What can lattices do for experimental designs?
In the area of psychological experiments, for several reasons, the need for a more systematic approach which might lead to some language for describing the designs should be encountered. First, the specification of a design in a concrete situation is not just a logical task: the nesting of the main questions in the environmental conditions has to be done under certain external constraints: thus, the construction is a process, a dialectic between the aims of the study and reality which together lead to some typical structures under the control of experimental aims and constraints. It is not plausible to find every good solution in a list of proposed designs: we need a language to formalize this process. Symmetrically, during the analysis we need to declare the design structure in computer programs, and to be able to ask our questions with compact formulae; this has been done since the early seventies by \textit{H. Rouanet} and \textit{D. Lepine} [Math. Sci. Hum. 56(1976), 5-46 (1977; Zbl 0392.62054)], in the line of so-called analysis of comparisons. The structure of designs which are admitted by the programs described by Rouanet and Lepine and by the author [Un programme de description de données. Cah. Psychol. 19, 109-118 (1976)], the so-called quasi complete designs, is rich enough to permit formalizing (even though partially) what is happening in many concrete situations. The grammar of these languages of formulae is based on the strong properties of distributive lattices. Throughout this paper, the standard notions of lattice theory as presented in \textit{G. Birkhoff}, Lattice theory. 3rd ed. (1967; Zbl 0153.025), are used. The effects ascribable to the factors are defined as comparators which are orthogonal projections associated with the underlying partitions of the data. The author develops the properties of locally orthogonal partitions which characterize the commutativity of comparators and is a basic condition for defining a canonical decomposition of comparators; complete local orthogonality leads to the definition of relative interaction associated with m partitions, which is shown to be a sum of m-ary linear products of partial comparators. Also, the paper contains a list of desirable properties of a general class of designs, the locally orthogonal designs, which are defined by partition sublattices whose elements are pairwise locally orthogonal. Their most important properties are to permit an additive orthogonal decomposition where all the terms are expressible as between- or irreducible-comparators or as interaction or relative interaction between such comparators. The calculus and the labelling process for the canonical decomposition are described, and a concrete example is given.
scientific article; zbMATH DE number 3986471
scientific article