Split-plot designs for robust product experimentation

Genichi Taguchi has emphasized the use of designed experiments in several novel and important applications. In this paper we focus on the use of statistical experimental designs in designingproducts to be robust to environmental conditions. The engineering concept of robust product design is very important because it is frequently impossible or prohibitively expensive to control or eliminate variation resulting from environmental conditions. Robust product design enablesthe experimenter to discover how to modify the design of the product to minimize the effect dueto variation from environmental sources. In experiments of this kind, Taguchi's total experimental arrangement consists of a cross-product of two experimental designs:an inner array containing the design factors and an outer array containing the environmental factors. Except in situations where both these arrays are small, this arrangement may involve a prohibitively large amount of experimental work. One of the objectives of this paper is to show how this amount of work can be reduced. In this paper we investigate the applicability of split-plot designs for thisparticular experimental situation. Consideration of the efficiency of split-plot designs and anexamination of several variants of split-plot designs indicates that experiments conductedin a split-plot mode can be of tremendous value in robust product design since they not only enable the contrasts of interest to be estimated efficiently but also the experiments can be considerably easier to conduct than the designs proposed by Taguchi.     

Selection and screening procedures to determine optimal product designs

To compare several promising product designs, manufacturers must measure their performance under multiple environmental conditions. In many applications, a product design is considered to be seriously flawed if its performance is poor for any level of the environmental factor. For example, if a particular automobile battery design does not function well under temperature extremes, then a manufacturer may not want to put this design into production. Thus, this paper considers the measure of a product's quality to be its worst performance over the levels of the environmental factor. We develop statistical procedures to identify (a near) optimal product design among a given set of product designs, i.e., the manufacturing design that maximizes the worst product performance over the levels of the environmental variable. We accomplish this by intuitive procedures based on the split-plot experimental design (and the randomized complete block design as a special case); split-plot designs have the essential structure of a product array and the practical convenience of local randomization. Two classes of statistical procedures are provided. In the first, the δ-best formulation of selection problems, we determine the number of replications of the basic split-plot design that are needed to guarantee, with a given confidence level, the selection of a product design whose minimum performance is within a specified amount, δ, of the performance of the optimal product design. In particular, if the difference between the quality of the best and second best manufacturing designs is δ or more, then the procedure guarantees that the best design will be selected with specified probability. For applications where a split-plot experiment that involves several product designs has been completed without the planning required of the δ-best formulation, we provide procedures to construct a ‘confidence subset’ of the manufacturing designs; the selected subset contains the optimal product design with a prespecified confidence level. The latter is called the subset selection formulation of selection problems. Examples are provided to illustrate the procedures.     

Optimum covariate designs in split-plot and strip-plot design set-ups

The problem considered is that of finding optimum covariate designs for estimation of covariate parameters in standard split-plot and strip-plot design set-ups with the levels of the whole-plot factor in r randomised blocks. Also an extended version of a mixed orthogonal array has been introduced, which is used to construct such optimum covariate designs. Hadamard matrices, as usual, play the key role for such construction.     

Generalized minimum aberration two-level split-plot designs

Split-plot experiments may arise when it is impractical to completely randomize the treatment combinations of a designed experiment. To provide more flexible design choices in the nonregular split-plot setting, we describe an approach for constructing minimum aberration orthogonal two-level split-plot designs having 12, 16, 20 and 24 runs. We consider five design scenarios that may be of importance to practitioners, and then propose an approach for assigning word lengths under these five scenarios. We then use the extended word length patterns to rank both regular and nonregular orthogonal split-plot designs. While most existing papers concerning orthogonal split-plot designs focus on regular orthogonal designs, we find that many minimum aberration split-plot designs are nonregular orthogonal designs.     

Optimal block sequences for blocked fractional factorial split-plot designs

It is known that for blocked 2^n-k$2^{n-k}$ designs a judicious sequencing of blocks may allow one to obtain early and insightful results regarding influential parameters in the experiment. Such findings may justify the early termination of the experiment thereby producing cost and time savings. This paper introduces an approach for selecting the optimal sequence of blocks for regular two-level blocked fractional factorial split-plot screening experiments. An optimality criterion is developed so as to give priority to the early estimation of low-order factorial effects. This criterion is then applied to the minimum aberration blocked fractional factorial split-plot designs tabled in McLeod and Brewster [2004. The design of blocked fractional factorial split-plot experiments. Technometrics 46, 135–146]. We provide a catalog of optimal block sequences for 16 and 32-run minimum aberration blocked fractional factorial split-plot designs run in either 4 or 8 blocks.     

A simple method to obtain minimum aberration split-plot designs

ABSTRACT

Split-plot designs have been utilized in factorial experiments with some factors applied to larger units and others to smaller units. Such designs with low aberration are preferred when the experimental size and the number of factors considered in both whole plot and subplot are determined. The minimum aberration split-plot designs can be obtained using either computer algorithms or the exhausted search. In this article, we propose a simple, easy-to-operate approach by using two ordered sequences of columns from two orthogonal arrays in obtaining minimum aberration split-plot designs for experiments of sizes 16 and 32.     

Individual control treatment in split-plot experiments

The paper deals with experiments laid out in a complete or an incomplete split-plot design in which one control (standard) treatment occurs in addition to the usual treatments. Usually the control (standard) treatment has been treated as one specific factor level. In this paper, in contrast to others in this area, the control (standard) may not be strictly connected with treatment combinations. The new incomplete split-plot designs with control satisfy all generally accepted methodological requirements, with special reference to the problems of randomisation. Moreover, tools are described which allow checking of the general balance or efficiency of the design, as well as performance of experiments with inference.     

On the efficiency of some non-orthogonal split-plot×split-block designs with control treatments

Many split-plot×split-block (SPSB) type experiments used in agriculture, biochemistry or plant protection are designed to study new crop plant cultivars or chemical agents. In these experiments it is usually very important to compare test treatments with the so-called control treatments. It happens yet that experimental material is limited and it does not allow using a complete (orthogonal) SPSB design. In the paper we propose a non-orthogonal SPSB design for consideration. Two cases of the design are presented here, i.e. when its incompleteness is connected with a crossed treatment structure only or with a nested treatment structure only. It is assumed the factors' levels connected with the incompleteness of the design are split into two groups: a set of test treatments and a set of control treatments. The method of constructions involves applying augmented block designs for some factors' levels. In a modelling data obtained from such experiments the structure of experimental material and appropriate randomization scheme of the different kinds of units before they enter the experiment are taken into account. With respect to the analysis of the obtained randomization model the approach typical to the multistratum experiments with orthogonal block structure is adapted. The proposed statistical analysis of linear model obtained includes estimation of parameters, testing general and particular hypotheses defined by the (basic) treatment contrasts with special reference to the notion of general balance.     

Papers That Evaluate Computer Programs

Commentaries are informative essays dealing with viewpoints of statistical practice, statistical education, and other topics considered to be of general interest to the broad readership of The American Statistician. Commentaries are similar in spirit to Letters to the Editor, but they involve longer discussions of background, issues, and perspectives. All commentaries will be refereed for their merit and compatibility with these criteria.     

The Religion of Statistics as Practiced in Medical Journals

Commentaries are informative essays dealing with viewpoints of statistical practice, statistical education, and other topics considered to be of general interest to the broad readership of The American Statistician. Commentaries are similar in spirit to Letters to the Editor, but they involve longer discussions of background, issues, and perspectives. All commentaries will be refereed for their merit and compatibility with these criteria.

The way in which hypothesis tests are often used as the sole tool of statistics in medical research is satirized. This is followed by a suggestion for reform.     

Group screening method for the statistical analysis of -optimal mixed-level supersaturated designs

In this paper, we propose the application of group screening methods for analyzing data using E(f_NOD)-optimal mixed-level supersaturated designs possessing the equal occurrence property. Supersaturated designs are a large class of factorial designs which can be used for screening out the important factors from a large set of potentially active variables. The huge advantage of these designs is that they reduce the experimental cost drastically, but their critical disadvantage is the high degree of confounding among factorial effects. Based on the idea of the group screening methods, the f factors are sub-divided into g “group-factors”. The “group-factors” are then studied using the penalized likelihood statistical analysis methods at a factorial design with orthogonal or near-orthogonal columns. All factors in groups found to have a large effect are then studied in a second stage of experiments. A comparison of the Type I and Type II error rates of various estimation methods via simulation experiments is performed. The results are presented in tables and discussion follows.     

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.     

Identification of factors influencing dispersion in split-plot experiments

As split-plot designs are commonly used in robust design it is important to identify factors in these designs that influence the dispersion of the response variable. In this article, the Bergman-Hynén method, developed for identification of dispersion effects in unreplicated experiments, is modified to be used in the context of split-plot experiments. The modification of the Bergman-Hynén method enables identification of factors that influence specific variance components in unreplicated two-level fractional factorial splitplot experiments. An industrial example is used to illustrate the proposed method.     

Incomplete Split-Plot Designs Supplemented by a Single Control

The article deals with the constructing methods for experiments carried out in an incomplete split-plot design supplemented by an additional treatment, called a single control. The control treatment has been treated usually as one specific factor level while not necessarily. The control cannot be connected with treatment combinations in an experiment. This distinguishes this article from others in the area considered. The proposed supplementation of whole incomplete split-plot designs leads to the designs with generally accepted methodological requirements, especially randomization. Moreover, we propose a few methods for constructing considered types of the designs with desirable statistical properties such as general balance and efficiency balance of the design with respect to treatment contrasts.     

A candidate-set-free algorithm for generating D-optimal split-plot designs

Summary.  We introduce a new method for generating optimal split-plot designs. These designs are optimal in the sense that they are efficient for estimating the fixed effects of the statistical model that is appropriate given the split-plot design structure. One advantage of the method is that it does not require the prior specification of a candidate set. This makes the production of split-plot designs computationally feasible in situations where the candidate set is too large to be tractable. The method allows for flexible choice of the sample size and supports inclusion of both continuous and categorical factors. The model can be any linear regression model and may include arbitrary polynomial terms in the continuous factors and interaction terms of any order. We demonstrate the usefulness of this flexibility with a 100-run polypropylene experiment involving 11 factors where we found a design that is substantially more efficient than designs that are produced by using other approaches.     

An extension of Banerjee and Rahim model in economic and economic statistical designs for multivariate quality characteristics under Burr XII distribution

The design parameters of the economic and economic statistical designs of control charts depend on the distribution of process failure mechanism or shock model. So far, only a small number of failure distributions, such as exponential, gamma, and Weibull with fixed or increasing hazard rates, have been used as a shock model in the economic and economic statistical designs of the Hotelling T² control charts. Due to both theoretical and practical aspects, the lifetime of the process under study may not follow a distribution with fixed or increasing hazard rate. A proper alternative for this situation may be the Burr distribution, in which the hazard rate can be fixed, increasing, decreasing, single mode, or even U-shaped. In this research article, economic and economic statistical designs of the Hotelling T² control charts under the Burr XII shock models under two uniform and non uniform sampling schemes were proposed, constructed, and compared. The obtained design models were implemented by a numerical example, and a sensitivity analysis was conducted to evaluate the effect of changing parameters of shock model distribution on the optimum values of the proposed design models. The results showed that first the proposed designs under non uniform sampling scheme perform better and second the optimum values of the designs are not significantly sensitive to changing of the Burr XII distribution parameters. We showed that the obtained design models are also true for the beta Burr XII shock model.     

A comparison of inference procedures in unbalanced split-plot designs

Several procedures for constructing confidence intervals and testing hypotheses about fixed effects in unbalanced split-plot experiments are described in this paper. These procedures can also be used for unbalanced repeated measures experiments when the repeated measures satisfy the Huyhn-Feldt (1970) conditions. A number of these procedures require that the whole plot error mean square has a distribution proportional to a chi-square distribution and that it be independent of estimators of the parameter functions. Often, neither of these conditions are met in unbalanced split-plot experiments. Simulation studies of a small design of eight observations and larger designs with 34 to 48 observations are used to investigate the performance of the different procedures.     

THE ANALYSIS OF MULTISTRATA DESIGNED EXPERIMENTS WITH INCOMPLETE DATA

Building upon early work of E. A. Cornish we show that G. N. Wilkinson's version of Yates' approach to the analysis of designed experiments with a single error stratum carries over completely to designs with an arbitrary non-singular covariance matrix, initially assumed known. We show that the equations, corrections, adjustments and algorithms all have their more general analogues and that these can be solved, computed or executed quite readily if the design has orthogonal block structure and satisfies Nelder's condition of general balance. The results are illustrated with a split-plot and a simple (square) lattice design.     

The robust economic statistical design of the Hotelling's T2 chart

ABSTRACT

Economic statistical designs aim at minimizing the cost of process monitoring when a specific scenario or a set of estimated process and cost parameters is given. But, in practice the process may be affected by more than one scenario which may lead to severe cost penalties if the wrong design is used. Here, we investigate the robust economic statistical design (RESD) of the T² chart in an attempt to reduce these cost penalties when there are multiple scenarios. Our method is to employ the genetic algorithm (GA) optimization method to minimize the total expected monitoring cost across all distinct scenarios. We illustrate the effectiveness of the method using two numerical examples. Simulation studies indicate that robust economic statistical designs should be encouraged in practice.     

Comparative Analyses of Pretest-Posttest Research Designs

Two common methods of analyzing data from a two-group pretest-posttest research design are (a) two-sample t test on the difference score between pretest and posttest and (b) repeated-measures/split-plot analysis of variance. The repeated-measures/split-plot analysis subsumes the t test analysis, although the former requires more assumptions to be satisfied. A numerical example is given to illustrate some of the equivalences of the two methods of analysis. The investigator should choose the method of analysis based on the research objective(s).