Miscellaneous Formulas
The rootmeansquare standard deviation of a cluster C_{K} is
The R^{2} statistic for a given level of the hierarchy is

R^{2} = 1  [(P_{G})/T]
The squared semipartial correlation for
joining clusters C_{K} and C_{L} is

semipartial R^{2} = [(B_{KL})/T]
The bimodality coefficient is

b = [(m_{3}^{2} + 1)/(m_{4} + [(3(n1)^{2})/((n2)(n3))])]
where m_{3} is skewness and m_{4} is kurtosis.
Values of b greater than 0.555 (the value
for a uniform population) may indicate
bimodal or multimodal marginal distributions.
The maximum of 1.0 (obtained for the Bernoulli distribution)
is obtained for a population with only two distinct values.
Very heavytailed distributions have small
values of b regardless of the number of modes.
Formulas for the cubicclustering criterion and
approximate expected R^{2} are given in Sarle (1983).
The pseudo F statistic for a given level is

pseudo F = [( [(T  P_{G})/(G  1)])/( [(P_{G})/(n  G)])]
The pseudo t^{2} statistic for joining C_{K} and C_{L} is

pseudo t^{2} = [(B_{KL})/([(W_{K} + W_{L})/(N_{K} + N_{L}  2)])]
The pseudo F and t^{2} statistics may be useful
indicators of the number of clusters, but they are not
distributed as F and t^{2} random variables.
If the data are independently sampled from a multivariate
normal distribution with a scalar covariance matrix and if
the clustering method allocates observations to clusters
randomly (which no clustering method actually does), then the
pseudo F statistic is distributed as an F random
variable with v(G  1) and v(n  G) degrees of freedom.
Under the same assumptions, the pseudo t^{2} statistic
is distributed as an F random variable with
v and v(N_{K} + N_{L}  2) degrees of freedom.
The pseudo t^{2} statistic differs computationally from
Hotelling's T^{2} in that the latter uses a general symmetric
covariance matrix instead of a scalar covariance matrix.
The pseudo F statistic was suggested
by Calinski and Harabasz (1974).
The pseudo t^{2} statistic is related to the
J_{e}(2)/J_{e}(1) statistic of Duda and Hart (1973) by

[(J_{e} (2))/(J_{e} (1))] = [(W_{K} + W_{L})/(W_{M})] = [1/(1 + [(t^{2})/(N_{K} + N_{L}  2)])]
See Milligan and Cooper (1985) and Cooper and Milligan
(1988) regarding the performance of these statistics
in estimating the number of population clusters.
Conservative tests for the number of clusters using the pseudo
F and t^{2} statistics can be obtained by the Bonferroni
approach
(Hawkins, Muller, and ten Krooden 1982, pp. 337 340).
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.