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. Author manuscript; available in PMC: 2022 Nov 1.
Published in final edited form as: J Expo Sci Environ Epidemiol. 2021 Jun 9;31(6):1057–1066. doi: 10.1038/s41370-021-00345-1

Table 2.

Results for case 2 in which a chi-squared distribution with two degrees of freedom was created for the outcome data with three repeated measures.

α = 0.3
α = 0.7
n % Censor LOD/2 MLE Quantile LOD/2 MLE Quantile
100 10 Bias 0.0197 0.0210 −0.0008 0.0073 0.0138 0.0006
MSE 0.0006 0.0006 0.0005 0.0003 0.0004 0.0007
RE 1.000 0.921 1.279 1.000 0.741 0.413
20 Bias 0.0186 0.0221 −0.0001 −0.0032 0.0166 0.0003
MSE 0.0006 0.0007 0.0005 0.0003 0.0005 0.0007
RE 1.000 0.823 1.194 1.000 0.588 0.424
30 Bias 0.0192 0.0231 −0.0020 −0.0155 0.0195 −0.0022
MSE 0.0006 0.0007 0.0005 0.0007 0.0006 0.0008
RE 1.000 0.884 1.324 1.000 1.087 0.873
40 Bias 0.0169 0.0236 −0.0057 −0.0477 0.0216 −0.0036
MSE 0.0010 0.0008 0.0006 0.0044 0.0007 0.0008
RE 1.000 1.261 1.679 1.000 6.211 5.668
500 10 Bias 0.0194 0.0207 −0.0010 0.0062 0.0127 −0.0007
MSE 0.0004 0.0005 0.0001 0.0001 0.0002 0.0001
RE 1.000 0.891 4.172 1.000 0.425 0.670
20 Bias 0.0185 0.0219 −0.0017 −0.0045 0.0156 −0.0014
MSE 0.0004 0.0005 0.0001 0.0001 0.0003 0.0001
RE 1.000 0.742 3.859 1.000 0.274 0.609
30 Bias 0.0192 0.0229 −0.0033 −0.0164 0.0185 −0.0031
MSE 0.0004 0.0006 0.0001 0.0004 0.0004 0.0002
RE 1.000 0.748 3.843 1.000 0.913 2.380
40 Bias 0.0188 0.0234 −0.0068 −0.0415 0.0205 −0.0064
MSE 0.0004 0.0006 0.0002 0.0021 0.0005 0.0002
RE 1.000 0.762 2.801 1.000 4.373 10.85
a

Bias - empirical bias.

b

MSE - empirical mean squared error.

c

RE - relative efficiency. These are the italicized ratios that, for each setting (n), compare the empirical MSE from the LOD/2 substitution method to the MSE from the use of MLE method or quantile regression model.