Accounting for confounding by time, early intervention adoption, and time-varying effect modification in the design and analysis of stepped-wedge designs: application to a proposed study design to reduce opioid-related mortality

Rennert, Lior; Heo, Moonseong; Litwin, Alain H.; Gruttola, Victor De

doi:10.1186/s12874-021-01229-6

Table 3 Simulation Results

From: Accounting for confounding by time, early intervention adoption, and time-varying effect modification in the design and analysis of stepped-wedge designs: application to a proposed study design to reduce opioid-related mortality

	Time effects				Scenario 1						Scenario 2.1						Scenario 2.2
Model	Main		Interaction		Standard						Confounding (+)						Confounding (-)
index	fixed	random	fixed	random	%bias	SD	SE	cov	pwr	α	%bias	SD	SE	cov	pwr	α	%bias	SD	SE	cov	pwr	α
1	disc				0.1	0.09	0.08	0.94	1	0.05	-0.7	0.11	0.08	0.83	1	0.22	0.1	0.11	0.09	0.91	1	0.11
2	disc	H/G			0.1	0.09	0.08	0.94	1	0.05	-0.8	0.11	0.09	0.92	1	0.08	0.1	0.11	0.1	0.93	1	0.07
3	disc	UN			0.5	0.1	0.11	0.96	1	0.05	-0.2	0.12	0.12	0.95	0.99	0.06	1.3	0.13	0.12	0.94	0.98	0.05
4	disc	lin			0.1	0.09	0.08	0.94	1	0.06	-0.3	0.11	0.08	0.82	1	0.22	0.1	0.11	0.09	0.89	1	0.11
5	disc	H/G	lin	H/G	0.8	0.23	0.24	0.95	0.6	0.06	0.6	0.29	0.26	0.92	0.52	0.07	6.1	0.28	0.26	0.93	0.55	0.08
6	disc	UN	lin	UN	0.3	0.27	0.31	0.97	0.38	0.03	0.7	0.32	0.33	0.95	0.34	0.05	8	0.33	0.35	0.97	0.35	0.04
7	disc	lin	lin	lin	1.3	0.24	0.24	0.96	0.59	0.06	1.8	0.29	0.23	0.86	0.58	0.15	6.3	0.28	0.26	0.93	0.57	0.1
8	lin	H/G	lin	H/G	-0.7	0.14	0.14	0.95	0.95	0.05	-6.2	0.17	0.16	0.9	0.84	0.11	4.9	0.17	0.16	0.93	0.91	0.08
9	lin	UN	lin	UN	-1.6	0.16	0.17	0.97	0.84	0.04	-4.9	0.19	0.19	0.94	0.75	0.06	3.2	0.19	0.21	0.95	0.77	0.06
10	lin	lin	lin	lin	-1	0.14	0.14	0.94	0.95	0.06	-10.8	0.18	0.13	0.81	0.87	0.24	2.5	0.17	0.15	0.91	0.92	0.12
	Time effects				Scenario 3						Scenario 4.1						Scenario 4.2
Model	Main		Interaction		Early adoption						Confounding (+)& Early adoption						Confounding (-)& Early adoption
index	fixed	random	fixed	random	%bias	SD	SE	cov	pwr	α	%bias	SD	SE	cov	pwr	α	%bias	SD	SE	cov	pwr	α
1	disc				-33.9	0.09	0.09	0.48	0.96	0.04	-34	0.12	0.08	0.45	0.93	0.21	-33.3	0.12	0.09	0.52	0.92	0.1
2	disc	H/G			-33.9	0.09	0.09	0.49	0.96	0.04	-34.3	0.11	0.1	0.57	0.89	0.1	-33.4	0.12	0.1	0.58	0.89	0.07
3	disc	UN			-33.6	0.11	0.11	0.68	0.85	0.03	-33.1	0.12	0.12	0.71	0.77	0.07	-32	0.14	0.13	0.73	0.74	0.05
4	disc	lin			-34.3	0.09	0.08	0.44	0.97	0.05	-34.1	0.12	0.08	0.45	0.94	0.21	-33.4	0.11	0.09	0.51	0.93	0.12
5	disc	H/G	lin	H/G	-3.4	0.26	0.24	0.93	0.53	0.06	-3.5	0.31	0.27	0.91	0.45	0.09	-0.9	0.3	0.27	0.93	0.47	0.08
6	disc	UN	lin	UN	0.1	0.3	0.33	0.96	0.33	0.03	1.1	0.34	0.35	0.94	0.33	0.05	2.5	0.33	0.36	0.96	0.28	0.04
7	disc	lin	lin	lin	-2.1	0.27	0.25	0.93	0.54	0.06	-0.5	0.32	0.24	0.86	0.56	0.17	0	0.3	0.26	0.92	0.5	0.08
8	lin	H/G	lin	H/G	-1.3	0.14	0.14	0.95	0.94	0.06	-7.4	0.18	0.16	0.91	0.81	0.09	1.6	0.18	0.16	0.93	0.87	0.08
9	lin	UN	lin	UN	-0.5	0.16	0.18	0.96	0.84	0.04	-5.2	0.19	0.2	0.94	0.72	0.06	1.7	0.2	0.2	0.96	0.74	0.05
10	lin	lin	lin	lin	-1.2	0.14	0.14	0.94	0.95	0.05	-11.7	0.19	0.13	0.8	0.85	0.22	-0.7	0.18	0.15	0.9	0.9	0.11

% bias \(\left (\frac {\widehat {\theta }-\theta }{\theta }\right)\), standard deviation (SD), estimated standard error (SE), coverage rate of 95% confidence interval (cov), power (pwr), and Type 1 error (α) of estimated intervention effect. Rows correspond to fitting models 1 through 10 under the scenarios described in Table 2; +/- imply positive/negative effect of confounding factors on outcome. Models include discrete (disc) or linear (lin) fixed time effects for all clusters (main effect) and/or control clusters (interaction effect). H/G, UN, and lin indicate Hooper/Girling, unstructured, and linear random effect structure for time effects

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