The Determination of Multiple Correlations between Several Variables, with Especial Reference to the Correlation of Physical Properties and Chemical Composition

M. H. Hey
Department of Mineralogy British Museum (Natural History)

Summary: If the true values of a number (n) of variables are exactly related by one or more (m) equations but all experimental values are liable to experimental error then, provided the errors of measurement can be assumed to be symmetrically distributed about a zero mean, provided the relations are linear, and provided the assessed probable errors of measurement of the several variables remain in constant ratio from one set of measurements to another, the best estimate of the relations derived from N sets of measurements will be given by the latent vectors corresponding to the m smallest latent roots of the n-square matrix whose terms are Σwi ξij ξik (summed from i = 1 to i = N) for all j and k from 1 to n, where wi is the weight of the ith set of measurements and ξij is the standardized value of the ith measurement of the jth variable measured from its weighted mean.

It is also shown that the alleged inconsistency of the maximum-likelihood estimate of the variance in such a case is simply a confusion of the root-mean-square residual with the root-mean-square normal residual.

An Autocode computer programme has been written to carry out the necessary operations for correlating the physical properties and chemical properties of an isomorphous series of minerals using this procedure.

Mineralogical Magazine; March 1969 v. 37; no. 285; p. 83-89; DOI: 10.1180/minmag.1969.037.285.09
© 1969, The Mineralogical Society
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