TABLE 4.

Formula-based sample size (N₄) and estimated Monte Carlo power $\left(\widehat{P}\right)$ at β = 0.2 and α = 0.025, based on hypothesis test of $H_0:{\mu }_{d_1}\le {\mu }_{d_2}\text{or }\cdots \text{or }{\mu }_{d_1}\le {\mu }_{d_8}$ versus $H_1:{\mu }_{d_1}>{\mu }_{d_2}\text{\&}\cdots \text{\& }{\mu }_{d_1}>{\mu }_{d_8}$, under varying average of absolute effect sizes $\left(\overline{|\delta |}={\sum }_{j=2}^8\left|{\delta }_{d_1-d_j}\right|/7\right)$ or standardized effect size $\left(\overline{\left|{\delta }^{*}\right|}={\sum }_{j=2}^8\left|{\delta }_{d_1-d_j}/{\sigma }_{d_1-d_j}\right|/7\right)$ and treatment “3” response rate $\left({\gamma }^{d_1}\right)$, skewness (λ) and degrees of freedom (ν), given treatment “8” response rate ${\gamma }^{d_5}=0.5$, σ₁ = 0.95, σ₁ = 1, λ = 0.975, τ = 0.85, expected % of available teeth per patient p_i = 80%

Absolute effect size	Skewness	Degrees of freedom	Response rate	Standardized effect size	Formula-based sample size	Monte Carlo power
|δ|	λ	v	${\gamma }^{d_1}$	|δ*|	N4	$\widehat{P}$
1.50	0	Inf	0.25	0.48	95.00	0.79
3.75				0.51	71.00	0.80
1.00			0.5	0.35	179.00	0.80
2.50				0.37	132.00	0.81
1.50		8	0.25	0.48	96.00	0.80
3.75				0.51	71.00	0.81
1.00			0.5	0.35	180.00	0.80
2.50				0.37	132.00	0.80
1.50		5	0.25	0.47	97.00	0.80
3.75				0.51	71.00	0.80
1.00			0.5	0.35	182.00	0.81
2.50				0.37	132.00	0.79
1.50		3	0.25	0.47	101.00	0.80
3.75				0.51	72.00	0.81
1.00			0.5	0.34	189.00	0.80
2.50				0.37	134.00	0.79
1.50	2	Inf	0.25	0.36	161.00	0.79
3.75				0.44	93.00	0.79
1.00			0.5	0.26	297.00	0.79
2.50				0.33	172.00	0.79
1.50		8	0.25	0.35	173.00	0.78
3.75				0.43	97.00	0.79
1.00			0.5	0.26	318.00	0.78
2.50				0.32	179.00	0.79
1.50		5	0.25	0.34	183.00	0.78
3.75				0.43	100.00	0.79
1.00			0.5	0.25	336.00	0.78
2.50				0.32	184.00	0.79
1.50		3	0.25	0.32	209.00	0.80
3.75				0.42	107.00	0.79
1.00			0.5	0.24	383.00	0.79
2.50				0.31	197.00	0.80
1.50	10	Inf	0.25	0.35	172.00	0.78
3.75				0.43	97.00	0.81
1.00			0.5	0.26	316.00	0.79
2.50				0.32	179.00	0.79
1.50		8	0.25	0.34	186.00	0.79
3.75				0.43	101.00	0.79
1.00			0.5	0.25	344.00	0.80
2.50				0.32	185.00	0.80
1.50		5	0.25	0.33	198.00	0.78
3.75				0.42	104.00	0.79
1.00			0.5	0.24	362.00	0.79
2.50				0.31	192.00	0.79
1.50		3	0.25	0.31	229.00	0.79
3.75				0.41	112.00	0.78
1.00			0.5	0.23	418.00	0.79
2.50				0.30	207.00	0.79