What can lattices do for experimental designs? - Duquenne, V. (Mathematical social sciences, 1986)
Let x 1:=((x 1)iϵI)ϵ R′ be a finite data set weighted by n 1:=((n 1)iϵI)ϵ R′ (reflecting a previous derivation or is constant), F: = {A,B,…} be a set of factors which index partitions of 1 (denoted by A, B…) together with ‘experimental dimensions’ (subjects' ages…), the experimental design be D F: = Im(I → X FF), ( P 1,⩽,∨,∧,I,1) and (Q ⊥,⩽,∨∨⊥, 0.1) denote the lattices of partition on I and of ortho-projections in R 1 〈x 1 | y 1〉: = Σ 1ϵ1n 1x 1y 1. F := {Fϵ P/ all Fϵ F∼ ; the effect ascribed to Aϵ F is evaluated through mA(x 1: (〈 x ̄ u〉 1) ja where each a is the A-class containing i, ( x a) 1: = 1 n a ∑ jϵa n jx 1, n a: = ∑ jϵa n. Let cA: = mA1 − m1 ϵQ ⊥, the map c: F · Q ⊥, A →cA established a connection P 1 = ⇆ [0,cI] which makes precise the link between the set theoretical description of an experiment and its analysis linear framework. A, B ϵ P 1 are said loccally orthogonal (LO) iff n a≥ b = n a n b , n u (all u ϵ A ∨ B, all a, b, ⊆ u); then the relative interaction ca · cB: = cA ∨ B acA ∧ cB) (called interaction, denoted by cA · cB whenever A ∧ B = 1 is a sum of partial interactions: cA · cB⊕⊥ uϵ A ∧ cA/ u · cB/ u with A/ u: = { a ϵ A/ a ⊆ u}. Any (pairwise) LO-subset A ⊆ P 1 induces (by c) a commutative Boolean subalgebra Q( X) ⊆ Q 1, whose atoms give the c∨ X canonical ⊕⊥- decomposition according to X. Let F generate an LO-design iff the generated sublattice (by ∨ and V) P( F) ⊆ P 1 is LO; using the Möbius function of if P ( F ), ⩾), the 9I canonical ⊕⊥-decomposition is easily calculated and labelled with formulae: its terms are expressible as n-ary(relative) interactions between ·-irreducible terms associated with the projective intervals of P( F) . A concrete example from psychology is discussed.