Skip to main content
. Author manuscript; available in PMC: 2018 Jul 10.
Published in final edited form as: Stat Med. 2017 Mar 28;36(15):2363–2377. doi: 10.1002/sim.7291

Table 2.

Simulation study: Presented results include the empirical bias (Bias) of the 500 estimated regression coefficients and there sample standard deviation (SD) obtained from analyzing data generated according to models M1–M5, for all considered pool sizes when J = 100 in the presence of measurement error (τ = 0.05). Also included are the average estimated standard errors (SE) and the empirical coverage probabilities (Cov) associated with 95% Wald confidence intervals. Two model fitting procedures were implemented, the proposed methodology (MCMLE) and the analytical approach described in Section 4 (MLE), with the latter technique only being applicable for model M1.

Model Measure c = 1 c = 2 c = 3 c = 4 c = 5
M1(MCMLE) β0 Bias(SD) 0.00(0.07) 0.00(0.05) 0.00(0.04) 0.00(0.04) 0.00(0.03)
Cov(SE) 0.93(0.07) 0.96(0.05) 0.94(0.04) 0.91(0.04) 0.94(0.03)
β1 Bias(SD) 0.00(0.05) 0.00(0.04) 0.00(0.03) 0.00(0.03) 0.00(0.02)
Cov(SE) 0.95(0.05) 0.94(0.04) 0.97(0.03) 0.93(0.03) 0.94(0.02)
β2 Bias(SD) 0.00(0.10) 0.00(0.07) 0.00(0.06) 0.00(0.05) 0.00(0.05)
Cov(SE) 0.94(0.10) 0.96(0.07) 0.94(0.06) 0.93(0.05) 0.92(0.05)
M1(MLE) β0 Bias(SD) 0.00(0.07) 0.00(0.05) 0.00(0.04) 0.00(0.04) 0.00(0.03)
Cov(SE) 0.94(0.07) 0.96(0.05) 0.94(0.04) 0.92(0.04) 0.95(0.03)
β1 Bias(SD) 0.00(0.05) 0.00(0.04) 0.00(0.03) 0.00(0.03) 0.00(0.02)
Cov(SE) 0.95(0.05) 0.95(0.04) 0.97(0.03) 0.93(0.03) 0.95(0.02)
β2 Bias(SD) 0.00(0.10) 0.00(0.07) 0.00(0.06) 0.00(0.05) 0.00(0.05)
Cov(SE) 0.94(0.10) 0.96(0.07) 0.95(0.06) 0.93(0.05) 0.93(0.05)
M2(MCMLE) β0 Bias(SD) 0.02(0.08) 0.00(0.06) 0.01(0.05) 0.01(0.05) 0.01(0.04)
Cov(SE) 0.94(0.08) 0.94(0.06) 0.94(0.05) 0.93(0.05) 0.94(0.04)
β1 Bias(SD) −0.01(0.05) 0.00(0.04) 0.00(0.03) 0.00(0.03) 0.00(0.03)
Cov(SE) 0.96(0.06) 0.93(0.04) 0.96(0.03) 0.94(0.03) 0.94(0.03)
β2 Bias(SD) −0.01(0.10) 0.00(0.08) 0.00(0.06) −0.01(0.06) 0.00(0.05)
Cov(SE) 0.95(0.10) 0.94(0.08) 0.94(0.07) 0.94(0.06) 0.94(0.05)
M3(MCMLE) β0 Bias(SD) 0.00(0.17) −0.01(0.13) 0.00(0.11) 0.00(0.09) 0.00(0.09)
Cov(SE) 0.93(0.17) 0.94(0.13) 0.94(0.11) 0.94(0.09) 0.94(0.08)
β1 Bias(SD) 0.00(0.12) 0.00(0.09) −0.01(0.08) 0.00(0.07) 0.00(0.06)
Cov(SE) 0.95(0.12) 0.94(0.09) 0.95(0.08) 0.93(0.07) 0.94(0.06)
β2 Bias(SD) −0.01(0.24) 0.01(0.19) 0.00(0.15) 0.00(0.13) 0.00(0.12)
Cov(SE) 0.95(0.24) 0.94(0.18) 0.95(0.15) 0.95(0.13) 0.95(0.12)
M4(MCMLE) β0 Bias(SD) 0.00(0.08) 0.00(0.06) 0.00(0.05) 0.00(0.05) 0.00(0.04)
Cov(SE) 0.95(0.08) 0.97(0.06) 0.95(0.05) 0.93(0.04) 0.96(0.04)
β1 Bias(SD) −0.01(0.06) −0.01(0.04) −0.01(0.04) 0.00(0.03) 0.00(0.03)
Cov(SE) 0.95(0.05) 0.95(0.04) 0.93(0.03) 0.92(0.03) 0.95(0.03)
β2 Bias(SD) −0.01(0.11) 0.00(0.08) 0.00(0.06) −0.01(0.06) 0.00(0.05)
Cov(SE) 0.92(0.10) 0.95(0.08) 0.94(0.06) 0.94(0.06) 0.94(0.05)
M5(MCMLE) β0 Bias(SD) 0.00(0.04) 0.00(0.03) 0.00(0.02) 0.00(0.02) 0.00(0.02)
Cov(SE) 0.96(0.04) 0.94(0.02) 0.94(0.02) 0.94(0.02) 0.93(0.02)
β1 Bias(SD) 0.00(0.03) 0.00(0.02) 0.00(0.01) 0.00(0.01) 0.00(0.01)
Cov(SE) 0.96(0.03) 0.96(0.02) 0.96(0.02) 0.94(0.01) 0.95(0.01)
β2 Bias(SD) 0.00(0.06) 0.00(0.04) 0.00(0.04) 0.00(0.03) 0.00(0.03)
Cov(SE) 0.95(0.06) 0.95(0.04) 0.93(0.03) 0.95(0.03) 0.94(0.03)