Extensions of D-optimal Minimal Designs for Symmetric Mixture Models.

The purpose of mixture experiments is to explore the optimum blends of mixture components, which will provide desirable response characteristics in finished products. D-optimal minimal designs have been considered for a variety of mixture models, including Scheffé's linear, quadratic, and cubic models. Usually, these D-optimal designs are minimally supported since they have just as many design points as the number of parameters. Thus, they lack the degrees of freedom to perform the Lack of Fit tests. Also, the majority of the design points in D-optimal minimal designs are on the boundary: vertices, edges, or faces of the design simplex. IN THIS PAPER EXTENSIONS OF THE D-OPTIMAL MINIMAL DESIGNS ARE DEVELOPED FOR A GENERAL MIXTURE MODEL TO ALLOW ADDITIONAL INTERIOR POINTS IN THE DESIGN SPACE TO ENABLE PREDICTION OF THE ENTIRE RESPONSE SURFACE: Also a new strategy for adding multiple interior points for symmetric mixture models is proposed. We compare the proposed designs with Cornell (1986) two ten-point designs for the Lack of Fit test by simulations.

Minimal plus one point designs for testing lack of fit for some sigmoid curve models.

D-optimal designs for nonlinear models are often minimally supported. They have been frequently criticized for their inability to test for lack of fit. We construct alternative designs to address this issue for some commonly used sigmoid curves, including logistic, probit, and Gompertz models with two, three, or four parameters. For each model, we compare five nonminimally supported designs in terms of their efficiency, and propose designs that are both statistically efficient and practically convenient for practitioners.

Balanced 2(n) factorial designs when observations are spatially correlated.

In this article we focus on the optimal factorial and fractional-factorial designs when observations within blocks are correlated. The topic was motivated by a problem when the pharmaceutical experimenter needed to develop a controlled release, once-daily tablet formulation. Typically, in order to compare different formulations, trials are conducted in healthy human volunteers where each formulation is administered and bioavailability is estimated. Since each subject is administered more than one formulation, the observations within subjects are correlated. Balanced designs for 2(n) factorial experiments when observations within blocks are spatially correlated, AR(1) with positive correlation (rho > 0), are characterized. An explicit construction and analytical proof of balanced designs for both 2(n) full and 2(n-1) fractional factorial experiments is provided. In order to illustrate the construction, two examples using a complete 2(3) factorial and a half replicate of 2(4) factorial experiment are provided. We consider the optimal or near-optimal designs provided by Cheng and Steinberg (1991), Martin et al. (1998c), and Elliott et al. (1999) as the starting point to obtain balanced designs. We compare the relative efficiencies of our balanced designs with these designs.

A note on kinetic modeling of stability data and implications on pooling.

In this note we discuss the relationship between the underlying kinetic model and the statistical (or analytic) model used to study degradation. For small degradation rates, the zeroth, first, and second order statistical models give approximately the same fits and predictions on either the original assay scale or the percent of label claim scale. However, It is shown that the zeroth and second order statistical models artificially induce differential degradation rates across strengths when the percent of label claim response data are analyzed and poolability is not allowed across strengths. The first order model is free of this problem when the true degradation kinetics are first order. We make some recommendations in pooling stability data across strengths.

Comparing distributions of shelf lives for drug products with two components under different designs.

To reduce the cost of stability testing in drug research and development, both bracketing and matrixing designs are recommended in the U.S. Food and Drug Administration's (FDA) guidelines for drug products with a single active ingredient (component). When the drug product contains multiple active components, the naïve approach is to take the minimum of shelf lives obtained from individual components as the labeled shelf life. The purpose of this article is to compare the distributions of the shelf life obtained as the minimum shelf life of two components for full FDA's sampling plan, matrixing, and bracketing designs. The results were obtained based on theoretical considerations and simulation studies.