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.     

Evaluation of Response Surface Designs in Presence of Errors in Factor Levels

Experimenters are often confronted with the problem that errors in setting factor levels cannot be measured. In the robust design scenario, the goal is to determine the design that minimizes the variability transmitted to the response from the variables’ errors. The prediction variance performance of response surface designs with errors is investigated using design efficiency and the maximum and minimum scaled prediction variance. The evaluation and comparison of response surface designs with and without errors in variables are developed for second order designs on spherical regions. The prediction variance and design efficiency results and recommendations for their use are provided.     

Graphical evaluation of robust parameter designs based on extended scaled prediction variance and extended spherical average prediction variance

For any response surface design, there are locations in the design region where responses are estimated well and locations where estimation is relatively poor. Consequently, graphical evaluation—such as variance dispersion graphs and the fraction of design space—is used as an alternative to a single-valued criterion. Such plots are used to investigate and compare the prediction capabilities of certain response surface designs currently available to the researcher. In this article, we propose the extended scaled prediction variance and extended spherical average prediction variance as prediction methods. We also illustrate how graphical methods can be employed to evaluate robust parameter designs.     

Finding maximum G-criterion values for central composite designs on the hypercube

For quadratic regression on the hypercube, G—efficiencies are often used in the selection process of an experimental design. To calculate a design's G—efficiency, it is necessary to maximize the prediction variance over the experimental design region. However, it is common to approximate a G—efficiency. This is achieved by calculating the prediction variances generated from a subset of points in the design space and taking the maximum to estimate the maximum prediction variance. This estimate is then applied to approximate the G—efficiency. In this paper, it will be shown that over the class of central composite designs (CCDs) on the hypercube. the prediction variance can be expressed in a closed-form. An exact value of the maximum prediction variance can then be determined by evaluating this closed-form expression over a finite subset of barycentric points. Tables of exact G—efficiencies will be presented. Design optimality criteria, quadratic regression on the hypercube, and the structures of the design matrix X, X'X, and (X'X)^?1 for any CCD will be discussed.     

ON THE DESIGN OF FIELD EXPERIMENTS UNDER A ONE-DIMENSIONAL NEIGHBOUR MODEL

This paper examines the effect of randomisation restrictions, either to satisfy conditions for a balanced incomplete block design or to attain a higher level of partial neighbour balance, on the average variance of pair-wise treatment contrasts under a neighbour model discussed by Gleeson & Cullis (1987). Results suggest that smaller average pairwise variances can be obtained by ignoring requirements for incomplete block designs and concentrating on achieving a higher level of partial neighbour balance. Field layout of the design, although often determined by practical constraints, e.g. size, shape of site, minimum plot size and experimental husbandry, may markedly affect average pairwise variance. For the one-dimensional (row-wise) neighbour model considered here, investigation of three different layouts suggests that for a rectangular array of plots, smaller average pairwise variances can generally be obtained from layouts with fewer rows and more plots per row.     

Designs for weighted least squares regression, with estimated weights

We study designs, optimal up to and including terms that are O(n ^?1), for weighted least squares regression, when the weights are intended to be inversely proportional to the variances but are estimated with random error. We take a finite, but arbitrarily large, design space from which the support points are to be chosen, and obtain the optimal proportions of observations to be assigned to each point. Specific examples of D- and I-optimal design for polynomial responses are studied. In some cases the same designs that are optimal under homoscedasticity remain so for a range of variance functions; in others there tend to be more support points than are required in the homoscedastic case. We also exhibit minimax designs, that minimize the maximum, over finite classes of variance functions, value of the loss. These also tend to have more support points, often resulting from the breaking down of replicates into clusters.     

Maximum expected entropy transformed Latin hypercube designs

Existing projection designs (e.g. maximum projection designs) attempt to achieve good space-filling properties in all projections. However, when using a Gaussian process (GP), model-based design criteria such as the entropy criterion is more appropriate. We employ the entropy criterion averaged over a set of projections, called expected entropy criterion (EEC), to generate projection designs. We show that maximum EEC designs are invariant to monotonic transformations of the response, i.e. they are optimal for a wide class of stochastic process models. We also demonstrate that transformation of each column of a Latin hypercube design (LHD) based on a monotonic function can substantially improve the EEC. Two types of input transformations are considered: a quantile function of a symmetric Beta distribution chosen to optimize the EEC, and a nonparametric transformation corresponding to the quantile function of a symmetric density chosen to optimize the EEC. Numerical studies show that the proposed transformations of the LHD are efficient and effective for building robust maximum EEC designs. These designs give projections with markedly higher entropies and lower maximum prediction variances (MPV''s) at the cost of small increases in average prediction variances (APV''s) compared to state-of-the-art space-filling designs over wide ranges of covariance parameter values.     

Comparison of Prediction Capabilities of Partially Replicated Central Composite Designs in Cuboidal Region

Three variations of partially replicated central composite designs have been compared. Their prediction capabilities with respect to their prediction variances are assessed using variance dispersion graphs in cuboidal region when the axial distance is α = 1. The results show that for n₀ = 1, 2, and 3 center points, the replicated star designs have better prediction potentials than the replicated cube designs when interest is in predicting responses in the face-centered cube. The face-centered cube is the second-order central composite design with axial distance, α = 1.     

Extrapolation designs with constraints

The author considers (asymptotically) minimax extrapolation designs for an approximately multiple linear model with the model contaminant f being restricted only by its L₂ norm. He splits the integrated mean squared prediction error (IMSPE) of the fitted value over the extrapolation space into two parts, namely the integrated prediction variance (IPV) and the integrated prediction bias (IPB). For a spherical design space and an annular extrapolation space, he constructs the design that minimizes the maximum value, over f, of IPB subject to bounding IPV. He also constructs the design that minimizes IPV subject to bounding the maximum IPB.     

Properties of maximum likelihood estimators of variance components in the one-way classification,balanced data

We studied properties of maximum likelihood estimators (MLEs) of the variance components obtained from balanced data of the one-way classification. Exact and asymptotic expected values and variances of these MLEs were derived under the usual normality assumptions. Numerical studies illustrate these expected values and variances, and also illustrate the probability of obtaining a negative solution to the maximum likelihood (ML) equation for the between-class variance component. Simulations were used to study the robustness of the ML estimators under non-normal distributions.     

Missing values in replicated Latin squares

Designs based on any number of replicated Latin squares are examined for their robustness against the loss of up to three observations randomly scattered throughout the design. The information matrix for the treatment effects is used to evaluate the average variances of the treatment differences for each design in terms of the number of missing values and the size of the design. The resulting average variances are used to assess the overall robustness of the designs. In general, there are 16 different situations for the case of three missing values when there are at least three Latin square replicates in the design. Algebraic expressions may be determined for all possible configurations, but here the best and worst cases are given in detail. Numerical illustrations are provided for the average variances, relative efficiencies, minimum and maximum variances and the frequency counts, showing the effects of the missing values for a range of design sizes and levels of replication.     

The exact confidence interval for the scale parameter and the mvue of the laplace distribution

The exact distribution of the sample median, and of the maximum likelihood estimator of the scale parameter of the Laplace distribution is derived. Tables of Teans, variances and the distribution functions of the corresponding dislributions are evaluacted. Exact ,solutions to the problem of confidence interval and hypothesrs testing for the scale paramrter are provided. The minimum variance unbiased estimator (MVUE) of the p.d.f. of the Laplace distribution when the location parameter is known is also given.     

The robustness of response surface designs with errors in factor levels

ABSRTACT

Since errors in factor levels affect the traditional statistical properties of response surface designs, an important question to consider is robustness of design to errors. However, when the actual design could be observed in the experimental settings, its optimality and prediction are of interest. Various numerical and graphical methods are useful tools for understanding the behavior of the designs. The D- and G-efficiencies and the fraction of design space plot are adapted to assess second-order response surface designs where the predictor variables are disturbed by a random error. Our study shows that the D-efficiencies of the competing designs are considerably low for big variance of the error, while the G-efficiencies are quite good. Fraction of design space plots display the distribution of the scaled prediction variance through the design space with and without errors in factor levels. The robustness of experimental designs against factor errors is explored through comparative study. The construction and use of the D- and G-efficiencies and the fraction of design space plots are demonstrated with several examples of different designs with errors.     

Extended scaled prediction variance optimality for modified central composite design

Robust parameter designs (RPDs) enable the experimenter to discover how to modify the design of the product to minimize the effect due to variation from noise sources. The aim of this article is to show how this amount of work can be reduced under modified central composite design (MCCD). We propose a measure of extended scaled prediction variance (ESPV) for evaluation of RPDs on MCCD. Using these measures, we show that we can check the error or bias associated with estimating the model parameters and suggest the values of α recommended for MCCS under minimum ESPV.     

Bayesian design for estimating the turning point of a quadratic regression

The problem of designing an experiment to estimate the point at which a quadratic regression is a maximum, or minimum. is studied. The efficiency of a design depends on the value of the unknown parameters and sequential design is, therefore, more efficient than non-sequential design. We use a Bayesian criterion which is a weighted trace of the inverse of the information matrix with the weights depending on a prior distribution. If design occurs sequentially the weights can be updated. Both sequential and non-sequential Bayesian designs are compared to non-Bayesian sequential designs. The comparison is both theoretical and by simulation.     

On the unbiased estimation of fixed effects in a mixed model for growth curves

A mixed model for growth curves is introduced. A method of estimating the variance components is indicated. It is shown that the generalised least squares estimates of the fixed effects using estimates of variance components are unbiased and an expression is obtained for the increase in variance due to the substitution of the variances by their estimates. These results are directly applied to designs with two-way elimination of heterogeneity.     

Minimum Variance and VOQL Chain Sampling Plans–ChSP-4(c 1, c 2)

In this article the outgoing quality and the total inspection for the chain sampling plan ChSP-4(c ₁, c ₂) are introduced as well-defined random variables. The probability distributions of outgoing quality and total inspection are stated based on total rectification of non conforming units. The variances of these random variables are studied. The aim of this article is to develop procedures for minimum variance ChSP-4(c ₁, c ₂) sampling plans and their determination. In addition to minimum variance sampling plans, a procedure is developed for designing plans with a designated maximum variance, a VOQL (Variance of Outgoing Quality Limit) plan. The VOQL concept is analogous to the AOQL (Average Outgoing Quality Limit) except in the VOQL plan, it is the maximum variance which is established instead of the usual maximum AOQ.     

Extension of Central Composite Design for Second-Order Response Surface Model Building

The central composite design (CCD) is perhaps the most popular class of second-order response surface designs. Even though the CCDs are popular for response surface designs, this class of design has some limitations such as it does not sometimes possess good statistical properties, and it does not fit complicated models well. In this article, we propose extended central composite designs (ECCDs) to overcome these limitations. We compare ECCDs with CCDs in terms of average prediction variance, and find that ECCDs are better than CCDs.     

Progressive type II censored order statistics from exponential distributions

In the model of progressive type II censoring, point and interval estimation as well as relations for single and product moments are considered. Based on two-parameter exponential distributions, maximum likelihood estimators (MLEs), uniformly minimum variance unbiased estimators (UMVUEs) and best linear unbiased estimators (BLUEs) are derived for both location and scale parameters. Some properties of these estimators are shown. Moreover, results for single and product moments of progressive type II censored order statistics are presented to obtain recurrence relations from exponential and truncated exponential distributions. These relations may then be used to compute all the means, variances and covariances of progressive type II censored order statistics based on exponential distributions for arbitrary censoring schemes. The presented recurrence relations simplify those given by Aggarwala and Balakrishnan (1996)     

General formulae for expectations, variances and covariances of the mean squares for staggered nested designs

Staggered nested experimental designs are the most popular class of unbalanced nested designs. Using a special notation which covers the particular structure of the staggered nested design, this paper systematically derives the canonical form for the arbitrary m-factors. Under the normality assumption for every random variable, a vector comprising m canonical variables from each experimental unit is normally independently and identically distributed. Every sum of squares used in the analysis of variance (ANOVA) can be expressed as the sum of squares of the corresponding canonical variables. Hence, general formulae for the expectations, variances and covariances of the mean squares are directly obtained from the canonical form. Applying the formulae, the explicit forms of the ANOVA estimators of the variance components and unbiased estimators of the ratios of the variance components are introduced in this paper. The formulae are easily applied to obtain the variances and covariances of any linear combinations of the mean squares, especially the ANOVA estimators of the variance components. These results are eff ectively applied for the standardization of measurement methods.