Experimentation order in factorial designs: new findings

Under some very reasonable hypotheses, it becomes evident that randomizing the run order of a factorial experiment does not always neutralize the effect of undesirable factors. Yet, these factors do have an influence on the response, depending on the order in which the experiments are conducted. On the other hand, changing the factor levels is many times costly; therefore it is not reasonable to leave to chance the number of changes necessary. For this reason, run orders that offer the minimum number of factor level changes and at the same time minimize the possible influence of undesirable factors on the experimentation have been sought. Sequences which are known to produce the desired properties in designs with 8 and 16 experiments can be found in the literature. In this paper, we provide the best possible sequences for designs with 32 experiments, as well as sequences that offer excellent properties for designs with 64 and 128 experiments. The method used to find them is based on a mixture of algorithmic searches and an augmentation of smaller designs.     

Detecting randomization restrictions caused by factors

In practice, randomization in factorial experiments has generally meant the sequential application of treatment combinations to experimental units determined by a random run order and the resetting of levels of factors only when levels change from one run to the next. Whenever factor levels are not reset for consecutive runs requiring the same level of a factor, the experiment is inadvertently split-plotted. The number and the size of the whole plots are formed randomly. The unbalanced split-plot experiment is assumed by experimenters to be completely randomized and analysis usually proceeds by the method of ordinary least squares. We show that once the experiment has been run it is often difficult, sometimes impossible, to determine the effect of such inadvertent split-plotting. Retrospectively, therefore, the correct analysis often cannot be performed. The aims of the paper are: (1) to urge experimenters and statisticians to recognize prospectively the difficulty in resetting factor levels so that split-plotting is deliberate but designed in a manner beneficial to both the conduct of the experiment and analysis of the data; (2) to make recommendations regarding more comprehensive reporting of data from randomized experiments.     

Experimentation order in factorial designs with 8 or 16 runs

Randomizing the order of experimentation in a factorial design does not always achieve the desired effect of neutralizing the influence of unknown factors. In fact, with some very reasonable assumptions, an important proportion of random orders afford the same degree of protection as that obtained by experimenting in the design matrix standard order. In addition, randomization can induce a big number of changes in factor levels and thus make experimentation expensive and difficult. This paper discusses this subject and suggests experimentation orders for designs with 8 or 16 runs that combine an excellent level of protection against the influence of unknown factors, with the minimum number of changes in factor levels.     

Experimentation order with good properties for 2 k factorial designs

Randomizing the order of experimentation in a factorial design does not always achieve the desired effect of neutralizing the influence of unknown factors. In fact, with some very reasonable assumptions, an important proportion of random orders achieve the same degree of protection as that obtained by experimenting in the design matrix standard order. In addition, randomization can induce a large number of changes in factor levels and thus make experimentation expensive and difficult. De Leon et al. [Experimentation order in factorial designs with 8 or 16 runs, J. Appl. Stat. 32 (2005), pp. 297–313] proposed experimentation orders for designs with eight or 16 runs that combine an excellent level of protection against the influence of unknown factors, with the minimum number of changes in factor levels. This article presents a new methodology to obtain experimentation orders with the desired properties for designs with any number of runs.     

A series of single array 2m factorial search designs for even m

By means of a search design one is able to search for and estimate a small set of non‐zero elements from the set of higher order factorial interactions in addition to estimating the lower order factorial effects. One may be interested in estimating the general mean and main effects, in addition to searching for and estimating a non‐negligible effect in the set of 2‐ and 3‐factor interactions, assuming 4‐ and higher‐order interactions are all zero. Such a search design is called a ‘main effect plus one plan’ and is denoted by MEP.1. Construction of such a plan, for 2^m factorial experiments, has been considered and developed by several authors and leads to MEP.1 plans for an odd number m of factors. These designs are generally determined by two arrays, one specifying a main effect plan and the other specifying a follow‐up. In this paper we develop the construction of search designs for an even number of factors m, m≠6. The new series of MEP.1 plans is a set of single array designs with a well structured form. Such a structure allows for flexibility in arriving at an appropriate design with optimum properties for search and estimation.     

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.     

Comparisons of search designs using search probabilities

Search designs are considered for searching and estimating one nonzero interaction from the two and three factor interactions under the search linear model. We compare three 12-run search designs D1, D2, and D3, and three 11-run search designs D4, D5, and D6, for a 2⁴ factorial experiment. Designs D2 and D3 are orthogonal arrays of strength 2, D1 and D4 are balanced arrays of full strength, D5 is a balanced array of strength 2, and D6 is obtained from D3 by deleting the duplicate run. Designs D4 and D5 are also obtained by deleting a run from D1 and D2, respectively. Balanced arrays and orthogonal arrays are commonly used factorial designs in scientific experiments. “Search probabilities” are calculated for the comparison of search designs. Three criteria based on search probabilities are presented to determine the design which is most likely to identify the nonzero interaction. The calculation of these search probabilities depends on an unknown parameter ρ which has a signal-to-noise ratio form. For a given value of ρ, Criteria I and II are newly proposed in this paper and Criteria III is given in Shirakura et al. (Ann. Statist. 24 (6) (1996) 2560). We generalize Criteria I–III for all values of ρ so that the comparison of search designs can be made without requiring a specific value of ρ. We have developed simplified methods for comparing designs under these three criteria for all values of ρ. We demonstrate, under all three criteria, that the balanced array D1 is more likely to identify the nonzero interaction than the orthogonal arrays D2 and D3, and the design D4 is more likely to identify the nonzero interaction than the designs D5 and D6.The methods of comparing designs developed in this paper are applicable to other factorial experiments for searching one nonzero interaction of any order.     

Mixed two- and four-level fractional factorial split-plot designs with clear effects

Mixed-level designs have become widely used in the practical experiments. When the levels of some factors are difficult to be changed or controlled, fractional factorial split-plot (FFSP) designs are often used. It is highly to know when a mixed-level FFSP design with resolution III or IV has clear effects. This paper investigates the conditions of a resolution III or IV FFSP design with both two-level and four-level factors to have various clear factorial effects, including two types of main effects and three types of two-factor interaction components. The structures of such designs are shown and illustrated with examples.     

Budget constrained run orders in optimum design

When experiments are to be performed in a time sequence, the observed responses may be affected by an unknown time trend. Run orders that are optimally balanced for time trends usually involve huge costs due to the large number of factor level changes. Therefore, trend-free run orders can be of low practical value in view of economical considerations. Tack and Vandebroek (J. Statist. Plann. Inference 98 (2001) 293) recently presented a design algorithm to construct trend-resistant run orders that give maximal information per unit cost. In this paper, the latter approach is extended to the construction of trendresistant run orders for which the total cost has to be lower than a specified budget. A new design algorithm is proposed that offers the experimenter a general method for solving a wide range of practical design problems.     

TESTING SUBHYPOTHESES AND ESTIMATING σ2 IN THE NONREPLICATED THREE-WAY MULTIPLICATIVE INTERACTION MODEL

ABSTRACT

Standard statistical techniques do not provide methods for analyzing data from nonreplicated factorial experiments. Such experiments occur for several reasons. Many experimenters may prefer conducting experiments having a large number of factor levels with no replications than conducting experiments with a few factor levels with replications particularly in pilot studies. Such experiments may allow one to identify factor combinations to be used in follow-up experiments. Another possibility is when the experimenter thinks that an experiment is replicated when in fact it is not. This occurs when a naive researcher believes that sub-samples are replicates when in reality they are not. Nonreplicated two-way experiments have been extensively studied. This paper discusses the analysis of nonreplicated three-way experiments. In particular, estimation of σ² is discussed and a test is derived for testing whether three-factor interaction is absent in sub-areas of three-way data using a nonreplicated three-way multiplicative interaction model with a single multiplicative term. Approximate null distribution of the derived test statistic is studied using Monte Carlo studies and results are illustrated through an example.     

Choosing appropriate covariance matrices in a nonparametric analysis of factorials in block designs

The standard nonparametric, rank-based approach to the analysis of dependent data from factorial designs is based on an estimated unstructured (UN) variance–covariance matrix, but the large number of variance–covariance terms in many designs can seriously affect test performance. In a simulation study for a factorial arranged in blocks, we compared estimates of type-I error probability and power based on the UN structure with the estimates obtained with a more parsimonious heterogeneous-compound-symmetry structure (CSH). Although tests based on the UN structure were anti-conservative with small number of factor levels, especially with four or six blocks, they became conservative at higher number of factor levels. Tests based on the CSH structure were anti-conservative, and results did not depend on the number of factor levels. When both tests were anti-conservative, tests based on the CSH structure were less so. Although use of the CSH structure is concluded to be more suitable than use of the UN structure for the small number of blocks typical in agricultural experiments, results suggest that further improvement of test statistics is needed for such situations.     

Some results on two-level factorial designs with dependent observations

There are many situations in which observations in factorial experiments may be dependent. When this is so, run orders are needed that result in efficient estimates of contrasts. The Cheng and Steinberg reverse foldover algorithm, which gives a maximal number of level changes, is known to produce very efficient main-effects two-level designs using the D-criterion, but less is known about other designs, models and criteria. We present some further theoretical results, and give another statistic of importance in predicting efficiency under strong dependence. The theory is illustrated using some 16-run designs.     

Computer-aided unbalanced supersaturated designs involving interactions

Supersaturated designs (SSDs) are defined as fractional factorial designs whose experimental run size is smaller than the number of main effects to be estimated. While most of the literature on SSDs has focused only on main effects designs, the construction and analysis of such designs involving interactions has not been developed to a great extent. In this paper, we propose a backward elimination design-driven optimization (BEDDO) method, with one main goal in mind, to eliminate the factors which are identified to be fully aliased or highly partially aliased with each other in the design. Under the proposed BEDDO method, we implement and combine correlation-based statistical measures taken from classical test theory and design of experiments field, and we also present an optimality criterion which is a modified form of Cronbach's alpha coefficient. In this way, we provide a new class of computer-aided unbalanced SSDs involving interactions, that derive directly from BEDDO optimization.     

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.     

Fixing a factor in the sequential design of two-level fractional factorial experiments

When designing two-level fractional factorial experiments sequentially, there is a wide choice of designs that could be used at each stage. Designs in which one of the factors is fixed at a particular level after the first experiment are studied in this paper. This sometimes allows all important effects to be estimated in fewer runs than would the standard sequences of designs, and effects can sometimes be estimated more efficiently. The properties of some sequences are presented, and extensions to fixing more than one factor and to factors with more than two levels are discussed.     

The effect of category choice on the odds ratio and several measures of association in case-control studies

In many case-control studies the risk factors are categorized in order to clarify the analysis and presentation of the data. However, inconsistent categorization of continuous risk factors may make interpretation difficult. This paper attempts to evaluate the effect of the categorization procedure on the odds ratio and several measures of association. Often the risk factor is dichotomized and the data linking the risk factor and the disease is presented in a 2 x 2 table. We show that the odds ratio obtained from the 2x2 table is usually considerably larger than the comparable statistic that would have been obtained had a large number of outpoints been used. Also, if 2 x 2, 2 x 3, or 2 x 4 tables are obtained by using a few outpoints on the risk factor, the measures of association for these tables are usually greater than the measure that would have been obtained had a large number of cntpoints been used. We propose an odds ratio measure that more closely approximates the odds ratio between the continuous risk factor and disease. A corresponding measure of association is also proposed for 2 x 2, 2x3, and 2x4 tables.     

On the inadequacy of the customary orthogonal arrays in quality control and general scientific experimentation,and the need of probing designs of higher revealing power ∗

This paper breaks new ground concerning the general problem of factorial experimentation, namely, the identification and estimation of nonnegligible factorial effects (with minimal number of runs), without making the usual unrealistic and artificial assumptions concerning the negligibility of higher order interactions. (In other words, we consider the general factor screening problem when interactions may be present.) Through an example, it is shown that the customary orthogonal arrays fall short of the need. New principles for sieving the set of factorial effects to determine the large ones, are introduced. The concept of ‘revealing power’of designs, i.e. of their ability to help identify nonnegligible parameters is developed, and the usefulness in this direction of balanced arrays of full strength is studied.     

Factorial experiments in the analysis of assembly systems

Assembly systems are a key tool for mass production and are increasingly being implemented in the manufacturing industry. Since the performance of such systems depends on the levels of many design variables, they are not well understood. In this paper, the performance of free transfer automatic assembly systems with closed inspection and repair loops is studied via factorial experiments of a simulated system. Five factors were identified that affect the throughput of the system: buffer size, number of pallets in the system, number of repair stations, repair time of jammed assembly machines, and subcomponent defect rate. Initially, two levels of each factor were considered, so that a full 2 5 factorial design of the experiment was used to study the system. Next, to develop a deeper understanding of the linear or non-linear effect of each factor, additional levels were investigated. Finally, a predictive model is proposed. Engineers and system designers can use this predictive model to estimate the performance of the system, given a combination of levels of each of the five factors that we studied.     

A New Method for the Analysis of Supersaturated Designs with Discrete Data

Supersaturated designs are factorial designs in which the number of potential effects is greater than the run size. They are commonly used in screening experiments, with the aim of identifying the dominant active factors with low cost. However, an important research field, which is poorly developed, is the analysis of such designs with non-normal response. In this article, we develop a variable selection strategy, through the modification of the PageRank algorithm, which is commonly used in the Google search engine for ranking Webpages. The proposed method incorporates an appropriate information theoretical measure into this algorithm and as a result, it can be efficiently used for factor screening. A noteworthy advantage of this procedure is that it allows the use of supersaturated designs for analyzing discrete data and therefore a generalized linear model is assumed. As it is depicted via a thorough simulation study, in which the Type I and Type II error rates are computed for a wide range of underlying models and designs, the presented approach can be considered quite advantageous and effective.     

D‐optimal minimax fractional factorial designs

The D‐optimal minimax criterion is proposed to construct fractional factorial designs. The resulting designs are very efficient, and robust against misspecification of the effects in the linear model. The criterion was first proposed by Wilmut & Zhou (2011); their work is limited to two‐level factorial designs, however. In this paper we extend this criterion to designs with factors having any levels (including mixed levels) and explore several important properties of this criterion. Theoretical results are obtained for construction of fractional factorial designs in general. This minimax criterion is not only scale invariant, but also invariant under level permutations. Moreover, it can be applied to any run size. This is an advantage over some other existing criteria. The Canadian Journal of Statistics 41: 325–340; 2013 © 2013 Statistical Society of Canada