Two new mixture models: Living with collinearity but removing its influence

Fitting equations to mixture data collected from highly constrained regions has challenged modelers for the past 35 years. Computer roundoff error, low precision of the estimates, and collinearity among the terms in the models are just a few of the problems encountered. It is well known that, due to collinearity, fitted models can contain coefficient estimates with magnitudes many hundreds or even a thousand times greater that the magnitude of the data values themselves. This can be very problematic to the modeler trying to convince a client the model is adequate. Fitting models in pseudocomponents with and without centering the terms has been one of the strategies used in an effort to reduce the effect of collinearity, but often to no avail. We introduce two additional model forms where the terms are scaled, and we illustrate the benefits of fitting the new models using two numerical examples. The equivalence among models of four different, but related, component systems is shown for the first time.