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.     

Orthogonal three-level parallel flats designs for user-specified resolution

Orthogonal factorial and fractional factorial designs are very popular in many experimental studies, particularly the two-level and three-level designs used in screening experiments. When an experimenter is able to specify the set of possibly nonnegligible factorial effects, it is sometimes possible to obtain an orthogonal design belonging to the class of parallel flats designs, that has a smaller run-size than a suitable design from the class of classical fractional factorial designs belonging to the class of single flat designs. Sri-vastava and Li (1996) proved a fundamental theorem of orthogonal s-level, s being a prime, designs of parallel flats type for the user-specified resolution. They also tabulated a series of orthogonal designs for the two-level case. No orthogonal designs for three-level case are available in their paper. In this paper, we present a simple proof for the theorem given in Srivastava and Li (1996) for the three-level case. We also give a dual form of the theorem, which is more useful for developing an algorithm for construction of orthogonal designs. Some classes of three-level orthogonal designs with practical run-size are given in the paper.     

Effect of missing plots in some response surface designs

The effect of one or more missing observations for response surface designs arranged in blocks are examined in this paper. The resu lts as applied to a central composite design with orthogonal blocking, and an equirdial design with orthogonal blocking, are reported.     

Randomized response multivariate designs for categorical data

Three approaches to multivariate estimation for categorical data using randomized response (RR) are described. In the first approach, practical only for 2×2 contingency tables, a multi-proportions design is used. In the second approach, a separate RR trial is used for each variate and it is noted that the multi­variate design matrix of conditional probabilities is given by the Kroneeker product of the univariate design matrices of each trial, provided that the trials are independent of each other in a certain sense. The third approach requires only a single randomization and thus may be viewed as the use of vector response. Finally, a special-purpose bivariate design is presented.     

A new model and class of designs for mixture experiments with process variables

Experiments that involve the blending of several components are known as mixture experiments. In some mixture experiments, the response depends not only on the proportion of the mixture components, but also on the processing conditions, A new combined model is proposed which is based on Taylor series approximation and is intended to be a compromise between standard mixture models and standard response surface models. Cost and/or time constraints often limit the size of industrial experiments. With this in mind, we present a new class of designs that will accommodate the fitting of the new combined model.     

Characterization of singular balanced fractional sm factorial designs derivable from balanced arrays with maximum number of constraints

An upper bound on the maximum number of constraints for s-symbol balanced arrays is derived. It is shown that balanced fractional s^m factorial designs derived from some balanced arrays with the maximum possible number of constraints become singular.     

Projection designs for mixture experiments in orthogonal blocks

Mixture experiments are often carried out in the presence of process variables, such as days of the week or different machines in a manufacturing process, or different ovens in bread and cake making. In such experiments it is particularly useful to be able to arrange the design in orthogonal blocks, so that the model in tue mixture vanauies may ue iitteu inucpenuentiy or tne UIOCK enects mtrouuceu to take account of the changes in the process variables. It is possible in some situations that some of the ingredients in the mixture, such as additives or flavourings, are present in soian quantities, pernaps as iuw a.s 5% ur even !%, resulting in the design space being restricted to only part of the mixture simplex. Hau and Box (1990) discussed the construction of experimental designs for situations where constraints are placed on the design variables. They considered projecting standard response surface designs, including factorial designs and central composite designs, into the restricted design space, and showed that the desirable property of block orthogonality is preserved by the projections considered. Here we present a number of examples of projection designs and illustrate their use when some of the ingredients are restricted to small values, such that the design space is restricted to a sub-region within the usual simplex in the mixture variables.     

On group divisible response surface (GDRS) designs

In this paper some experimental situations are identified corresponding to which suitable response surface designs do not exist. A class of response surface designs is introduced to cope with these situations. Their analysis with and without blocking and methods of construction is discussed.     

The performance of subset response surface designs for estimating third order terms

Response surface designs are widely used in industries like chemicals, foods, pharmaceuticals, bioprocessing, agrochemicals, biology, biomedicine, agriculture and medicine. One of the major objectives of these designs is to study the functional relationship between one or more responses and a number of quantitative input factors. However, biological materials have more run to run variation than in many other experiments, leading to the conclusion that smaller response surface designs are inappropriate. Thus designs to be used in these research areas should have greater replication. Gilmour (2006) introduced a wide class of designs called “subset designs” which are useful in situations in which run to run variation is high. These designs allow the experimenter to fit the second order response surface model. However, there are situations in which the second order model representation proves to be inadequate and unrealistic due to the presence of lack of fit caused by third or higher order terms in the true response surface model. In such situations it becomes necessary for the experimenter to estimate these higher order terms. In this study, the properties of subset designs, in the context of the third order response surface model, are explored.     

Mixture designs with orthogonal blocks

Constructions of blocked mixture designs are considered in situations where BLUEs of the block effect contrasts are orthogonal to the BLUEs of the regression coefficients. Orthogonal arrays (OA), Balanced Arrays (BAs), incidence matrices of balanced incomplete block designs (BIBDs), and partially balanced incomplete block designs (PBIBDs) are used. Designs with equal and unequal block sizes are considered. Also both cases where the constants involved in the orthogonality conditions depend and do not depend on the factors have been taken into account. Some standard (already available) designs can be obtained as particular cases of the designs proposed here.     

Near-optimal designs for dual channel microarray studies

Summary.  Much biological and medical research employs microarray studies to monitor gene expression levels across a wide range of organisms and under many experimental conditions. Dual channel microarrays are a common platform and allow two samples to be measured simultaneously. A frequently used design uses a common reference sample to make conditions across different arrays comparable. Our aim is to formulate microarray experiments in the experimental design context and to use simulated annealing to search for near-optimal designs. We identify a subclass of designs, the so-called interwoven loop designs, that seems to have good optimality properties compared with the near-optimal designs that are found by simulated annealing. Commonly used reference designs and dye swap designs are shown to be highly inefficient.     

Further exploratory analysis of split-plot experiments to study certain stratified effects

Designed experiments are a key component in many companies' improvement strategies. Because completely randomized experiments are not always reasonable from a cost or physical perspective, split-plot experiments are prevalent. The recommended analysis accounts for the different sources of variation affecting whole-plot and split-plot error. However experiments on industrial processes must be run and, consequently analyzed quite differently from ones run in a controlled environment. Such experiments are typically subject to a wide array of uncontrolled, and barely understood, variation. In particular, it is important to examine the experimental results for additional, unanticipated sources of variation. In this paper, we consider how unanticipated, stratified effects may influence a split-plot experiment and discuss further exploratory analysis to indicate the presence of stratified effects. Examples of such experiments are provided, additional tests are suggested and discussed in light of their power, and recommendations given.     

On d- and e- minimax optimal designs for estimating the axial slopes of a second-order response surface over hypercubic regions

Design of experiments for estimating the slopes of a response surface is considered. Design criteria analogous to the traditional ones but based upon the variance-covariance matrix of the estimated slopes along factor axes are proposed. Optimal designs under the proposed criteria are derived for second-order polynomial regression over hypercubic regions. Best de¬signs within some commonly used classes of designs are also obtained and their efficiencies are investigated.     

Orthogonal arrays of strength three from regular 3-wise balanced designs

The construction given in Kreher, J Combin Des 4 (1996) 67 is extended to obtain new infinite families of orthogonal arrays of strength 3. Regular 3-wise balanced designs play a central role in this construction.     

A note on an upper bound for the constraints of balancedarrays with strength t

In this paper we obtain an upper bound for the number of constraints of some balanced arrays (B-arrays) of strength t(t≥3) and with two symbols.     

New 3-level response surface designs constructed from incomplete block designs

Box and Behnken [1958. Some new three level second-order designs for surface fitting. Statistical Technical Research Group Technical Report No. 26. Princeton University, Princeton, NJ; 1960. Some new three level designs for the study of quantitative variables. Technometrics 2, 455–475.] introduced a class of 3-level second-order designs for fitting the second-order response surface model. These 17 Box–Behnken designs (BB designs) are available for 3–12 and 16 factors. Although BB designs were developed nearly 50 years ago, they and the central-composite designs of Box and Wilson [1951. On the experimental attainment of optimum conditions. J. Royal Statist. Soc., Ser. B 13, 1–45.] are still the most often recommended response surface designs. Of the 17 aforementioned BB designs, 10 were constructed from balanced incomplete block designs (BIBDs) and seven were constructed from partially BIBDs (PBIBDs). In this paper we show that these seven BB designs constructed from PBIBDs can be improved in terms of rotatability as well as average prediction variance, D$D$- and G$G$-efficiency. In addition, we also report new orthogonally blocked solutions for 5, 8, 9, 11 and 13 factors. Note that an 11-factor BB design is available but cannot be orthogonally blocked. All new designs can be found at http://www.math.montana.edu/∼$\sim$jobo/bbd/.     

The plan of enrichment designs for dealing with high placebo response

To deal with high placebo response in clinical trials for psychiatric and other diseases, different enrichment designs, such as the sequential parallel design, two‐way enriched design, and sequential enriched design, have been proposed and implemented recently. Depending on the historical trial information and the trial sponsors' resources, detailed design elements are needed for determining which design to adopt. To assist in making more suitable decisions, we perform evaluations for selecting required design elements in terms of power optimization and sample size planning. We also discuss the implementation of the interim analysis related to its applicability.     

On the construction of orthogonal factorial designs of resolution 11

A new method of construction of orthogonal resolution IV designs for symmetrical and asymmetrical factorials has been presented. Many new series of orthogonal factorial designs of resolution IV can be obtained by the above general method.     

Difference variance dispersion graphs for comparing response surface designs with applications in food technology

Variance dispersion graphs have become a popular tool in aiding the choice of a response surface design. Often differences in response from some particular point, such as the expected position of the optimum or standard operating conditions, are more important than the response itself. We describe two examples from food technology. In the first, an experiment was conducted to find the levels of three factors which optimized the yield of valuable products enzymatically synthesized from sugars and to discover how the yield changed as the levels of the factors were changed from the optimum. In the second example, an experiment was conducted on a mixing process for pastry dough to discover how three factors affected a number of properties of the pastry, with a view to using these factors to control the process. We introduce the difference variance dispersion graph (DVDG) to help in the choice of a design in these circumstances. The DVDG for blocked designs is developed and the examples are used to show how the DVDG can be used in practice. In both examples a design was chosen by using the DVDG, as well as other properties, and the experiments were conducted and produced results that were useful to the experimenters. In both cases the conclusions were drawn partly by comparing responses at different points on the response surface.