PARAMETRIC STATISTICS AND LEVELS OF MEASUREMENT - FACTORIAL-DESIGNS AND MULTIPLE-REGRESSION

Let Y be a continuous, ordinal measure of a latent variable Θ. In general, for factorial designs, an analysis of variance of the observed variable Y cannot be used to draw inferences about main effects and interactions on the latent variable Θ even when the standard normality and equality of variance assumptions hold. If Y is a continuous, ordinal measure of a latent variable Θ; X1, ..., Xn are continuous, ordinal measures of latent variables Φ1, ..., Φn; and the observed measures have a multivariate normal distribution, then a multiple regression analysis of the observed criterion measure Y and predictors X1, ..., Xn can be used to test hypotheses about multivariate associations among the latent variables. Furthermore, the predicted values Y′ are unbiased estimates of quantities that are monotonically related to predicted values on the latent criterion variable Θ.