Tests and diagnostic plots for detecting lack-of-fit for circular-linear regression models

Regression diagnostics and lack-of-fit tests mainly focus on linear-linear regression models. When the design points are distributed on the circumference of a circle, difficulties arise as there is no natural starting point or origin. Most classical lack-of-fit tests require an arbitrarily chosen origin, but different choices may result in different conclusions. We propose a graphical diagnostic tool and a closely related lack-of-fit test, which does not require a natural starting point. The method is based on regional residuals which are defined on arcs of the circle. The graphical method formally locates and visualizes subsets of poorly fitting observations on the circle. A data example from the food technology is used to point out the before-mentioned problems with conventional lack-of-fit tests and to illustrate the strength of the methodology based on regional residuals in detecting and localizing departures from the no-effect hypothesis. A small simulation study shows a good performance of the regional residual test in case of both global and local deviations from the null model. Finally, the ideas are extended to the case of more than one predictor variable.