A family of multivariate discrete distributions

In this short paper, the multivariate Poisson-Gamma, the multinomial N-mixture and the negative multinomial distributions are shown to have probability mass functions of the same form and thus to share, broadly, the same distributional properties. The three distributions are, however, fundamentally very different in nature, that is, in terms of genesis, interpretation and model building, and these differences are highlighted and discussed

Bayesian and maximin optimal designs for heteroscedastic regression models

The problem of constructing standardized maximin D-optimal designs for weighted polynomial regression models is addressed. In particular it is shown that, by following the broad approach to the construction of maximin designs introduced recently by Dette, Haines and Imhof (2003), such designs can be obtained as weak limits of the corresponding Bayesian Φq-optimal designs. The approach is illustrated for two specific weighted polynomial models and also for a particular growth model. --

Maximin and Bayesian optimal designs for regression models

For many problems of statistical inference in regression modelling, the Fisher information matrix depends on certain nuisance parameters which are unknown and which enter the model nonlinearly. A common strategy to deal with this problem within the context of design is to construct maximin optimal designs as those designs which maximize the minimum value of a real valued (standardized) function of the Fisher information matrix, where the minimum is taken over a specified range of the unknown parameters. The maximin criterion is not differentiable and the construction of the associated optimal designs is therefore difficult to achieve in practice. In the present paper the relationship between maximin optimal designs and a class of Bayesian optimal designs for which the associated criteria are differentiable is explored. In particular, a general methodology for determining maximin optimal designs is introduced based on the fact that in many cases these designs can be obtained as weak limits of appropriate Bayesian optimal designs. --maximin optimal designs,Bayesian optimal designs,nonlinear regression models,parameter estimation,least favourable prior

Bayesian and Maximum Optimal Designs for Heteroscedastic Regression Models

The problem of constructing standardized maximin D-optimal designs for weighted polynomial regression models is addressed. In particular it is shown that, by following the broad approach to the construction of maximin designs introduced recently by Dette, Haines and Imhof (2003), such designs can be obtained as weak limits of the corresponding Bayesian Φ_q-optimal designs. The approach is illustrated for two specific weighted polynomial models and also for a particular growth model

Exact Likelihoods for N-mixture models with Time-to-Detection Data

This paper is concerned with the formulation of $N$-mixture models for estimating the abundance and probability of detection of a species from binary response, count and time-to-detection data. A modelling framework, which encompasses time-to-first-detection within the context of detection/non-detection and time-to-each-detection and time-to-first-detection within the context of count data, is introduced. Two observation processes which depend on whether or not double counting is assumed to occur are also considered. The main focus of the paper is on the derivation of explicit forms for the likelihoods associated with each of the proposed models. Closed-form expressions for the likelihoods associated with time-to-detection data are new and are developed from the theory of order statistics. A key finding of the study is that, based on the assumption of no double counting, the likelihoods associated with times-to-detection together with count data are the product of the likelihood for the counts alone and a term which depends on the detection probability parameter. This result demonstrates that, in this case, recording times-to-detection could well improve precision in estimation over recording counts alone. In contrast, for the double counting protocol with exponential arrival times, no information was found to be gained by recording times-to-detection in addition to the count data. An R package and an accompanying vignette are also introduced in order to complement the algebraic results and to demonstrate the use of the models in practice.Comment: 21 pages, 1 figur

A- and D-optimal row-column designs for two-colour cDNA microarray experiments using linear mixed effects models

Microarray experiments help scientists to study the expression level of thousands of genes simultaneously. These experiments have many design challenges, such as, for example, whichmRNA samples should be co-hybridized together and which treatments should be labelled with which fluorescent dye. Therefore a carefully designed microarray experiment to obtain efficient and reliable data so as to ensure the precise estimate of comparisons of interest is required. The present paper is concerned with A- and D-optimal row-column designs for two-colour microarray experiments, with the array and dye effects treated as the column and row effects, respectively. Linear mixed effects models were used to describe experiments for which a comparison of all possible pairs of treatments is of particular interest by taking the arrays as random column effects. The results of this study show that the optimal row-column designs under the linear fixed effects model are not necessarily optimal under the linear mixed effects model setting

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 dZ3 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.Financial support from the following : University of Pretoria, through a research grant provided by the Research Development Programme and the The University of KwaZulu-Natal, the University of Cape Town and the National Research Foundation, South Africa.www.elsevier.com/locate/jsp

D-optimal population designs for the simple linear random coefficients regression model

In this paper D-optimal population designs for the simple linear random coefficients regression model with values of the explanatory variable taken from a set of equally spaced, non-repeated time points are considered. The D-optimal designs depend on the values of the variance components and locally optimal designs are therefore considered. It is shown that, if the time points are linearly transformed, then the D-optimal population designs for both the fixed effects and the variance components do not necessarily map onto one another. This result is illustrated numerically by means of a simple example.http://www.sastat.org.za/journal.htmnf201

A- and D-optional row-column designs for two-colour cDNA microarray experiments using linear mixed effects models

Microarray experiments help scientists to study the expression level of thousands of genes simultaneously. These experiments have many design challenges, such as, for example, which mRNA samples should be co-hybridized together and which treatments should be labelled with which fluorescent dye. Therefore a carefully designed microarray experiment to obtain efficient and reliable data so as to ensure the precise estimate of comparisons of interest is required. The present paper is concerned with A- and D-optimal row-column designs for two-colour microarray experiments, with the array and dye effects treated as the column and row effects, respectively. Linear mixed effects models were used to describe experiments for which a comparison of all possible pairs of treatments is of particular interest by taking the arrays as random column effects. The results of this study show that the optimal row-column designs under the linear fixed effects model are not necessarily optimal under the linear mixed effects model setting.Grants from the National Research Foundation (NRF) through the Competitive Support for Unrated Researchers Programme (Reference: SUR20110629000019843 and Grant No: 80407) and the Incentive Funding for Rated Researchers Programme (Grant No: UID 85456), the University of Pretoria and the University of Cape Town.http://www.sastat.org.za/journal/informationhttp://reference.sabinet.co.za/sa_epublication/sasjam201