Analysis of variance (ANOVA) is one of the most frequently used statistical analysis in several research areas, namely in medical research. Despite its wide use, it has been applied assuming that sample dimensions are known. In this work we aim to carry out ANOVA like analysis of one-way fixed effects models, to situations where the samples sizes may not be previously known. Assuming that the samples were generated by Pois- son counting processes we obtain the unconditional distribution of the test statistic, under the assumption that we have random sample sizes. The applicability of the pro- posed approach is illustrated considering a real data example on cancer registries. The results obtained suggested that false rejections may be avoid by applying our approach.

Analysis of variance (ANOVA) is one of the most frequently used statistical analyses in several research areas, namely in medical research. Despite its wide use, it has been applied assuming that sample dimensions are known. In this work we aim to carry out ANOVA like analysis of one-way fixed effects models, to situations where the samples sizes may not be previously known. In these situations it is more appropriate to consider the sample sizes as realizations of independent random variables. This approach must be based on an adequate choice of the distributions of the samples sizes. We assume the Poisson distribution when the occurrence of observations corresponds to a counting process. The Binomial distribution is the proper choice if we have observations failures and there exist an upper bound for the sample sizes. We also show how to carry out our main goal by computing correct critical values. The applicability of the proposed approach is illustrated considering a real data example on cancer registries. The results obtained suggested that false rejections may be avoided by applying our approach.

The article addresses alternative estimators for estimable vectors. The relative performances of the proposed estimators are studied through numerical analysis. The major motivation behind this article is providing an optimal estimator through applying simple modifications in models with Orthogonal Block Structure. In addition, an illustrative example is presented to demonstrate the efficiency and accuracy of the proposed method.

Mixed linear models are a versatile and powerful tool for analysing data collected in experiments in several areas. Amixed model is a model with orthogonal block structure, OBS, when its variance–covariance matrix is ofall the positive semi-definite linear combinations of known pairwise orthogo-nal orthogonal projection matrices that add up to the identity matrix. Models with commutative orthogonal block structure, COBS, are a special case of OBS in which the orthogonal projection matrix on the space spanned by the mean vector commutes with the variance–covariance matrix. Using the algebraic structure of COBS, based on Commuta-tive Jordan algebras of symmetric matrices, and the Carte-sian product we build up complex models from simpler ones through joining, in order to analyse together models obtained independently. This commutativity condition of COBS is a necessary and sufficient condition for the least square esti-mators, LSE, to be best linear unbiased estimators, BLUE, whatever the variance components. Since joining COBS we obtain new COBS, the good properties of estimators hold for the joined models.

Mixed models whose variance–covariance matrices are the positive definite linear combinations of pairwise orthogonal orthogonal projection matrices have orthogonal block structure. Here, we will obtain uniformly minimum-variance unbiased estimators for the relevant parameters when normality is assumed and we show that those for estimable vectors are, in general, uniformly best linear unbiased estimators. This is, they are best linear unbiased estimators whatever the variance components.

Let T be a tree with n≥2 vertices. Set S(T) for the set of all real symmetric matrices whose graph is T. Let A∈S(T) and i∈{1,…,n} . We denote by A(i) the principal submatrix of A obtained after deleting the row and column i. We set mA(i)(0)=mA(0)+1, we say that i is a P-vertex of A. As usual, M(T) denotes the maximum nullity occurring of B∈S(T). In this paper we determine an upper bound and a lower bound for the number of P-vertices in a matrix A∈S(T)with nullity M(T). We also prove that if the integer b is between these two bounds, then there is a matrix E∈S(T) with b P-vertices and maximum nullity.

This work presents ToBLoOM – Triple Bottom Line Optimization Modeling, a decision support tool for the design and planning of sustainable supply chains. It consists of a multi-objective mixed integer linear programming model which integrates several interconnected decisions: facility location and capacity determination; supplier selection and purchase levels definition; technology selection and allocation; transportation network definition including both unimodal and intermodal options; supply planning; product recovery and remanufacturing. The three pillars of sustainability are addressed as objective functions: economic, through Net Present Value; environmental through the Life Cycle Analysis methodology ReCiPe; and social through a developed GDP-based metric. Uncertainty is considered using a stochastic ToBloOM. This applied to a case of a European based company with markets in Europe and South America. This work contributes to the literature by building on several identified research gaps such as the need for an integrated approach that allows simultaneous assessment of different interacting supply chain decisions, the need to explicitly assess the environmental impact in closed-loop supply chains, the need to assess the impact of supply chains on society, and the need for a multi-objective tool that includes all the three pillars of sustainability. Strategies towards a more sustainable supply chain are also derived from this work.