Confidence Areas for Fixed-Effects PCA

TL;DR

This paper develops inferential methods for PCA in fixed-effects models, providing confidence regions and variability assessments through bootstrap, jackknife, and approximations, validated by simulation studies.

Contribution

It introduces new approaches for assessing PCA variability in fixed-effects models, including a cell-wise jackknife and bootstrap methods, with visualization techniques.

Findings

Bootstrap method effectively captures variability.

Jackknife provides a computationally cheaper alternative.

Simulation shows method performance varies with data size and relationship strength.

Abstract

PCA is often used to visualize data when the rows and the columns are both of interest. In such a setting there is a lack of inferential methods on the PCA output. We study the asymptotic variance of a fixed-effects model for PCA, and propose several approaches to assessing the variability of PCA estimates: a method based on a parametric bootstrap, a new cell-wise jackknife, as well as a computationally cheaper approximation to the jackknife. We visualize the confidence regions by Procrustes rotation. Using a simulation study, we compare the proposed methods and highlight the strengths and drawbacks of each method as we vary the number of rows, the number of columns, and the strength of the relationships between variables.