Robust variance estimation for covariate-adjusted unconditional treatment effect in randomized clinical trials with binary outcomes

Figures & data

Table 1. Simulation mean and standard deviation (SD) of ${\hat{\theta }}_2-{\hat{\theta }}_1$, average standard error (SE), and coverage probability (CP) of 95% asymptotic confidence interval for ${\theta }_2-{\theta }_1$ under Cases I–II and simple randomization.

			${\hat{\theta }}_2-{\hat{\theta }}_1$		Robust SE in (1(1) $\begin{array}{r}{n}^{-1}\left\{\mathrm{\nabla }f\right(\hat{\mathit{\theta }}\left){\}}^{\mathrm{\top }}\hat{\mathit{V}}\right\{\mathrm{\nabla }f\left(\hat{\mathit{\theta }}\right)\},\hat{\mathit{V}}=\left(\begin{array}{cccc}{\hat{v}}_{11}& {\hat{v}}_{12}& \dots & {\hat{v}}_{1k}\\ ⋮& ⋮& \ddots & ⋮\\ {\hat{v}}_{1k}& {\hat{v}}_{2k}& \dots & {\hat{v}}_{kk}\end{array}\right),\end{array}$(1) )		SE in Ge et al. (2011)
Case	${\theta }_2-{\theta }_1$	n	Mean	SD	SE	CP (%)	SE	CP (%)
I	0.5227	200	0.5228	0.0464	0.0464	94.44	0.0415	91.27
		500	0.5227	0.0295	0.0294	94.70	0.0264	91.94
II	0.4467	200	0.4469	0.0457	0.0458	94.56	0.0404	91.08
		500	0.4463	0.0289	0.0290	94.90	0.0257	91.77

Table 2. Simulation mean and standard deviation (SD) of g-computation estimators, average standard error (SE), and coverage probability (CP) of 95% asymptotic confidence interval based on robust SE (1(1) $\begin{array}{r}{n}^{-1}\left\{\mathrm{\nabla }f\right(\hat{\mathit{\theta }}\left){\}}^{\mathrm{\top }}\hat{\mathit{V}}\right\{\mathrm{\nabla }f\left(\hat{\mathit{\theta }}\right)\},\hat{\mathit{V}}=\left(\begin{array}{cccc}{\hat{v}}_{11}& {\hat{v}}_{12}& \dots & {\hat{v}}_{1k}\\ ⋮& ⋮& \ddots & ⋮\\ {\hat{v}}_{1k}& {\hat{v}}_{2k}& \dots & {\hat{v}}_{kk}\end{array}\right),\end{array}$(1) ) under Case III and simple randomization.

		n = 200				n = 500
Parameter	Truth	Mean	SD	SE	CP	Mean	SD	SE	CP
${\theta }_2-{\theta }_1$	0.2177	0.2176	0.0578	0.0573	94.34	0.2170	0.0366	0.0363	94.82
$\log \left({\theta }_2/{\theta }_1\right)$	0.5711	0.5798	0.1701	0.1664	94.50	0.5726	0.1053	0.1042	94.59
$\log \frac{{\theta }_2/(1-{\theta }_2)}{{\theta }_1/(1-{\theta }_1)}$	0.9328	0.9440	0.2620	0.2586	94.63	0.9341	0.1637	0.1624	94.79
${\theta }_3-{\theta }_1$	0.4346	0.4348	0.0581	0.0568	94.15	0.4347	0.0360	0.0360	94.84
$\log \left({\theta }_3/{\theta }_1\right)$	0.9311	0.9432	0.1653	0.1611	94.43	0.9353	0.1018	0.1009	94.92
$\log \frac{{\theta }_3/(1-{\theta }_3)}{{\theta }_1/(1-{\theta }_1)}$	1.8621	1.8852	0.2920	0.2851	94.57	1.8712	0.1791	0.1788	95.01

Table 3. Simulation mean and standard deviation (SD) of ${\hat{\theta }}_2-{\hat{\theta }}_1$, average standard error (SE) and coverage probability (CP) of 95% asymptotic confidence interval for ${\theta }_2-{\theta }_1$ under Cases I–II and complete randomization that fixes $n_t=n{\pi }_t$.

			${\hat{\theta }}_2-{\hat{\theta }}_1$		Robust SE in (1(1) $\begin{array}{r}{n}^{-1}\left\{\mathrm{\nabla }f\right(\hat{\mathit{\theta }}\left){\}}^{\mathrm{\top }}\hat{\mathit{V}}\right\{\mathrm{\nabla }f\left(\hat{\mathit{\theta }}\right)\},\hat{\mathit{V}}=\left(\begin{array}{cccc}{\hat{v}}_{11}& {\hat{v}}_{12}& \dots & {\hat{v}}_{1k}\\ ⋮& ⋮& \ddots & ⋮\\ {\hat{v}}_{1k}& {\hat{v}}_{2k}& \dots & {\hat{v}}_{kk}\end{array}\right),\end{array}$(1) )		SE in Ge et al. (2011)
Case	${\theta }_2-{\theta }_1$	n	Mean	SD	SE	CP (%)	SE	CP (%)
I	0.5227	200	0.5231	0.0464	0.0462	94.62	0.0414	91.54
	500	0.5230	0.0298	0.0294	94.67	0.0264	91.70
II	0.4467	200	0.4469	0.0457	0.0456	94.68	0.0403	91.29
	500	0.4471	0.0290	0.0290	94.74	0.0257	91.45

Table 4. Simulation mean and standard deviation (SD) of g-computation estimators, average standard error (SE), and coverage probability (CP) of 95% asymptotic confidence interval based on robust SE (1(1) $\begin{array}{r}{n}^{-1}\left\{\mathrm{\nabla }f\right(\hat{\mathit{\theta }}\left){\}}^{\mathrm{\top }}\hat{\mathit{V}}\right\{\mathrm{\nabla }f\left(\hat{\mathit{\theta }}\right)\},\hat{\mathit{V}}=\left(\begin{array}{cccc}{\hat{v}}_{11}& {\hat{v}}_{12}& \dots & {\hat{v}}_{1k}\\ ⋮& ⋮& \ddots & ⋮\\ {\hat{v}}_{1k}& {\hat{v}}_{2k}& \dots & {\hat{v}}_{kk}\end{array}\right),\end{array}$(1) ) under Case III and complete randomization that fixes $n_t=n{\pi }_t$.

		n = 200				n = 500
Parameter	Truth	Mean	SD	SE	CP	Mean	SD	SE	CP
${\theta }_2-{\theta }_1$	0.2177	0.2177	0.0580	0.0570	94.25	0.2182	0.0364	0.0362	94.61
$\log \left({\theta }_2/{\theta }_1\right)$	0.5711	0.5790	0.1687	0.1652	94.60	0.5755	0.1046	0.1040	95.02
$\log \frac{{\theta }_2/(1-{\theta }_2)}{{\theta }_1/(1-{\theta }_1)}$	0.9328	0.9437	0.2616	0.2567	94.78	0.9391	0.1627	0.1621	94.73
${\theta }_3-{\theta }_1$	0.4346	0.4349	0.0579	0.0567	93.96	0.4354	0.0364	0.0360	94.23
$\log \left({\theta }_3/{\theta }_1\right)$	0.9311	0.9424	0.1640	0.1602	94.02	0.9371	0.1017	0.1008	94.99
$\log \frac{{\theta }_3/(1-{\theta }_3)}{{\theta }_1/(1-{\theta }_1)}$	1.8621	1.8853	0.2910	0.2844	94.31	1.8751	0.1811	0.1788	94.48

Ge, M., Durham, L. K., Meyer, R. D., Xie, W., & Thomas, N. (2011). Covariate-adjusted difference in proportions from clinical trials using logistic regression and weighted risk differences. Drug Information Journal: DIJ/Drug Information Association, 45(4), 481–493. https://doi.org/10.1177/009286151104500409

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