Response surface designs for the detection of model inadequacy
Jones, Edward Rex
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In many experiments, it is tentatively assumed that the experimental response is related to some independent variables, x, by η₁(x) = x₁’ β₁. However, there is frequently some doubt whether this model adequately approximates the true response function, so a lack of fit test is used as part of the analysis. We suppose that the true response function is η₁(x) = x₁’ β₁ + x₂’ β₂. The power of the usual lack of fit test is a monotone increasing function of the non-centrality parameter λ = σ⁻² β₂’ L β₂, where the positive semi-definite matrix L is determined by the experimental design. We use λ as a measure of a design's ability to detect the higher-order parameters, β₂. This investigation has concentrated on the development and evaluation of criteria for the selection of experimental designs that have good properties for the detection of model inadequacy. We suppose that the inadequacy of our proposed model is measured by the positive definite quadratic form τ = σ⁻² β₂’ T β₂, where T is specified by the experimenter according to his own interests. Two choices for τ are proposed. τ₁ is proposed as a measure of the inherent departure of a response surface from an assumed class of models. Another choice for τ, τ₂, is the ratio of the average squared bias to the sampling variance, σ². Whereas τ₁ measures the inherent departure of a response surface, τ₂ measures the departure of the fitted model from the response surface. As a result, τ₁ is independent of the design, whereas τ₂ depends upon the experimental design. We examine the following criteria: (1) maximize the minimum value of λ for τ = δ, (2) maximize the average value of λ for τ = δ, (3) minimize the maximum value of τ for λ = ρ, and (4) minimize the average value of τ for λ = ρ, where δ and ρ are any positive constants. We show that criteria (1) and (3) are equivalent. They select designs that maximize the minimum characteristic root of [T⁻¹ L]. We also show that criterion (2) selects designs that maximize Tr[T⁻¹ L], and that criterion (4) selects designs that minimize Tr[L⁻¹ T]. In addition, we propose a modification of (2) to allow the experimenter to favor designs that afford greater "protection" in the sense of minimizing τ, and a modification of (4) to allow the experimenter to favor designs that afford greater "detection" in the sense of maximizing λ. We show that all of these criteria, referred to as the Λ(T) criteria, are invariant under non-singular linear transformations of the independent variables provided that τ is invariant to such transformations, and we show that τ₁ and τ₂ are invariant to such transformations. In addition, we obtain several results for rotations of D-optimal and Λ(T)-optimal designs. Optimal designs for all of these criteria are obtained and evaluated for a variety of cases. Primary consideration is given to the use of τ₁ and τ₂ for one and two factor, first-order vs. second-order and second-order vs. third-order, polynomial models. We have found that the Λ(T)-optimal designs generally have much better variance properties than minimum bias designs, lower bias than D-optimal designs, and greater power than either for detecting lack of fit. Criterion (2) was also selected and used as the basis for a more extensive investigation of the types of designs generated by our approach.
- Doctoral Dissertations