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. Author manuscript; available in PMC: 2022 Jan 6.

Published in final edited form as: J Biopharm Stat. 2020 Sep 24;31(2):191–206. doi: 10.1080/10543406.2020.1814795

Appendix Table 1.

Cluster size distributions

Distribution	Cluster frequencies	Cluster sizes
Uniform	$f_{a} = f_{b} = f_{c} = \frac{m}{3}$	$g_{a} = n - \frac{R}{2}$	g_b = n	$g_{c} = n + \frac{R}{2}$
Unimodal	$f_{a} = \frac{m}{6},$ $f_{b} = \frac{2 m}{3},$ $f_{c} = \frac{m}{6}$	$g_{a} = n - \frac{R}{2}$	g_b = n	$g_{c} = n + \frac{R}{2}$
Bimodal	$f_{a} = \frac{5 m}{12},$ $f_{b} = \frac{m}{6},$ $f_{c} = \frac{5 m}{12}$	$g_{a} = n - \frac{R}{2}$	g_b = n	$g_{c} = n + \frac{R}{2}$
Positively Skewed	$f_{a} = \frac{m}{2},$ $f_{b} = \frac{m}{3},$ $f_{c} = \frac{m}{6}$	$g_{a} = n - \frac{R}{3}$	$g_{b} = n + \frac{R}{6}$	$g_{c} = n + \frac{2 R}{3}$
Negatively Skewed	$f_{a} = \frac{m}{6},$ $f_{b} = \frac{m}{3},$ $f_{c} = \frac{m}{2}$	$g_{a} = n - \frac{2 R}{3}$	$g_{b} = n - \frac{R}{6}$	$g_{c} = n + \frac{R}{3}$

m = # of clusters; n = cluster sizes;

f_a = # of clusters with cluster size g_a; f_b = # of clusters with cluster size g_b; f_c = # of clusters with cluster size g_c;

R = range of cluster sizes = g_c − g_a.