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L’Analyse statistique des protocoles multidimensionnels: Analyse des comparaisons (Nuage pondéré sur le croisement de deux facteurs). (French) Zbl 0571.62001

Given a finite set J and an observation space \({\mathcal O}\) which is a vector space or, more generally, an affine space (since in the context of this paper only differences of the observations are important). The authors define a ”protocol” or a ”cloud” \(M^ J\) to be a set of observations \(\{\) (j, \(M^ j):j\in J\}\), with \(M^ j\in {\mathcal O}\). A cloud may be ”weighted” by a set of positive integers \(n_ j.\)
A ”contrast” \(c_ J\) on J is a signed measure on J of total mass 0, i.e., \(c_ J=\{c_ j:j\in J\}\) with \(\Sigma_ jc_ j=0\). Its ”effect” on a cloud \(M^ J\) is \(\Sigma_ jc_ jM^ j\) and its ”moment of inertia” is defined as \(\| \Sigma_ jc_ jM^ j\|^ 2/\Sigma_ j(c^ 2_ j/n_ j).\)
These notions are applied to the crossing of two factors, A and B, in which case there are three sets J of interest: A, B, and \(A\times B\) (the elements of A being the levels of A and similarly B).
Various concepts are defined in terms of measures, contrasts, and clouds on these three sets; for instance, the cloud induced on A by a cloud on \(A\times B\) together with a contrast on B. The interrelations between these notions are studied, including the moments of inertia. The approach is data-analytic and no attempt at inference is made.
Reviewer: R.A.Wijsman

MSC:

62-07 Data analysis (statistics) (MSC2010)
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