Darcy P. Mays

Testing for Additivity in Chemical Mixtures Using a Fixed-Ratio Ray Design and Statistical Equivalence Testing Methods

Fixed-ratio ray designs have been used for detecting and characterizing interactions of large numbers of chemicals in combination. Single-chemical dose-response data are used to predict an “additivity curve” along an environmentally relevant ray. A “mixture curve” is estimated from the mixture dose-response data along the ray. A test of additivity is equivalent to a test of coincidence of these two curves, which is based on the traditional hypothesis testing framework that assumes additivity in the null hypothesis and rejects with evidence of interaction. However, failure to reject may be due to lack of statistical power, making the claim of additivity problematic. As a solution we have developed rigorous methodology to test for additivity using statistical equivalence testing logic in which additivity is claimed based on pre-specified biologically important additivity margins, if the data support such a claim. Using the principle of confidence interval inclusion, a confidence region about the difference of meaningful functions of model parameters from the mixture model and that predicted under additivity is computed. When the confidence region is completely contained within the additivity margins then additivity is claimed with a Type I error rate chosen a priori to be some acceptably small value. The method is illustrated using an environmentally relevant fixed-ratio mixture of nine haloacetic acids where cytotoxic response is measured.

Optimal Response Surface Designs in the Presence of Dispersion Effects

Standard response surface design procedures typically assume homogeneous variance throughout the design region, and the designs used in practice usually reflect this assumption. With two design variables, a replicated two-level factorial design is D-and..

Bayesian approach for the design and analysis of a two level factorial experiment in the presence of dispersion effects

With linear dispersion effects, the standard factorial designs are not optimal estimation of a mean model. A sequential two-stage experimental design procedure has been proposed that first estimates the variance structure, and then uses the variance estimates and the variance optimality criterion to develop a second stage design that efficiency estimates the mean model. This procedure has been compared to an equal replicate design analyzed by ordinary least squares, and found to be a superior procedure in many situations. However with small first stage sample sizes the variance estiamtes are not reliable, and hence an alternative procedure could be more beneficial. For this reason a Bayesian modification to the two-stage procedure is proposed which will combine the first stage variance estiamtes with some prior variance information that will produce a more efficient procedure. This Bayesian procedure will be compared to the non-Bayesian twostage procedure and to the two one-stage alternative procedures li...

Design and analysis for a two-level factorial experiment in the presence of variance heterogeneity

With heterogeneous variance, the standard factorial designs are not optimal for estimation of a mean model. A sequential two-stage experimental design procedure is proposed that first allows estimation of the variance structure, and then uses the variance estimates and the Q-optimality criterion to develop a second stage design that efficiently estimates the mean model. Other procedures considered include the “equal replicate” design analyzed by ordinary least squares, and the same design analyzed by weighted least squares. Finally, the three procedures are compared for various models and variance structures, and for each case a recommendation is made as to which procedure is preferred.

Two-stage central composite designs with dispersion effects

Traditional response surface methods assume homogeneous variance throughout the design region. To estimate a second order location model, a central composite design is often used. In situations with few design variables, it is possible to have replications at the design locations. Mays (1996) illustrated that an equal replicate central composite design is not optimal if the true heterogeneous variances are known. However, the true heterogeneous variances are rarely known. Hence the variance structure must be estimated, and a weighted least squares type approach is beneficial. This analysis considers a two-stage experimental design procedure in which the first stage estimates the variance structure, and then the second stage augments this first stage design with additional runs that produces the most efficient total design. The integrated variance optimality criterion is used to analyze the two-stage procedure and compare it to a standard equal replicate one-stage design analyzed by ordinary least squares ...

A Study of the Authorship of the Books of Oz Using Nested Linear Models

Abstract The variability of a writers style has been studied in a number of ways by researchers interested in authorship attribution. However, to date, the analysis of variance has not emerged as a technique for distinguishing between writers. This article first employs nested linear models to find lexical variables which discriminate the two authors of the books of Oz. In the second-stage nested model, the chosen variables are used to ascribe the correct author of books treated as anonymous. Having been found to have sharp discriminating capability, the method is then applied to identify the author of the 15th book. The result supports prevailing consensus on the books authorship.

THE IMPACT OF CORRELATED RESPONSES AND DISPERSION EFFECTS ON OPTIMAL THREE LEVEL FACTORIAL DESIGNS

Many experiments require that the experimental runs at specific design locations be completed in sequence, therefore reducing the randomization of the experimental process. Such experiments can create correlated responses, and the goal of this paper is to analyze the effect that such correlated responses together with heterogeneous variance have on optimal designs. Various dispersion and correlation cases are considered, the D- and I-optimal designs determined in each case, and the results analyzed to examine the worth of a standard equal replicate design that is used under ideal conditions.

Near-saturated two-stage designs with dispersion effects

Many experimental processes are quite expensive, and traditional factorial or central composite design experiments are beyond the scope of the experimental budget. Situations in which calibrated equipment must be purchased or batches of material created are two examples. In situations like these, experimenters prefer to use saturated or near-saturated response surface designs, such as Koshal designs, hybrid designs, or small composite designs, to reduce the cost of the experimental process. Applications of such near-saturated designs are abundant, but most are applied to processes that assume homogeneous process variance throughout the design region. However, in many situations the assumption of homogeneous process variance is violated, and instead a unknown heterogeneous variance structure must be considered. Traditional statistical procedures involve the application of weighted least squares to analyze the data from such an experiment. This analysis considers the application of a two-stage experimental ...

Bayesian application to the two-stage near-saturated experimental design method with dispersion effects

Experimental designs are critical in the quality improvement of manufacturing processes and to the development of new processes. Experimental methods are often used to identify the most important controllable variables to the product, and the relationship of the product response to the controllable variables can be determined by experimental data. Traditionally, regression analysis is used to establish an empirical mean model for the product response, and competing experimental designs are evaluated with regard to estimating coefficients in the regression model. However, most traditional procedures involve the assumption of homogeneous process variance, which in many processes is inappropriate. A two-stage experimental design procedure developed by Mays and Myers [Mays, D.P. and Myers, R.H., 1997, Design and analysis for a two-level factorial experiment in the presence of variance heterogeneity. Computational Statistics and Data Analysis, 26, 219–233.] has proven to be beneficial in many heterogeneous variance situations. In the procedure, the first stage estimates the heterogeneous variance structure, and the second stage augments the first stage design to create an efficient design for estimating the mean. However, the first stage variance estimation is not reliable for small first stage experiment sizes, and hence a modified Bayesian two-stage procedure has been proposed for such situations [Mays, D.P. and Myers, R.H., 1996, Bayesian approach for the design and analysis of a two level factorial experiment in the presence of dispersion effects. Communications in Statistics—Theory and Methods, 25, 1409–1428.]. This study applies the Bayesian two-stage procedure to Koshal designs, hybrid designs, and small composite designs and examines the variance estimation in the first stage in order to find the optimal number of replicates to make at each design location. The Bayesian procedure is compared with the non-Bayesian two-stage procedure and with two competing one-stage equal replicate procedures. The effects of the various variance information on the procedures are also investigated.