Robust first-order rotatable lifetime improvement experimental designs

Experimental designs are widely used in predicting the optimal operating conditions of the process parameters in lifetime improvement experiments. The most commonly observed lifetime distributions are log-normal, exponential, gamma and Weibull. In the present article, invariant robust first-order rotatable designs are derived for autocorrelated lifetime responses having log-normal, exponential, gamma and Weibull distributions. In the process, robust first-order D-optimal and rotatable conditions have been derived under these situations. For these lifetime distributions with correlated errors, it is shown that robust first-order D-optimal designs are always robust rotatable but the converse is not true. Moreover, it is observed that robust first-order D-optimal and rotatable designs depend on the respective error variance–covariance structure but are independent from these considered lifetime response distributions.     

On D-optimal robust second order slope-rotatable designs

Das and Park (2006) introduced slope-rotatable designs overall directions for correlated observations which is known as A-optimal robust slope-rotatable designs. This article focuses D-optimal slope-rotatable designs for second-order response surface model with correlated observations. It has been established that robust second-order rotatable designs are also D-optimal robust slope-rotatable designs. A class of D-optimal robust second-order slope-rotatable designs has been derived for special correlation structures of errors.     

D- and V-optimal population designs for the quadratic regression model with a random intercept term

In this paper D- and V-optimal population designs for the quadratic regression model with a random intercept term and with values of the explanatory variable taken from a set of equally spaced, non-repeated time points are considered. D-optimal population designs based on single-point individual designs were readily found but the derivation of explicit expressions for designs based on two-point individual designs was not straightforward and was complicated by the fact that the designs now depend on ratio of the variance components. Further algebraic results pertaining to d-point D-optimal population designs where d≥3 and to V-optimal population designs proved elusive. The requisite designs can be calculated by careful programming and this is illustrated by means of a simple example.     

An algorithm for calculating D-optimal designs for polynomial regression through a fixed point

Optimal designs are required to make efficient statistical experiments. By using canonical moments, in 1980, Studden found D_s-optimal designs for polynomial regression models. On the other hand, integrable systems are dynamical systems whose solutions can be written down concretely. In this paper, polynomial regression models through a fixed point are discussed. In order to calculate D-optimal designs for these models, a useful relationship between canonical moments and discrete integrable systems is introduced. By using canonical moments and discrete integrable systems, a new algorithm for calculating D-optimal designs for these models is proposed.     

Optimal designs for tumor regrowth models

Most growth curves can only be used to model the tumor growth under no intervention. To model the growth curves for treated tumor, both the growth delay due to the treatment and the regrowth of the tumor after the treatment need to be taken into account. In this paper, we consider two tumor regrowth models and determine the locally D- and c-optimal designs for these models. We then show that the locally D- and c-optimal designs are minimally supported. We also consider two equally spaced designs as alternative designs and evaluate their efficiencies.     

Modified Robust Second-Order Slope-Rotatable Designs

Generally it is very difficult to construct robust slope-rotatable designs along axial directions. Present paper focuses on modified second-order slope-rotatable designs (SOSRDs) with correlated errors. Modified robust second-order slope-rotatability conditions are derived for a general variance–covariance structure of errors. These conditions get simplified for intraclass correlation structure. A few robust second-order slope-rotatable designs (over all directions, or with equal maximum directional variance slope, or D-optimal slope) are examined with respect to modified robust slope-rotatability. It is observed that robust second-order slope-rotatable designs over all directions, or with equal maximum directional variance slope, or D-optimal slope are not generally modified robust second-order slope-rotatable designs.     

Optimal designs for the Michaelis–Menten model with correlated observations

In this paper we investigate the problem of designing experiments for generalized least-squares analysis in the Michaelis–Menten model. We study the structure of exact D-optimal designs in a model with an autoregressive error structure. Explicit results for locally D-optimal designs are derived for the case where two observations can be taken per subject. Additionally standardized maximin D-optimal designs are obtained in this case. The results illustrate the enormous difficulties to find exact optimal designs explicitly for nonlinear regression models with correlated observations.     

Rank-order choice-based conjoint experiments: Efficiency and design

In a rank-order choice-based conjoint experiment, the respondent is asked to rank a number of alternatives of a number of choice sets. In this paper, we study the efficiency of those experiments and propose a D-optimality criterion for rank-order experiments to find designs yielding the most precise parameter estimators. For that purpose, an expression of the Fisher information matrix for the rank-ordered conditional logit model is derived which clearly shows how much additional information is provided by each extra ranking step. A simulation study shows that, besides the Bayesian D-optimal ranking design, the Bayesian D-optimal choice design is also an appropriate design for this type of experiments. Finally, it is shown that considerable improvements in estimation and prediction accuracy are obtained by including extra ranking steps in an experiment.     

Statistical properties of Rechtschaffner designs

Rechtschaffner designs are saturated designs of resolution V   in which main effects and two-factor interactions are estimable if three-factor and higher order interactions are negligible. Statistical properties of Rechtschaffner designs are studied in this paper. Best linear unbiased estimators of main effects and two-factor interactions are given explicitly and asymptotic properties of correlations between these estimators are studied as well. It is shown that designs recommended by Rechtschaffner [1967. Saturated fractions of 2ⁿ$2^n$ and 3ⁿ$3^n$ factorial designs, Technometrics 9, 569–576] are not only A-optimal but also D-optimal. Comparisons of Rechtschaffner designs with other A- and D-optimal designs of resolution V are also discussed.     

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.     

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.     

D-optimal designs for combined polynomial and trigonometric regression on a partial circle

Consider the D-optimal designs for a combined polynomial and trigonometric regression on a partial circle. It is shown that the optimal design is equally supported and the structure of the optimal design depends only on the length of the design interval and the support points are analytic functions of this parameter. Moreover, the Taylor expansion of the optimal support points can be determined efficiently by a recursive procedure. Examples are presented to illustrate the procedures for computing the optimal designs.     

The analytic construction of D-optimal designs for the two-variable binary logistic regression model without interaction

Candidate locally D-optimal designs for the binary two-variable logistic model with no interaction, which comprise 3 and 4 support points lying in the first quadrant of the two-dimensional Euclidean space, were introduced by Haines et al. (D-optimal designs for logistic regression in two variables. In: Lopez-Fidalgo J, Rodrigez-Diaz JM, Torsney B, editors. MODA8 – advances in model-oriented designs and analysis. Heidelberg: Physica-Verlag; 2007. p. 91–98). The authors proved algebraically the global D-optimality of the 3-point design for the special case in which the intercept parameter is equal to?1.5434. However for other selected values of the intercept parameter, the global D-optimality of the proposed 3- and 4-point designs was only demonstrated numerically. In this paper, we provide analytical proofs of the D-optimality of these 3- and 4-point designs for all negative and zero intercept parameters of the binary two-variable logistic model with no interaction. The results are extended to the construction of D-optimal designs on a rectangular design space and illustrated by means of two examples of which one is a real example taken from the literature.     

On representing maximin efficient designs by Taylor series

This paper considers exponential and rational regression models that are nonlinear in some parameters. Recently, locally D-optimal designs for such models were investigated in [Melas, V. B., 2005. On the functional approach to optimal designs for nonlinear models. J. Statist. Plann. Inference 132, 93–116] based upon a functional approach. In this article a similar method is applied to construct maximin efficient D-optimal designs. This approach allows one to represent the support points of the designs by Taylor series, which gives us the opportunity to construct the designs by hand using tables of the coefficients of the series. Such tables are provided here for models with two nonlinear parameters. Furthermore, the recurrent formulas for constructing the tables for arbitrary numbers of parameters are introduced.     

Effect of choice complexity on design efficiency in conjoint choice experiments

Conjoint choice experiments have become a powerful tool to explore individual preferences. The consistency of respondents' choices depends on the choice complexity. For example, it is easier to make a choice between two alternatives with few attributes than between five alternatives with several attributes. In the latter case it will be much harder to choose the preferred alternative which is reflected in a higher response error. Several authors have dealt with this choice complexity in the estimation stage but very little attention has been paid to set up designs that take this complexity into account. The core issue of this paper is to find out whether it is worthwhile to take this complexity into account in the design stage. We construct efficient semi-Bayesian D-optimal designs for the heteroscedastic conditional logit model which is used to model the across respondent variability that occurs due to the choice complexity. The degree of complexity is measured by the entropy, as suggested by Swait and Adamowicz (2001). The proposed designs are compared with a semi-Bayesian D-optimal design constructed without taking the complexity into account. The simulation study shows that it is much better to take the choice complexity into account when constructing conjoint choice experiments.     

Constructing D-optimal symmetric stated preference discrete choice experiments

We give new constructions for DCEs in which all attributes have the same number of levels. These constructions use several combinatorial structures, such as orthogonal arrays, balanced incomplete block designs and Hadamard matrices. If we assume that only the main effects of the attributes are to be used to explain the results and that all attribute level combinations are equally attractive, we show that the constructed DCEs are D-optimal.     

Optimal designs for additive mixture model with heteroscedastic errors

This paper presents a study of D- and A-optimality of direct sum designs for additive mixture models when the errors are heteroscedastic. Sufficient conditions are given so that D- and A-optimal designs for additive mixture models can be constructed from the D- and A-optimal designs for homogeneous models in sub-mixture systems.     

A new characterization of Elfving's method for high dimensional computation

We give a new characterization of Elfving's (1952) method for computing c-optimal designs in k dimensions which gives explicit formulae for the k unknown optimal weights and k unknown signs in Elfving's characterization. This eliminates the need to search over these parameters to compute c-optimal designs, and thus reduces the computational burden from solving a family of optimization problems to solving a single optimization problem for the optimal finite support set. We give two illustrative examples: a high dimensional polynomial regression model and a logistic regression model, the latter showing that the method can be used for locally optimal designs in nonlinear models as well.     

D-optimal block designs with at most six varieties

For given positive integers v, b, and k (all of them ≥2) a block design is a k × b array of the variety labels 1,…,v with blocks as columns. For the usual one-way heterogeneity model in standard form the problem is studied of finding a D-optimal block design for estimating the variety contrasts, when no balanced block design (BBD) exists. The paper presents solutions to this problem for v≤6. The results on D-optimality are derived from a graph-theoretic context. Block designs can be considered as multigraphs, and a block design is D-optimal iff its multigraph has greatest complexity (=number of spanning trees).     

Algorithmic and analytical construction of efficient designs in small blocks for comparing consecutive pairs of treatments

Optimal block designs in small blocks are explored under the A-, E- and D-criteria when the treatments have a natural ordering and interest lies in comparing consecutive pairs of treatments. We first formulate the problem via approximate theory which leads to a convenient multiplicative algorithm for obtaining A-optimal design measures. This, in turn, yields highly efficient exact designs, under the A-criterion, even when the number of blocks is rather small. Moreover, our approach is seen to allow nesting of such efficient exact designs which is an advantage when the resources for the experiment are available in possibly several stages. Illustrative examples are given and tables of A-optimal design measures are provided. Approximate theory is also seen to yield analytical results on E- and D-optimal design measures.