ANOVA (analysis of variance) in the quantum linguistic formulation of statistics

TL;DR

This paper reformulates ANOVA within the framework of quantum language, a linguistic interpretation of quantum mechanics, providing new insights into the role of probability theory in statistical analysis.

Contribution

It introduces a quantum language formulation of ANOVA, bridging quantum mechanics concepts with classical statistical methods for clearer theoretical understanding.

Findings

ANOVA can be expressed in quantum language

Kolmogorov's probability theory is used for Gaussian integrals

Clarifies the role of probability theory in statistical analysis

Abstract

Recently, we proposed quantum language (or, measurement theory), which is characterized as the linguistic turn of the Copenhagen interpretation of quantum mechanics. We believe that this language has a great powet of description, and therefore, even statistics can be described by quantum language. In this paper, we show that ANOVA (analysis of variance (one-way and two-way)) can be formulated in quantum language. Since quantum language is suited for theoretical arguments, we believe that our results are visible and understandable. For example, we can answer the question "What kind of role does Kolmogorov's probability theory play in ANOVA?" That is, the readers find that Kolmogorov's probability theory is merely used in order to calculate multi-dimenstional Gauss integrals, and thus, they can avoid to confuse the relation between Kolmogorov's probability theory and statistics.