Measuring Latent Quantities
Emeritus lecture by Roderick P. McDonald, University of Sydney, Australia
Chair: Brian Junker, Thursday 23rd July, 14.00 - 14.45, Palmeston Lecture Theatre, Fisher Building.
The topic of “estimating” latent quantities - factor scores, latent traits, disturbances, multilevel components - is still subject to confusion and authoritative contradiction. The issues are foundational and difficult to resolve. The confusion arises from failure to distinguish between measurement and prediction, and also from failure to define what is measured. By definition a measurement consists of the quantity to be measured and an (additive) error of measurement. In the standard linear measurement model, it follows that measures of a vector of latent variables are given by the conditional inverses of the common factor loading matrix. These include
In contrast, Thomson’s “estimator” is given by the particular solution of the factor equations, i.e., from the conditional inverse of the full set of common and unique factor loadings. Thomson’s solution is a regression predictor, and, under assumptions, an Empirical Bayes predictor. However, it is not a measure and, contrary to modern practice, should not be used as such. A sufficient condition will be given for “pure” measurement of each latent quantity. A definition will be offered for the latent quantities. Parallel conclusions for item response models, disturbances, and multilevel components will be outlined. (253)