A Biologically Meaningful Approach to Response Surface Analysis

Abstract

Response surface analysis is commonly used to summarize the results of a complex biological experiment. The technique produces a quadratic function that relates experimental conditions (factors) to biological response, where the quadratic behavior allows for a possible optimum. Typically, the variables in this function are transformed to give the model greater flexibility. In fisheries literature, exponential transformations (with exponents presumed unknown) have been widely applied. Although properties of the quadratic model are well documented, the surface features that result from exponential transformations are not. Frequently, the practitioner obtains parameter estimates with widely varying magnitudes and contour plots that exhibit strange and confusing distortions. The parameters may bear no apparent relationship to the surface and, consequently, may appear to be useless pieces of information. This paper defines new parameters for the quadratic model with exponential transformations and demonstrates precisely the role of each parameter in determining the shape of the surface. The parameters have the added advantage that numerical methods to estimate them perform efficiently, thus avoiding convergence problems sometimes encountered in the past. Statistical error is discussed, analytically and intuitively in terms of model parameters. Evidence is given that one of the transformations may be wrong in fisheries applications, and a reasonable alternative is proposed. Worked examples from historical literature illustrate all aspects of the new approach and show how both quantitative and qualitative errors can be avoided through biological understanding of the significance of every parameter.