Image Theory for the Structure of Quantitative Variates

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Abstract
A universe of infinitely many quantitative variables is considered, from which a sample of n variables is arbitrarily selected. Only linear least-squares regressions are considered, based on an infinitely large population of individuals or respondents. In the sample of variables, the predicted value of a variable x from the remaining n − 1 variables is called the partial image of x, and the error of prediction is called the partial anti-image of x. The predicted value of x from the entire universe, or the limit of its partial images as n → ∞, is called the total image of x, and the corresponding error is called the total anti-image. Images and anti-images can be used to explain “why” any two variables xj and xk are correlated with each other, or to reveal the structure of the intercorrelations of the sample and of the universe. It is demonstrated that image theory is related to common-factor theory but has greater generality than common-factor theory, being able to deal with structures other than those describable in a Spearman-Thurstone factor space. A universal computing procedure is suggested, based upon the inverse of the correlation matrix.

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