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arxiv: 2606.03880 · v1 · pith:RPZLTUMCnew · submitted 2026-06-02 · 📊 stat.ME · stat.AP

Principal Components Decomposition of Fraction of Variance Explained in High Dimensional Linear Models with Strong Correlation

Man Luo , Chun Chieh Fan , David Azriel , Armin Schwartzman This is my paper

Pith reviewed 2026-06-28 08:42 UTC · model grok-4.3

classification 📊 stat.ME stat.AP
keywords fraction of variance explainedhigh-dimensional linear modelsprincipal componentsstrong correlationbias reductionGWASHLMM-REMLbrain imaging
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The pith

Decomposing fraction of variance explained into strong and weak correlation parts reduces bias for high-dimensional estimators.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

High-dimensional linear models often have predictors with strong correlations that bias standard estimators of the fraction of variance explained. The paper proposes splitting the FVE into a low-dimensional piece captured by dominant principal components, which low-dimensional methods can estimate directly, and a remaining high-dimensional piece with only weak residual correlations, which specialized estimators can then handle. Simulations show this split produces lower bias than applying GWASH or LMM-REML directly to the full model. Performance stays consistent as both the number of predictors and the number of observations grow. The approach is illustrated on brain imaging data to recover heritability signals for cognitive outcomes.

Core claim

The fraction of variance explained can be partitioned into a low-dimensional component that captures strong correlations among predictors via their dominant principal components and a high-dimensional component consisting of weak residual correlations, allowing separate estimation that reduces overall bias.

What carries the argument

Principal components decomposition of the fraction of variance explained that separates strong from weak correlations among predictors.

If this is right

  • Simulations demonstrate improved bias reduction when dominant PCs are decomposed before applying GWASH or LMM-REML.
  • The estimator remains consistent as both the number of predictors and the number of samples increase.
  • Application to the ABCD brain imaging dataset captures nuanced heritability signals in the FVE of cognitive measures.
  • The low-dimensional component is estimable by ordinary low-dimensional methods while the high-dimensional component uses existing high-dimensional tools.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same split could be tested with other high-dimensional FVE estimators beyond GWASH and LMM-REML.
  • Fields such as genomics that also feature blocks of strong predictor correlation might see similar bias reductions.
  • If the number of dominant PCs required grows with dimension, the separation benefit could shrink.

Load-bearing premise

The dominant principal components capture essentially all strong correlations, leaving only weak residual correlations that high-dimensional estimators can handle without bias.

What would settle it

A simulation or dataset in which removing the dominant principal components still leaves residual correlations strong enough to bias GWASH or LMM-REML estimates, or where the decomposed FVE shows no bias reduction relative to direct application of those estimators.

Figures

Figures reproduced from arXiv: 2606.03880 by Armin Schwartzman, Chun Chieh Fan, David Azriel, Man Luo.

Figure 1
Figure 1. Figure 1: Panel A shows ABCD cortical surface area data in [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The gray histograms show the eigenvalues (brown solid lines) after rescaling by the right edge [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Monte Carlo simulation results of estimators for the second spectral moments as a function of [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Monte Carlo simulation results for FVE estimation under a one-block exchangeable correlation [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Simulation results under a one-block exchangeable correlation structure. The first and second [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Simulation results under a two-block exchangeable correlation structure with [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FVE estimation after decomposition for crystallized intelligence using cortical surface area data [PITH_FULL_IMAGE:figures/full_fig_p019_7.png] view at source ↗
read the original abstract

The fraction of variance explained (FVE) in a linear model quantifies the extent to which predictors account for outcome variability. In high-dimensional settings, where traditional FVE estimators do not apply, modern FVE estimators such as GWASH or linear mix-effect model estimated through the restricted maximum likelihood (LMM-REML) struggle with strong correlation among predictors, often found, for example, in brain imaging data. We propose a decomposition framework that partitions the FVE into two components: a low-dimensional component capturing the strong correlation, estimable by low dimensional methods, and a high-dimensional component with remaining weak correlation, estimable by high dimensional methods. Simulations demonstrate that decomposing dominant principal components (PCs) and estimating the high-dimensional FVE using GWASH or LMM-REML leads to improved bias reduction compared to directly applying standard approaches such as GWASH and LMM-REML. Our method shows consistent performance asymptotically as both the number of predictors and the number of samples increase. We illustrate the method in an analysis of the Adolescent Brain Cognitive Development (ABCD) brain imaging dataset, capturing nuanced heritability signals in the FVE of cognitive measures predicted by high-resolution brain imaging data.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 4 minor

Summary. The paper proposes a principal components decomposition framework for estimating the fraction of variance explained (FVE) in high-dimensional linear models where predictors exhibit strong correlations. It partitions the total FVE into a low-dimensional component captured by dominant principal components (addressing strong correlations via standard low-dimensional methods) and a residual high-dimensional component with weak correlations (estimated via GWASH or LMM-REML). Simulations are used to demonstrate bias reduction relative to direct application of GWASH or LMM-REML, along with asymptotic consistency as both p and n grow; the approach is illustrated on the ABCD brain imaging dataset for heritability signals in cognitive measures.

Significance. If the partitioning isolates strong correlations without introducing new bias, the framework could meaningfully extend reliable FVE estimation to correlated high-dimensional regimes common in neuroimaging and genomics. The explicit use of established high-dimensional estimators on the residual, combined with simulation evidence of bias reduction and consistency, and the real-data application, constitute concrete strengths. The approach is presented as a methodological framework rather than a tautological identity.

minor comments (4)
  1. [§3.1] §3.1: The precise definition of the two FVE components (low-dimensional vs. high-dimensional) should include an explicit equation showing how the total FVE is recovered as their sum; the current description leaves the additivity implicit.
  2. [Figure 2] Figure 2: The simulation panels comparing bias for the proposed method versus direct GWASH/LMM-REML would benefit from error bars or standard errors across replicates to allow visual assessment of variability.
  3. [§4.2] §4.2: The criterion used to choose the number of dominant PCs is described only qualitatively; a short algorithmic statement or pseudocode would improve reproducibility.
  4. [Table 1] Table 1: The asymptotic consistency claim is stated for p,n → ∞, but the table reports finite-sample results only; a brief note on the rate or conditions would clarify the link to theory.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, the accurate summary of the proposed PC-based FVE decomposition framework, and the recommendation for minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper proposes a decomposition of FVE into low-dimensional (strong correlation via dominant PCs) and high-dimensional (weak residual correlation) components, with claims of bias reduction validated via simulations and asymptotic consistency as p and n grow. No equations, derivations, or self-referential definitions appear in the provided text that reduce any prediction or result to a fitted quantity defined by the same inputs. The framework is presented as a methodological partitioning rather than a self-definitional identity, and no load-bearing self-citations or ansatzes are invoked. This is a standard non-circular finding for a simulation-supported proposal.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are identifiable from the abstract; the method description does not introduce new fitted constants or postulated objects.

pith-pipeline@v0.9.1-grok · 5746 in / 1131 out tokens · 23107 ms · 2026-06-28T08:42:32.880684+00:00 · methodology

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Reference graph

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