Confidence Intervals for the Number of Components in Factor Analysis and Principal Components Analysis via Subsampling

TL;DR

This paper introduces a subsampling-based method to construct confidence intervals for the number of components in factor analysis and PCA, addressing uncertainty in estimating the component count.

Contribution

It proposes a novel data-driven approach with theoretical guarantees for confidence intervals of component numbers in FA and PCA, including the first Edgeworth expansion for spiked eigenvalues.

Findings

The method achieves accurate coverage probabilities in simulations.

It provides reliable confidence intervals for real genotyping data.

Theoretical analysis supports the method's validity.

Abstract

Factor analysis (FA) and principal component analysis (PCA) are popular statistical methods for summarizing and explaining the variability in multivariate datasets. By default, FA and PCA assume the number of components or factors to be known \emph{a priori}. However, in practice the users first estimate the number of factors or components and then perform FA and PCA analyses using the point estimate. Therefore, in practice the users ignore any uncertainty in the point estimate of the number of factors or components. For datasets where the uncertainty in the point estimate is not ignorable, it is prudent to perform FA and PCA analyses for the range of positive integer values in the confidence intervals for the number of factors or components. We address this problem by proposing a subsampling-based data-intensive approach for estimating confidence intervals for the number of components in FA and PCA. We study the coverage probability of the proposed confidence intervals and provide non-asymptotic theoretical guarantees concerning the accuracy of the confidence intervals. As a byproduct, we derive the first-order \emph{Edgeworth expansion} for spiked eigenvalues of the sample covariance matrix when the data matrix is generated under a factor model. We also demonstrate the usefulness of our approach through numerical simulations and by applying our approach for estimating confidence intervals for the number of factors of the genotyping dataset of the Human Genome Diversity Project.