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. Author manuscript; available in PMC: 2017 Mar 7.
Published in final edited form as: Stat Med. 2013 Oct 24;33(7):1205–1221. doi: 10.1002/sim.6020

Table 6.

Simulation results: Marginal structural GLMs for skewed outcome data

A. Poisson Outcome
1) Poisson Regression
Variable True Value % Bias Monte Carlo SE Model-based SE rMSE Coverage
M1 M2 M3 M4 M1 M2 M3 M4 M1 M2 M3 M4 M1 M2 M3 M4 M1 M2 M3 M4
A1 0.0653 0.5% 22.8% 1.5% 2.9% 0.029 0.029 0.029 0.028 0.028 0.028 0.028 0.028 0.029 0.033 0.029 0.028 95.0% 92.2% 95.0% 95.4%
A2 −0.0315 7.8% 42.2% 2.8% 8.8% 0.03 0.03 0.03 0.031 0.031 0.031 0.031 0.031 0.03 0.033 0.030 0.031 96.0% 94.0% 95.8% 94.2%
A3 −0.0097 19.1% 5.7% 5.2% 1.2% 0.022 0.022 0.022 0.022 0.022 0.022 0.022 0.022 0.022 0.022 0.022 0.022 94.0% 94.2% 94.2% 95.0%
2) Negative Binomial Regression
Variable True Value % Bias Monte Carlo SE Model-based SE rMSE Coverage
M1 M2 M3 M4 M1 M2 M3 M4 M1 M2 M3 M4 M1 M2 M3 M4 M1 M2 M3 M4
A1 0.0653 1.2% 21.0% 0.2% 0.8% 0.029 0.029 0.029 0.028 0.028 0.028 0.028 0.028 0.029 0.032 0.029 0.028 94.2% 90.4% 94.0% 95.8%
A2 −0.0315 7.0% 41.2% 1.8% 4.8% 0.031 0.031 0.031 0.03 0.031 0.031 0.031 0.031 0.031 0.033 0.031 0.03 94.0% 93.2% 93.8% 95.2%
A3 −0.0097 24.4% 0.8% 0.4% 2.8% 0.022 0.022 0.022 0.022 0.022 0.022 0.022 0.022 0.022 0.022 0.022 0.022 95.0% 95.0% 95.2% 95.0%
3) Zero-Inflated Poisson Regression
Variable True Value % Bias Monte Carlo SE Model-based SE rMSE Coverage
M1 M2 M3 M4 M1 M2 M3 M4 M1 M2 M3 M4 M1 M2 M3 M4 M1 M2 M3 M4
A1 0.0652 6.2% 18.3% 6.2% 5.6% 0.033 0.033 0.034 0.035 0.034 0.034 0.033 0.035 0.034 0.035 0.034 0.035 92.4% 89.0% 92.2% 91.0%
A2 −0.0332 13.9% 37.6% 8.2% 7.0% 0.036 0.035 0.037 0.035 0.037 0.036 0.037 0.038 0.036 0.037 0.037 0.035 93.2% 90.8% 92.8% 93.4%
A3 −0.0092 15.1% 13.8% 14.8% 2.7% 0.027 0.025 0.027 0.029 0.027 0.025 0.027 0.029 0.028 0.026 0.027 0.029 92.4% 92.2% 92.0% 93.6%
B. Log-Normal Outcome
1) Poisson Regression
Variable True Value % Bias Monte Carlo SE Model-based SE rMSE Coverage
M1 M2 M3 M4 M1 M2 M3 M4 M1 M2 M3 M4 M1 M2 M3 M4 M1 M2 M3 M4
A1 0.0883 1.8% 25.3% 1.1% 1.1% 0.023 0.023 0.023 0.023 0.023 0.023 0.023 0.023 0.023 0.032 0.023 0.023 93.6% 84.8% 93.6% 93.6%
A2 −0.0445 2.4% 44.9% 1.6% 1.4% 0.025 0.025 0.025 0.025 0.026 0.026 0.026 0.026 0.025 0.032 0.025 0.025 95.4% 90.0% 95.2% 95.2%
A3 −0.0119 24.1% 2.5% 2.8% 3.0% 0.018 0.018 0.018 0.018 0.018 0.018 0.018 0.018 0.018 0.018 0.018 0.018 93.6% 95.0% 94.8% 94.8%
2) Negative Binomial Regression
Variable True Value % Bias Monte Carlo SE Model-based SE rMSE Coverage
M1 M2 M3 M4 M1 M2 M3 M4 M1 M2 M3 M4 M1 M2 M3 M4 M1 M2 M3 M4
A1 0.0883 1.8% 25.2% 1.1% 1.0% 0.023 0.023 0.023 0.023 0.023 0.023 0.023 0.023 0.023 0.032 0.023 0.023 93.8% 84.2% 93.4% 93.6%
A2 −0.0445 2.7% 45.4% 1.4% 1.2% 0.025 0.025 0.025 0.025 0.026 0.026 0.026 0.026 0.025 0.032 0.025 0.025 96.0% 90.0% 95.6% 95.6%
A3 −0.0119 24.9% 3.9% 3.3% 3.5% 0.018 0.018 0.018 0.018 0.018 0.018 0.018 0.018 0.019 0.018 0.018 0.018 93.0% 94.0% 94.8% 95.0%
3) Zero-Inflated Poisson Regression
Variable True Value % Bias Monte Carlo SE Model-based SE rMSE Coverage
M1 M2 M3 M4 M1 M2 M3 M4 M1 M2 M3 M4 M1 M2 M3 M4 M1 M2 M3 M4
A1 0.0883 0.3% 26.6% 0.5% 0.6% 0.024 0.024 0.024 0.024 0.023 0.023 0.023 0.023 0.024 0.034 0.024 0.024 92.6% 82.6% 92.6% 92.4%
A2 −0.0445 7.1% 49.3% 3.3% 3.5% 0.026 0.026 0.026 0.026 0.026 0.026 0.026 0.026 0.026 0.034 0.026 0.026 95.4% 85.8% 95.4% 95.4%
A3 −0.0119 27.1% 5.7% 6.1% 6.3% 0.018 0.018 0.018 0.018 0.018 0.018 0.018 0.018 0.018 0.018 0.018 0.018 94.8% 95.4% 95.6% 95.6%
C. Mixture Outcome (Zero-Inflated Poisson)
1) Poisson Regression
Variable True Value % Bias Monte Carlo SE Model-based SE rMSE Coverage
M1 M2 M3 M4 M1 M2 M3 M4 M1 M2 M3 M4 M1 M2 M3 M4 M1 M2 M3 M4
A1 0.0376 1.2% 232.0% 3.8% 13.8% 0.035 0.039 0.034 0.032 0.038 0.038 0.039 0.038 0.035 0.095 0.034 0.033 96.2% 35.8% 96.8% 97.2%
A2 −0.0072 224.0% 1098.0% 37.8% 83.2% 0.04 0.043 0.039 0.037 0.042 0.041 0.043 0.042 0.043 0.09 0.039 0.038 95.2% 52.4% 97.4% 96.8%
A3 −0.0473 40.9% 7.8% 3.3% 7.3% 0.029 0.028 0.025 0.024 0.029 0.029 0.03 0.029 0.035 0.029 0.025 0.024 90.4% 94.4% 98.8% 99.4%
2) Negative Binomial Regression
Variable True Value % Bias Monte Carlo SE Model-based SE rMSE Coverage
M1 M2 M3 M4 M1 M2 M3 M4 M1 M2 M3 M4 M1 M2 M3 M4 M1 M2 M3 M4
A1 0.0376 5.5% 278.0% 9.6% 21.2% 0.039 0.04 0.036 0.036 0.04 0.039 0.039 0.04 0.039 0.112 0.037 0.037 95.0% 23.2% 96.6% 96.0%
A2 −0.0071 285.0% 1428.0% 37.7% 143.0% 0.042 0.044 0.04 0.039 0.044 0.043 0.043 0.044 0.047 0.111 0.040 0.040 94.4% 34.6% 96.4% 97.4%
A3 −0.0473 52.4% 47.5% 6.7% 16.3% 0.032 0.03 0.026 0.025 0.031 0.03 0.03 0.031 0.04 0.038 0.026 0.026 87.2% 89.4% 97.8% 97.6%
3) Zero-Inflated Poisson Regression
Variable True Value % Bias Monte Carlo SE Model-based SE rMSE Coverage
M1 M2 M3 M4 M1 Value M3 M4 M1 M2 M3 M4 M1 M2 M3 M4 M1 M2 M3 M4
A1 0.0890 2.0% 45.5% 0.1% 0.2% 0.03 0.026 0.031 0.03 0.031 0.027 0.031 0.031 0.03 0.048 0.031 0.03 95.0% 67.8% 94.4% 95.6%
A2 −0.0543 9.8% 69.2% 0.5% 0.8% 0.034 0.03 0.035 0.034 0.034 0.03 0.035 0.034 0.034 0.048 0.035 0.034 95.2% 78.8% 95.2% 95.6%
A3 −0.0041 228.0% 160.0% 30.6% 49.9% 0.025 0.023 0.025 0.025 0.024 0.021 0.024 0.024 0.027 0.024 0.025 0.025 91.6% 91.4% 94.2% 94.6%

Model 1: Regression adjusting for time-varying treatment (A1–A3) and baseline covariate only (L1).

Model 2: Regression adjusting for time-varying treatment (A1–A3) and all time-varying covariates (L1–L3).

Model 3: Marginal structural model weights added in the regression adjusting for time-varying treatment only (A1–A3).

Model 4: Marginal structural model weights added in the regression adjusting for time-varying treatment and baseline covariates (A1–A3 and L1).