Optimal blocked main effects plans

In this paper, we consider the construction of optimal blocked main effects designs where m two-level factors are to be studied in N runs which are partitioned into b blocks of equal size. For N ≡ 2±od4 sufficient conditions are derived for a design to be Φ _f optimal among all designs having main effects occurring equally often at their high and low levels within blocks and then this result is extended to the class of all designs for the case when the block size is two. Methods of constructing designs satisfying the sufficient conditions derived are also given.     

Über das asymptotische verhalten von bayesschen und maximum-likelihood-schätzungen

The paper deals with the asymptotic equivalence of the maximum-likelihood-estimator with sums of independent and identically distributed random variables for a family of distribution functions, the density of which may vanish depending on the parameter. Further the asymptotic equivalence of the maximum-likelihood-estimator with BAYES estimators is shown.     

Minimization of model inadequacy when observations subject to shift in means and variances

In this article optimality of experimental design for fitting a lower-order polynomial to a higher order response function for the situation in which observations may be subject to shift in means as well as in variances is considered. It is found that Karson, Manson and Hader‘s (1969) optimum designs provide pro-tection, in some sense, against model inadequacies even when observations are subject to shift in means and variances.     

On a paper by kageyama and mohan

Kageyama Mohan (1984) have presented three methods of constructing new incomplete block designs from balanced incomplete block designs, They raise questions about the designs which come from each of their methods, These questions are answered, Another series of group divisible designs is derived as a special case of their second method.     

Isomorphism check in fractional factorial designs via letter interaction pattern matrix

Two fractional factorial designs are considered isomorphic if one can be obtained from the other by relabeling the factors, reordering the runs, and/or switching the levels of factors. To identify the isomorphism of two designs is known as an NP hard problem. In this paper, we propose a three-dimensional matrix named the letter interaction pattern matrix (LIPM) to characterize the information contained in the defining contrast subgroup of a regular two-level design. We first show that an LIPM could uniquely determine a design under isomorphism and then propose a set of principles to rearrange an LIPM to a standard form. In this way, we can significantly reduce the computational complexity in isomorphism check, which could only take O(2^p)+O(3k³)+O(2^k) operations to check two 2^k−p designs in the worst case. We also find a sufficient condition for two designs being isomorphic to each other, which is very simple and easy to use. In the end, we list some designs with the maximum numbers of clear or strongly clear two-factor interactions which were not found before.     

Spacing of information in polynomial regression:a simple solution

Consider a k polynomial regression on a single real variable. If n uncorrelated observations are to be taken in a design with support on more than k+1 points, there is an approximate experiment, ν, with support on k+1 points and n observations such that both designs have the same information matrix for the model. A proof of this result is provided. A method to obtain the approximate design ν is given and illustrated by an example. The source of disagreement between Kiefer (1959) and De La Garza (1954) in the solution of this problem is clarified.     

Supersaturated design including an orthogonal base

Recently, many supersaturated designs have been proposed. A supersaturated design is a fractional factorial design in which the number of factors is greater than the number of experimental runs. The main thrust of the previous studies has been to generate more columns while avoiding large values of squared inner products among all design columns. These designs would be appropriate if the probability for each factor being active is uniformly distributed. When factors can be partitioned into two groups, namely, with high and low probabilities of each factor being active, it is desirable to maintain orthogonality among columns to be assigned to the factors in the high-probability group. We discuss a supersaturated design including an orthogonal base which is suitable for this common situation. Mathematical results on the existence of the supersaturated designs are shown, and the construction of supersaturated designs is presented. We next discuss some properties of the proposed supersaturated designs based on the squared inner products.     

Blocking in two-level non-regular fractional factorial designs

In this paper we consider screening experiments where a two-level fractional factorial design is to be used to identify significant factors in an experimental process and where the runs in the experiment are to occur in blocks of equal size. A simple method based on the foldover technique is given for constructing resolution IV orthogonal and non-orthogonal blocked designs and examples are given to illustrate the process.     

Modified D-optimal design for logistic model

Asymptotic theory of using the Fisher information matrix may provide poor approximation to the exact variance matrix of maximum likelihood estimation in nonlinear models. This may be due to not obtaining an efficient D-optimal design. In this article, we propose a modified D-optimality criterion, using a more accurate information matrix, based on the Bhattacharyya matrix. The proposed information matrix and its properties are given for two parameters simple logistic model. It is shown that the resulted modified locally D-optimal design is more efficient than the previous one; particularly, for small sample size experiments.     

The nearest proportional to size sampling design

The construction of a proportional to size sampling design nearest to a given design is introduced. The realization of the new design can be easily made with the help of the original one.     

BALANCED ARRAYS AND WEIGHING DESIGNS

Dey (19711, Saha (1975), Kageyama & Saha (1983) and others have shown how optimum chemical balance weighing designs can be constructed from the incidence matrices of balanced incomplete block (BIB) designs. In this paper, it is shown that weighing designs can be constructed from some suitably chosen two-symbol balanced arrays of strength two, which need not always be incidence matrices of BIB designs. The findings lead us to construct new optimum chemical balance weighing designs from incidence matrices of BIB designs.     

Design effect of randomized systematic sampling

In statistical practice, systematic sampling (SYS) is used in many modifications due to its simple handling. In addition, SYS may provide efficiency gains if it is well adjusted to the structure of the population under study. However, if SYS is based on an inappropriate picture of the population a high decrease of efficiency, i.e. a high increase in variance may result by changing from simple random sampling to SYS. In the context of two-stage designs SYS so far seems often in use for subsampling within the primary units. As an alternative to this practice, we propose to randomize the order of the primary units, then to select systematically a number of primary units and, thereafter, to draw secondary units by simple random sampling without replacement within the primary units selected. This procedure is more efficient than simple random sampling with replacement from the whole population of all secondary units, i.e. the variance of an adequate estimator for a total is never increased by changing from simple random sampling to randomized SYS whatever be the values associated by a characteristic with the secondary units, while there are values for which the variance decreases for the change mentioned. This result should hold generally, even if our proof, so far, is not complete for general sample sizes.     

A method of selecting the levels of regressor variables in lognormal regression model

Kupper and Meydrech and Myers and Lahoda introduced the mean squared error (MSE) approach to study response surface designs, Duncan and DeGroot derived a criterion for optimality of linear experimental designs based on minimum mean squared error. However, minimization of the MSE of an estimator maxr renuire some knowledge about the unknown parameters. Without such knowledge construction of designs optimal in the sense of MSE may not be possible. In this article a simple method of selecting the levels of regressor variables suitable for estimating some functions of the parameters of a lognormal regression model is developed using a criterion for optimality based on the variance of an estimator. For some special parametric functions, the criterion used here is equivalent to the criterion of minimizing the mean squared error. It is found that the maximum likelihood estimators of a class of parametric functions can be improved substantially (in the sense of MSE) by proper choice of the values of regressor variables. Moreover, our approach is applicable to analysis of variance as well as regression designs.     

Improved conditions for the robustness of binary block designs against the loss of whole blocks

Several authors have investigated conditions for a binary block design, D, to be maximally robust such that every eventual design obtained from D by eliminating r_[υ]−1 blocks is connected, where r_[υ] is the smallest treatment replication. Four new results for the maximal robustness of D with superior properties are given. An extension of these results to widen the assessment of robustness of the planned design is also presented.     

E-optimal design in irregular BIBD settings

When the necessary conditions for a BIBD are satisfied, but no BIBD exists, there is no simple answer for the optimal design problem. This paper identifies the E-optimal information matrices for any such irregular BIBD setting when the number of treatments is no larger than 100. A- and D-optimal designs are typically not E-optimal. An E-optimal design for 15 treatments in 21 blocks of size 5 is found.     

Optimal designs for estimating a choice hierarchy by a general nested multinomial logit model

Abstract

In choice experiments the process of decision-making can be more complex than the proposed by the Multinomial Logit Model (MNL). In these scenarios, models such as the Nested Multinomial Logit Model (NMNL) are often employed to model a more complex decision-making. Understanding the decision-making process is important in some fields such as marketing. Achieving a precise estimation of the models is crucial to the understanding of this process. To do this, optimal experimental designs are required. To construct an optimal design, information matrix is key. A previous research by others has developed the expression for the information matrix of the two-level NMNL model with two nests: Alternatives nest (J alternatives) and No-Choice nest (1 alternative). In this paper, we developed the likelihood function for a two-stage NMNL model for M nests and we present the expression for the information matrix for 2 nests with any amount of alternatives in them. We also show alternative D-optimal designs for No-Choice scenarios with similar relative efficiency but with less complex alternatives which can help to obtain more reliable answers and one application of these designs.     

Approximate E-optimal designs for the model of spring balance weighing with a constant bias

The present paper analyzes the linear regression model with a nonzero intercept term on the vertices of a d-dimensional unit cube. This setting may be interpreted as a model of weighing d objects on a spring balance with a constant bias. We give analytic formulas for E-optimal designs, as well as their minimal efficiencies under the class of all orthogonally invariant optimality criteria, proving the criterion-robustness of the E-optimal designs. We also discuss the D- and A-optimal designs for this model.     

Construction and analysis of edge designs from skew-symmetric supplementary difference sets

The purpose of screening experiments is to identify the dominant variables from a set of many potentially active variables which may affect some characteristic y. Edge designs were recently introduced in the literature and are constructed by using conferences matrices and were proved to be robust. We introduce a new class of edge designs which are constructed from skew-symmetric supplementary difference sets. These designs are particularly useful since they can be applied for experiments with an even number of factors and they may exist for orders where conference matrices do not exist. Using this methodology, examples of new edge designs for 6, 14, 22, 26, 38, 42, 46, 58, and 62 factors are constructed. Of special interest are the new edge designs for studying 22 and 58 factors because edge designs with these parameters have not been constructed in the literature since conference matrices of the corresponding order do not exist. The suggested new edge designs achieve the same model-robustness as the traditional edge designs. We also suggest the use of a mirror edge method as a test for the linearity of the true underlying model. We give the details of the methodology and provide some illustrating examples for this new approach. We also show that the new designs have good D-efficiencies when applied to first order models.     

Two-level Orthogonal and Saturated Designs for Estimating Main Effects and Certain Important Interactions

In a two-level factorial experiment, we consider orthogonal designs that allow joint estimation of the grand mean, all main effects, and certain classes of two-level interactions, assuming that the remaining effects are all negligible. Based on a judicious allocation of the factorial effects of interest to the columns of a Hadamard matrix, we propose some general classes of orthogonal and saturated designs which include some existing orthogonal main-effect plans of asymmetric factorials as special cases.     

On the E-optimality of complete block designs under a mixed interference model

In experiments in which the response to a treatment can be affected by other treatments, the interference model with neighbor effects is usually used. It is known that circular neighbor balanced designs (CNBDs) are universally optimal under such a model if the neighbor effects are fixed (Druilhet, 1999) or random (4 and 7). However, such designs cannot exist for every combination of design parameters. In the class of block designs with the same number of treatments as experimental units per block, a CNBD cannot exist if the number of blocks, b  , is equal to p(t−1)±1$p(t-1)\pm 1$, where p is a positive integer and t is the number of treatments. Filipiak et al. (2008) gave the structure of the left-neighboring matrix of E-optimal complete block designs with p  =1 under the model with fixed neighbor effects. The purpose of this paper is to generalize E-optimality results for designs with p∈N$p\in \mathbb{N}$ assuming random neighbor effects.