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.     

Some adjusted orthogonal,variance balanced,multidimensional block designs

A multidimensional block design (MBD) is an experimental design with d > 1 blocking criteria geometrically represented as a d-dimensional lattice with treatment varieties assigned to some or all nodes of the lattice. Intrablock analysis of variance tables for some special classes of two- and three-dimensional block designs with some empty nodes are given. Design plans and efficiencies for 31 two-dimensional designs, each universally optimal in defined classes of designs, and 7 three-dimensional designs, each nearly optimal in defined classes of designs, are listed in the appendices. A need for such designs is apparent when the blocking criteria are implemented successively and empty nodes do not represent wasted experimental units.     

A non-iterative least squares estimation of missing values in experimental designs

In this paper an attempt has been made to obtain a systematic method of estimating the missing values in experimental designs. When the observations are missing in a particular pattern (in RBD and LSD) explicit expressions are given for the estimators of the missing values. This procedure is compared with Yate's iterative procedure by numerical examples.     

Optimal change-over designs for correlated observations

Design implications of an autoregressive model for change-over experiments are investigated. In this model, the residual effect due to the previous treatment is assumed to be proportional to the response in the previous period. In addition, the errors from the same experimental subject are assumed to be correlated according to a first-order autoregressive model. Models with fixed and random subject effects are discussed separately. An attempt has been made to identify and construct optimal or nearly optimal designs in various situations. Empirical conclusions of Taka and Armitage [Commun. Statist. Theor. Meth. (1983)12, 865-876] regarding the efficiency of some designs have also been confirmed.     

Cross-over Designs in the Presence of Higher Order Carry-overs

In cross-over experiments, where different treatments are applied successively to the same experimental unit over a number of time periods, it is often expected that a treatment has a carry-over effect in one or more periods following its period of application. The effect of interaction between the treatments in the successive periods may also affect the response. However, it seems that all systematic studies of the optimality properties of cross-over designs have been done under models where carry-over effects are assumed to persist for only one subsequent period. This paper proposes a model which allows for the possible presence of carry-over effects up to k subsequent periods, together with all the interactions between treatments applied at k + 1 successive periods. This model allows the practitioner to choose k for any experiment according to the requirements of that particular experiment. Under this model, the cross-over designs are studied and the class of optimal designs is obtained. A method of constructing these optimal designs is also given.     

Measures of non orthogonality in block designs

Orthogonality is an important concept in block design. Necessary and sufficient condition for a connected block design to be orthogonal is well known. However, when a design is not orthogonal, it is not clear how much it deviates from orthogonality. In this paper, an attempt has been made to first define the measures of or indices to non orthogonality in block design and then to characterize designs possessing minimum non orthogonality indices. It is shown that a Balanced Incomplete Block Design (BIBD) and a Balanced Block Design (BBD), if exist, possess this property.     

Optimal k-circulant supersaturated designs

Optimal k-circulant supersaturated designs have been constructed in literature using computer intensive methods. A systematic method of construction for multi-level experiments based on balanced incomplete block designs is presented in this paper. The method is also applicable to two-level experiments. Illustrative examples are also given.     

Balanced Factorial Designs for cDNA Microarray Experiments

Balanced factorial designs are introduced for cDNA microarray experiments. Single replicate designs obtained using the classical method of confounding are shown to be particularly useful for deriving suitable balanced designs for cDNA microarrays. Classical factorial designs obtained using methods other than the method of confounding are also shown to be useful. The paper provides a systematic method of deriving designs for microarray experiments as opposed to algorithmic and ad-hoc methods and generalizes several of the microarray designs given recently in the literature.     

Construction of nested space-filling designs using difference matrices

Experiments that study complex real world systems in business, engineering and sciences can be conducted at different levels of accuracy or sophistication. Nested space-filling designs are suitable for such multi-fidelity experiments. In this paper, we propose a systematic method to construct nested space-filling designs for experiments with two levels of accuracy. The method that makes use of nested difference matrices can be easily performed, many nested space-filling designs for experiments with two levels of accuracy can thus be constructed, and the resulting designs achieve stratification in low dimensions. In addition, the proposed method can also be used to obtain sliced space-filling designs for conducting computer experiments with both qualitative and quantitative factors.     

ON DERIVATION OF INITIAL BLOCKS OF B.I.B. DESIGNS WITH MORE THAN ONE INITIAL BLOCK

A good amount of work has been done on the construction of balanced incomplete block (B.I.B.) designs by Bose (1939, 1942), Sprott (1954, 1956), Rao (1961), Takeuchi (1962) and others. Sprott (1954, 1956) obtained several series of B.I.B. designs through difference sets. The main purpose of the present investigation is to provide two methods of construction of B.I.B. designs obtainable through more than one initial block. The first method derives initial blocks of a series of designs from some specified blocks of a B.I.B. design obtainable by developing one or more initial blocks. The second method attempts to obtain one of the initial blocks (the basic initial block) through the different powers of an element of a finite field; then an appropriate method for generating the other initial blocks from it is discussed. A table showing the basic initial block for different designs has been presented. By these methods several solutions of some B.I.B. designs could be obtained from different initial blocks. An examination was therefore made to see if these designs were all isomorphic.     

Bayes a-opttmal and efficient block designs for comparing test treatments with a standard treatment

A sufficient condition for the Bayes A-optimality of block designs when comparing a standard treatment with v test treatments is given by Majumdar. (In:Optimal Design and Analysis of Experiments, Y. Dodge, V. V. Fedorov and H. P. Wynn (Eds.), 15-27, North-Holland, 1988). The priors that he considers depend on a constant α ε [0, ∞), with α - 0 corresponding to no prior information at all. The given sufficient condition, consequently, also depends on a. Large families of optimal and highly efficient designs are only known for the case α - 0. We will show how some of the results for α - 0 can be extended to obtain large families of optimal and highly efficient designs for arbitrary values of α. In addition, these results are useful when considering design robustness against an improper choice of α.     

ALL-BIAS DESIGNS FOR POLYNOMIAL SPLINE REGRESSION MODELS

Polynomial spline regression models of low degree have proved useful in modeling responses from designed experiments in science and engineering when simple polynomial models are inadequate. Where there is uncertainty in the number and location of the knots, or breakpoints, of the spline, then designs that minimize the systematic errors resulting from model misspecification may be appropriate. This paper gives a method for constructing such all‐bias designs for a single variable spline when the distinct knots in the assumed and true models come from some specified set. A class of designs is defined in terms of the inter‐knot intervals and sufficient conditions are obtained for a design within this class to be all‐bias under linear, quadratic and cubic spline models. An example of the construction of all‐bias designs is given.     

Equally spaced design points in polynomial regression: A comparison of systematic sampling methods with the optimal design of experiments

In some situations an experimenter may desire to have equally spaced design points. Three methods of obtaining such points on the interval [—1,1]—namely systematic random sampling, centrally located systematic sampling, and a purposive systematic sampling method which includes the endpoints - 1 and 1 as two of the design points-are evaluated under the D-optimal and G-optimal criteria. These methods are also compared to the optimal designs in polynomial regression and to the limiting designs of Kiefer and Studden (1976).     

Constructions of partially efficiency-balanced designs and their analysis

Partially Efficiency-Balanced (PEB) designs were introduced by Puri and Nigam $year:1977. These designs may be particularly useful for bio-assays and factorial experiments and have a very simple analysis. In the present paper, we give several new systematic procedures of constructing PEB designs with varying replications. A number of practical series of PEB designs are presented. We shall also pre- sent the methods of analysing these designs produced here.     

Construction on non-adaptive hypergeometric group testing desings for identifying at most two defectives

In the literature a systematic method of obtaining a group testing design is not available at present. Weideman and Raghavarao (1987a, b) gave methods for the construction of non - adaptive hypergeometric group testing designs for identifying at most two defectives by using a dual method. In the present investigation we have developed a method of construction of group testing designs from (i) Hypercubic Designs for t ≡ 3 (mod 6) and (ii) Balanced Incomplete Block Designs for t ≡ 1 (mod 6) and t ≡ 3 (mod 6). These constructions are accomplished by the use of dual designs. The designs so constructed satisfy specified properties and attained an optimal bound as discussed by Weidman and Raghavarao (1987a,b). Here it is also shown that the condition for pairwise disjoint sets of BIBD for t ≡ 1 (mod 6) given by Weideman and Raghavarao (1987b) is not true for all such designs.     

ON INCOMPLETE BLOCK DESIGNS FOR SYMMETRIC PARALLEL LINE ASSAYS

Incomplete block designs for symmetric parallel line assays, which estimate L_p, L₁and L1/1 with full efficiency, are constructed in a unified manner. Necessary and sufficient conditions are given for existence. A complete solution is presented when the number of doses is even. A general systematic method of construction is given when the number of doses is odd, and all possible designs for up to 15 doses of each preparation are tabulated.     

One-eighth- and one-sixteenth-fraction quaternary code designs with high resolution

The development of a general methodology for the construction of good two-level nonregular designs has received significant attention over the last 10 years. Recent works by Phoa and Xu (2009) and Zhang et al. (2011) indicate that quaternary code (QC) designs are very promising in this regard. This paper explores a systematic construction for 1/8th and 1/16th fraction QC designs with high resolution for any number of factors. The 1/8th fraction QC designs often have larger resolution than regular designs of the same size. A majority of the 1/16th fraction QC designs also have larger resolution than comparable two-level regular designs.     

Efficient designs based on orthogonal arrays of type I and type II for experiments using units ordered over time or space

For a wide variety of applications, experiments are based on units ordered over time or space. Models for these experiments generally may include one or more of: correlations, systematic trends, carryover effects and interference effects. Since the standard optimal block designs may not be efficient in these situations, orthogonal arrays of type I and type II, which were introduced in 1961 by C.R. Rao [Combinatorial arrangements analogous to orthogonal arrays, Sankhya A 23 (1961) 283–286], have been recently used to construct optimal and efficient designs for many of these experiments. Results in this area are unified and the salient features are outlined.     

A comparative study on detection of influential observations in linear regression

A large number of statistics are used in the literature to detect outliers and influential observations in the linear regression model. In this paper comparison studies have been made for determining a statistic which performs better than the other. This includes: (i) a detailed simulation study, and (ii) analyses of several data sets studied by different authors. Different choices of the design matrix of regression model are considered. Design A studies the performance of the various statistics for detecting the scale shift type outliers, and designs B and C provide information on the performance of the statistics for identifying the influential observations. We have used cutoff points using the exact distributions and Bonferroni's inequality for each statistic. The results show that the studentized residual which is used for detection of mean shift outliers is appropriate for detection of scale shift outliers also, and the Welsch's statistic and the Cook's distance are appropriate for detection of influential observations.