Célia Maria da Silva Fernandes

Models with Stair Nesting

When we work with balanced nested designs, the usual approach forces us to divide repeatedly the plots, so we have few degrees of freedom for the first levels. As an alternative we have the stair nested designs. Stair nesting leads to very light models since we can work with fewer observations and the amount of information for the different factors is more evenly distributed. This is in fact a big advantage when comparing with balanced nesting, since stair nesting will allow experiments that will become cheaper, due to the less number of observations involved, or with the same resources produce more experiments. For models with stair nesting it is easy to carry out inference, as we shall see. It is interesting to point out that stair nested designs although being orthogonal, with the advantages associated to that condition, are not balanced.

Crossing balanced and stair nested designs

Balanced nesting is the most usual form of nesting and originates, when used singly or with crossing of such sub-models, orthogonal models. In balanced nesting we are forced to divide repeatedly the plots and we have few degrees of freedom for the first levels. If we apply stair nesting we will have plots all of the same size rendering the designs easier to apply. The stair nested designs are a valid alternative for the balanced nested designs because we can work with fewer observations, the amount of information for the different factors is more evenly distributed and we obtain good results. The inference for models with balanced nesting is already well studied. For models with stair nesting it is easy to carry out inference because it is very similar to that for balanced nesting. Furthermore stair nested designs being unbalanced have an orthogonal structure. Other alternative to the balanced nesting is the staggered nesting that is the most popular unbalanced nested design which also has the advantage of requiring fewer observations. However staggered nested designs are not orthogonal, unlike the stair nested designs. In this work we start with the algebraic structure of the balanced, the stair and the staggered nested designs and we finish with the structure of the cross between balanced and stair nested designs.

A Fundamental Partition in Models with Commutative Orthogonal Block Structure

Models with commutative orthogonal block structure, COBS, constitute an interesting class of models with orthogonal block structure, OBS, in which the orthogonal projection matrix on the space Ω spanned by the mean vectors commute with the known pairwise orthogonal projection matrices Q1,…, Qm that figure in the expression of the variance‐covariance matrix V =  ∑ j = 1mγjQj of the model. We discuss the importance of the orthogonal partition Y = YΩ+YΩ⊥ where Y, YΩ and YΩ⊥ are the observation vectors and its orthogonal projection on Ω. and Ω⊥, the orthogonal complement on parameters estimation.

Algebraic Structure for Interaction on Mixed Models

Abstract Binary operations on commutative Jordan algebras, CJA, can be used to study interactions between sets of factors belonging to a pair of models in which one nests the other. It should be noted that from two CJA we can, through these binary operations, build CJA. So when we nest the treatments from one model in each treatment of another model, we can study the interactions between sets of factors of the first and the second models.

Study of the interactions in a three-way crossed classification model

Crossed classification models are applied in many investigations taking in consideration the existence of interaction between all factors or, in alternative, excluding all interactions, and in this case only the effects and the error term are considered. In this work we use commutative Jordan algebras in the study of the algebraic structure of these designs and we use them to obtain similar designs where only some of the interactions are considered. We finish presenting the expressions of the variance componentes estimators.

Inference with Inducer Pivot Variables, an Application to the One‐Way ANOVA

Having in mind complete and sufficient statistics for a relevant set of parameters of a given model, we show how to induce probability measures in the parameter spaces, which may be used to obtain confidence intervals. Hypothesis testing for these parameters can be carried out trough duality. Since the computations to construct induced measures tend to be heavy, we explain how they can be constructed through Monte‐Carlo methods.We give the random effects linear model as an example, showing how to obtain complete sufficient statistics for such models from which UMVUE are obtained for the variance components. We derive explicit formulas for the One‐Way ANOVA model.