Required sample size to detect mediation in 3-level implementation studies

Table 1 Frequency of study designs with statistical power ≥ 0.8 by study characteristic (N = 17,496 designs)

Study characteristic	Parameter value	Total N of designs	N of adequately powered designs (≥ .8)	Proportion of adequately powered designs (% ≥.8)
Total		17,496	463	2.6%
a₃ (standardized X→M coefficient)	0.14	5832	0	0.0%
	0.39	5832	46	0.8%
	0.59	5832	417	7.2%
b₃ (standardized M→Y coefficient)	0.14	5832	0	0.0%
	0.39	5832	32	0.5%
	0.59	5832	431	7.4%
c’₃ (standardized direct effect)	0.14	8748	161	1.8%
c’₃ (standardized direct effect)	0.39	8748	302	3.5%
ICC_M3 (level-3 intraclass correlation coefficient for mediator M)	0.05	5832	36	0.6%
	0.10	5832	125	2.1%
	0.20	5832	302	5.2%
ICC_Y2 (level-2 intraclass correlation coefficient for outcome Y)	0.05	5832	136	2.3%
	0.10	5832	148	2.5%
	0.20	5832	179	3.1%
ICC_Y3 (level-3 intraclass correlation coefficient for outcome Y)	0.05	5832	140	2.4%
	0.10	5832	161	2.8%
	0.20	5832	162	2.8%
N₃ (level-3 sample size; N of highest-level units/clusters, e.g., organizations)	10	4374	0	0.0%
	20	4374	0	0.0%
	40	4374	110	2.5%
	60	4374	353	8.1%
N₂ (level-2 sample size; N of nested intermediate-level units per cluster, e.g., providers)	5	5832	42	0.7%
	10	5832	129	2.2%
	20	5832	292	5.0%
N₁ (level-1 sample size; N of nested lowest-level units per intermediate unit, e.g., patients)	3	5832	131	2.2%
	6	5832	159	2.7%
	12	5832	173	3.0%
Total sample size (N₃* N₂* N₁)	150	486	0	0.0%
	300	1458	0	0.0%
	600	2916	0	0.0%
	900	486	9	1.9%
	1200	3402	8	0.2%
	1800	972	43	4.4%
	2400	2916	33	1.1%
	3600	1458	112	7.7%
	4800	1458	39	2.7%
	7200	972	116	11.9%
	9600	486	30	6.2%
	14,400	486	73	15.0%

Note: Power was calculated for N = 17,496 designs based on Monte Carlo simulations (500 replications per design) conducted in Mplus 8. All models represent 3-2-1 mediation designs estimated using maximum likelihood with robust standard errors based on a linear multilevel model with manifest variables (MVM). For each design, power was calculated as the proportion of replications (out of 500) for which the null hypothesis, H₀: a₃*b₃ = 0, was rejected based on the Sobel test, assuming α = 0.05 (two-tailed)