Invariant functions on measurement structures
type: Item
instance of: scholarly article
title: Invariant functions on measurement structures
zbMATH Open document ID: 0833.92029
author: Ching-Yuan Chiang
published in: Journal of Mathematical Psychology
publication date: 1996-03-20T00:00:00Z
review text: Since the publication of \textit{S. Stevens}' celebrated articles [Science 103, 677-680 (1946); S. Stevens (ed.), Handb. Exp. Psychol., 1-49 (1951)]\ the relationship between measurement structures and appropriate statistics has been an outstanding problem. In an attempt to explicate Stevens' idea of ``permissible statistics'', \textit{E. W. Adams} et al. [Psychometrika 30, 99-127 (1967)]\ have studied absolute invariance, reference invariance, and comparison invariance of real-valued functions called ``generalized operations on measurements'' with respect to measurement structures (called ``numerical assignment systems''). \textit{J. Pfanzagl} [Theory of measurement. (1968; Zbl 0186.536)], while developing a theory of ``meaningful relations'', uses the concept of comparison invariance to define ``meaningful statistics'', which are real-valued functions on measurement structures. And \textit{I. Klein} [Stat. Hefte 26, 313-320 (1985; Zbl 0592.62003)]\ has studied absolute invariance, reference invariance, and comparison invariance of real- valued functions on statistical distributions. We extend the concepts of absolute invariance, reference invariance, and comparison invariance to vector-valued functions on relational systems which are range systems of measurement structures. We also study the characterization and interrelationships of these concepts of invariance. The results can serve as a basis for a more general treatment of the problem of appropriate (or ``meaningful'') statistics which include multidimensional statistics.
Mathematics Subject Classification ID: 91C05
Mathematics Subject Classification ID: 08A02
zbMATH DE Number: 792518
zbMATH Keywords: absolute invariance
zbMATH Keywords: reference invariance
zbMATH Keywords: comparison invariance
zbMATH Keywords: vector- valued functions
zbMATH Keywords: relational systems
zbMATH Keywords: measurement structures
MaRDI profile type: Publication
full work available at URL: jmps.1995.1010
OpenAlex ID: W2057746191
DOI: 10.1006/JMPS.1995.1010
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description: scientific article; zbMATH DE number 792518