Date of Award
Campus Access Dissertation
Low-order polynomial models often do not fit curvilinear relationships well. Even if these models provide adequate fit, there are often problems with error assumptions and, with quadratic models, perhaps a spurious minimum or maximum in the fitted surface. With limited data, nonparametric regression is not a realistic option; non-linear models may provide a better fit than polynomials, but in the absence of any guiding theory may be very labor-intensive. Here, we study the use of simple power transformations of the response and the predictors as originally proposed by Box and Tidwell (1962). Using these transformed variables, four types of fractional polynomial models are considered: a first-order model, a first-order model with cross-product terms, a second-order model, and a second-order model with cross-product terms. It is shown that minor identifiability problems for second-order models can be circumvented by restricting the values of the power parameter. A user-friendly R (R Development Core Team, 2010) function was written to perform the maximum likelihood estimation of the transformation parameters. This function allows the user to fix any of the transformation parameters; it produces model diagnostics and optionally includes a back-transformed plot of the fitted surface. In each of the many examples used as test cases, the best fractional polynomial model was a clear improvement over the untransformed polynomial model.
While performing the steepest ascent procedure in a sequential experimentation framework, the researcher is searching for a maximum (or minimum) in the response surface. Traditionally the search stops when overall curvature is detected; unfortunately, the presence of curvature in the response surface does not necessarily imply that we are close to an optimizer. Here, two new tests for "bumpature" are introduced, called the Delta test and Max test. These two new tests are compared to conventional tests used in sequential experimentation through Monte Carlo simulation for surfaces with 1 or 2 predictors. Using the classical sequential experimentation techniques (the test for overall curvature and the lack-of-fit test) on a curved but monotonic surface, the testing shows that the researcher will consistently be mislead into believing that the desired maximum is within the range of their current experimental design. The two new tests of "bumpature" do not have this drawback, and are able to detect "bumps" when these are present. However, they are quite conservative. They should be used in conjunction with the standard techniques to improve the success of steepest ascent sequential experimentation processes.
Chalmers, N.(2011). Fractional Polynomial Response Surfaces and Tests for "Bumpature". (Doctoral dissertation). Retrieved from https://scholarcommons.sc.edu/etd/2643