On the Nature of the Stationary Point of a Quadratic Response Surface: A Bayesian Simulation-Based Approach

In response-surface methodology, when the data are fitted using a quadratic model, it is important to make inference about the eigenvalues of the matrix of pure and mixed second-order coefficients, since they contain information on the nature of the stationary point and the shape of the surface. In this article, we propose a Bayesian simulation-based approach to explore the behavior of the posterior distributions of these eigenvalues. Highest posterior density (HPD) intervals for the ordered eigenvalues are then computed and their empirical coverage probabilities are evaluated. A user-friendly software tool has been developed to get the kernel density plots of these simulated posterior distributions and to obtain the corresponding HPD intervals. It is provided online as supplementary materials to this article.