What can lattices do for experimental designs?
type: Item
instance of: scholarly article
title: What can lattices do for experimental designs?
zbMATH Open document ID: 0609.62118
DOI: 10.1016/0165-4896(86)90028-4
published in: Mathematical Social Sciences
publication date: 1986-01-01T00:00:00Z
review text: 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 com…
Mathematics Subject Classification ID: 62K99
Mathematics Subject Classification ID: 06D99
Mathematics Subject Classification ID: 62J10
Mathematics Subject Classification ID: 62P15
Mathematics Subject Classification ID: 91E99
zbMATH DE Number: 3986471
zbMATH Keywords: permutable partitions
zbMATH Keywords: Möbius function
zbMATH Keywords: direct meet representation
zbMATH Keywords: analysis of variance
zbMATH Keywords: modular lattices
zbMATH Keywords: projective interval
zbMATH Keywords: analysis of comparisons
zbMATH Keywords: quasi complete designs
zbMATH Keywords: locally orthogonal partitions
zbMATH Keywords: commutativity of comparators
zbMATH Keywords: canonical decomposition of comparators
zbMATH Keywords: relative interaction
zbMATH Keywords: locally orthogonal designs
zbMATH Keywords: partition sublattices
zbMATH Keywords: labelling process
author: Vincent Duquenne
MaRDI profile type: Publication
full work available at URL: 0165-4896(86)90028-4
OpenAlex ID: W2040131513
cites work: Q5685119
cites work: Q5331549
cites work: Q3241542
cites work: The Möbius function of a lattice
cites work: Q5545191
cites work: Q3322108
cites work: Q3952143
cites work: Q3293417
cites work: Q3875983
cites work: Q5623714
cites work: The analysis of randomized experiments with orthogonal block structure. I. Block structure and the null analysis of variance
cites work: Q5820917
cites work: Theory of equivalence relations
cites work: On the foundations of combinatorial theory I. Theory of M�bius Functions
description: scientific article; zbMATH DE number 3986471