BMC Medical Research Methodology

Table 2 Estimated required number of clusters for HTE test by the direct inflation (\({n}_{0}\)), the proposed formula under MCAR (\({n}_{1}\)), and the proposed procedure under MAR (\({n}_{2}\)), the empirical type I error rate of the Wald test for HTE (\(\psi\)), and predicted power (\({\varphi }_{pre}^{,MAR}\)) and empirical power (\({\varphi }_{emp}^{MAR}\)) of the HTE test with a continuous individual-level effect modifier under MAR. The effect size is \({\updelta }=\{0.1, 0.25\}\). The missingness ICC is \(\tau =0.05\)

From: Accounting for expected attrition in the planning of cluster randomized trials for assessing treatment effect heterogeneity

				\(\delta =0.10\)						\(\delta =0.25\)
\(m\)	\({\rho }_{x}\)	\({\rho }_{y\|x}\)	\(\pi\)	\({n}_{0}\)	\({n}_{1}\)	\({n}_{2}\)	\({\varphi }_{pre}^{MAR}\)	\({\varphi }_{emp}^{MAR}\)	\(\psi\)	\({n}_{0}\)	\({n}_{1}\)	\({n}_{2}\)	\({\varphi }_{pre}^{MAR}\)	\({\varphi }_{emp}^{MAR}\)	\(\psi\)
20	0.1	0.01	0.7	228	228	238	0.802	0.807	0.053	38	38	38	0.802	0.793	0.050
		0.01	0.9	178	178	182	0.803	0.804	0.051	30	30	30	0.814	0.809	0.052
		0.10	0.7	226	226	234	0.801	0.781	0.052	36	36	38	0.807	0.808	0.051
		0.10	0.9	176	176	178	0.800	0.794	0.048	28	28	30	0.820	0.816	0.051
	0.5	0.01	0.7	244	240	250	0.803	0.799	0.059	40	40	40	0.802	0.765	0.043
		0.01	0.9	190	190	194	0.805	0.791	0.050	32	32	32	0.815	0.796	0.056
		0.10	0.7	318	302	312	0.802	0.796	0.057	52	50	50	0.802	0.800	0.048
		0.10	0.9	248	244	246	0.801	0.792	0.048	40	40	40	0.807	0.792	0.053
50	0.1	0.01	0.7	94	92	96	0.800	0.789	0.048	16	16	16	0.816	0.812	0.056
		0.01	0.9	72	72	74	0.804	0.815	0.056	12	12	12	0.808	0.802	0.042
		0.10	0.7	90	90	94	0.802	0.791	0.048	16	16	16	0.826	0.831	0.051
		0.10	0.9	70	70	72	0.804	0.814	0.044	12	12	12	0.820	0.813	0.051
	0.5	0.01	0.7	108	104	108	0.804	0.807	0.045	18	18	18	0.821	0.785	0.057
		0.01	0.9	84	84	84	0.802	0.795	0.048	14	14	14	0.818	0.760	0.051
		0.10	0.7	144	138	142	0.804	0.798	0.052	24	22	24	0.826	0.815	0.058
		0.10	0.9	112	110	112	0.804	0.808	0.053	18	18	18	0.806	0.807	0.053
100	0.1	0.01	0.7	48	48	50	0.812	0.811	0.049	8	8	8	0.811	0.798	0.051
		0.01	0.9	38	38	38	0.809	0.801	0.055	6	6	6	0.803	0.800	0.051
		0.10	0.7	46	46	48	0.810	0.809	0.050	8	8	8	0.826	0.812	0.052
		0.10	0.9	36	36	36	0.805	0.812	0.047	6	6	6	0.820	0.824	0.047
	0.5	0.01	0.7	60	58	60	0.809	0.809	0.048	10	10	10	0.826	0.791	0.057
		0.01	0.9	48	46	48	0.813	0.805	0.057	8	8	8	0.828	0.784	0.062
		0.10	0.7	76	74	76	0.804	0.814	0.048	14	12	14	0.856	0.851	0.053
		0.10	0.9	60	60	60	0.808	0.816	0.047	10	10	10	0.824	0.814	0.057

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