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A Psychophysics Paradigm for the Collection and Analysis of Similarity Judgments

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Reliability as Projection in Operator-Theoretic Test Theory: Conditional Expectation, Hilbert Space Geometry, and

Bruno D Zumbo¹

¹The University of British Columbia, Vancouver, Canada.

Educational and Psychological Measurement

|November 17, 2025

Summary

This summary is machine-generated.

This study redefines reliability using Hilbert space geometry, viewing it as a theorem, not an assumption. This new framework connects psychometric reliability with regression and statistical models for enhanced measurement understanding.

Keywords:

Hilbert space conditional expectation operator theory projection theorem psychometrics reliability

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Area of Science:

Psychometrics
Mathematical Statistics
Geometric Analysis

Background:

Classical test theory relies on assumptions for reliability.
Existing frameworks lack a unified mathematical foundation for reliability.

Purpose of the Study:

To reconceptualize reliability using Hilbert space projection geometry.
To provide a rigorous, theorem-based definition of reliability.
To unify reliability with related statistical concepts.

Main Methods:

Defined true score as conditional expectation (orthogonal projection).
Expressed reliability as the variance of the projected true score divided by the observed score variance.
Utilized operator-theoretic perspective and measure-theoretic proofs.

Main Results:

Reliability is derived as a theorem from projection geometry.
Reliability quantifies projection efficiency (squared cosine).
Unified reliability with regression R-squared, communality, and predictive accuracy.

Conclusions:

The Hilbert space framework offers a rigorous mathematical foundation for reliability.
This approach connects psychometric theory with modern statistical and geometric analysis.
Measurement error is the orthogonal complement of the projection; reliability measures score alignment.