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

Table 2 Minimum sample sizes required for adequate statistical power to detect mediation

ICC_m3	ICC_y3	Standardized effect sizes for a₃ path (X→M) and b₃ path (M→Y)
ICC_m3	ICC_y3	SS	SM	SL	MS	MM	ML	LS	LM	LL
S	S	-	-	-	-	-	-	-	-	c_s: 60 [20 [6]]
S	S	-	-	-	-	-	-	-	-	c_m: 60 [20 [3]]
S	M	-	-	-	-	-	-	-	-	c_s: 60[20[6]]
S	M	-	-	-	-	-	-	-	-	c_m: 60[20[3]]
S	L	-	-	-	-	-	-	-	-	c_s: 60[20[6]]
S	L	-	-	-	-	-	-	-	-	c_m: 60[20[3]]
M	S	-	-	-	-	-	-	-	c_s: -	c_s: 40[20[12]]
M	S	-	-	-	-	-	-	-	c_m: 60[20[6]]	c_m: 40[20[3]]
M	M	-	-	-	-	-	-	-	c_s: -	c_s: 40[20[6]]
M	M	-	-	-	-	-	-	-	c_m: 60[20[6]]	c_m: 40[20[3]]
M	L	-	-	-	-	-	-	-	-	c_s: 40[20[12]]
M	L	-	-	-	-	-	-	-	-	c_m: 40[20[3]]
L	S	-	-	-	-	-	c_s: 60[20[3]]	-	c_s: -	c_s: 40[20[3]]
L	S	-	-	-	-	-	c_m: 60[20[3]]	-	c_m: 60[10[12]]	c_m: 40[10[3]]
L	M	-	-	-	-	-	c_s: 60[20[3]]	-	c_s: -	c_s: 40[10[12]]
L	M	-	-	-	-	-	c_m: 60[10[12]]	-	c_m: 60[10[6]]	c_m: 40[10[3]]
L	L	-	-	-	-	-	c_s: 60[20[3]]	-	c_s: -	c_s: 40[10[12]]
L	L	-	-	-	-	-	c_m: 60[10[12]]	-	c_m: 60[10[6]]	c_m: 40[5[6]]

Note: Sample sizes shown are the smallest sample size required to achieve statistical power ≥ 0.8 to reject the null hypothesis a₃*b₃ = 0 given the design parameters shown. Within each cell, two sample sizes are provided, one assuming a small direct effect (c_s) and the other assuming a medium direct effect (c_m). Sample sizes are presented as N₃[N₂[N₁]] where N₃ = number of highest-level clusters (level 3), N₂ = number of intermediate nested units (level 2) per cluster, and N₁ = number of lowest-level nested observations (level 1) per level-2 unit. Blank cells (-) indicate there were no sample sizes that achieved adequate power for that design. Required sample sizes were generated using linear multilevel modeling with manifest variables assuming α=0.05 (2-tailed). ICC_m3 level-3 intraclass correlation coefficient of the mediator, ICC_y3 level-3 intraclass correlation coefficient of the outcome. ICCs were evaluated at S=0.05, M=.1, and L=.2. Standardized effect sizes indicate the size of the a₃ path followed by the size of the b₃ path, where S=.14, M=.39, and L=.59

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