Variance Dispersion Graphs Research Articles

Evaluation of Statistical Designs for Experiments Involving Noise Variables

In process robustness studies, it is desirable to simultaneously minimize the influence of noise factors on the system and to determine the levels of controllable factors that will optimize the overall response or outcome. A methodology for evaluating designed experiments that involve both controllable and uncontrollable, or noise, factors is outlined and presented in this paper. Two variance expressions are developed for evaluating competing experimental design strategies. The maximum, average, and minimum scaled prediction error variances resulting from the models developed are displayed visually on variance dispersion graphs. The scaled prediction error variances account for mean model errors as well as variation transmitted to the process by noise variables.

Difference Variance Dispersion Graphs for Comparing Response Surface Designs with Applications in Food Technology

SUMMARY Variance dispersion graphs have become a popular tool in aiding the choice of a response surface design. Often differences in response from some particular point, such as the expected position of the optimum or standard operating conditions, are more important than the response itself. We describe two examples from food technology. In the first, an experiment was conducted to find the levels of three factors which optimized the yield of valuable products enzymatically synthesized from sugars and to discover how the yield changed as the levels of the factors were changed from the optimum. In the second example, an experiment was conducted on a mixing process for pastry dough to discover how three factors affected a number of properties of the pastry, with a view to using these factors to control the process. We introduce the difference variance dispersion graph (DVDG) to help in the choice of a design in these circumstances. The DVDG for blocked designs is developed and the examples are used to show how the DVDG can be used in practice. In both examples a design was chosen by using the DVDG, as well as other properties, and the experiments were conducted and produced results that were useful to the experimenters. In both cases the conclusions were drawn partly by comparing responses at different points on the response surface.

Variance Dispersion Graphs for Comparing Blocked Response Surface Designs

In this paper, we extend the use of the variance dispersion graph (VDG) to experiments in which the response surface (RS) design must be blocked. Through several examples we evaluate the prediction performances of RS designs in non-orthogonal block designs compared with the equivalent unblocked designs and orthogonally blocked designs. These examples illustrate that good prediction performance of designs in small blocks can be expected in practice. Most importantly, we show that the allocation of the treatment set to blocks can seriously affect the prediction properties of designs; thus, much care is needed in performing this allocation.

Quantile plots of the prediction variance for response surface designs

Giovannitti-Jensen and Myers (1989) used variance dispersion graphs (VDG's) to assess the overall prediction capability of a response surface design inside a region of interest, R. The VDG's consist of plots of the maximum and minimum prediction variance values on concentric spheres inside R against their radii. These plots provide useful information about the amount of dispersion in the prediction variance throughout the region R. They do not, however, account for values of the prediction variance on a sphere other than its maximum and minimum. Quite often, information concerning the prediction variance distribution on a sphere will provide a more comprehensive and accurate assessment of the quality of prediction than can be obtained from the VDG's. In this article, we describe such a distribution by using a plot of its quantiles. A method for deriving these quantiles is given. Two examples are presented to illustrate the utility of the proposed approach.

Quantile plots of the average slope variance for response surface designs

This article is concerned with the comparison of designs on the basis of their capabilities to estimate the slope of a response function. Jang and Park (1993) used slope variance dispersion graphs (SVDG's) to assess the so-called slope rotatability over all directions of a response surface design. Although these graphs provide useful information about the dispersion in the average slope variance (ASV) throughout a region of interest, their utility is somewhat limited since they only account for the maximum and minimum values of the ASV on a hypersphere inside the region of interest. In this article, a method is proposed for estimating the ASV distribution on a hypersphere by using a plot of its quantiles. This method provides more information than can be obtained from considering only the maximum and minimum values of the ASV. Two examples are given to illustrate the usefulness of the proposed method.

Spherical Prediction-Variance Properties of Central Composite and Box-Behnken Designs

For quadratic regression on the hypercube, a single-number criterion, such as a G efficiency that is based on the prediction variance, is often included as one of the criteria when selecting a response surface design, As an alternative to the single-number-criterion approach, the variance dispersion graph, presented by Giovannitti-Jensen and Myers, is a graphical technique for evaluating prediction-variance properties throughout the experimental region. Three properties of interest are the maximum, minimum, and average spherical prediction variances, given the spherical radius. As an alternative to the computer-based approach requiring an optimization algorithm to evaluate these properties, the maximum, minimum, and spherical prediction variances for central composite and Box–Behnken designs can be determined analytically and are functions only of the radius and the design parameters. These functions yield the exact values of the spherical prediction-variance properties of central composite and Box–Behnke...

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 resea...

Corrigenda

Click to increase image sizeClick to decrease image sizeThis article refers to:Confidence Bounds for Capability IndicesStudentDistributional and Inferential Properties of Process Capability IndicesA Review and Analysis of Cause-Selecting Control ChartsA Computer Program for Generating Variance Dispersion Graphs

A Computer Program for Generating Variance Dispersion Graphs

The variance dispersion graph has been shown to be a powerful tool for evaluating experimental designs. A FORTRAN computer program is given that generates these plots. The program finds the maximum...

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.

Variance Dispersion Properties of Second-Order Response Surface Designs

Often, second-order response surface designs are chosen on the basis of a single-valued criterion such as D- or G-optimality. While such criteria provide a useful basis for selecting designs, they often fail to convey the true nature of the design's support of the fitted model in terms of prediction properties over a region of interest. As an alternative to a single-valued criterion, we propose variance dispersion graphs to compare various second-order response surface designs. This graphical procedure is used to examine the relative strengths and weaknesses of such popular designs as the central composite, Box-Behnken, and small composite. Areas in the region of interest where prediction is relatively good and relatively poor are discussed, and designs which have good overall performance are highlighted. A summary comparison of such saturates or near-saturates designs as the hybrid, Box-Draper, and Notz designs is given together with recommendations concerning the proper choice among these designs. Finally, we show how the use of the D-optimality criterion can lead to misleading results when the user is truly interested in response surface prediction.