Generalized staggered nested designs for variance components estimation

Staggered nested experimental designs are the most popular class of unbalanced nested designs in practical fields. The most important features of the staggered nested design are that it has a very simple open-ended structure and each sum of squares in the analysis of variance has almost the same degrees of freedom. Based on the features, a class of unbalanced nested designs that is a generalization of the staggered nested design is proposed in this paper. Formulae for the estimation of variance components and their sums are provided. Comparing the variances of the estimators to the staggered nested designs, it is found that some of the generalized staggered nested designs are more efficient than the traditional staggered nested design in estimating some of the variance components and their sums. An example is provided for illustration.     

Comparison of designs for the three-fold nested random model

The quality of estimation of variance components depends on the design used as well as on the unknown values of the variance components. In this article, three designs are compared, namely, the balanced, staggered, and inverted nested designs for the three-fold nested random model. The comparison is based on the so-called quantile dispersion graphs using analysis of variance (ANOVA) and maximum likelihood (ML) estimates of the variance components. It is demonstrated that the staggered nested design gives more stable estimates of the variance component for the highest nesting factor than the balanced design. The reverse, however, is true in case of lower nested factors. A comparison between ANOVA and ML estimation of the variance components is also made using each of the aforementioned designs.     

Description and evaluation of a nested cube experimental design

An incomplete factorial design based on an extension of the Fawiliar 2^kfactorial called a nested cube is proposed for use in response surface investigationso The simplicity and general efficiency of the nested cube suggest its suitability to many areas of research, especially that repeated at many locations orconductea over a long period. Comparisons to potentially competing designs are provided for bias in response estination due to fitting an ioappropriate model and for profiles of variance. merits of the nested cube are (1) a level of relative bias and variance judged to be favorable though not optimal, (2) an ability to utilize a minimum blag estimator not available to competing designs, and (3) a simplicity associated with use of equal spacing

and nearly equal replication on the margin for each factor level.     

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.     

Generalized case–cohort sampling

A class of cohort sampling designs, including nested case–control, case–cohort and classical case–control designs involving survival data, is studied through a unified approach using Cox's proportional hazards model. By finding an optimal sample reuse method via local averaging, a closed form estimating function is obtained, leading directly to the estimators of the regression parameters that are relatively easy to compute and are more efficient than some commonly used estimators in case–cohort and nested case–control studies. A semiparametric efficient estimator can also be found with some further computation. In addition, the class of sampling designs in this study provides a variety of sampling options and relaxes the restrictions of sampling schemes that are currently available.     

Optimal two-point designs for the michaelis-menten model with heteroscedastic errors

We construct D-optimal designs for the Michaelis-Menten model when the variance of the response depends on the independent variable. However, this dependence is only partially known. A Bayesian approacn is used to find an optimal design by incorporating the prior lnformation about the variance structure. We demonstrate the method for a class of error variance structures and present efficiencies of these optimal designs under prior mis-specifications. In particular, we show that an erroneous assumption on the variance structure for the Michaelis-Menten model can have serious consequences.     

Minimax A-, c-, and I-optimal regression designs for models with heteroscedastic errors

It is well known that it is difficult to construct minimax optimal designs. Furthermore, since in practice we never know the true error variance, it is important to allow small deviations and construct robust optimal designs. We investigate a class of minimax optimal regression designs for models with heteroscedastic errors that are robust against possible misspecification of the error variance. Commonly used A-, c-, and I-optimality criteria are included in this class of minimax optimal designs. Several theoretical results are obtained, including a necessary condition and a reflection symmetry for these minimax optimal designs. In this article, we focus mainly on linear models and assume that an approximate error variance function is available. However, we also briefly discuss how the methodology works for nonlinear models. We then propose an effective algorithm to solve challenging nonconvex optimization problems to find minimax designs on discrete design spaces. Examples are given to illustrate minimax optimal designs and their properties.     

Interblock information in multidimensional block designs

In many experimental situations, d-way heterogeneity among experimental units may be controlled through use of multiple blocking criteria. In some cases it is reasonable to regard some or all of the block effects as random. Then the model is mixed and observations within blocks are correlated. Very general estimators of treatment effects and their dispersion matrix with recovery of interblock information are provided. They apply to designs with d > 1 blocking criteria that may be crossed, nested, or a combination thereof. These general results may be specialized to provide analyses of new classes of MBD's or used directly for numerical analyses of designs in the general class, perhaps through use as the basis for very general computer programs. Estimation of variance components is discussed, and an example is provided to illustrate adaptation of the general results.     

Robustness properties of minimally‐supported Bayesian D‐optimal designs for heteroscedastic models

Bayesian D‐optimal designs supported on a fixed number of points were found by Dette & Wong (1998) for estimating parameters in a polynomial model when the error variance depends exponentially on the explanatory variable. The present authors provide optimal designs under a broader class of error variance structures and investigate the robustness properties of these designs to model and prior distribution assumptions. A comparison of the performance of the optimal designs relative to the popular uniform designs is also given. The authors' results suggest that Bayesian D‐optimal designs suported on a fixed number of points are more likely to be globaly optimal among all designs if the prior distribution is symmetric and is concentrated around its mean.     

Linear Regression With Nested Errors Using Probability‐Linked Data

Probabilistic matching of records is widely used to create linked data sets for use in health science, epidemiological, economic, demographic and sociological research. Clearly, this type of matching can lead to linkage errors, which in turn can lead to bias and increased variability when standard statistical estimation techniques are used with the linked data. In this paper we develop unbiased regression parameter estimates to be used when fitting a linear model with nested errors to probabilistically linked data. Since estimation of variance components is typically an important objective when fitting such a model, we also develop appropriate modifications to standard methods of variance components estimation in order to account for linkage error. In particular, we focus on three widely used methods of variance components estimation: analysis of variance, maximum likelihood and restricted maximum likelihood. Simulation results show that our estimators perform reasonably well when compared to standard estimation methods that ignore linkage errors.     

Robust estimation and design procedures for the random effects model

The authors discuss two robust estimators for estimating variance components in the random effects model, and they obtain finite‐sample breakdown points for the estimators. Based on the finite‐sample breakdown point, they propose a criterion for selecting robust designs. With robust designs, one can get efficient and reliable estimates for variance components regardless of outliers which may happen in the experiment. The authors give examples to show the performance of robust estimators and to compare robust designs with optimal designs based on the traditional analysis of variance estimation method.     

Nearly optimal orthogonally blocked desings for a quadratic mixture model with q components

In experiments with mixtures involving process variables, orthogonal block designs may be used to allow estimation of the parameters of the mixture components independently of estimation of the parameters of the process variables. In the class of orthogonally blocked designs based on pairs of suitably chosen Latin squares, the optimal designs consist primarily of binary blends of the mixture components, regardless of how many ingredients are available for the mixture. This paper considers ways of modifying these optimal designs so that some or all of the runs used in the experiment include a minimum proportion of each mixture ingredient. The designs considered are nearly optimal in the sense that the experimental points are chosen to follow ridges of maxima in the optimality criteria. Specific designs are discussed for mixtures involving three and four components and distinctions are identified for different designs with the same optimality properties. The ideas presented for these specific designs are readily extended to mixtures with q>4 components.     

Optimal Designs for the Carryover Model with Random Interactions Between Subjects and Treatments

The paper considers a model for crossover designs with carryover effects and a random interaction between treatments and subjects. Under this model, two observations of the same treatment on the same subject are positively correlated and therefore provide less information than two observations of the same treatment on different subjects. The introduction of the interaction makes the determination of optimal designs much harder than is the case for the traditional model. Generalising the results of Bludowsky's thesis, the present paper uses Kushner's method to determine optimal approximate designs. We restrict attention to the case where the number of periods is less than or equal to the number of treatments. We determine the optimal designs in the important special cases that the number of periods is 3, 4 or 5. It turns out that the optimal designs depend on the variance of the random interactions and in most cases are not binary. However, we can show that neighbour balanced binary designs are highly efficient, regardless of the number of periods and of the size of the variance of the interaction effects.     

A method of selecting the levels of regressor variables in lognormal regression model

Kupper and Meydrech and Myers and Lahoda introduced the mean squared error (MSE) approach to study response surface designs, Duncan and DeGroot derived a criterion for optimality of linear experimental designs based on minimum mean squared error. However, minimization of the MSE of an estimator maxr renuire some knowledge about the unknown parameters. Without such knowledge construction of designs optimal in the sense of MSE may not be possible. In this article a simple method of selecting the levels of regressor variables suitable for estimating some functions of the parameters of a lognormal regression model is developed using a criterion for optimality based on the variance of an estimator. For some special parametric functions, the criterion used here is equivalent to the criterion of minimizing the mean squared error. It is found that the maximum likelihood estimators of a class of parametric functions can be improved substantially (in the sense of MSE) by proper choice of the values of regressor variables. Moreover, our approach is applicable to analysis of variance as well as regression designs.     

Robust Linear Calibration

We regard the simple linear calibration problem where only the response y of the regression line y = β₀ + β₁ t is observed with errors. The experimental conditions t are observed without error. For the errors of the observations y we assume that there may be some gross errors providing outlying observations. This situation can be modeled by a conditionally contaminated regression model. In this model the classical calibration estimator based on the least squares estimator has an unbounded asymptotic bias. Therefore we introduce calibration estimators based on robust one-step-M-estimators which have a bounded asymptotic bias. For this class of estimators we discuss two problems: The optimal estimators and their corresponding optimal designs. We derive the locally optimal solutions and show that the maximin efficient designs for non-robust estimation and robust estimation coincide.     

Graphical technique for comparing designs for random models

Methods for comparing designs for a random (or mixed) linear model have focused primarily on criteria based on single-valued functions. In general, these functions are difficult to use, because of their complex forms, in addition to their dependence on the model's unknown variance components. In this paper, a graphical approach is presented for comparing designs for random models. The one-way model is used for illustration. The proposed approach is based on using quantiles of an estimator of a function of the variance components. The dependence of these quantiles on the true values of the variance components is depicted by plotting the so-called quantile dispersion graphs (QDGs), which provide a comprehensive picture of the quality of estimation obtained with a given design. The QDGs can therefore be used to compare several candidate designs. Two methods of estimation of variance components are considered, namely analysis of variance and maximum-likelihood estimation.     

Equivalence theorem for Schur optimality of experimental designs

An experimental design is said to be Schur optimal, if it is optimal with respect to the class of all Schur isotonic criteria, which includes Kiefer's criteria of _Φp${\Phi }_p$-optimality, distance optimality criteria and many others. In the paper we formulate an easily verifiable necessary and sufficient condition for Schur optimality in the set of all approximate designs of a linear regression experiment with uncorrelated errors. We also show that several common models admit a Schur optimal design, for example the trigonometric model, the first-degree model on the Euclidean ball, and the Berman's model.     

Optimum designs for estimating EDp based on raw optical density data

For raw optical density (ROD) data, such as those generated in biological assays employing an ELISA plate reader, ED_p-optimal designs are identified for a family of homogeneous non-linear models with two parameters. In every case, the theoretical ED_p-optimal design is a design with one or two support points. These theoretical optimal designs might not be suitable for many practical applications. To overcome this shortcoming, we have specified ED_p-optimal designs within the class of k-point equally spaced and uniform designs. The efficiency robustness of these designs with respect to initial nominal values of the parameters have been investigated.     

An experimental design criterion for minimizing meta-model prediction errors applied to die casting process design

Summary. We propose the expected integrated mean-squared error (EIMSE) experimental design criterion and show how we used it to design experiments to meet the needs of researchers in die casting engineering. This criterion expresses in a direct way the researchers' goal to minimize the expected meta-model prediction errors, taking into account the effects of both random experimental errors and errors deriving from our uncertainty about the true model form. Because we needed to make assumptions about the prior distribution of model coefficients to estimate the EIMSE, we performed a sensitivity analysis to verify that the relative prediction performance of the design generated was largely insensitive to our assumptions. Also, we discuss briefly the general advantages of EIMSE optimal designs, including lower expected bias errors compared with popular response surface designs and substantially lower variance errors than certain Box–Draper all-bias designs.     

Construction of nested space-filling designs using difference matrices

Experiments that study complex real world systems in business, engineering and sciences can be conducted at different levels of accuracy or sophistication. Nested space-filling designs are suitable for such multi-fidelity experiments. In this paper, we propose a systematic method to construct nested space-filling designs for experiments with two levels of accuracy. The method that makes use of nested difference matrices can be easily performed, many nested space-filling designs for experiments with two levels of accuracy can thus be constructed, and the resulting designs achieve stratification in low dimensions. In addition, the proposed method can also be used to obtain sliced space-filling designs for conducting computer experiments with both qualitative and quantitative factors.