Split-plot designs for multistage experimentation

Most of today’s complex systems and processes involve several stages through which input or the raw material has to go before the final product is obtained. Also in many cases factors at different stages interact. Therefore, a holistic approach for experimentation that considers all stages at the same time will be more efficient. However, there have been only a few attempts in the literature to provide an adequate and easy-to-use approach for this problem. In this paper, we present a novel methodology for constructing two-level split-plot and multistage experiments. The methodology is based on the Kronecker product representation of orthogonal designs and can be used for any number of stages, for various numbers of subplots and for different number of subplots for each stage. The procedure is demonstrated on both regular and nonregular designs and provides the maximum number of factors that can be accommodated in each stage. Furthermore, split-plot designs for multistage experiments with good projective properties are also provided.     

Split-plot designs with mirror image pairs as sub-plots

In this article we investigate two-level split-plot designs where the sub-plots consist of only two mirror image trials. Assuming third and higher order interactions negligible, we show that these designs divide the estimated effects into two orthogonal sub-spaces, separating sub-plot main effects and sub-plot by whole-plot interactions from the rest. Further we show how to construct split-plot designs of projectivity P≥3. We also introduce a new class of split-plot designs with mirror image pairs constructed from non-geometric Plackett-Burman designs. The design properties of such designs are very appealing with effects of major interest free from full aliasing assuming that 3rd and higher order interactions are negligible.     

Model robust extrapolation designs

We seek designs which are optimal in some sense for extrapolation when the true regression function is in a certain class of regression functions. More precisely, the class is defined to be the collection of regression functions such that its (h + 1)-th derivative is bounded. The class can be viewed as representing possible departures from an ‘ideal’ model and thus describes a model robust setting. The estimates are restricted to be linear and the designs are restricted to be with minimal number of points. The design and estimate sought is minimax for mean square error. The optimal designs for cases X = [0, ∞] and X = [-1, 1], where X is the place where observations can be taken, are discussed.     

Some robust designs for polynomial regression models

The author identifies static optimal designs for polynomial regression models with or without intercept. His optimality criterion is an average between the D‐optimality criterion for the estimation of low‐degree terms and the D₈‐optimality criterion for testing the significance of higher degree terms. His work relies on classical results concerning canonical moments and the theory of continued fractions.     

Mixture designs for product improvement studies

Many products are mixtures of several components (ingredient). Characteristics of the products such as the strength of steel, the efficacy of a chemical pesticide, or the viscosity of a liquid detergent, depend only on the relative proportions of the components in the mixture. Studying changes in a product' properties caused by varying the ingredient proportions is the objective of performing mixture experiments. The inherent restriction that the sum of the component proportions equal unity creates different design strategies than are usually employed with independent factors where factorial arrangements are quite common. Experimental designs for exploring the entire mixture simplex region as well as for exploring only a subregion of the simplex are presented. In those cases where four or more components are considered and a subregion is to be investigated, computer-aided designs are the rule rather than the exception. Design criterion based on the properties (variance and bias) of the prediction equation are mentioned briefly and some suggestions are made for future research in mixture experiments.     

Constructing non-regular robust parameter designs

In recent years there has been considerable attention paid to robust parameter design as a strategy for variance reduction. Of particular concern is the selection of a good experimental plan in light of the two different types of factors in the experiment (control and noise) and the asymmetric manner in which effects of the same order are treated. Recent work has focussed on the selection of regular fractional factorial designs in this setting. In this article, we consider the construction and selection of optimal non-regular experiment plans for robust parameter design. Our approach defines the word-length pattern for non-regular fractional factorial designs with two different types of factors which allows for the choice of optimal design to emphasize the estimation of the effects of interest. We use this new word-length pattern to rank non-regular robust parameter designs. We show that one can easily find minimum aberration robust parameter designs from existing orthogonal arrays. The methodology is demonstrated by finding optimal assignments for control and noise factors for 12, 16 and 20-run orthogonal arrays.     

Restricted minimax robust designs for misspecified regression models

The authors propose and explore new regression designs. Within a particular parametric class, these designs are minimax robust against bias caused by model misspecification while attaining reasonable levels of efficiency as well. The introduction of this restricted class of designs is motivated by a desire to avoid the mathematical and numerical intractability found in the unrestricted minimax theory. Robustness is provided against a family of model departures sufficiently broad that the minimax design measures are necessarily absolutely continuous. Examples of implementation involve approximate polynomial and second order multiple regression.     

Equi-information robust designs: Which designs are possible?

The effects of missing observations in a designed experiment are reviewed. Conditions are determined for a design to retain equal information, i.e. the same generalized variance of unknown parameters, when either any single or any pair of observations is lost. Some examples of designs with this property are given. Although there are many designs which retain equal information for the loss of exactly t observations, where t = 1,2,3,…, it is shown that it is not possible to obtain any design which retains equal information when any one and any two and also any three observations are missing.     

On D-optimal robust designs for lifetime improvement experiments

Locating the optimal operating conditions of the process parameters is critical in a lifetime improvement experiment. For log-normal lifetime distribution with compound error structure (i.e., symmetry, inter-class and intra-class correlation error structures), we have developed methods for construction of D-optimal robust first order designs. It is shown that D-optimal robust first order designs are always robust first order rotatable but the converse is not always true.     

Some robust parameter designs from orthogonal arrays

Robust parameter design, originally proposed by Taguchi [System of Experimental Design, Vols. I and II, UNIPUB, New York, 1987], is an offline production technique for reducing variation and improving a product's quality by using product arrays. However, the use of the product arrays results in an exorbitant number of runs. To overcome this drawback, several scientists proposed the use of combined arrays, where the control and noise factors are combined in a single array. In this paper, we use non-isomorphic orthogonal arrays as combined arrays, in order to identify a model that contains all the main effects (control and noise), their control-by-noise interactions and their control-by-control interactions with high efficiency. Some cases where the control-by-control-noise are of interest are also considered.     

Information in an observation in robust designs

We consider the Information contained 1n each observation in a given design robust with respect to the estlmability of parameters and against the unavailability of observations. We compare the observations in various 1-, 2- and 3- dimensional designs on the basis of their informations.     

A measure of robust slope-rotatability for second-order response surface experimental designs

In response surface methodology, rotatability and slope-rotatability are natural and highly desirable properties for second-order regression models. In this paper a measure of robust slope-rotatability for second-order response surface designs with a general correlated error structure is developed and illustrated with different examples for autocorrelated error structure.     

Model robust optimal designs for comparing test treatments with a control

Families of designs are obtained for comparing test treatments with a control. They are simultaneously A- and MV-optimal for either one-way or two-way elimination of heterogeneity when the model of response is homoscedastic and linear additive. These designs can be easily cataloged.     

Block designs robust against the presence of positional effects

The problem considered is to find optimum designs for treatment effects in a block design (BD) setup, when positional effects are also present besides treatment and block effects, but they are ignored while formulating the model. In the class of symmetric balanced incomplete block designs, the Youden square design is shown to be optimal in the sense of minimizing the bias term in the mean squared error (MSE) of the best linear unbiased estimators of the full set of orthonormal treatment contrasts, irrespective of the value of the positional effects.     

An orthogonal arrays approach to robust parameter designs methodology

Robust parameter design methodology was originally introduced by Taguchi [14 Taguchi, G. 1986. Introduction to Quality Engineering: Designing Quality Into Products and Process, Tokyo: Asian Productivity Organization.  [Google Scholar]] as an engineering methodology for quality improvement of products and processes. A robust design of a system is one in which two different types of factors are varied; control factors and noise factors. Control factors are variables with levels that are adjustable, whereas noise factors are variables with levels that are hard or impossible to control during normal conditions, such as environmental conditions and raw-material properties. Robust parameter design aims at the reduction of process variation by properly selecting the levels of control factors so that the process becomes insensitive to changes in noise factors. Taguchi [14 Taguchi, G. 1986. Introduction to Quality Engineering: Designing Quality Into Products and Process, Tokyo: Asian Productivity Organization.  [Google Scholar] 15 Taguchi, G. 1987. System of Experimental Design, Vol. I and II, New York: UNIPUB.  [Google Scholar]] proposed the use of crossed arrays (inner–outer arrays) for robust parameter design. A crossed array is the cross-product of an orthogonal array (OA) involving control factors (inner array) and an OA involving noise factors (outer array). Objecting to the run size and the flexibility of crossed arrays, several authors combined control and noise factors in a single design matrix, which is called a combined array, instead of crossed arrays. In this framework, we present the use of OAs in Taguchi's methodology as a useful tool for designing robust parameter designs with economical run size.     

Optimal robust parameter designs with qualitative and quantitative factors

Robust parameter design is an effective methodology for reducing variance and improving the quality of a product and a process. Recent work has mainly concentrated on two‐level robust parameter designs. We consider general robust parameter designs with factors having two or more or mixed levels these levels being either qualitative or quantitative. We propose a methodology and develop a generalised minimum aberration optimality criterion for selecting optimal robust parameter designs. A catalogue of 18‐run optimal designs is constructed and tabulated.     

On D-optimal robust second order slope-rotatable designs

Das and Park (2006) introduced slope-rotatable designs overall directions for correlated observations which is known as A-optimal robust slope-rotatable designs. This article focuses D-optimal slope-rotatable designs for second-order response surface model with correlated observations. It has been established that robust second-order rotatable designs are also D-optimal robust slope-rotatable designs. A class of D-optimal robust second-order slope-rotatable designs has been derived for special correlation structures of errors.     

Exactly optimal sampling designs for processes with a product covariance structure

The author considers the problem of finding exactly optimal sampling designs for estimating a second‐order, centered random process on the basis of finitely many observations. The value of the process at an unsampled point is estimated by the best linear unbiased estimator. A weighted integrated mean squared error or the maximum mean squared error is used to measure the performance of the estimator. The author presents a set of necessary and sufficient conditions for a design to be exactly optimal for processes with a product covariance structure. Expansions of these conditions lead to conditions for asymptotic optimality.     

A robust I-sample analysis of means type randomization test for variances for unbalanced designs

Two analysis of means type randomization tests for testing the equality of I variances for unbalanced designs are presented. Randomization techniques for testing statistical hypotheses can be used when parametric tests are inappropriate. Suppose that I independent samples have been collected. Randomization tests are based on shuffles or rearrangements of the (combined) sample. Putting each of the I samples ‘in a bowl’ forms the combined sample. Drawing samples ‘from the bowl’ forms a shuffle. Shuffles can be made with replacement (bootstrap shuffling) or without replacement (permutation shuffling). The tests that are presented offer two advantages. They are robust to non-normality and they allow the user to graphically present the results via a decision chart similar to a Shewhart control chart. A Monte Carlo study is used to verify that the permutation version of the tests exhibit excellent power when compared to other robust tests. The Monte Carlo study also identifies circumstances under which the popular Levene's test fails.