Norman R. Draper

D-Optimality for Regression Designs: A Review

After stating the model and the design problem, we briefly present the results for regression design prior to the work of Kiefer and Wolfowitz. We then review the major results of Kiefer and Wolfowitz, particularly those on the theory of design, as well as the way the criterion has been extended to non-linear models. Finally, we discuss algorithms for constructing D-optimum designs.

Factorial Designs, the |X′X| Criterion, and Some Related Matters

Two of the basic approaches to choosing an n-point experimental design in many industrial situations are (i) to set down a simple factorial or fractional factorial design in the factors being studied, or (ii) to choose a design based on the well-known |X′X| criterion. Experimenters often prefer (i) due to its simplicity; our viewpoint here is that (ii) is much better. We first indicate some situations for which (when all the factors are restricted to a cuboidal region) the factorial approach is optimal, as judged by the |X′X| criterion, but the assumed models are often not sensible ones in practical work. We then examine what (similarly restricted) designs are optimal under the |X′X| criterion for the standard linear models of first and second order; because of the very rapid increase in computational difficulties, we consider only “cube plus star” type designs for k ≥ 3 (except for k = 3, n = 10). In spite of computational requirements, we recommend use of the |X′X| criterion in general rather than the i...

Ridge Regression and James-Stein Estimation: Review and Comments

The literature of ridge regression and James-Stein estimation is broadly reviewed, and critical comments are interpolated on a number of papers. The authors also express their viewpoints on ridge regression and their antipathy to its mechanical use.

Projection properties of Plackett and Burman designs

The projection properties of the 2 R q–p fractional factorials are well known and have been used effectively in a number of published examples of experimental investigations. The Plackett and Burman designs also have interesting projective properties, knowledge of which allows the experimenter to follow up an initial Plackett and Burman design with runs that increase the initial resolution for the factors that appear to matter and thus permit efficient separation of effects of interest. Projections of designs into 2–5 dimensions are discussed, and the 12-run case is given in detail. A numerical example illustrates the practical uses of these projections.

Response-surface designs for quantitative and qualitative variables

When some variables are quantitative and some qualitative in a response-surface context, standard designs may not be suitable. The reasons for this are illuminated. Some alternative designs are discussed. Every n-point design provides n linearly independent estimation contrasts. Some of these, p say, are needed to estimate the p parameters of the postulated model. The remaining n — p linearly independent estimation contrasts are available to estimate pure error (if used) and to test for lack of fit, either overall or in particular ways. The key to choosing a good design is to use the available degrees of freedom well, given certain assumptions about the model to be fitted. When there is also uncertainty about the model assumptions, dogmatic design advice is not possible. Sound guidelines are available, however, and these are presented and illustrated.

Small response-surface designs

Standard composite designs for fitting second-order response surfaces typically have a fairly large number of points, especially when k is large. In some circumstances, it is desirable to reduce the number of runs as much as possible while maintaining the ability to estimate all of the terms in the model. We first review prior work on small composite designs and then suggest some alternatives for k ≤ 10 factors. In some cases, even minimal-point designs are possible.

Influential Observations and Outliers in Regression

Statistics offered by Cook (1977) and Andrews and Pregibon (1978) purport to reveal influential observations in a regression analysis. Detailed examination of these statistics shows that two different types of influence are being measured and this is illustrated with examples derived from a set of data given by Mickey, Dunn, and Clark (1967). Recommendations are given for obtaining the best use of the statistics available.

“Ridge Analysis” of Response Surfaces

In a 1959 paper, A. E. Hoerl discussed a method for examining a second order response surface. Thii paper provides a mathematically simpler derivation of the technique and proofs of some stated properties.

Transformations: Some Examples Revisited

Modern computing equipment is extremely fast and can also provide graphical out-put. Prior to the computer era, problems were often formulated in terms of a single numerical criterion which could be handled conveniently on a desk calculator. Now several aspects of a problem or several criteria can be considered simultaneously and a more flexible attitude adopted. The situation then can often be easily understood by the experimenter, and compromise decisions can be made by him. In this paper we consider the particular problem of transforming data from this viewpoint by re-examining some specific published examples.

Small Composite Designs

Small second-order composite designs were suggested by Hartley (1959). Westlake (1965) provided even smaller designs for k = 5, 7, and 9 factors, for which intricate construction methods were needed. Here, simple designs formed using Plackett and Burman (1946) designs are offered for k = 5,7, and 9. Designs with one run fewer than Westlakes for k = 5 and 7 and three fewer for k = 9 are feasible by deleting repeat points that occur in some of the designs.