Row-column designs with minimal units

A new class of row-column designs is proposed. These designs are saturated in terms of eliminating two-way heterogeneity with an additive model. The proposed designs are treatment-connected, i.e., all paired comparisons of treatments in the designs are estimable in spite of the existence of row and column effects. The connectedness of the designs is justified from two perspectives: linear model and contrast estimability. Comparisons with other designs are studied in terms of A-, D-, E-efficiencies as well as design balance.     

A class of saturated row–column designs

A new class of row–column designs is proposed. These designs are saturated in terms of eliminating two-way heterogeneity with an additive model. The (m,s)-criterion is used to select optimal designs. It turns out that all (m,s)-optimal designs are binary. Square (m,s)-optimal designs are constructed and they are treatment-connected. Thus, all treatment contrasts are estimable regardless of the row and column effects.     

m-Associate PBIB designs using Youden-m squares

In this paper, we have proposed a type of arrangement that we call Youden-m square and is similar to the usual Youden square but generates PBIB designs instead of BIB designs when its columns are taken as blocks. We have also discussed its construction methodologies, introduced two new m-associate class association schemes, and also constructed some series of Youden-m square type PBIB designs.     

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.     

An easy method of constructing latin square designs balanced for the immediate residual and other order effects

Bradley (1958) proposed a very simple procedure for constructing latin square designs to counterbalance the immediate sequential effect for an even number of treatments. When the number of treatments is odd, balance in a single latin square is not possible. In the present note we have developed an analogous method for the construction of such designs which may be used for an even or odd number of treatments. A proof has also been offered to assure the general validity of the procedure.     

Low-order spatially-distinct latin squares

The order three to five spatially-distinct Latin squares, and the order three to six spatially-distinct Latin square treatment designs are listed. Some statistical results are given. Designs for 4, 5 and 6 treatments that were found previously to be robust to a linear by linear interacrion are shown to be optimal. Designs with good neighbour balanced are also considered.     

Paired permutation testing in 2<Superscript><Emphasis Type="Italic">k</Emphasis></Superscript> unreplicated factorials

Exact permutation testing of effects in unreplicated two-level multifactorial designs is developed based on the notion of realigning observations and on paired permutations. This approach preserves the exchangeability of error components for testing up tok effects. Advantages and limitations of exact permutation procedures for unreplicated factorials are discussed and a simulation study on paired permutation testing is presented.     

Analyzing type II censored data obtained from repetitious experiments

Experimental design and Taguchi's parameter design are widely employed by industry to optimize the process/product. However, censored data are often observed in product lifetime testing during the experiments. After implementing a repetitious experiment with type II censored data, the censored data are usually estimated by establishing a complex statistical model. However, using the incomplete data to fit a model may not accurately estimates the censored data. Moreover, the model fitting process is complicated for a practitioner who has only limited statistical training. This study proposes a less complex approach to analyze censored data, using the least square estimation method and Torres's analysis of unreplicated factorials with possible abnormalities. This study also presents an effective method to analyze the censored data from Taguchi's parameter design using least square estimation method. Finally, examples are given to illustrate the effectiveness of the proposed methods.     

Randomization of neighbour balanced generalized Youden designs

If a crossover design with more than two treatments is carryover balanced, then the usual randomization of experimental units and periods would destroy the neighbour structure of the design. As an alternative, Bailey [1985. Restricted randomization for neighbour-balanced designs. Statist. Decisions Suppl. 2, 237–248] considered randomization of experimental units and of treatment labels, which leaves the neighbour structure intact. She has shown that, if there are no carryover effects, this randomization validates the row–column model, provided the starting design is a generalized Latin square. We extend this result to generalized Youden designs where either the number of experimental units is a multiple of the number of treatments or the number of periods is equal to the number of treatments. For the situation when there are carryover effects we show for so-called totally balanced designs that the variance of the estimates of treatment differences does not change in the presence of carryover effects, while the estimated variance of this estimate becomes conservative.     

On the equivalence of three estimators for dispersion effects in unreplicated two-level factorial designs

Box and Meyer [1986. Dispersion effects from fractional designs. Technometrics 28(1), 19–27] were the first to consider identifying both location and dispersion effects from unreplicated two-level fractional factorial designs. Since the publication of their paper a number of different procedures (both iterative and non-iterative) have been proposed for estimating the location and dispersion effects. An overview and a critical analysis of most of these procedures is given by Brenneman and Nair [2001. Methods for identifying dispersion effects in unreplicated factorial experiments: a critical analysis and proposed strategies. Technometrics 43(4), 388–405]. Under a linear structure for the dispersion effects, non-iterative estimation methods for the dispersion effects were proposed by Brenneman and Nair [2001. Methods for identifying dispersion effects in unreplicated factorial experiments: a critical analysis and proposed strategies. Technometrics 43(4), 388–405], Liao and Iyer [2000. Optimal 2^n-p$2^{n-p}$ fractional factorial designs for dispersion effects under a location-dispersion model. Comm. Statist. Theory Methods 29(4), 823–835] and Wiklander [1998. A comparison of two estimators of dispersion effects. Comm. Statist. Theory Methods 27(4), 905–923] (see also Wiklander and Holm [2003. Dispersion effects in unreplicated factorial designs. Appl. Stochastic. Models Bus. Ind. 19(1), 13–30]). We prove that for two-level factorial designs the proposed estimators are different representations of a single estimator. The proof uses the framework of Seely [1970a. Linear spaces and unbiased estimation. Ann. Math. Statist. 41, 1725–1734], in which quadratic estimators are expressed as inner products of symmetric matrices.     

Optimal Circular Block Designs for Neighbouring Competition Effects

Competition or interference occurs when the responses to treatments in experimental units are affected by the treatments in neighbouring units. This may contribute to variability in experimental results and lead to substantial losses in efficiency. The study of a competing situation needs designs in which the competing units appear in a predetermined pattern. This paper deals with optimality aspects of circular block designs for studying the competition among treatments applied to neighbouring experimental units. The model considered is a four-way classified model consisting of direct effect of the treatment applied to a particular plot, the effect of those treatments applied to the immediate left and right neighbouring units and the block effect. Conditions have been obtained for the block design to be universally optimal for estimating direct and neighbour effects. Some classes of balanced and strongly balanced complete block designs have been identified to be universally optimal for the estimation of direct, left and right neighbour effects and a list of universally optimal designs for v<20 and r<100 has been prepared.     

Balanced incomplete Latin square designs

Latin squares have been widely used to design an experiment where the blocking factors and treatment factors have the same number of levels. For some experiments, the size of blocks may be less than the number of treatments. Since not all the treatments can be compared within each block, a new class of designs called balanced incomplete Latin squares (BILS) is proposed. A general method for constructing BILS is proposed by an intelligent selection of certain cells from a complete Latin square via orthogonal Latin squares. The optimality of the proposed BILS designs is investigated. It is shown that the proposed transversal BILS designs are asymptotically optimal for all the row, column and treatment effects. The relative efficiencies of a delete-one-transversal BILS design with respect to the optimal designs for both cases are also derived; it is shown to be close to 100%, as the order becomes large.     

The efficiency and optimality characterization of certain six-Treatment incomplete-Block designs emanating from some quasi-Semi-Latin squares

Incomplete-block designs were constructed from three types of three-factor block-structured combinatorial designs which are not necessarily semi-Latin squares but whose treatments are ab initio arranged as in the semi-Latin square. The basic consideration for the construction involves the exploitation of the different block structures of the experimental units of the Quasi-semi-Latin Squares (QSLS) for the same treatments. All the emanating designs are found to be connected, only the designs in nine blocks are both variance- and efficiency-balanced and offer gain in efficiency but lead to about 46 percent loss of information.     

Designs for two-colour microarray experiments

Summary.  Designs for two-colour microarray experiments can be viewed as block designs with two treatments per block. Explicit formulae for the A- and D-criteria are given for the case that the number of blocks is equal to the number of treatments. These show that the A- and D-optimality criteria conflict badly if there are 10 or more treatments. A similar analysis shows that designs with one or two extra blocks perform very much better, but again there is a conflict between the two optimality criteria for moderately large numbers of treatments. It is shown that this problem can be avoided by slightly increasing the number of blocks. The two colours that are used in each block effectively turn the block design into a row–column design. There is no need to use a design in which every treatment has each colour equally often: rather, an efficient row–column design should be used. For odd replication, it is recommended that the row–column design should be based on a bipartite graph, and it is proved that the optimal such design corresponds to an optimal block design for half the number of treatments. Efficient row–column designs are given for replications 3–6. It is shown how to adapt them for experiments in which some treatments have replication only 2.     

Repeated measurements designs with unequal period sizes

Summary This paper is concerned with the designs in which each experimental unit is assigned more than once to a treatment, either different or identical. An easy method of constructing balanced minimal repeated measurements designs with unequal period sizes is presented whenever the number of periods is less than the number of treatments. Strongly balanced minimal repeated measurements designs with unequal period sizes are also constructed whenever the number of periods is less than the number of treatments.     

Designing fractional factorial split-plot experiments with few whole-plot factors

Summary.  When it is impractical to perform the experimental runs of a fractional factorial design in a completely random order, restrictions on the randomization can be imposed. The resulting design is said to have a split-plot, or nested, error structure. Similarly to fractional factorials, fractional factorial split-plot designs can be ranked by using the aberration criterion. Techniques that generate the required designs systematically presuppose unreplicated settings of the whole-plot factors. We use a cheese-making experiment to demonstrate the practical relevance of designs with replicated settings of these factors. We create such designs by splitting the whole plots according to one or more subplot effects. We develop a systematic method to generate the required designs and we use the method to create a table of designs that is likely to be useful in practice.     

Generalized linear models and transformations for unreplicated factorial experiments in the presence of dispersion effects

For the analysis of replicated designs, many different methods have been suggested. These allow for the estimation of functional dependencies between mean and variance as well as possible dispersion effects within the same model framework. However, in the situation of unreplicated designs, most methods known so far rely on the assumption of constant variances, or a functional relationship between mean and variance as the only source of heteroscedasticity. In this paper, we propose two methods for dealing with unreplicated data, when dispersion effects might also be of importance. One of these is an extension of the Box–Cox-method [Box, G.E.P., Cox, D.R., 1964. An analysis of transformations. Journal of the Royal Statistical Society B 26, 211–252], the other is based on double generalized linear models. Both these methods turn out to yield approximately equivalent results in the case of comparable assumptions, whereas the double generalized linear model is the more general one and allows further extensions. If this class of models is assumed, consistency, asymptotic efficiency and normality of the resulting estimates are shown.     

Doubly adaptive biased coin designs with delayed responses

In clinical studies, patients are usually accrued sequentially. Response‐adaptive designs are then useful tools for assigning treatments to incoming patients as a function of the treatment responses observed thus far. In this regard, doubly adaptive biased coin designs have advantageous properties under the assumption that their responses can be obtained immediately after testing. However, it is a common occurrence that responses are observed only after a certain period of time. The authors examine the effect of delayed responses on doubly adaptive biased coin designs and derive some of their asymptotic properties. It turns out that these designs are relatively insensitive to delayed responses under widely satisfied conditions. This is illustrated with a simulation study.     

Clustering Effects in Unreplicated Factorial Experiments

The analysis of unreplicated factorial designs concentrates much attention since there are no degrees of freedom left to estimate the error variance. In this article, we propose clustering the factorial estimates in two groups, one containing the active effects and one containing the inactive effects. The powerfulness of the proposed method is revealed via a comparative simulation study.     

Some important classes of generalized neighbor designs for linear blocks

In this paper, some infinite series of generalized neighbor designs are constructed for the linear blocks which are useful to balance out the neighbor effects for the cases where (a) one of the v treatments has some neighbor effects with other treatments, while remaining (v – 1) treatments have half of that neighbor effect among selves, (b) some of the v treatments have some neighbor effect with other treatments, while remaining treatments have half of that neighbor effect among themselves, (c) one of the v treatments has some neighbor effect with other treatments, while remaining (v – 1) treatments have double of that effect among themselves, and (d) some of the v treatments have some neighbor effect with other, while remaining treatments have double of it among themselves.