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. Author manuscript; available in PMC: 2014 May 1.
Published in final edited form as: Stat Med. 2012 Sep 17;32(9):1494–1508. doi: 10.1002/sim.5613

Table I.

Parameter estimates for simulated data with a binary primary phenotype and a binary secondary trait.

Methods θ1 = 0.5
θ2 = −0.5
θ12 = −0.3
Bias A.std E.std Bias A.std E.std Bias A.std E.std
r0 approximates 1% (or β0 = −4.6)
Controls −0.039 0.242 0.243 −0.007 0.151 0.152 −0.034 0.212 0.211
Marginal 0.073 0.122 0.124 0.116 0.091 0.094 0.018 0.105 0.109
Condit. −0.025 0.125 0.127 −0.006 0.097 0.099 −0.029 0.109 0.111
Correct. −0.008 0.124 0.126 0.011 0.095 0.097 −0.019 0.108 0.111
Comp 0.002 0.124 0.127 0.001 0.096 0.098 −0.002 0.109 0.112
MLE 0.002 0.124 0.126 0.001 0.096 0.098 −0.002 0.109 0.112
r0 approximates 5% (or β0 = −2.95)
Controls −0.081 0.267 0.265 −0.033 0.159 0.157 −0.076 0.216 0.227
Marginal 0.033 0.129 0.129 0.085 0.096 0.095 −0.018 0.112 0.117
Condit. −0.066 0.132 0.132 −0.035 0.100 0.100 −0.067 0.116 0.119
Correct. −0.004 0.129 0.130 0.028 0.096 0.096 −0.029 0.113 0.117
Comp. 0.008 0.130 0.133 0.003 0.097 0.100 −0.006 0.115 0.119
MLE 0.008 0.130 0.131 0.003 0.097 0.097 −0.006 0.115 0.120
r0 approximates 10% (or β0 = −2.2)
Controls −0.104 0.295 0.286 −0.071 0.157 0.162 −0.072 0.243 0.241
Marginal 0.001 0.127 0.135 0.050 0.096 0.097 −0.022 0.121 0.123
Condit. −0.096 0.130 0.138 −0.070 0.101 0.101 −0.071 0.125 0.125
Correct. −0.003 0.128 0.136 0.026 0.096 0.097 −0.014 0.123 0.123
Comp. 0.000 0.128 0.138 −0.004 0.098 0.101 −0.016 0.125 0.126
MLE 0.000 0.129 0.136 0.004 0.098 0.098 −0.015 0.125 0.124
r0 approximates 15% (or β0 = −1.73)
Controls −0.144 0.314 0.305 −0.093 0.165 0.167 −0.086 0.257 0.253
Marginal −0.003 0.135 0.141 0.030 0.099 0.099 −0.027 0.127 0.127
Condit. −0.100 0.137 0.144 −0.090 0.103 0.103 −0.076 0.129 0.130
Correct. 0.011 0.136 0.141 −0.023 0.099 0.099 −0.009 0.129 0.128
Comp. 0.003 0.136 0.144 −0.007 0.101 0.103 −0.008 0.130 0.130
MLE 0.003 0.136 0.142 −0.008 0.101 0.099 −0.008 0.130 0.129

Baseline case-prevalence rate r0 is assumed known in Correct., Comp., and MLE.

Bias, estimate-truth; A.std, square root of the averaged estimated variance; E.std, empirical standard deviation from the repeated simulations; θ1, genetic effect on the secondary phenotype; θ2, environment effect on the secondary trait; θ12, gene–environment interaction on the secondary trait; Controls, analysis based on controls only; Marginal, marginal analysis; Condit., conditional analysis; Correct., corrected joint analysis based on the approximate odds ratio model; Comp., joint analysis by the compensator approach; MLE, the maximum likelihood approach.