Experimental Design and Power for Moderation in Multisite Cluster Randomized Trials

Figures & data

Figure 1. A conceptual framework for investigating moderation effects of professional development.

Table 1. List of design and software modules of three-level multisite cluster randomized trials (MCRTs).

1	2	3	4	5	6	7
Model Number	Level of Moderator	Slope of Treatment or Moderation	Binary Moderator		Continuous Moderator
			MDESD Calculation	Power Calculation	MDESD Calculation	Power Calculation
MRT3-2R-1	1	Random
MRT3-2R-2	2	Random	MRT32R_MDESD	MRT32R_Power	MRT32Rc_MDESD	MRT32Rc_Power
MRT3-2R-3	3	Random
MRT3-2N-1	1	Nonrandomly Varying
MRT3-2N-2	2	Nonrandomly Varying	MRT32N_MDESD	MRT32N_Power	MRT32Nc_MDESD	MRT32Nc_Power
MRT3-2N-3	3	Nonrandomly Varying

Table 2. Summary of standardized noncentrality parameters, MDESD and 100*(1-α)% confidence intervals for three-level MCRTs.

Model Number	HLM	Standardized Noncentrality Parameter (λ)	MDESD and 100*(1-α)% Confidence Interval	Degree of Freedom (v)
MCRT3-2R-1	L1: $Y_{\mathit{\text{ijk}}}={\pi }_{0\mathit{\text{jk}}}+{\pi }_{1\mathit{\text{jk}}}{M}_{\mathit{\text{ijk}}}^{(1)}+{\pi }_{2\mathit{\text{jk}}}{X}_{\mathit{\text{ijk}}}+e_{\mathit{\text{ijk}}},$ $e_{\mathit{\text{ijk}}}\sim N(0,{\sigma }_{1|M,X}^2).$ L2: ${\pi }_{0\mathit{\text{jk}}}={\beta }_{00k}+{\beta }_{01k}{T}_{\mathit{\text{jk}}}+{\beta }_{02k}{W}_{\mathit{\text{jk}}}+r_{0\mathit{\text{jk}}}$ ${\pi }_{1\mathit{\text{jk}}}={\beta }_{10k}+{\beta }_{11k}{T}_{\mathit{\text{jk}}}+r_{1\mathit{\text{jk}}}$ ${\pi }_{2\mathit{\text{jk}}}={\beta }_{20k}$ $\left(\begin{array}{c}{r}_{0\mathit{\text{jk}}}\\ r_{1\mathit{\text{jk}}}\end{array}\right)\sim N\left[\left(\begin{array}{c}0\\ 0\end{array}\right),\left(\begin{array}{cc}{\tau }_{00|T,W}^2& {\tau }_{01|T,W}\\ & {\tau }_{11|T}^2\end{array}\right)\right].$ L3: ${\beta }_{00k}={\gamma }_{000}+u_{00k}$ ${\beta }_{01k}={\gamma }_{010}+u_{01k}$ ${\beta }_{02k}={\gamma }_{020}$ ${\beta }_{10k}={\gamma }_{100}+u_{10k}$ ${\beta }_{11k}={\gamma }_{110}+u_{11k}$ ${\beta }_{20k}={\gamma }_{200}$ $\left(\begin{array}{c}\begin{array}{c}{u}_{00k}\\ u_{01k}\\ u_{10k}\end{array}\\ u_{11k}\end{array}\right)\sim N\left[\left(\begin{array}{c}\begin{array}{c}0\\ 0\\ 0\end{array}\\ 0\end{array}\right),\left(\begin{array}{cccc}{\tau }_{0000}^2& {\tau }_{0001}& {\tau }_{0010}& {\tau }_{0011}\\ & {\tau }_{0101}^2& {\tau }_{0110}& {\tau }_{0111}\\ & & {\tau }_{1010}^2& {\tau }_{1011}\\ & & & {\tau }_{1111}^2\end{array}\right)\right].$	Binary Moderator: ${\widehat{\delta }}_{1b}/\sqrt{\frac{{\omega }_{3T{M}^{(1)}}^2}{K}+\frac{{\omega }_{2M^{(1)}}^2}{P(1-P)\mathit{\text{KJ}}}+\frac{(1-{\rho }_3-{\rho }_2)(1-R_1^2)}{P(1-P)Q_1(1-Q_1)\mathit{\text{KJn}}}}$ Continuous Moderator: ${\widehat{\delta }}_{1c}/\sqrt{\frac{{\omega }_{3T{M}^{(1)}}^2}{K}+\frac{{\omega }_{2M^{(1)}}^2}{P(1-P)\mathit{\text{KJ}}}+\frac{(1-{\rho }_3-{\rho }_2)(1-R_1^2)}{P(1-P)\mathit{\text{KJn}}}}$	Binary Moderator: $M_v\sqrt{\frac{{\omega }_{3T{M}^{(1)}}^2}{K}+\frac{{\omega }_{2M^{(1)}}^2}{P(1-P)\mathit{\text{KJ}}}+\frac{(1-{\rho }_3-{\rho }_2)(1-R_1^2)}{P(1-P)Q_1(1-Q_1)\mathit{\text{KJn}}}}$ ${(M}_v\pm t_{\alpha /2})\sqrt{\frac{{\omega }_{3T{M}^{(1)}}^2}{K}+\frac{{\omega }_{2M^{(1)}}^2}{P(1-P)\mathit{\text{KJ}}}+\frac{(1-{\rho }_3-{\rho }_2)(1-R_1^2)}{P(1-P)Q_1(1-Q_1)\mathit{\text{KJn}}}}$ Continuous Moderator: $M_v\sqrt{\frac{{\omega }_{3T{M}^{(1)}}^2}{K}+\frac{{\omega }_{2M^{(1)}}^2}{P(1-P)\mathit{\text{KJ}}}+\frac{(1-{\rho }_3-{\rho }_2)(1-R_1^2)}{P(1-P)\mathit{\text{KJn}}}}$ ${(M}_v\pm t_{\alpha /2})\sqrt{\frac{{\omega }_{3T{M}^{(1)}}^2}{K}+\frac{{\omega }_{2M^{(1)}}^2}{P(1-P)\mathit{\text{KJ}}}+\frac{(1-{\rho }_3-{\rho }_2)(1-R_1^2)}{P(1-P)\mathit{\text{KJn}}}}$	K − 1
MCRT3-2R-2	L1: $Y_{\mathit{\text{ijk}}}={\pi }_{0\mathit{\text{jk}}}+{\pi }_{1\mathit{\text{jk}}}{X}_{\mathit{\text{ijk}}}+e_{\mathit{\text{ijk}}}$, $e_{\mathit{\text{ijk}}}\sim N(0,{\sigma }_{1|X}^2)$. L2: ${\pi }_{0\mathit{\text{jk}}}={\beta }_{00k}+{\beta }_{01k}{T}_{\mathit{\text{jk}}}+{\beta }_{02k}{T}_{\mathit{\text{jk}}}{M}_{\mathit{\text{jk}}}^{(2)}+{\beta }_{03k}{M}_{\mathit{\text{jk}}}^{(2)}+{\beta }_{04k}{W}_{\mathit{\text{jk}}}+r_{0\mathit{\text{jk}}}$, $r_{0\mathit{\text{jk}}}\sim N(0,{\tau }_{00|T,M,W}^2$), ${\pi }_{1\mathit{\text{jk}}}={\beta }_{10k}$, L3: ${\beta }_{00k}={\gamma }_{000}+u_{00k}$ ${\beta }_{01k}={\gamma }_{010}+u_{01k}$ ${\beta }_{02k}={\gamma }_{020}+u_{02k}$ ${\beta }_{03k}={\gamma }_{030}$ ${\beta }_{04k}={\gamma }_{040}$ ${\beta }_{10k}={\gamma }_{100}$ $\left(\begin{array}{c}{u}_{00k}\\ u_{01k}\\ u_{02k}\end{array}\right)\sim N\left[\left(\begin{array}{c}0\\ 0\\ 0\end{array}\right),\left(\begin{array}{ccc}{\tau }_{0000}^2& {\tau }_{0001}& {\tau }_{0002}\\ & {\tau }_{0101}^2& {\tau }_{0102}\\ & & {\tau }_{0202}^2\end{array}\right)\right]$.	Binary Moderator: ${\widehat{\delta }}_{2b}/\sqrt{\frac{{\omega }_{3T{M}^{(2)}}^2}{K}+\frac{{\rho }_2(1-R_2^2)}{P(1-P)Q_2(1-Q_2)\mathit{\text{KJ}}}+\frac{(1-{\rho }_3-{\rho }_2)(1-R_1^2)}{P(1-P)Q_2(1-Q_2)\mathit{\text{KJn}}}}$ Continuous Moderator: ${\widehat{\delta }}_{2c}/\sqrt{\frac{{\omega }_{3T{M}^{(2)}}^2}{K}+\frac{{\rho }_2(1-R_2^2)}{P(1-P)\mathit{\text{KJ}}}+\frac{(1-{\rho }_3-{\rho }_2)(1-R_1^2)}{P(1-P)\mathit{\text{KJn}}}}$	Binary Moderator: $M_v\sqrt{\frac{{\omega }_{3T{M}^{(2)}}^2}{K}+\frac{{\rho }_2(1-R_2^2)}{P(1-P)Q_2(1-Q_2)\mathit{\text{KJ}}}+\frac{(1-{\rho }_3-{\rho }_2)(1-R_1^2)}{P(1-P)Q_2(1-Q_2)\mathit{\text{KJn}}}}$ ${(M}_v\pm t_{\alpha /2})\sqrt{\frac{{\omega }_{3T{M}^{(2)}}^2}{K}+\frac{{\rho }_2(1-R_2^2)}{P(1-P)Q_2(1-Q_2)\mathit{\text{KJ}}}+\frac{(1-{\rho }_3-{\rho }_2)(1-R_1^2)}{P(1-P)Q_2(1-Q_2)\mathit{\text{KJn}}}}$ Continuous Moderator: $M_v\sqrt{\frac{{\omega }_{3T{M}^{(2)}}^2}{K}+\frac{{\rho }_2(1-R_2^2)}{P(1-P)\mathit{\text{KJ}}}+\frac{(1-{\rho }_3-{\rho }_2)(1-R_1^2)}{P(1-P)\mathit{\text{KJn}}}}$ ${(M}_v\pm t_{\alpha /2})\sqrt{\frac{{\omega }_{3T{M}^{(2)}}^2}{K}+\frac{{\rho }_2(1-R_2^2)}{P(1-P)\mathit{\text{KJ}}}+\frac{(1-{\rho }_3-{\rho }_2)(1-R_1^2)}{P(1-P)\mathit{\text{KJn}}}}$	K − 1
MCRT3-2R-3	L1: $Y_{\mathit{\text{ijk}}}={\pi }_{0\mathit{\text{jk}}}+{\pi }_{1\mathit{\text{jk}}}{X}_{\mathit{\text{ijk}}}+e_{\mathit{\text{ijk}}}$, $e_{\mathit{\text{ijk}}}\sim N(0,{\sigma }_{1|X}^2)$. L2: ${\pi }_{0\mathit{\text{jk}}}={\beta }_{00k}+{\beta }_{01k}{T}_{\mathit{\text{jk}}}+{\beta }_{02k}{W}_{\mathit{\text{jk}}}+r_{0\mathit{\text{jk}}}$, $r_{0\mathit{\text{jk}}}\sim N(0,{\tau }_{00|T,W}^2$), ${\pi }_{1\mathit{\text{jk}}}={\beta }_{10k}$, L3: ${\beta }_{00k}={\gamma }_{000}+{\gamma }_{001}{M}_k^{\left(3\right)}+u_{00k}$ ${\beta }_{01k}={\gamma }_{010}+{\gamma }_{011}{M}_k^{\left(3\right)}+u_{01k}$ ${\beta }_{02k}={\gamma }_{020}$ ${\beta }_{10k}={\gamma }_{100}$ $\left(\begin{array}{c}{u}_{00k}\\ u_{01k}\end{array}\right)\sim N\left[\left(\begin{array}{c}0\\ 0\end{array}\right),\left(\begin{array}{cc}{\tau }_{0000|M}^2& {\tau }_{0001|M}\\ & {\tau }_{0101|M}^2\end{array}\right)\right]$.	Binary Moderator: ${\widehat{\delta }}_{3b}/\sqrt{\frac{{\omega }_{3T}^2-{\widehat{\delta }}_{3b}^2{Q}_3(1-Q_3)}{K{Q}_3(1-Q_3)}+\frac{{\rho }_2(1-R_2^2)}{P(1-P)Q_3(1-Q_3)\mathit{\text{KJ}}}+\frac{(1-{\rho }_3-{\rho }_2)(1-R_1^2)}{P(1-P)Q_3(1-Q_3)\mathit{\text{KJn}}}}$ Continuous Moderator: ${\widehat{\delta }}_{3c}/\sqrt{\frac{{\omega }_{3T}^2-{\widehat{\delta }}_{3c}^2}{K}+\frac{{\rho }_2(1-R_2^2)}{P(1-P)\mathit{\text{KJ}}}+\frac{(1-{\rho }_3-{\rho }_2)(1-R_1^2)}{P(1-P)\mathit{\text{KJn}}}}$	Binary Moderator: $M_v\sqrt{\left(\frac{{\omega }_{3T}^2}{K{Q}_3(1-Q_3)}+\frac{{\rho }_2(1-R_2^2)}{P(1-P)Q_3(1-Q_3)\mathit{\text{KJ}}}+\frac{(1-{\rho }_3-{\rho }_2)(1-R_1^2)}{P(1-P)Q_3(1-Q_3)\mathit{\text{KJn}}}\right)/(1+\frac{M_v^2}{K})}$ ${(M}_v\pm t_{\alpha /2})\sqrt{\left(\frac{{\omega }_{3T}^2}{K{Q}_3(1-Q_3)}+\frac{{\rho }_2(1-R_2^2)}{P(1-P)Q_3(1-Q_3)\mathit{\text{KJ}}}+\frac{(1-{\rho }_3-{\rho }_2)(1-R_1^2)}{P(1-P)Q_3(1-Q_3)\mathit{\text{KJn}}}\right)/(1+\frac{M_v^2}{K})}$ Continuous Moderator: $M_v\sqrt{\left(\frac{{\omega }_{3T}^2}{K}+\frac{{\rho }_2(1-R_2^2)}{P(1-P)\mathit{\text{KJ}}}+\frac{(1-{\rho }_3-{\rho }_2)(1-R_1^2)}{P(1-P)\mathit{\text{KJn}}}\right)/(1+\frac{M_v^2}{K})}$ ${(M}_v\pm t_{\alpha /2})\sqrt{\left(\frac{{\omega }_{3T}^2}{K}+\frac{{\rho }_2(1-R_2^2)}{P(1-P)\mathit{\text{KJ}}}+\frac{(1-{\rho }_3-{\rho }_2)(1-R_1^2)}{P(1-P)\mathit{\text{KJn}}}\right)/(1+\frac{M_v^2}{K})}$	K − 2
MCRT3-2N-1	L1: $Y_{\mathit{\text{ijk}}}={\pi }_{0\mathit{\text{jk}}}+{\pi }_{1\mathit{\text{jk}}}{M}_{\mathit{\text{ijk}}}^{(1)}+{\pi }_{2\mathit{\text{jk}}}{X}_{\mathit{\text{ijk}}}+e_{\mathit{\text{ijk}}}$, $e_{\mathit{\text{ijk}}}\sim N(0,{\sigma }_{1|M,X}^2)$. L2: ${\pi }_{0\mathit{\text{jk}}}={\beta }_{00k}+{\beta }_{01k}{T}_{\mathit{\text{jk}}}+{\beta }_{02k}{W}_{\mathit{\text{jk}}}+r_{0\mathit{\text{jk}}}$ ${\pi }_{1\mathit{\text{jk}}}={\beta }_{10k}+{\beta }_{11k}{T}_{\mathit{\text{jk}}}$ ${\pi }_{2\mathit{\text{jk}}}={\beta }_{20k}$ $r_{0\mathit{\text{jk}}}\sim N(0,{\tau }_{00|T,W}^2)$. L3: ${\beta }_{00k}={\gamma }_{000}+u_{00k}$ ${\beta }_{01k}={\gamma }_{010}$ ${\beta }_{02k}={\gamma }_{020}$ ${\beta }_{10k}={\gamma }_{100}$ ${\beta }_{11k}={\gamma }_{110}$ ${\beta }_{20k}={\gamma }_{200}$ $u_{00k}\sim N(0,{\tau }_{0000}^2)$	Binary Moderator: ${\widehat{\delta }}_{1b}/\sqrt{\frac{(1-{\rho }_3-{\rho }_2)(1-R_1^2)}{P(1-P)Q_1(1-Q_1)\mathit{\text{KJn}}}}$ Continuous Moderator: ${\widehat{\delta }}_{1c}/\sqrt{\frac{(1-{\rho }_3-{\rho }_2)(1-R_1^2)}{P(1-P)\mathit{\text{KJn}}}}$	Binary Moderator: $M_v\sqrt{\frac{(1-{\rho }_3-{\rho }_2)(1-R_1^2)}{P(1-P)Q_1(1-Q_1)\mathit{\text{KJn}}}}$ ${(M}_v\pm t_{\alpha /2})\sqrt{\frac{(1-{\rho }_3-{\rho }_2)(1-R_1^2)}{P(1-P)Q_1(1-Q_1)\mathit{\text{KJn}}}}$ Continuous Moderator: $M_v\sqrt{\frac{(1-{\rho }_3-{\rho }_2)(1-R_1^2)}{P(1-P)\mathit{\text{KJn}}}}$ ${(M}_v\pm t_{\alpha /2})\sqrt{\frac{(1-{\rho }_3-{\rho }_2)(1-R_1^2)}{P(1-P)\mathit{\text{KJn}}}}$	KJ(n-1)-3
MCRT3-2N-2	L: $Y_{\mathit{\text{ijk}}}={\pi }_{0\mathit{\text{jk}}}+{\pi }_{1\mathit{\text{jk}}}{X}_{\mathit{\text{ijk}}}+e_{\mathit{\text{ijk}}}$, $e_{\mathit{\text{ijk}}}\sim N(0,{\sigma }_{1|X}^2)$. L2: ${\pi }_{0\mathit{\text{jk}}}={\beta }_{00k}+{\beta }_{01k}{T}_{\mathit{\text{jk}}}+{\beta }_{02k}{T}_{\mathit{\text{jk}}}{M}_{\mathit{\text{jk}}}^{(2)}+{\beta }_{03k}{M}_{\mathit{\text{jk}}}^{(2)}+{\beta }_{04k}{W}_{\mathit{\text{jk}}}+r_{0\mathit{\text{jk}}}$, $r_{0\mathit{\text{jk}}}\sim N(0,{\tau }_{00|T,M,W}^2$), ${\pi }_{1\mathit{\text{jk}}}={\beta }_{10k}$, L3: ${\beta }_{00k}={\gamma }_{000}+u_{00k}$ ${\beta }_{01k}={\gamma }_{010}$ ${\beta }_{02k}={\gamma }_{020}$ ${\beta }_{03k}={\gamma }_{030}$ ${\beta }_{04k}={\gamma }_{040}$ ${\beta }_{10k}={\gamma }_{100}$ $u_{00k}\sim N(0,{\tau }_{0000}^2)$	Binary Moderator: ${\widehat{\delta }}_{2b}/\sqrt{\frac{{\rho }_2(1-R_2^2)}{P(1-P)Q_2(1-Q_2)\mathit{\text{KJ}}}+\frac{(1-{\rho }_3-{\rho }_2)(1-R_1^2)}{P(1-P)Q_2(1-Q_2)\mathit{\text{KJn}}}}$ Continuous Moderator: ${\widehat{\delta }}_{2c}/\sqrt{\frac{{\rho }_2(1-R_2^2)}{P(1-P)\mathit{\text{KJ}}}+\frac{(1-{\rho }_3-{\rho }_2)(1-R_1^2)}{P(1-P)\mathit{\text{KJn}}}}$	Binary Moderator: $M_v\sqrt{\frac{{\rho }_2(1-R_2^2)}{P(1-P)Q_2(1-Q_2)\mathit{\text{KJ}}}+\frac{(1-{\rho }_3-{\rho }_2)(1-R_1^2)}{P(1-P)Q_2(1-Q_2)\mathit{\text{KJn}}}}$ ${(M}_v\pm t_{\alpha /2})\sqrt{\frac{{\rho }_2(1-R_2^2)}{P(1-P)Q_2(1-Q_2)\mathit{\text{KJ}}}+\frac{(1-{\rho }_3-{\rho }_2)(1-R_1^2)}{P(1-P)Q_2(1-Q_2)\mathit{\text{KJn}}}}$ Continuous Moderator: $M_v\sqrt{\frac{{\rho }_2(1-R_2^2)}{P(1-P)\mathit{\text{KJ}}}+\frac{(1-{\rho }_3-{\rho }_2)(1-R_1^2)}{P(1-P)\mathit{\text{KJn}}}}$ ${(M}_v\pm t_{\alpha /2})\sqrt{\frac{{\rho }_2(1-R_2^2)}{P(1-P)\mathit{\text{KJ}}}+\frac{(1-{\rho }_3-{\rho }_2)(1-R_1^2)}{P(1-P)\mathit{\text{KJn}}}}$	K(J-1)-4
MCRT3-2N-3	L1: $Y_{\mathit{\text{ijk}}}={\pi }_{0\mathit{\text{jk}}}+{\pi }_{1\mathit{\text{jk}}}{X}_{\mathit{\text{ijk}}}+e_{\mathit{\text{ijk}}}$, $e_{\mathit{\text{ijk}}}\sim N(0,{\sigma }_{1|X}^2)$. L2: ${\pi }_{0\mathit{\text{jk}}}={\beta }_{00k}+{\beta }_{01k}{T}_{\mathit{\text{jk}}}+{\beta }_{02k}{W}_{\mathit{\text{jk}}}+r_{0\mathit{\text{jk}}}$, $r_{0\mathit{\text{jk}}}\sim N(0,{\tau }_{00|T,W}^2$), ${\pi }_{1\mathit{\text{jk}}}={\beta }_{10k}$, L3: ${\beta }_{00k}={\gamma }_{000}+{\gamma }_{001}{M}_k^{\left(3\right)}+u_{00k}$ ${\beta }_{01k}={\gamma }_{010}+{\gamma }_{011}{M}_k^{\left(3\right)}$ ${\beta }_{02k}={\gamma }_{020}$ ${\beta }_{10k}={\gamma }_{100}$ $u_{00k}\sim N(0,{\tau }_{0000|M}^2)$	Binary Moderator: ${\widehat{\delta }}_{3b}/\sqrt{\frac{{\rho }_2(1-R_2^2)}{P(1-P)Q_3(1-Q_3)\mathit{\text{KJ}}}+\frac{(1-{\rho }_3-{\rho }_2)(1-R_1^2)}{P(1-P)Q_3(1-Q_3)\mathit{\text{KJn}}}}$ Continuous Moderator: ${\widehat{\delta }}_{3c}/\sqrt{\frac{{\rho }_2(1-R_2^2)}{P(1-P)\mathit{\text{KJ}}}+\frac{(1-{\rho }_3-{\rho }_2)(1-R_1^2)}{P(1-P)\mathit{\text{KJn}}}}$	Binary Moderator: $M_v\sqrt{\frac{{\rho }_2(1-R_2^2)}{P(1-P)Q_3(1-Q_3)\mathit{\text{KJ}}}+\frac{(1-{\rho }_3-{\rho }_2)(1-R_1^2)}{P(1-P)Q_3(1-Q_3)\mathit{\text{KJn}}}}$ ${(M}_v\pm t_{\alpha /2})\sqrt{\frac{{\rho }_2(1-R_2^2)}{P(1-P)Q_3(1-Q_3)\mathit{\text{KJ}}}+\frac{(1-{\rho }_3-{\rho }_2)(1-R_1^2)}{P(1-P)Q_3(1-Q_3)\mathit{\text{KJn}}}}$ Continuous Moderator: $M_v\sqrt{\frac{{\rho }_2(1-R_2^2)}{P(1-P)\mathit{\text{KJ}}}+\frac{(1-{\rho }_3-{\rho }_2)(1-R_1^2)}{P(1-P)\mathit{\text{KJn}}}}$ ${(M}_v\pm t_{\alpha /2})\sqrt{\frac{{\rho }_2(1-R_2^2)}{P(1-P)\mathit{\text{KJ}}}+\frac{(1-{\rho }_3-{\rho }_2)(1-R_1^2)}{P(1-P)\mathit{\text{KJn}}}}$	K(J-1)-3

Note. MCRT3-2N-1, MCRT3-2N-2, and MCRT3-2N-3 stand for three-level MCRTs where treatment is at level 2 with a level-1, -2, -3 moderator with nonrandomly varying slopes, respectively. MCRT3-2R-1, MCRT3-2R-2, and MCRT3-2R-3 stand for three-level MRTs where treatment is at level 2 with a level-1, -2, -3 moderator with random slopes, respectively.

Figure 2. Power from simulation and formulas.

Note. Simulation results were based on 2,000 replications. Under the assumptions: The intraclass correlation coefficients at Level-2 and 3: ${\rho }_2$ = 0.1, and ${\rho }_3$ = 0.2; The proportions of variances at level 1 and level 2 explained by covariates: $R_1^2$ = $R_2^2$ = 0.50; The proportion of clusters assigned to the treatment group, P = 0.5; $Q_1$ = $Q_2$ = $Q_3$ = 0.5 for binary moderators. For random slope models, the standardized effect variability of the moderation across sites for Level-1 moderator (${\omega }_{{3\mathit{\text{TM}}}^{(1)}}^2$) and Level-2 moderator (${\omega }_{{3\mathit{\text{TM}}}^{(2)}}^2$) are 0.05, the standardized variability of the treatment effect across sites for Level-3 moderator (${\omega }_{3T}^2$) is 0.09, the standardized effect variability of Level-1 moderator across Level-2 units (${\omega }_{{2M}^{(1)}}^2$) is 0.05, the moderator effect size = 0.20, sample size per level-2 unit (n) is 20, sample size per site (J) is 10, and total sample size of sites (K) is 20. For nonrandom slope models, the moderator effect size = 0.10, sample size per level-2 unit (n) is 20, sample size per site (J) is 4 for level 1 moderator and 10 for levels-2 and 3 moderators, and total sample size of sites (K) is 20 (except for binary level-1 moderator, K = 40).