André I. Khuri

Response surfaces: designs and analyses

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Response surface methodology: 1966–1988

Response sarfxe methodology (RSM) is a collection of tools developed in the 1950s for the purpose of determining optimum operating conditions in applications in the chemical industry. This article reviews the progrrss of RSM in the general areas of experimental design and analysis and indicates how its role has been affected by advanccs in other fields of applied statistics. Current areas of research in RSM are highlighted. and areas for future research are discussed.

Simultaneous Optimization of Multiple Responses Represented by Polynomial Regression Functions

An algorithm is developed for the simultaneous optimization of several response functions that depend on the same set of controllable variables and are adequately represented by polynomial regression models of the same degree. The data are first checked for linear dependencies among the responses. If such dependencies exist, a basic set of responses among which no linear functional relationships exist is chosen and used in developing a function that measures the distance of the vector of estimated responses from the estimated “ideal” optimum. This distance function permits the user to account for the variances and covariances of the estimated responses and for the random error variation associated with the estimated ideal optimum. Suitable operating conditions for the simultaneous optimization of the responses are specified by minimizing the prescribed distance function over the experimental region. An extension of the optimization procedure to mixture experiments is also given and the method is illustrat...

Response Surface Alternatives to the Taguchi Robust Parameter Design Approach

Abstract This department publishes articles of interest to statistical practitioners. Innovative applications of known methodology may be suitable, but sizable case studies should be submitted to other journals. Brief descriptions and illustrations of new developments that are potentially useful in statistical practice are appropriate. Acceptable articles should appeal to a substantial number of practitioners. In the decade of the 1980s much attention was given to the data and analytic and experimental design efforts of Genichi Taguchi. Methodology advocated by Taguchi, often called robust parameter design, gained the interest of practitioners working in industry in quality improvement. Many statisticians in the West have pointed out apparent flaws in the Taguchi approach. As the controversy surrounding Taguchi matures, several investigators have embraced important aspects of parameter design and the result is a collection of alternatives to the Taguchi approach. Some of these alternatives highlight the u...

Design Issues for Generalized Linear Models: A Review

Generalized linear models (GLMs) have been used quite effectively in the modeling of a mean response under nonstandard conditions, where discrete as well as continuous data distributions can be accommodated. The choice of design for a GLM is a very important task in the development and building of an adequate model. However, one major problem that handicaps the construction of a GLM design is its dependence on the unknown parameters of the fitted model. Several approaches have been proposed in the past 25 years to solve this problem. These approaches, however, have provided only partial solutions that apply in only some special cases, and the problem, in general, remains largely unresolved. The purpose of this article is to focus attention on the aforementioned dependence problem. We provide a survey of various existing techniques dealing with the dependence problem. This survey includes discussions concerning locally optimal designs, sequential designs, Bayesian designs and the quantile dispersion graph approach for comparing designs for GLMs.

Variance Components Analysis: A Selective Literature Survey

Summary A survey is given of developments in the area of variance components during the last three decades. The survey covers mainly point estimation, interval estimation, and hypothesis testing concerning the variance components for both balanced and unbalanced models.

Response Surface Methodology and Related Topics

Two-Level Factorial and Fractional Factorial Designs in Blocks of Size Two. Part 2 (Y J Yang & N R Draper) Response Surface Experiments on Processes with High Variation (S G Gilmour & L A Trinca) Random Run Order, Randomization and Inadvertent Split-Plots in Response Surface Experiments (J Ganju & J M Lucas) Statistical Inference for Response Surface Optima (D K J Lin & J J Peterson) A Search Method for the Exploration of New Regions in Robust Parameter Design (G Mero-Quesada & E del Castillo) Response Surface Approaches to Robust Parameter Design (T J Robinson & S S Wulff) Response Surface Methods and Their Application in the Treatment of Cancer with Drug Combinations: Some Reflections (K S Dawson et al.) Generalized Linear Models and Response Transformation (A C Atkinson) GLM Designs: The Dependence on Unknown Parameters Dilemma (A I Khuri & S Mukhopadhyay) Design for a Trinomial Response to Dose (S K Fan & K Chaloner) Evaluating the Performance of Non-Standard Designs: The San Cristobal Design (L M Haines) 50 Years of Mixture Experiment Research: 1955-2004 (G F Piepel) Graphical Methods for Comparing Response Surface Designs for Experiments with Mixture Components (H B Goldfarb & D C Montgomery) Graphical Methods for Assessing the Prediction Capability of Response Surface Designs (J J Borkowski) Using Fraction of Design Space Plots for Informative Comparisons between Designs (C M Anderson-Cook & A Ozol-Godfrey) Concepts of Slope-Rotatability for Second Order Response Surface Designs (S H Park) Design of Experiments for Estimating Differences between Responses and Slopes of the Response (S Huda).

A measure of rotatability for response-surface designs

A measure that quantifies the amount of rotatability in a given response-surface design is introduced in this article. The measure, which is expressible as a percentage, takes the value 100 if and only if the design is rotatable. One of the main advantages of this measure is that it can be used to “repair” a nonrotatable design by the addition of experimental runs that maximize the percent rotatability over a spherical region of interest. Four numerical examples are given to illustrate the applications of this measure.

Response surface models with random block effects

In many experimental situations, a response surface design is divided into several blocks to control an extraneous source of variation. The traditional approach in most response surface applications is to treat the block effect as fixed in the assumed model. There are, however, situations in which it is more appropriate to consider the block effect as random. This article is concerned with inference about a response surface model in the presence of a random block effect. Since this model also contains fixed polynomial effects, it is considered to be a mixed-effects model. The main emphasis of the proposed analysis is on estimation and testing of the fixed effects. A two-stage mixed-model procedure is developed for this purpose. The variance components due to the random block effect and the experimental error are first estimated and then used to obtain the generalized least squares estimator of the fixed effects. This procedure produces the so-called Yates combined intra- and inter-block estimator. By cont...

Robust Parameter Design Using Generalized Linear Mixed Models

In robust parameter design, it is often the case that the quality characteristic is nonnormal. An example in semiconductor manufacturing is resistivity, which typically follows a gamma distribution. It is also common to have models that contain, in addition to fixed polynomial effects, a random effect representing an extraneous source of variation. In this article, we demonstrate the use of generalized linear mixed models (GLMM) for situations in which the response is nonnormal and in which the noise variable is a random effect. We discuss the analysis of the random effects as well as the fixed effects in a fitted model using GLMM techniques. A numerical example from semiconductor manufacturing is provided for illustration.