Among the numerous design evaluation criteria for response surface designs is the IV-criterion that is based on the average prediction variance (APV). In this paper the various APV measures generated by computing packages and their relationships to the IV-criterion are summarized and critiqued. The computergenerated APV criteria require averaging over either a fixed set or a random set of evaluation points. When a fixed set of points is used, the APV measure will be larger than the corresponding IV-criterion value and could lead to the selection of an inferior design based on the IV-criterion. A simple approach that provides exact evaluation of the IV-criterion value for response surface designs on the hypercube is discussed.
Key Words: Average Prediction Variance, Integrated Prediction Variance.
By JOHN J. BORKOWSKI, Montana State University, Bozeman, MT 59717
IN many research projects, experiments are run to describe relationships between design variables x1, x2, ..., xk and the response y of interest. In many industrial situations, a response surface design is implemented that will enable the experimenter to fit the second-order model given by
After considering temporal, economic, and physical constraints, experimenters often use design optimality criteria to evaluate a design prior to running a proposed experiment. For a design having an expanded design matrix X, the D, A, E, G, and IV design evaluation criteria are based on properties of Dr. Borkowski is an Associate Professor of Statistics in the Department of Mathematical Sciences. His email address (X'X)-1. For a detailed discussion of these five criteria, see Atkinson and Donev (1992, Chapter 10). In this paper, only the integrated average prediction variance (or IV) criterion is studied. The IV-criterion is based on the prediction variance
This paper is organized as follows. First, the MPCA method is outlined along with a brief explanation of PCA. Then the existing methods for predicting future observations are reviewed and the proposed method is presented. Next, a case study on a PVC batch process is described to demonstrate the proposed method. Finally, the performance of the proposed method is discussed, and concluding remarks are given. See Box and Draper (1959), Myers (1971, Chapter 9), and Meyer and Nachtsheim (1995) for discussions of this criterion. A design which minimizes IV has been referred to as IV-optimal, as well as Q-optimal (Myers and Montgomery (1995)), V-optimal (Welch (1984), Atkinson (1988), and Atkinson and Donev (1992)), and I-optimal (Haines (1987), Nachtsheim (1987), and SAS (1995)).
While published academic research may give exact IV-criterion values, statistical computing packages do not. Many packages, however, do include an average prediction variance (APV) measure in the software output. Yet little or no documentation regarding their APV measures is provided. For designs in the hypercube that are used to fit the second-order model in Equation (1), the integration to generate an exact evaluation of the IV-criterion is not complicated because V(x) will be a quartic polynomial in k-variables. It is important to note both the conspicuous absence of exact IV-criterion values and the variety of different APV measures adopted by different software packages.
Although some statisticians might know that statistical packages provide APV measures, some researchers and many practitioners in industry may not be familiar with the specific criteria provided as output. For those who believe that the measures provided by statistical packages are approximations of the IV-criterion, it is shown in this paper how inaccurate these measures can be as approximations. It will be shown that (i) most of these measures will be greater than the IV-criterion value, (ii) the differences between these measures and the true IV-criterion value can remain very large even when the approximation is based on very large sets of points, and (iii) the measure provided is dependent on which software package is used.
The common software approach is to provide an APV measure that is the average of V(x) over a subset of points in the design space. This is discussed in the next two sections followed by the generation of exact IV values using Matlab software (The Mathworks 2000). For comparison purposes, four different types of composite designs will be studied for 3, 4, and 5 design factors:
In this study, one centerpoint CCDs, PBCDs, and SCDs and the saturated Notz designs are considered. For a discussion of response surface methodology and these designs, see Box and Draper (1987, Chapter 7), Myers and Montgomery (1995, Chapters 7 and 8), and Khuri and Cornell (1996, Chapter 4).
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