Extended scaled prediction variance optimality for modified central composite design

Robust parameter designs (RPDs) enable the experimenter to discover how to modify the design of the product to minimize the effect due to variation from noise sources. The aim of this article is to show how this amount of work can be reduced under modified central composite design (MCCD). We propose a measure of extended scaled prediction variance (ESPV) for evaluation of RPDs on MCCD. Using these measures, we show that we can check the error or bias associated with estimating the model parameters and suggest the values of α recommended for MCCS under minimum ESPV.     

The robustness of response surface designs with errors in factor levels

ABSRTACT

Since errors in factor levels affect the traditional statistical properties of response surface designs, an important question to consider is robustness of design to errors. However, when the actual design could be observed in the experimental settings, its optimality and prediction are of interest. Various numerical and graphical methods are useful tools for understanding the behavior of the designs. The D- and G-efficiencies and the fraction of design space plot are adapted to assess second-order response surface designs where the predictor variables are disturbed by a random error. Our study shows that the D-efficiencies of the competing designs are considerably low for big variance of the error, while the G-efficiencies are quite good. Fraction of design space plots display the distribution of the scaled prediction variance through the design space with and without errors in factor levels. The robustness of experimental designs against factor errors is explored through comparative study. The construction and use of the D- and G-efficiencies and the fraction of design space plots are demonstrated with several examples of different designs with errors.     

Evaluation of Response Surface Designs in Presence of Errors in Factor Levels

Experimenters are often confronted with the problem that errors in setting factor levels cannot be measured. In the robust design scenario, the goal is to determine the design that minimizes the variability transmitted to the response from the variables’ errors. The prediction variance performance of response surface designs with errors is investigated using design efficiency and the maximum and minimum scaled prediction variance. The evaluation and comparison of response surface designs with and without errors in variables are developed for second order designs on spherical regions. The prediction variance and design efficiency results and recommendations for their use are provided.     

Minimum,maximum, and average spherical prediction variances for central composite and box-behnken designs

Single value design optimality criteria are often considered when selecting a response surface design. An alternative to a single value criterion is to evaluate prediction variance properties throughout the experimental region and to graphically display the results in a variance dispersion graph (VDG) (Giovannitti-Jensen and Myers (1989)). Three properties of interest are the spherical average, maximum, and minimum prediction variances. Currently, a computer-intensive optimization algorithm is utilized to evaluate these prediction variance properties. It will be shown that the average, maximum, and minimum spherical prediction variances for central composite designs and Box-Behnken designs can be derived analytically. These three prediction variances can be expressed as functions of the radius and the design parameters. These functions provide exact spherical prediction variance values eliminating the implementation of extensive computing involving algorithms which do not guarantee convergence. This research is concerned with the theoretical development of these analytical forms. Results are presented for hyperspherical and hypercuboidal regions.     

Extension of Central Composite Design for Second-Order Response Surface Model Building

The central composite design (CCD) is perhaps the most popular class of second-order response surface designs. Even though the CCDs are popular for response surface designs, this class of design has some limitations such as it does not sometimes possess good statistical properties, and it does not fit complicated models well. In this article, we propose extended central composite designs (ECCDs) to overcome these limitations. We compare ECCDs with CCDs in terms of average prediction variance, and find that ECCDs are better than CCDs.     

Finding maximum G-criterion values for central composite designs on the hypercube

For quadratic regression on the hypercube, G—efficiencies are often used in the selection process of an experimental design. To calculate a design's G—efficiency, it is necessary to maximize the prediction variance over the experimental design region. However, it is common to approximate a G—efficiency. This is achieved by calculating the prediction variances generated from a subset of points in the design space and taking the maximum to estimate the maximum prediction variance. This estimate is then applied to approximate the G—efficiency. In this paper, it will be shown that over the class of central composite designs (CCDs) on the hypercube. the prediction variance can be expressed in a closed-form. An exact value of the maximum prediction variance can then be determined by evaluating this closed-form expression over a finite subset of barycentric points. Tables of exact G—efficiencies will be presented. Design optimality criteria, quadratic regression on the hypercube, and the structures of the design matrix X, X'X, and (X'X)^?1 for any CCD will be discussed.     

Fraction of design space plots for generalized linear models

The use of graphical methods for comparing the quality of prediction throughout the design space of an experiment has been explored extensively for responses modeled with standard linear models. In this paper, fraction of design space (FDS) plots are adapted to evaluate designs for generalized linear models (GLMs). Since the quality of designs for GLMs depends on the model parameters, initial parameter estimates need to be provided by the experimenter. Consequently, an important question to consider is the design's robustness to user misspecification of the initial parameter estimates. FDS plots provide a graphical way of assessing the relative merits of different designs under a variety of types of parameter misspecification. Examples using logistic and Poisson regression models with their canonical links are used to demonstrate the benefits of the FDS plots.     

Slope-rotatable designs with equal maximum directional variance for second order response surface models

The concept of sloperotaiability with equal maximum directional vari ance for second order response surface models is introduced as a new design property. This requires that the maximum variance of the estimated slope over all possible directions be only a function of p, which is the distance from the design originif is shown that a rotatable design satisfies this property Also, minimization of tiie maximum variance of the estimated slope over all possible directions is proposed as a new design optirnality criterion, and op¬timal designs are called slope-directional minirnax designs. For the class of cquiradial designs, the slope-directional minirnax designs are compared with D— optimal designs.     

Comparison of designs for multivariate generalized linear models

The purpose of this paper is to discuss response surface designs for multivariate generalized linear models (GLMs). Such models are considered whenever several response variables can be measured for each setting of a group of control variables, and the response variables are adequately represented by GLMs. The mean-squared error of prediction (MSEP) matrix is used to assess the quality of prediction associated with a given design. The MSEP incorporates both the prediction variance and the prediction bias, which results from using maximum likelihood estimates of the parameters of the fitted linear predictor. For a given design, quantiles of a scalar-valued function of the MSEP are obtained within a certain region of interest. The quantiles depend on the unknown parameters of the linear predictor. The dispersion of these quantiles over the space of the unknown parameters is determined and then depicted by the so-called quantile dispersion graphs. An application of the proposed methodology is presented using the special case of the bivariate binary distribution.     

Response surface design evaluation and comparison

Designing an experiment to fit a response surface model typically involves selecting among several candidate designs. There are often many competing criteria that could be considered in selecting the design, and practitioners are typically forced to make trade-offs between these objectives when choosing the final design. Traditional alphabetic optimality criteria are often used in evaluating and comparing competing designs. These optimality criteria are single-number summaries for quality properties of the design such as the precision with which the model parameters are estimated or the uncertainty associated with prediction. Other important considerations include the robustness of the design to model misspecification and potential problems arising from spurious or missing data. Several qualitative and quantitative properties of good response surface designs are discussed, and some of their important trade-offs are considered. Graphical methods for evaluating design performance for several important response surface problems are discussed and we show how these techniques can be used to compare competing designs. These graphical methods are generally superior to the simplistic summaries of alphabetic optimality criteria. Several special cases are considered, including robust parameter designs, split-plot designs, mixture experiment designs, and designs for generalized linear models.     

Graphical methods for evaluating ridge regression estimator in mixture experiments

When the component proportions in mixture experiments are restricted by lower and upper bounds, multicollinearity appears all too frequently. Thus, we can suggest the use of ridge regression as a mean for stabilizing the coefficient estimates in the fitted model. We propose graphical methods for evaluating the effect of ridge regression estimator with respect to the predicted response value and the prediction variance.     

Additional comparisons of designs to estimate variance components in a two-way classification model

In this paper, the research of Muse and Anderson is extended to include additional comparisons of designs, featuring planned unbalance, for the estimation of variance components in a two-way cross classification model. Their results are extended to Include the following: (i) a small sample study of the original off-diagonal (OD) design and (ii) an asymptotic maximum likelihood investigation of three modifica-tions of the balanced diagonal rectangles (BD) design and one modification of the 01) design to permit the estimation of row, column, interaction and error variance components. Also a general iterative least.     

Test and prediction in factorial models with independent variance estimates

The multiple inference character of several tests in the same application is usually taken into consideration by requiring that the tests have a multiple level of significance. Also, a prediction problem in an application with several possible predictor variables requires that the multiple inference character of the problem be considered. This is not being done in the methods commonly used to choose predictor variables. Here, we discuss both the test and prediction methods in two-level factorial designs and suggest a principle for choosing variables which is based on multiple inference thinking. By an example use demonstrated that the principle proposed leads to the use of fewer prediction variables than does the Akaike method.     

Some adjusted orthogonal,variance balanced,multidimensional block designs

A multidimensional block design (MBD) is an experimental design with d > 1 blocking criteria geometrically represented as a d-dimensional lattice with treatment varieties assigned to some or all nodes of the lattice. Intrablock analysis of variance tables for some special classes of two- and three-dimensional block designs with some empty nodes are given. Design plans and efficiencies for 31 two-dimensional designs, each universally optimal in defined classes of designs, and 7 three-dimensional designs, each nearly optimal in defined classes of designs, are listed in the appendices. A need for such designs is apparent when the blocking criteria are implemented successively and empty nodes do not represent wasted experimental units.     

Some remarks about the gasser-sroka-jennen-steinmetz variance estimator

This article proposes some simplifications of the residual variance estimator of Gasset, Sroka, and Jeneen-Steinmetz (GSJ, 1986) which is often used in conjunction with non parametric regression. The GSJ estimator is a quadratic form of the data, which depends on the relative spacings of the design points. When the errors are independent, identically distributed Gaussian variables, and the true regression curve is flat, the estimate is distributed as a weighted sum of x² variables. By matching the first two moments, the distribution can be approximated by a x² with degrees of freedom determined by the coefficients of the. quadratic form. Computation of the estimated degrees of freedom requires computing the trace of the square of an n x n matrix, where n is the number of design points. In this article, (n-2)/3 is shown to be a conservative estimate of the approximate degrees of freedom, and (n-2)/2 is shown to be conservative for many designs. In addition, a simplified version of the estimator is shown to be asymptotically equivalent, under many conditions.     

Nonproper variance balanced designs and optimality

This communication deals with the construction and optimality of non-proper (unequal block sized) variance balanced (VB) designs obtainable under linear homoscedastic normal model. Several methods of construction of non-proper VB designs have been given. Some constructed designs are universally optimal non-proper variance balanced designs.     

A measure and a graphical method for evaluating slope rotatability in response surface designs

A measure for evaluating slope rotatability over all directions in response surface designs, is proposed. This measure is used to form slope variance dispersion graph evaluating the overall slope rotatability and the slope estimation capability of an experimental design throughout the region of interest. This graph allows for an easy comparison of competing designs.     

An experimental design criterion for minimizing meta-model prediction errors applied to die casting process design

Summary. We propose the expected integrated mean-squared error (EIMSE) experimental design criterion and show how we used it to design experiments to meet the needs of researchers in die casting engineering. This criterion expresses in a direct way the researchers' goal to minimize the expected meta-model prediction errors, taking into account the effects of both random experimental errors and errors deriving from our uncertainty about the true model form. Because we needed to make assumptions about the prior distribution of model coefficients to estimate the EIMSE, we performed a sensitivity analysis to verify that the relative prediction performance of the design generated was largely insensitive to our assumptions. Also, we discuss briefly the general advantages of EIMSE optimal designs, including lower expected bias errors compared with popular response surface designs and substantially lower variance errors than certain Box–Draper all-bias designs.     

Using quantile plots of the prediction variance for comparing designs for a constrained mixture region: an application involving a fertilizer experiment

Summary. Vining and co-workers have used plots of the prediction variance trace (PVT) along the so-called prediction rays to compare mixture designs in a constrained region R . In the present paper, we propose a method for describing the distribution of the prediction variance within the region R by using quantile plots. More comprehensive comparisons between mixture designs are possible through the proposed plots than with the PVT plots. The utility of the quantile plots is illustrated with a four-component fertilizer experiment that was initiated in São Paulo, Brazil.