Ronald D. Snee

Validation of Regression Models: Methods and Examples

Methods to determine the validity of regression models include comparison of model predictions and coefficients with theory, collection of new data to check model predictions. comparison of results with theoretical model calculations, and data splitting or cross-validation in which a portion of the data is used to estimate the model coefficients, and the remainder of the data is used to measure the prediction accuracy of the model. An expository review of these methods is presented. It is concluded that data splitting is an effective method of model validation when it is not practical to collect new data to test the model. The DUPLEX algorithm, developed by R. W. Kennard, is recommended for dividing the data into the estimation set and prediction set when there is no obvious variable such as time to use as a basis to split the data. Several examples are included to illustrate the various methods of model validation.

Graphical Display of Means

Abstract A graphical procedure for the display of treatment means that enables one to determine the statistical significance of the observed differences is presented. It is shown that the widely used least significant difference and honestly significant difference statistics can be used to construct plots in which any two means whose uncertainty intervals do not overlap are significantly different at the assigned probability level. It is argued that these plots, because of their straightforward decision rules, are more effective than those that show the observed means with standard errors or confidence limits. Several examples of the proposed displays are included to illustrate the procedure.

Some Aspects of Nonorthogonal Data Analysis: Part I. Developing Prediction Equations

It is known that regression results can be misleading when the predictor variables (xs) are highly correlated (nonorthogonal). The objective of this paper is to present some guidelines for deciding when the correlations among the xs are so large that ..

Test Statistics for Mixture Models

Regression models of the forms proposed by Scheffe and by Becker have been widely and usefully applied to describe the response surfaces of mixture systems. These models do not contain a constant term. It has been common practice to test the statistical significance of these mixture models by the same statistical procedures used for other regression models whose constant term is absent (e.g., because the regression must pass through the origin). In this paper we show that the common practice produces misleading reslllts for mixtures. The mixture models require a different set of F, R 2, and R A 2 statistics. The correct mixture statistics correspond to a physically consistent null hypothesis and are also consistent with the expression of the mixture model in the older “slack-variable” form. An illustrative example is included.

Experimental Designs for Quadratic Models in Constrained Mixture Spaces

The development of experimental designs for constrained mixtttre systems in which the response can be described by a qlladratic model is discussed. It is shown that efficient designs can be constructed from a srtbset of the vertices and centroids of the feasible region. For systems with five or more components, it is recommended that the Wynn and exchange algorithms be used to select a design from the available vertires and centroids. Illustrative examples are included.

Techniques for the Analysis of Mixture Data

Techniques for the analysis of mixture data are discussed. It is shown that mixture models often can be simplified by observing near equalities among the estimated coefficients. Expressing the components in terms of pseudocomponents will sometimes improve the fit of Beckers models [1] and improve the numerical analysis. Scatter plots are useful in detecting dominant components in a mixture system. The ratio model is shown to be a useful alternative to the polynomial and Beckers models. Examples are presented to illustrate these methods of analysis.

Design and Analysis of Mixture Experiments

Techniques for the design and analysis of mixture experiments are discussed. Emphasis is on the practical aspects of selecting the appropriate mixture design, development and interpretation of mixture models, and analysis of mixture data. That the class..

Screening Concepts and Designs for Experiments with Mixtures

A strategy for determining the most important components in a mixture system is developed. The proposed screening designs are usefrd in those situations where the number of candidate components, q, is large. Our simplex screening designs, which contain 2q + 1 or 3q + 1 points, are recommended when it is possible to experiment over the total composition range of all components (0–100%) or the experimental region can be expressed as a simplex in terms of pseudocomponents. We recommend extreme vertices screening designs, which contain approximately q + 10 points, when some or all of the components are subject to upper and lower constraints. Examples are included to illustrate the proposed designs and associated analyses.

Design of a statistical method for the analysis of mutagenesis at the hypoxanthine-guanine phosphoribosyl transferase locus of cultured Chinese hamster ovary cells

Mutagenesis data collected in the mammalian cell CHO/HGPRT assay were analyzed to study the distribution of the experimental errors associated with the test. The data neither followed the widely assumed Poisson distribution nor satisfied the usual statistical assumptions of normality and homogeneous variance of experimental errors. We transformed the data by using the power formula Y = (X + A) gamma where X is the observed mutation frequency, Y is the transformed frequency, and A and gamma are constants determined by the procedure of Box and Cox. Setting A = 1 and gamma = 0.15 we produced transformed values for which the assumptions of homogeneous variance and normal distribution were satisfied. This transformation enables one to properly use Students t-test and dose-response analysis of variance to analyze CHO/HGPRT results. The experimental design for CHO/HGPRT mutagenesis assays is also discussed.

Developing Blending Models for Gasoline and Other Mixtures

The construction of gasoline blending models is discussed to illustrate some of the practical problems encountered in mixture experimentation. Attention is focused on the use and modification of the simplex and extreme vertices designs in the development of blending models. The XVERT, WYNN, EXCHANGE, CONSIM, and CADEX algorithms are shown to be useful aids in constructing linear and quadratic model designs when the region of feasible blends is restricted by single-component and multiple-component constraints. The evaluation of competing models and the use of the quadratic blending model in conjunction with linear programming calculations are also discussed. The methodology is general and can be used in all types of mixture experiments and product formulation studies, Examples are included to illustrate the use of the design algorithms and models.