Latent Variables and Indices: Herman Wold’s Basic Design and Partial Least Squares

Abstract

In this chapter it is shown that the PLS-algorithms typically converge if the covariance matrix of the indicators satisfies (approximately) the “basic design”, a factor analysis type of model. The algorithms produce solutions to fixed point equations; the solutions are smooth functions of the sample covariance matrix of the indicators. If the latter matrix is asymptotically normal, the PLS-estimators will share this property. The probability limits, under the basic design, of the PLS-estimators for loadings, correlations, multiple R’s, coefficients of structural equations et cetera will differ from the true values. But the difference is decreasing, tending to zero, in the “quality” of the PLS estimators for the latent variables. It is indicated how to correct for the discrepancy between true values and the probability limits. We deemphasize the “normality”-issue in discussions about PLS versus ML: in employing either method one is not required to subscribe to normality; they are “just” different ways of extracting information from second-order moments.

We also propose a new “back-to-basics” research program, moving away from factor analysis models and returning to the original object of constructing indices that extract information from high-dimensional data in a predictive, useful way. For the generic case we would construct informative linear compounds, whose constituent indicators have non-negative weights as well as non-negative loadings, satisfying constraints implied by the path diagram. Cross-validation could settle the choice between various competing specifications. In short: we argue for an upgrade of principal components and canonical variables analysis.