Eigenvalue Significance Testing for Genetic Association

Abstract

Genotype eigenvectors are widely used as covariates for control of spurious stratification in genetic association. Significance testing for the accompanying eigenvalues has typically been based on a standard Tracy-Widom limiting distribution for the largest eigenvalue, derived under white-noise assumptions. It is known that even modest local correlation among markers inflates the largest eigenvalues, even in the absence of true stratification. In addition, a few sample eigenvalues may be extreme, creating further complications in accurate testing. We explore several methods to identify appropriate null eigenvalue thresholds, while remaining sensitive to eigenvalues corresponding to population stratification. We introduce a novel block permutation approach, designed to produce an appropriate null eigenvalue distribution by eliminating long-range genomic correlation while preserving local correlation. We also propose a fast approach based on eigenvalue distribution modeling, using a simple fit criterion and the general Marčenko-Pastur equation under a simple discrete eigenvalue model. Block permutation and the model-based approach work well for pure simulations and for data resampled from the 1000 Genomes project. In contrast, we find that the standard approach of computing an “effective” number of markers does not perform well. The performance of the methods is also demonstrated for a motivating example from the International Cystic Fibrosis Consortium.

A note on cyclic shift permutation testing for large eigenvalues

“…Block permutation, introduced by Zhou et al (), identifies blocks of rows and permutes them across samples/columns. Block permutation is intended to preserve the local feature block structure, while eliminating the long‐range correlation between blocks.…”

“…This approach can be justified from a perspective of exact conditional inference, assuming that the blocks have been identified correctly and are exchangeable under the null hypothesis of no‐ancestry variation. Beyond the setting of genomics used for Zhou et al (), a much larger number of applications involve time‐series data. If the data appear in discrete temporal blocks, then block permutation may be justified.…”

“…Simulations by Sahoo et al () indicated that their test had appropriate false‐positive rates under the simulation conditions, including scenarios in which the data included an AR(1) autoregressive structure across time. However, as detailed in Zhou et al (), such tests can be highly sensitive to row–row correlations in real data that might remain, even after explicit time‐series modelling. Our approach can be used to generate an appropriate null distribution by preserving row–row correlation, while removing the column–column correlation.…”

“…Testing for large sample eigenvalues in the presence of modest variation of true eigenvalues is a more difficult problem, even in conceptualizing the meaning of the null. Zhou, Marron, and Wright () developed methods specifically for genetic ancestry testing, where large eigenvalues correspond to components of genetic covariance matrices computed for thousands of single‐nucleotide polymorphisms. The modest variation in true eigenvalues is produced by local genomic correlation structures, which are considered endemic to the data and not produced by the ancestral variation of primary interest.…”

“…Much of Zhou et al () was concerned with direct modelling of the true eigenvalue distribution using the general Marc̆enko–Pastur (MP) law, for which a difficult computational inverse problem was made faster by the SPECTRODE algorithm (Dobriban, ). Once an appropriate fit to the MP distribution was determined, the corresponding TW approximation was used for testing significance of sample eigenvalues.…”