Whole-Plot Exchange Algorithms for Constructing D-Optimal Multistratum Designs

Multistratum experiments contain several different sizes of experimental units. Examples include split-plot, strip-plot designs, and randomized block designs. We propose a strategy for constructing a D-optimal multistratum design by improving a randomly generated design through a sequence of whole-plot exchanges. This approach preserves the design structure and simplifies updates to the information and is applicable to any multistratum design where the largest-sized experimental unit is either a whole plot or a block. Two whole-plot exchange algorithms inspired by the point-exchange strategies of Fedorov (1972 Fedorov , V. V. ( 1972 ). Theory of Optimal Experiments . New York : Academic Press . [Google Scholar]) and Wynn (1972 Wynn , H. P. ( 1972 ). Results in the theory and construction of D-optimum experimental designs . Journal of the Royal Statistical Society Series B Statistics Methodology 34 : 133 – 147 . [Google Scholar]) are described. The application of the algorithms to several design problems is discussed.     

Maximum and Minimum Prediction Variance for Spherical and Cuboidal Split Plot Designs

The impact of restricted randomization on the information matrix has created challenges for the computation of design optimality criteria. This article focuses on the computation of the maximum and minimum prediction variance for Central Composite (CCD) and Box–Behnken (BBD) split plot designs (SPD). The approach is to analytically determine the exact maximum and minimum prediction variance for both spherical and cuboidal second-order SPD. A particular feature of these analytical functions is that they are functions of the design parameters. Finally, the application of these analytical functions is demonstrated for a CCD SPD.     

Implementation of ridge analysis on quadratic response surfaces

A computer program that performs ridge analysis on quadratic response surfaces is presented in this paper, the primary goal of which is to seek the estimated optimum operating conditions inside a spherical region of experimentation during the stage of process optimization. The computational algorithm is developed based upon the trust-region methods in nonlinear optimization and guarantees the resulting operating conditions to be globally optimal without any priori assumption on the structure of response functions. Under a particular condition termed the "hard case" arising from the trust region literature, the conventional ridge analysis procedure fails to provide a set of acceptable optimum operating settings, yet the proposed algorithm has the capability of locating a pair of non-unique global solutions achieved on an identical estimated response value. Two illustrative examples taken from the response surface methodology (RSM) literature are given to demonstrate the effectiveness and efficiency of the method addressed in the paper.     

The performance of subset response surface designs for estimating third order terms

Response surface designs are widely used in industries like chemicals, foods, pharmaceuticals, bioprocessing, agrochemicals, biology, biomedicine, agriculture and medicine. One of the major objectives of these designs is to study the functional relationship between one or more responses and a number of quantitative input factors. However, biological materials have more run to run variation than in many other experiments, leading to the conclusion that smaller response surface designs are inappropriate. Thus designs to be used in these research areas should have greater replication. Gilmour (2006) introduced a wide class of designs called “subset designs” which are useful in situations in which run to run variation is high. These designs allow the experimenter to fit the second order response surface model. However, there are situations in which the second order model representation proves to be inadequate and unrealistic due to the presence of lack of fit caused by third or higher order terms in the true response surface model. In such situations it becomes necessary for the experimenter to estimate these higher order terms. In this study, the properties of subset designs, in the context of the third order response surface model, are explored.     

Robustness of subset response surface designs to missing observations

Experiments designed to investigate the effect of several factors on a process have wide application in modern industrial and scientific research. Response surface designs allow the researcher to model the effects of the input variables on the response of the process. Missing observations can make the results of a response surface experiment quite misleading, especially in the case of one-off experiments or high cost experiments. Designs robust to missing observations can attract the user since they are comparatively more reliable. Subset designs are studied for their robustness to missing observations in different experimental regions. The robustness of subset designs is also improved for multiple levels by using the minimax loss criterion.     

Direction dependence in a regression line

In this paper, we derive some simple formulae to express the association between two random variables in the case of a linear relationship, One of these representations, the cube of the correlation coefficient, is given as the ratio of the skewness of the response variable to that of the explanatory variable. This result, along with other expressions of the correlation coefficient presented in this paper, has implications for choosing the response variable in a linear regression modelling.     

Robust and diagnostic regression analyses

Graphical methods of diagnostic regression analysis are applied to three examples in which least squares and robust regression analyses give substantially different results. The diagnostic tools lead to the identification of data deficiencies and model inadequacies. The analyses serve as a reminder that robust regressions depend upon the linear model and upon the scale in whicli the response is analysed. The robust analysis may also be sensitive to gross errors in one or more explanatory variables     

Variance Reduction Using Nonlinear Controls and Transformations

Nonlinear regression-adjusted control variables are investigated for improving variance reduction in statistical and system simulations. To this end, simple control variables are piecewise sectioned and then transformed using linear and nonlinear transformations. Optimal parameters of these transformations are selected using linear or nonlinear least-squares regression algorithms. As an example, piecewise power-transformed variables are used in the estimation of the mean for the twovariable Anderson-Darling goodness-of-fit statistic W ₂ ². Substantial variance reduction over straightforward controls is obtained. These parametric transformations are compared against optimal, additive nonparametric transformations obtained by using the ACE algorithm and are shown, in comparison to the results from ACE, to be nearly optimal.