Table 1.

Results of Simulation Studies to Compare Sequential Conditional Mean Models with Inverse Probability Weighted Estimation of Marginal Structural Models

Modela	Independence			Unstructured
	Biasb	95% CIc	SDd	Biasb	95% CIc	SDd
SCMM
Form of $E\left(Y_t|{\overline{X}}_t,{\overline{Y}}_{t-1}\right)$
i) ${\mathrm{\beta }}_0+{\mathrm{\beta }}_{X1}{X}_t$	0.425	0.420, 0.430	0.081	0.256	0.251, 0.262	0.087
ii) ${\mathrm{\beta }}_0+{\mathrm{\beta }}_{X1}{X}_t+{\mathrm{\beta }}_Y{Y}_{t-1}$	0.151	0.146, 0.156	0.080	0.050	0.045, 0.055	0.086
iii) ${\mathrm{\beta }}_0+{\mathrm{\beta }}_{X1}{X}_t+{\mathrm{\beta }}_{X2}{X}_{t-1}$	0.115	0.109, 0.120	0.092	−0.002	0.008, 0.004	0.095
iv) ${\mathrm{\beta }}_0+{\mathrm{\beta }}_{X1}{X}_t+{\mathrm{\beta }}_{X2}{X}_{t-1}+{\mathrm{\beta }}_Y{Y}_{t-1}$	−0.001	−0.007, 0.005	0.095	0.001	−0.004, 0.007	0.095
SCMM using propensity scores
Form of $E\left(Y_t|{\overline{X}}_t,{\overline{Y}}_{t-1}\right)$
i) ${\mathrm{\beta }}_0+{\mathrm{\beta }}_{X1}{X}_t+{\mathrm{\beta }}_{PS}{\widehat{PS}}_t$	0.001	−0.005, 0.007	0.096	0.001	−0.005, 0.007	0.095
ii) ${\mathrm{\beta }}_0+{\mathrm{\beta }}_{X1}{X}_t+{\mathrm{\beta }}_Y{Y}_{t-1}+{\mathrm{\beta }}_{PS}{\widehat{PS}}_t$	0.001	−0.005, 0.007	0.096	0.006	0.000, 0.012	0.097
iii) ${\mathrm{\beta }}_0+{\mathrm{\beta }}_{X1}{X}_t+{\mathrm{\beta }}_{X2}{X}_{t-1}+{\mathrm{\beta }}_{PS}{\widehat{PS}}_t$	0.003	−0.002, 0.009	0.096	−0.002	−0.008, 0.004	0.095
iv) ${\mathrm{\beta }}_0+{\mathrm{\beta }}_{X1}{X}_t+{\mathrm{\beta }}_{X2}{X}_{t-1}+{\mathrm{\beta }}_Y{Y}_{t-1}+{\mathrm{\beta }}_{PS}{\widehat{PS}}_t$	−0.001	−0.007, 0.005	0.096	0.001	−0.005, 0.007	0.096
IPW estimation of MSMs
Unstabilized weights
i) $E\left(Y_t^{x_t}\right)={\mathrm{\omega }}_0^{⁎}+{\mathrm{\omega }}_{X1}^{⁎}{x}_t$	0.022	0.001, 0.043	0.340	0.046	−0.137, 0.230	2.959
ii) $E\left(Y_t^{{\overline{x}}_t}\right)={\mathrm{\omega }}_0+{\mathrm{\omega }}_{X1}{x}_t+{\mathrm{\omega }}_{X2}{x}_{t-1}$	0.007	−0.012, 0.026	0.306	3.635	−3.208, 10.478	110.4
Stabilized weights
i) $E\left(Y_t^{x_t}\right)={\mathrm{\omega }}_0^{⁎}+{\mathrm{\omega }}_{X1}^{⁎}{x}_t$	0.297	0.291, 0.302	0.090	0.187	0.180, 0.194	0.110
ii) $E\left(Y_t^{{\overline{x}}_t}\right)={\mathrm{\omega }}_0+{\mathrm{\omega }}_{X1}{x}_t+{\mathrm{\omega }}_{X2}{x}_{t-1}$	−0.002	−0.009, 0.004	0.107	−0.060	−0.067, −0.053	0.114
Stabilized weights: truncated at the 1st and 99th percentiles
i) $E\left(Y_t^{x_t}\right)={\mathrm{\omega }}_0^{⁎}+{\mathrm{\omega }}_{X1}^{⁎}{x}_t$	0.309	0.304, 0.315	0.087	0.196	0.190, 0.202	0.098
ii) $E\left(Y_t^{{\overline{x}}_t}\right)={\mathrm{\omega }}_0+{\mathrm{\omega }}_{X1}{x}_t+{\mathrm{\omega }}_{X2}{x}_{t-1}$	0.018	0.012, 0.024	0.101	−0.051	−0.058, −0.045	0.106
Stabilized weights: truncated at the 5th and 95th percentiles
i) $E\left(Y_t^{x_t}\right)={\mathrm{\omega }}_0^{⁎}+{\mathrm{\omega }}_{X1}^{⁎}{x}_t$	0.325	0.320, 0.330	0.086	0.214	0.209, 0.220	0.092
ii) $E\left(Y_t^{{\overline{x}}_t}\right)={\mathrm{\omega }}_0+{\mathrm{\omega }}_{X1}{x}_t+{\mathrm{\omega }}_{X2}{x}_{t-1}$	0.025	0.019, 0.032	0.099	−0.043	−0.049, −0.037	0.102
Stabilized weights: truncated at the 10th and 90th percentiles
i) $E\left(Y_t^{x_t}\right)={\mathrm{\omega }}_0^{⁎}+{\mathrm{\omega }}_{X1}^{⁎}{x}_t$	0.341	0.335, 0.346	0.085	0.225	0.219, 0.230	0.091
ii) $E\left(Y_t^{{\overline{x}}_t}\right)={\mathrm{\omega }}_0+{\mathrm{\omega }}_{X1}{x}_t+{\mathrm{\omega }}_{X2}{x}_{t-1}$	0.044	0.038, 0.050	0.097	−0.032	−0.039, −0.026	0.100
Stabilized weights: truncated at the 20th and 80th percentiles
i) $E\left(Y_t^{x_t}\right)={\mathrm{\omega }}_0^{⁎}+{\mathrm{\omega }}_{X1}^{⁎}{x}_t$	0.364	0.359, 0.370	0.083	0.236	0.231, 0.242	0.088
ii) $E\left(Y_t^{{\overline{x}}_t}\right)={\mathrm{\omega }}_0+{\mathrm{\omega }}_{X1}{x}_t+{\mathrm{\omega }}_{X2}{x}_{t-1}$	0.067	0.061, 0.073	0.094	−0.021	−0.027, −0.015	0.097

Abbreviations: CI, confidence interval; GEE, generalized estimating equation; IPW, inverse probability weight; MSM, marginal structural model; SCMM, sequential conditional mean model; SD, standard deviation.

^a All models were fitted using GEEs with an independence working correlation matrix and an unstructured working correlation matrix.

^b Bias in the estimated short-term causal effect of $X_t$ on $Y_t$ averaged over 1,000 simulations.

^c Monte Carlo 95% confidence interval corresponding to the bias.

^d Empirical standard deviation of the estimates.