Miscellaneous Formulas

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The CLUSTER Procedure

Miscellaneous Formulas

The root-mean-square standard deviation of a cluster C_K is

${RMSSTD} = \sqrt{\frac{W_K}{v(N_K - 1)}}$

The R² statistic for a given level of the hierarchy is

R² = 1 - [(P_G)/T]

The squared semipartial correlation for joining clusters C_K and C_L is

semipartial R² = [(B_KL)/T]

The bimodality coefficient is

b = [(m₃² + 1)/(m₄ + [(3(n-1)²)/((n-2)(n-3))])]

where m₃ is skewness and m₄ 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 heavy-tailed distributions have small values of b regardless of the number of modes.

Formulas for the cubic-clustering criterion and approximate expected R² 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 t² statistic for joining C_K and C_L is

pseudo t² = [(B_KL)/([(W_K + W_L)/(N_K + N_L - 2)])]

The pseudo F and t² statistics may be useful indicators of the number of clusters, but they are not distributed as F and t² 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 t² statistic is distributed as an F random variable with v and v(N_K + N_L - 2) degrees of freedom. The pseudo t² statistic differs computationally from Hotelling's T² 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 t² 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 + [(t²)/(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 t² statistics can be obtained by the Bonferroni approach (Hawkins, Muller, and ten Krooden 1982, pp. 337 -340).

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