D-Optimal Split-plot Designs With Random Whole Plot Factor and Fixed Sub-plot Factor

This study examines design preference in Completely Randomized (CR) split-plot experiments involving random whole plot factor effect and fixed sub-plot factor effect. Many previous works on optimally designing split-plot experiments assumed only factors with fixed levels. However, the cases where interests are on random factors have received little attention. These problems have similarities with optimal design of experiments for fixed parameters of non-linear models because the solution rely on the unknown parameters.  Design Space (DS) containing exhaustive list of balanced designs for a fixed sample size were compared for optimality using the product of determinants of derived information matrices of the Maximum Likelihood (ML) estimators equivalent to random and fixed effect in the model. Different magnitudes of components of variance configurations where variances of factor effects are larger than variances of error term were empirically used for the comparisons. The results revealed that the D-optimal designs are those with whole plot factor levels greater than replicates within each level of whole plot.

Download Full-text

The properties of partial trace and block trace operators of partitioned matrices

Electronic Journal of Linear Algebra ◽

10.13001/1081-3810.3688 ◽

2018 ◽

Vol 33 ◽

pp. 3-15 ◽

Cited By ~ 5

Author(s):

Katarzyna Filipiak ◽

Daniel Klein ◽

Erika Vojtková

Keyword(s):

Linear Models ◽

Positive Definiteness ◽

Linear Operators ◽

Optimal Designs ◽

Trace Operator ◽

Multivariate Models ◽

Unknown Parameters ◽

Partial Trace ◽

Multi Level

The aim of this paper is to give the properties of two linear operators defined on non-square partitioned matrix: the partial trace operator and the block trace operator. The conditions for symmetry, nonnegativity, and positive-definiteness are given, as well as the relations between partial trace and block trace operators with standard trace, vectorizing and the Kronecker product operators. Both partial trace as well as block trace operators can be widely used in statistics, for example in the estimation of unknown parameters under the multi-level multivariate models or in the theory of experiments for the determination of an optimal designs under the linear models.

Download Full-text

Algebraic geometry of Poisson regression

Journal of Algebraic Statistics ◽

10.18409/jas.v7i1.43 ◽

2016 ◽

Vol 7 (1) ◽

Cited By ~ 1

Author(s):

Thomas Kahle ◽

Kai-Friederike Oelbermann ◽

Rainer Schwabe

Keyword(s):

Algebraic Geometry ◽

Generalized Linear Models ◽

Poisson Regression ◽

Linear Models ◽

Optimal Designs ◽

Local Optimality ◽

Unknown Parameters ◽

Explanatory Variables ◽

Saturated Design ◽

Poisson Counts

Designing experiments for generalized linear models is difficultbecause optimal designs depend on unknown parameters. Here weinvestigate local optimality. We propose to study for a given designits region of optimality in parameter space. Often these regions aresemi-algebraic and feature interesting symmetries. We demonstratethis with the Rasch Poisson counts model. For any given interactionorder between the explanatory variables we give a characterization ofthe regions of optimality of a special saturated design. This extendsknown results from the case of no interaction. We also give analgebraic and geometric perspective on optimality of experimentaldesigns for the Rasch Poisson counts model using polyhedral andspectrahedral geometry.

Download Full-text

Asymptotic optimality of experimental designs in estimating a product of means

Journal of Applied Mathematics and Stochastic Analysis ◽

10.1155/s104895339000003x ◽

1990 ◽

Vol 3 (1) ◽

pp. 15-25 ◽

Cited By ~ 1

Author(s):

Kamel Rekab

Keyword(s):

Linear Models ◽

Asymptotic Optimality ◽

Optimal Designs ◽

Bayes Risk ◽

Necessary And Sufficient Condition ◽

Unknown Parameters ◽

Normal Populations ◽

True Value ◽

Estimation Problems ◽

Necessary And Sufficient

In nonlinear estimation problems with linear models, one difficulty in obtaining optimal designs is their dependence on the true value of the unknown parameters. A Bayesian approach is adopted with the assumption the means are independent apriori and have conjuguate prior distributions. The problem of designing an experiment to estimate the product of the means of two normal populations is considered. The main results determine an asymptotic lower bound for the Bayes risk, and a necessary and sufficient condition for any sequential procedure to achieve the bound.

Download Full-text