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Table 3 Impact of Auxiliary Relationship Under Conditions 1, 2, and 3.

From: The use of complete-case and multiple imputation-based analyses in molecular epidemiology studies that assess interaction effects

a.condition 1
Scenario Variable Method Mean β Mean SE Mean Bias MSE RelMSE Coverage
A: No Auxiliary         
  X1 (X2 = 0)
(True effect = 1.0)
Full 0.998 0.211 -0.002 0.043 0.403 95.2 (93.9,96.5)
   CC 0.999 0.323 -0.001 0.106 1.000 95.3 (94.0,96.6)
   MI 1.106 0.321 0.106 0.101 0.951 95.4 (94.1,96.7)
  X1 (X2 = 1)
(True effect = 2.5)
Full 2.542 0.378 0.042 0.161 0.031 95.2 (94.0,96.6)
   CC 2.890 11.419 0.390 5.271 1.000 95.7 (94.4,97.0)
   MI 2.330 0.615 -0.170 0.317 0.060 92.8 (91.2,94.4)
  Interaction (True effect = 1.5) Full 1.544 0.434 0.044 0.195 0.037 96.0 (94.8,97.2)
   CC 1.891 11.500 0.391 5.321 1.000 96.4 (95.2,97.6)
   MI 1.223 0.690 -0.277 0.382 0.072 94.6 (93.2,96.0)
B: Moderate
Auxiliary
        
  X1 (X2 = 0)
(True effect = 1.0)
Full 0.986 0.211 -0.014 0.043 0.410 96.1 (94.9,97.3)
   CC 0.984 0.323 -0.016 0.105 1.000 95.7 (94.4,97.0)
   MI 1.157 0.312 0.157 0.110 1.039 94.9 (93.5,96.3)
  X1 (X2 = 1) (True effect = 2.5) Full 2.53 0.376 0.03 0.149 0.066 95.9 (94.7,97.1)
   CC 2.698 4.867 0.198 2.28 1.000 96.1 (94.9,97.3)
   MI 2.438 0.600 -0.062 0.251 0.110 96.3 (95.1,97.5)
  Interaction (True effect = 1.5) Full 1.544 0.432 0.044 0.195 0.080 95.7 (94.4,97.0)
   CC 1.714 4.949 0.214 2.437 1.000 96.8 (95.7,97.9)
   MI 1.281 0.671 -0.219 0.317 0.130 96.7 (95.6,97.8)
C: Strong Auxiliary         
  X1 (X2 = 0) (True effect = 1.0) Full 0.997 0.211 -0.003 0.046 0.403 95.4 (94.1,96.7)
   CC 0.999 0.324 -0.001 0.113 1.000 94.4 (93.0,95.8)
   MI 1.027 0.223 0.027 0.051 0.447 95.3 (94.0,96.6)
  X1 (X2 = 1)
True effect = 2.5)
Full 2.546 0.378 0.046 0.158 0.027 96 (94.8,97.2)
   CC 2.968 12.65 0.468 5.941 1.000 96.9 (95.8,98.0)
   MI 2.555 0.407 0.055 0.176 0.030 96.5 (95.0,97.4)
  Interaction (True effect = 1.5) Full 1.549 0.434 0.049 0.200 0.034 96.1 (94.9,97.3)
   CC 1.969 12.730 0.469 5.971 1.000 96.6 (95.5,97.7)
   MI 1.528 0.462 0.028 0.205 0.034 96.2 (95.0,97.4)
b. Condition 2
Scenario Variable Method Mean β Mean SE Mean Bias MSE RelMSE Coverage
D: No Auxiliary         
  X1 (X2 = 0) (True effect = 1.0) Full 1.008 0.108 0.008 0.011 0.496 95.6 (94.3,96.9)
   CC 1.013 0.146 0.013 0.023 1.000 93.8 (92.3,95.3)
   MI 1.090 0.145 0.090 0.029 1.254 92.0 (90.3,93.7)
  X1 (X2 = 1) (True effect = 2.5) Full 2.551 0.280 0.051 0.083 0.869 95.6 (94.3,96.9)
   CC 2.550 0.299 0.050 0.096 1.000 95.4 (94.1,96.7)
   MI 2.357 0.293 -0.143 0.085 0.888 93.1 (91.5,94.7)
  Interaction (True effect = 1.5) Full 1.543 0.300 0.043 0.093 0.804 94.9 (93.5,96.3)
   CC 1.537 0.333 0.037 0.116 1.000 95.5 (94.2,96.8)
   MI 1.267 0.325 -0.233 0.126 1.084 90.2 (88.4,92.0)
E: Moderate
Auxiliary
        
  X1 (X2 = 0)
(True effect = 1.0)
Full 1.011 0.108 0.010 0.012 0.533 95.4 (94.1,96.7)
   CC 1.009 0.146 0.009 0.023 1.000 94.5 (93.1,95.9)
   MI 1.163 0.142 0.163 0.046 2.039 81.4 (79.0,83.8)
  X1 (X2 = 1)
(True effect = 2.5)
Full 2.558 0.280 0.058 0.090 0.911 94.2 (92.8,95.6)
   CC 2.555 0.298 0.055 0.099 1.000 94.5 (93.5,96.3)
   MI 2.571 0.296 0.071 0.085 0.855 96.7 (95.6,97.8)
  Interaction
(True effect = 1.5)
Full 1.547 0.300 0.047 0.103 0.845 94.9 (93.5,96.3)
   CC 1.545 0.333 0.045 0.122 1.000 94.7 (93.3,96.1)
   MI 1.408 0.328 -0.092 0.102 0.832 95.1 (93.8,96.4)
F: Strong
Auxiliary
        
  X1 (X2 = 0) (True effect = 1.0) Full 1.002 0.108 0.002 0.012 0.518 94.5 (93.1,95.9)
   CC 1.004 0.146 0.004 0.024 1.000 94.2 (92.8,95.6)
   MI 1.053 0.114 0.053 0.016 0.693 93.0 (91.4,94.6)
  X1 (X2 = 1) (True effect = 2.5) Full 2.530 0.278 0.030 0.079 0.860 95.2 (93.9,96.5)
   CC 2.530 0.297 0.030 0.092 1.000 95.0 (93.6,96.4)
   MI 2.573 0.280 0.073 0.084 0.908 95.7 (93.8,96.4)
  Interaction (True effect = 1.5) Full 1.527 0.298 0.027 0.092 0.788 95.0 (93.6,96.4)
   CC 1.526 0.331 0.026 0.117 1.000 94.9 (93.5,96.3)
   MI 1.520 0.303 0.020 0.092 0.791 95.1 (93.8,96.4)
c. Condition 3
Scenario Variable Method Mean β Mean SE Mean Bias MSE RelMSE Coverage
G: No Auxiliary         
  X1 (X2 = 0) (True effect = 1.0) Full 1.006 0.108 0.006 0.011 0.066 96.2 (95.0,97.4)
   CC 0.608 0.119 -0.392 0.168 1.000 10.2 (8.3,12.1)
   MI 0.680 0.119 -0.320 0.116 0.691 23.9 (21.3,26.5)
  X1 (X2 = 1)
(True effect = 2.5)
Full 2.549 0.279 0.049 0.085 0.693 94.6 (93.2,96.0)
   CC 2.322 0.288 -0.178 0.122 1.000 85.3 (83.1,87.5)
   MI 1.974 0.274 -0.526 0.324 2.654 49.9 (46.8,53.0)
  Interaction (True effect = 1.5) Full 1.543 0.299 0.043 0.097 0.640 95.1 (93.8,96.4)
   CC 1.714 0.312 0.214 0.152 1.000 92.3 (90.6,94.0)
   MI 1.294 0.297 -0.206 0.093 0.609 93.5 (92.0,95.0)
H: Moderate
Auxiliary
        
  X1 (X2 = 0) (True effect = 1.0) Full 1.003 0.108 0.004 0.012 0.073 94.5 (93.1,95.9)
   CC 0.610 0.119 -0.390 0.167 1.000 11.9 (9.9,13.9)
   MI 0.793 0.116 -0.207 0.057 0.340 54.4 (51.3,57.5)
  X1 (X2 = 1) (True effect = 2.5) Full 2.535 0.277 0.035 0.085 0.657 93.5 (92.0,95.0)
   CC 2.309 0.287 -0.191 0.129 1.000 85.1 (82.9,87.3)
   MI 2.222 0.279 -0.278 0.141 1.093 80.9 (78.5,83.3)
  Interaction (True effect = 1.5) Full 1.531 0.298 0.031 0.096 0.658 94.3 (92.9,95.7)
   CC 1.699 0.310 0.198 0.146 1.000 91.4 (89.7,93.1)
   MI 1.428 0.301 -0.072 0.074 0.508 95.4 (94.1,96.7)
I: Strong Auxiliary         
  X1 (X2 = 0) (True effect = 1.0) Full 0.999 0.108 -0.001 0.012 0.069 95.4 (94.1,96.7)
   CC 0.603 0.118 -0.397 0.172 1.000 11.1 (9.2,13.0)
   MI 0.986 0.109 -0.014 0.012 0.072 94.8 (93.4,96.2)
  X1 (X2 = 1) (True effect = 2.5) Full 2.533 0.277 0.033 0.081 0.635 95.0 (93.6,96.4)
   CC 2.303 0.287 -0.197 0.128 1.000 84.2 (81.9,86.5)
   MI 2.533 0.278 0.033 0.081 0.628 95.2 (93.9,96.5)
  Interaction (True effect = 1.5) Full 1.533 0.298 0.033 0.091 0.638 95.3 (94.0,96.6)
   CC 1.700 0.310 0.200 0.142 1.000 92.8 (91.2,94.4)
   MI 1.547 0.299 0.047 0.091 0.640 95.8 (94.6,97.0)
  1. Results From Fitting Full, Complete-Case, and Multiple Imputation Models to 1000 Simulated Data Sets With a Sample Size of 1000 Where the Covariate of Interest and as a Result the Interaction Term Were Missing for Approximately 20% of Subjects.