Image theory for the structure of quantitative variates

Abstract

A universe of infinitely many quantitative variables is considered, from which a sample ofn 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 variablex from the remainingn − 1 variables is called the partial image ofx, and the error of prediction is called the partial anti-image ofx. The predicted value ofx from the entire universe, or the limit of its partial images asn → ∞, is called the total image ofx, and the corresponding error is called the total anti-image. Images and anti-images can be used to explain “why” any two variablesx _j andx _k 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.