A geometrical perspective on parametric psychometric models

Abstract

Psychometrics and quantitative psychology rely strongly on statistical models to measure psychological processes. As a branch of mathematics, geometry is inherently connected to measurement and focuses on properties such as distance and volume. However, despite the common root of measurement, geometry is currently not used a lot in psychological measurement. In this paper, my aim is to illustrate how ideas from non-Euclidean geometry may be relevant for psychometrics.

Figures (11)

• Figure 1: Graphical illustration of a smooth two-dimensional statistical manifold $\mathcal{M}$ together with a coordinate space $\Omega$. Each location on the manifold corresponds to a distribution of some family and the $(\theta_1,\theta_2)$ is a pair of coordinates indexing the distributions.
• Figure 2: Graphical illustration of estimation (Panel (a)) and model selection (Panel (b)).
• Figure 3: Illustration of the tangent space $T_{p_0}\mathcal{M}$ (red plane) to the manifold $\mathcal{M}$ (blue surface) at the point $p_0 \equiv p(y|\theta_0)$ (red dot).
• Figure 4: Three possible curves between $N(0,1)$ and $N\left(2,(\sqrt{2})^2 \right)$: A straight line (in blue), a circular arc (in orange) and an ellipse (in red, also the geodesic curve). The distances along the three curves are 1.744, 1.697, and 1.656, respectively. See text for the parametric equations and the calculations.
• Figure 5: The left panel shows 50 geodesic curves (as rays) all starting at $N(0,1)$ and ending at the distribution that is $d_{\min}=1.656$ away. The right panel shows equidistant points at 0.01 from the center for various combinations of $\mu$ and $\sigma$. These figures are created using the Python package "geomstats" (Miolane et al., 2020).

Theorems & Definitions (7)

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