | 0.0440
| CP | 0.9600 | 0.9580 | 0.9370 | 0.9570 | 0.9580 | 0.9550 | 0.9570 | 0.9520 | 0.9590
Observed selection probabilities: $(0.36,0.24,0.24,0.16)$;
$\bm{\alpha}=(1,0.6,0.6)^{\texttt{T}}$
$\beta_{0}$ | Bias | 0.0104 | 0.0987 | 0.0106 | 0.0204 | 0.0263 | 0.0204 | 0.0263 | 0.2643 | 0.0087
| SD | 0.1532 | 0.2598 | 0.2208 | 0.1927 | 0.1811 | 0.1927 | 0.1811 | 0.1564 | 0.1796
| ASE | 0.1521 | 0.2567 | 0.1985 | 0.1622 | 0.1637 | 0.1733 | 0.1728 | 0.1788 | 0.1764
| MSE | 0.0236 | 0.0772 | 0.0489 | 0.0376 | 0.0335 | 0.0376 | 0.0335 | 0.0943 | 0.0323
| CP | 0.9520 | 0.9480 | 0.9310 | 0.8970 | 0.9210 | 0.9220 | 0.9320 | 0.7270 | 0.9490
$\beta_{1}$ | Bias | 0.0197 | 0.0468 | 0.0494 | 0.0174 | 0.0081 | 0.0174 | 0.0081 | -0.3518 | 0.0374
| SD | 0.2187 | 0.3867 | 0.4124 | 0.3330 | 0.3099 | 0.3330 | 0.3099 | 0.1799 | 0.3008
| ASE | 0.2108 | 0.3701 | 0.3479 | 0.2385 | 0.2455 | 0.2696 | 0.2772 | 0.2487 | 0.2821
| MSE | 0.0482 | 0.1517 | 0.1725 | 0.1112 | 0.0961 | 0.1112 | 0.0961 | 0.1561 | 0.0919
| CP | 0.9490 | 0.9460 | 0.9110 | 0.8510 | 0.8860 | 0.8870 | 0.9170 | 0.7550 | 0.9480
$\beta_{2}$ | Bias | -0.0041 | 0.0258 | 0.0319 | 0.0030 | -0.0077 | 0.0030 | -0.0077 | -0.3637 | 0.0190
| SD | 0.2012 | 0.3776 | 0.4078 | 0.3288 | 0.3016 | 0.3288 | 0.3016 | 0.1751 | 0.2888
| ASE | 0.2099 | 0.3685 | 0.3458 | 0.2392 | 0.2450 | 0.2684 | 0.2755 | 0.2480 | 0.2808
| MSE | 0.0405 | 0.1432 | 0.1673 | 0.1081 | 0.0910 | 0.1081 | 0.0910 | 0.1629 | 0.0838
| CP | 0.9560 | 0.9530 | 0.9250 | 0.8580 | 0.8950 | 0.9100 | 0.9320 | 0.7240 | 0.9580
$\beta_{3}$ | Bias | 0.0072 | 0.1678 | 0.0933 | 0.0081 | 0.0063 | 0.0081 | 0.0063 | -0.0128 | 0.0161
| SD | 0.2085 | 0.4101 | 0.2899 | 0.2121 | 0.2109 | 0.2121 | 0.2109 | 0.2076 | 0.2124
| ASE | 0.2104 | 0.3955 | 0.2237 | 0.2121 | 0.2118 | 0.2029 | 0.2112 | 0.2093 | 0.2137
| MSE | 0.0435 | 0.1963 | 0.0928 | 0.0450 | 0.0445 | 0.0450 | 0.0445 | 0.0433 | 0.0454
| CP | 0.9600 | 0.9500 | 0.8760 | 0.9570 | 0.9570 | 0.9420 | 0.9560 | 0.9540 | 0.9640
$\widehat{\bm{\beta}}_{M1}$: MI1 estimator whose value is the solution of
$\bm{U}_{M1}(\bm{\beta})=\bm{0}$ in Equation (18) with Rubin’s type
estimated variance in Equation (26)
$\widehat{\bm{\beta}}_{M2}$: MI2 estimator whose value is the solution of
$\bm{U}_{M2}(\bm{\beta})=\bm{0}$ in Equation (19) with Rubin’s type
estimated variance in Equation (27)
$\widehat{\bm{\beta}}_{M1n}$: MI1 estimator whose value is the solution of
$\bm{U}_{M1}(\bm{\beta})=\bm{0}$ in Equation (18) with proposed
estimated variance in Equation (40)
$\widehat{\bm{\beta}}_{M2n}$: MI2 estimator whose value is the solution of
$\bm{U}_{M2}(\bm{\beta})=\bm{0}$ in Equation (19) with proposed
estimated variance in Equation (41)

Table 8: Relative efficiencies in Study 4 ($M=15$; $n=1,500$);
$\bm{\beta}=(1.2,1,1,1)^{\texttt{T}}$;
$\bm{\gamma}=(0.7,-0.2,0.1,-1.2)^{\texttt{T}}$ Relative efficiency
---
$\bm{\beta}$ | $C1$ | $W1$ | $M11$ | $M21$ | $RF1$ | $EM1$ | $C2$ | $W2$ | $M12$ | $M22$ | $RF2$ | $EM2$ | $M12n$
Observed selection probabilities: $(0.70,0.12,0.12,0.06)$;
$\bm{\alpha}=(2.4,0.6,0.6)^{\texttt{T}}$
$\beta_{0}$ | 1.143 | 1.040 | 0.987 | 0.988 | 1.025 | 1.005 | 1.124 | 1.022 | 0.970 | 0.972 | 1.008 | 0.989 | 0.983
$\beta_{1}$ | 1.091 | 1.084 | 0.972 | 0.978 | 0.974 | 1.008 | 1.043 | 1.036 | 0.929 | 0.935 | 0.931 | 0.963 | 0.956
$\beta_{2}$ | 1.091 | 1.082 | 0.972 | 0.978 | 0.974 | 1.008 | 1.044 | 1.036 | 0.930 | 0.936 | 0.933 | 0.965 | 0.957
$\beta_{3}$ | 1.265 | 1.011 | 1.006 | 1.005 | 1.000 | 1.009 | 1.257 | 1.004 | 0.999 | 0.998 | 0.993 | 1.002 | 0.993
Observed selection probabilities: $(0.50,0.19,0.19,0.12)$;
$\bm{\alpha}=(1.6,0.6,0.6)^{\texttt{T}}$
$\beta_{0}$ | 1.286 | 1.084 | 0.963 | 0.967 | 1.031 | 1.009 | 1.267 | 1.068 | 0.949 | 0.952 | 1.016 | 0.994 | 0.985
$\beta_{1}$ | 1.197 | 1.168 | 0.927 | 0.942 | 0.941 | 1.019 | 1.141 | 1.113 | 0.883 | 0.897 | 0.896 | 0.971 | 0.953
$\beta_{2}$ | 1.203 | 1.172 | 0.931 | 0.944 | 0.948 | 1.022 | 1.145 | 1.116 | 0.886 | 0.899 | 0.902 | 0.973 | 0.952
$\beta_{3}$ | 1.544 | 1.038 | 1.019 | 1.018 | 1.009 | 1.024 | 1.515 | 1.019 | 1.000 | 0.999 | 0.990 | 1.005 | 0.981
Observed selection probabilities: $(0.36,0.24,0.24,0.16)$;
$\bm{\alpha}=(1.0,0.6,0.6)^{\texttt{T}}$
$\beta_{0}$ | 1.482 | 1.146 | 0.936 | 0.945 | 1.032 | 1.018 | 1.485 | 1.148 | 0.939 | 0.947 | 1.034 | 1.021 | 1.002
$\beta_{1}$ | 1.373 | 1.291 | 0.885 | 0.911 | 0.923 | 1.047 | 1.335 | 1.255 | 0.860 | 0.885 | 0.897 | 1.018 | 0.972
$\beta_{2}$ | 1.373 | 1.288 | 0.891 | 0.913 | 0.924 | 1.046 | 1.338 | 1.255 | 0.868 | 0.889 | 0.900 | 1.020 | 0.974
$\beta_{3}$ | 1.949 | 1.103 | 1.045 | 1.044 | 1.031 | 1.053 | 1.872 | 1.059 | 1.004 | 1.003 | 0.991 | 1.012 | 0.961
$Ar$, $A\in\\{C,W,M1,M2,RF,EM\\}$, $r=1,2$, are the relative efficiencies of
CC, SIPW, MI1, MI2,
RFMI, and SAEM estimators to MI$r$n estimators for each $\beta_{i}$,
$i=0,1,2,3$.
$Ar$ is the ratio of ASE of $A$ estimator to ASE of MI$r$n estimator for each
$\beta_{i}$.
$Ar=ASE_{A}/ASE_{MIrn}$, $M12n=ASE_{MI1n}/ASE_{MI2n}$
MI1: MI1 estimation method with Rubin’s type estimated variance in Equation
(26)
MI2: MI2 estimation method with Rubin’s type estimated variance in Equation
(27)
MI1n: MI1 estimation method with proposed estimated variance in Equation (40)
MI2n: MI2 estimation method with proposed estimated variance in Equation (41)

Table 9: Results of logistic regression analysis of Feng Chia night market (FCNM) survey data with original missing data, i.e., selection probabilities: $(0.666,0.334,0,0)$ Parameter |  | $\widehat{\bm{\beta}}_{C}$ | $\widehat{\bm{\beta}}_{W}$ | $\widehat{\bm{\beta}}_{M1}$ | $\widehat{\bm{\beta}}_{M2}$ | $\widehat{\bm{\beta}}_{M1n}$ | $\widehat{\bm{\beta}}_{M2n}$ | $\widehat{\bm{\beta}}_{RF}$ | $\widehat{\bm{\beta}}_{EM}$
---|---|---|---|---|---|---|---|---|---
$\beta_{0}$ | est | -2.5586 | -2.6680 | -2.7531 | -2.7927 | -2.7531 | -2.7927 | -2.7613 | -2.7427
| ASE | 0.1628 | 0.1497 | 0.1426 | 0.1470 | 0.1448 | 0.1535 | 0.1468 | 0.1446
|  $p$-value | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000
$\beta_{1}$ | est | 0.5423 | 0.5463 | 0.5456 | 0.6302 | 0.5456 | 0.6302 | 0.5546 | 0.5091
| ASE | 0.1674 | 0.1656 | 0.1521 | 0.1664 | 0.1644 | 0.1638 | 0.1744 | 0.1669
|  $p$-value | 0.0012 | 0.0010 | 0.0003 | 0.0002 | 0.0009 | 0.0001 | 0.0015 | 0.0023
$\beta_{2}$ | est | 1.7978 | 1.8184 | 1.8649 | 1.8789 | 1.8649 | 1.8789 | 1.8820 | 1.8771
| ASE | 0.1756 | 0.1713 | 0.1483 | 0.1477 | 0.1472 | 0.1745 | 0.1475 | 0.1487
|  $p$-value | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000
$\beta_{3}$ | est | 0.1201 | 0.2294 | 0.2459 | 0.2397 | 0.2459 | 0.2397 | 0.2446 | 0.2479
| ASE | 0.1108 | 0.1048 | 0.0996 | 0.1005 | 0.0991 | 0.1046 | 0.0921 | 0.0918
|  $p$-value | 0.2783 | 0.0286 | 0.0135 | 0.0170 | 0.0131 | 0.0231 | 0.0079 | 0.0069
Estimate (est), asymptotic standard error (ASE), and $p$-value for testing
parameter = 0
$\texttt{X}_{1}$ ($1=\\{1,2,3\\}$ times; $0=$ more than three times): response
to the question
“What is the number of times you went shopping in the FCNM in the last half
year?”.
$\texttt{X}_{2}$ ($1=$ Neighborhood of Taichung and Others City, $0=$ Taichung
City):
response to the question “Which county or city is your current place of
residence?”.
$Z$ ($Z=\\{0.1,0.35,0.75,1.25,1.75,3\\}$ after taking the median and being
divided by $1,000$):
response to question “How much do you spend each time you visit FCNM?”.
$\widehat{\bm{\beta}}_{M1}$: MI1 estimator whose value is the solution of
$\bm{U}_{M1}(\bm{\beta})=\bm{0}$ in Equation (18) with
Rubin’s type estimated variance in Equation (26)
$\widehat{\bm{\beta}}_{M2}$: MI2 estimator whose value is the solution of
$\bm{U}_{M2}(\bm{\beta})=\bm{0}$ in Equation (19) with
Rubin’s type estimated variance in Equation (27)
$\widehat{\bm{\beta}}_{M1n}$: MI1 estimator whose value is the solution of
$\bm{U}_{M1}(\bm{\beta})=\bm{0}$ in Equation (18) with
proposed estimated variance in Equation (40)
$\widehat{\bm{\beta}}_{M2n}$: MI2 estimator whose value is the solution of
$\bm{U}_{M2}(\bm{\beta})=\bm{0}$ in Equation (19) with
proposed estimated variance in Equation (41)

Table 10: Results of logistic regression analysis of Feng Chia night market (FCNM) survey data with artificial missing data, i.e., selection probabilities: $(0.485,0.255,0.181,0.079)$ Parameter |  | $\widehat{\bm{\beta}}_{C}$ | $\widehat{\bm{\beta}}_{W}$ | $\widehat{\bm{\beta}}_{M1}$ | $\widehat{\bm{\beta}}_{M2}$ | $\widehat{\bm{\beta}}_{M1n}$ | $\widehat{\bm{\beta}}_{M2n}$ | $\widehat{\bm{\beta}}_{RF}$ | $\widehat{\bm{\beta}}_{EM}$
---|---|---|---|---|---|---|---|---|---
$\beta_{0}$ | est | -3.0189 | -2.8525 | -2.7711 | -2.8419 | -2.7711 | -2.8419 | -2.5365 | -2.7117
| ASE | 0.2221 | 0.1923 | 0.1486 | 0.1562 | 0.1580 | 0.1698 | 0.1601 | 0.1531
|  $p$-value | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000
$\beta_{1}$ | est | 0.6458 | 0.6781 | 0.5828 | 0.7056 | 0.5828 | 0.7056 | 0.6027 | 0.6297
| ASE | 0.2192 | 0.2073 | 0.1563 | 0.1693 | 0.1640 | 0.1657 | 0.1633 | 0.1697
|  $p$-value | 0.0032 | 0.0011 | 0.0002 | 0.0000 | 0.0004 | 0.0000 | 0.0002 | 0.0002
$\beta_{2}$ | est | 1.9736 | 2.0838 | 1.9205 | 1.9396 | 1.9205 | 1.9396 | 1.4151 | 1.7793
| ASE | 0.2346 | 0.2227 | 0.1634 | 0.1685 | 0.1790 | 0.1784 | 0.2424 | 0.1799
|  $p$-value | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000
$\beta_{3}$ | est | 0.0452 | 0.1255 | 0.2277 | 0.2202 | 0.2277 | 0.2202 | 0.3436 | 0.2212
| ASE | 0.1478 | 0.1002 | 0.0938 | 0.0958 | 0.0897 | 0.0934 | 0.0952 | 0.0945
|  $p$-value | 0.7599 | 0.2103 | 0.0152 | 0.0215 | 0.0111 | 0.0184 | 0.0003 | 0.0193
Estimate (est), asymptotic standard error (ASE), and $p$-value for testing
parameter = 0
$\texttt{X}_{1}$ ($1=\\{1,2,3\\}$ times; $0=$ more than three times): response
to the question
“What is the number of times you went shopping in the FCNM in the last half
year?”.
$\texttt{X}_{2}$ ($1=$ Neighborhood of Taichung and Others City, $0=$ Taichung
City):
response to the question “Which county or city is your current place of
residence?”.
$Z$ ($Z=\\{0.1,0.35,0.75,1.25,1.75,3\\}$ after taking the median and dividing
by $1,000$):
response to question “How much do you spend each time you visit FCNM?”.
$\widehat{\bm{\beta}}_{M1}$: MI1 estimator whose value is the solution of
$\bm{U}_{M1}(\bm{\beta})=\bm{0}$ in Equation (18) with
Rubin’s type estimated variance in Equation (26)
$\widehat{\bm{\beta}}_{M2}$: MI2 estimator whose value is the solution of
$\bm{U}_{M2}(\bm{\beta})=\bm{0}$ in Equation (19) with
Rubin’s type estimated variance in Equation (27)
$\widehat{\bm{\beta}}_{M1n}$: MI1 estimator whose value is the solution of
$\bm{U}_{M1}(\bm{\beta})=\bm{0}$ in Equation (18) with
proposed estimated variance in Equation (40)
$\widehat{\bm{\beta}}_{M2n}$: MI2 estimator whose value is the solution of
$\bm{U}_{M2}(\bm{\beta})=\bm{0}$ in Equation (19) with
proposed estimated variance in Equation (41)