Nonlinearity effects in multidimensional scaling

When multidimensional scaling of n cases is derived from dissimilarities that are functions of p basic continuous variables, the question arises of how to relate the values of the variables to the configuration of n points. We provide a methodology based on nonlinear biplots that expresses nonlinearity in two ways: (i) each variable is represented by a nonlinear trajectory and (ii) each trajectory is calibrated by an irregular scale. Methods for computing, calibrating and interpreting these trajectories are given and exemplified. Not only are the tools of immediate practical utility but the methodology established assists in a critical appraisal of the consequences of using nonlinear measures in a variety of multidimensional scaling methods.