A PLS algorithm version working with ordinal variables

A linear Structural Equation Model with Latent Variables (SEM-LV) consists of two sets of equations: the structural or inner model describing the relationships among latent variables, and the measurement or outer model, representing the relationships among latent (unobservable) variables and appropriate corresponding manifest variables. It often happens that manifest indicators are measured on ordinal scales, e.g. when responses given to a questionnaire are on Likert type scales assuming a unique common finite set of possible categories. This type of scale requires appropriate methods to be applied. Observe that, in most research and applied works, averages, linear transformations, covariances and Pearson correlation coefficients are conventionally computed also on the ordinal variables coming from surveys. This practice can be theoretically justified by invoking the so-called operational and pragmatic approaches to statistical measurement: according to the latter approach ’the precise property being measured is defined simultaneously with the procedure for measuring it’, so when defining the scale the measuring instrument is also defined, and the kind of scale can be, in a certain sense, chosen by the researcher. In order to treat manifest indicators of the ordinal type, a more traditional approach would require appropriate procedures to be adopted for parameter estimation in SEM-LV. Estimation procedures within a covariance-based framework are availale, which are based on the assumption that for each manifest indicator there corresponds a further underlying continuous latent variable. In case of the Partial Least Squares (PLS) framework some different approaches have been proposed: some are based on generalized linear models, or on Alternating Least Squares and based on the Hayashi first quantification method. Our proposal considers the traditional psychometric approach, by applying a method related to ordinal measures according to the well-known Thurstone scaling procedure.