Bias in genome-wide association test statistics due to omitted interactions

This paper demonstrates that omitting epistatic interactions in standard linear genome-wide association studies can lead to biased test statistics and spurious significant findings, particularly in anti-conservative regimes, thereby urging caution when interpreting results from models that assume purely additive genetic effects.

The Gist

The Big Picture: The "One-Size-Fits-All" Trap

Imagine you are a detective trying to solve a mystery: Why do some people have blue eyes and others have brown eyes? You have a massive list of suspects (genetic variants) and a huge database of clues (data from hundreds of thousands of people).

For the last 20 years, scientists have been using a very simple, reliable tool to solve this: The Linear Model. Think of this tool as a straight ruler. It assumes that every suspect contributes to the final result in a simple, additive way.

• Suspect A adds 1 inch of height.
• Suspect B adds 2 inches of height.
• Total height = 1 + 2 = 3 inches.

This works great if the world is simple. But biology is messy. Sometimes, Suspects don't just add up; they interact. Maybe Suspect A only adds height if Suspect B is also present. If Suspect B is missing, Suspect A does nothing. This is called epistasis (gene-gene interaction).

The Problem: Ignoring the Teamwork

This paper asks a scary question: What happens if we keep using our "straight ruler" to measure a world that is actually full of "teamwork" (interactions)?

The authors found that when you ignore these interactions, your ruler doesn't just give a slightly wrong answer; it starts hallucinating. It points the finger at innocent suspects and says, "You did it!" when they actually did nothing.

The Analogy: The "Ghost" in the Machine

Imagine you are trying to measure the speed of a car (the trait) based on the size of its engine (the gene).

• The Reality: The car's speed depends on the engine size AND whether the driver is pressing the gas pedal (the interaction).
• The Mistake: You only look at the engine size. You ignore the gas pedal.

Now, imagine a specific scenario:

1. The gas pedal is pressed hard (high interaction effect).
2. By pure chance, the cars with the "wrong" engine size happen to be the ones where the driver is pressing the gas hard.

Because you aren't measuring the gas pedal, your math gets confused. It sees the car speeding up and thinks, "Aha! It must be because of the engine size!" It creates a false connection.

In the paper's language:

• The Engine is the "Target SNP" (the gene you are testing).
• The Gas Pedal is the "Interaction Term" (the hidden teamwork between genes).
• The False Connection is the "Spurious Association" (a statistical result that looks real but isn't).

The Mathematical Discovery: The "Inflation" Effect

The authors did some heavy math to prove exactly how this happens. They found that when you ignore interactions:

1. The Average Shifts: Your "ruler" gets pushed off-center. Instead of the average result being zero (no effect), it drifts away.
2. The Tail Gets Fat: In statistics, the "tail" represents extreme results (the ones that make headlines). The authors found that the tail gets "inflated." This means you get way more "significant" results than you should.

The "Anti-Conservative" Regime:
In science, we usually want to be "conservative" (safe). We'd rather miss a real discovery than falsely accuse an innocent person.

• Conservative: The ruler is too strict; it rarely finds anything.
• Anti-Conservative (The Danger Zone): The ruler is too loose. It screams "FIRE!" every time a match is struck.

The paper shows that with modern, massive datasets (like the Estonian Biobank with 200,000+ people), we are deep in the Anti-Conservative zone. Even if the interactions are small, the sheer size of the data makes the "ghost" signals look incredibly strong.

The "Strict No-Path" Test

To prove this wasn't just a fluke, the authors set up a very strict test. They created a scenario where the "Target Gene" was completely disconnected from the "Interaction Team."

• Analogy: They made sure the engine size had absolutely nothing to do with the gas pedal.
• Result: Even with this strict separation, the math still produced false alarms. Why? Because of Linkage Disequilibrium (LD).

LD Analogy: Imagine the "Target Gene" is a red car, and the "Interaction Gene" is a blue car. They aren't the same car, but they are always parked next to each other in the same parking lot. If the blue car (interaction) causes the speed, the red car (target) gets blamed because it's always right there.

The Takeaway: A Warning for the Future

The paper concludes with a major warning for the scientific community:

1. Bigger isn't always better: As we gather more data (millions of people), these false alarms get louder, not quieter.
2. Linear models have limits: Assuming genes just "add up" is becoming a dangerous oversimplification.
3. Spurious Hits: Many of the "significant" genes we have found in the last decade might actually be ghosts—statistical illusions caused by ignoring gene interactions.

In short: If you are looking for a needle in a haystack, and you ignore the fact that the needles are magnetically stuck to the hay, you might end up picking up a clump of hay and thinking it's a needle. This paper tells us to stop using the simple ruler and start building tools that can measure the complex, messy teamwork of our DNA.

Technical Summary

1. Problem Statement

Genome-wide association studies (GWAS) have successfully identified thousands of genetic variants associated with complex human traits. However, the standard methodology relies heavily on linear models (specifically Linear Mixed Models, LMMs) that assume genetic effects are predominantly additive.

• The Gap: Biological systems often involve epistasis (gene-gene interactions) and gene-environment interactions. When these nonlinear interactions exist but are omitted from the statistical model, they act as omitted variables.
• The Consequence: The paper investigates whether omitting interaction terms in GWAS leads to spurious associations (false positives). Specifically, it asks if standard linear models can produce statistically significant signals for SNPs that have no direct causal effect on the phenotype, simply because they are correlated with the omitted interaction terms.
• Current Knowledge: While "synthetic associations" (where a SNP tags a combination of other variants) are known, there has been no rigorous mathematical modeling to determine if the omission of epistasis alone can inflate test statistics to the point of generating false positives under realistic parameter settings.

2. Methodology

The authors employ a combination of algebraic derivation and large-scale phenotype simulation using real genotype data.

A. Mathematical Framework

1. True Data Generating Process (DGP):
   The true phenotype $y$ is modeled as:
   $$y = X\beta + \alpha g + u + \varepsilon$$
   Where:

   • $g$: Target SNP.
   • $u$: The realized contribution of interaction terms (epistasis), defined as $u = Z\beta_u$.
   • $\varepsilon$: Noise including environmental and additive polygenic effects.
   • $\alpha$: The true additive effect of $g$ (set to 0 for the null hypothesis).

2. Fitted (Misspecified) Model:
   A standard LMM is fitted assuming no interaction:
   $$y = X\beta + \alpha g + \varepsilon$$
   The goal is to analyze the behavior of the test statistic for $H_0: \alpha = 0$.

3. Preprocessing and Derivation:

   • The authors apply a linear operator $T$ to whiten the data and remove covariates (age, sex, PCs), transforming the model into a space where noise is spherical.
   • They derive the expected value and variance of the Ordinary Least Squares (OLS) estimator $\hat{\alpha}$ under the null hypothesis.
   • Key Insight: The omission of $u$ causes a shift in the mean and variance of the test statistic. The statistic $t$ follows a distribution $N(\mu, \sigma^2_{res})$ rather than the nominal $N(0, 1)$.
   • Parameters: The bias is driven by three factors:
     • $\lambda$: The fraction of variance explained by the interaction term.
     • $\rho$: The correlation between the target SNP ($g$) and the interaction term ($u$).
     • $n$: The sample size.

4. Conservativeness Ratio ($R$):
   The authors define a ratio $R(x) = p_{true}(x) / p_{nom}(x)$ to quantify bias.

   • $R > 1$: Anti-conservative (inflated tail, increased false positives).
   • $R < 1$: Conservative (deflated tail).
   • They analyze this at the standard GWAS significance threshold ($x = \pm 5.45$, corresponding to $p = 5 \times 10^{-8}$).

5. Strict No-Path Null ($H^NP_0$):
   To ensure bias is purely due to correlation and not a direct causal pathway, they define a strict null where the target SNP is excluded from the interaction features ($u$). This isolates the bias caused by Linkage Disequilibrium (LD) between the target SNP and the loci involved in the interaction.

B. Simulation Analysis

• Data: Estonian Biobank genotypes (approx. 210,000 samples).
• Design:
  • Simulated 10,000 phenotypes with varying non-additive variance fractions ($\lambda$ from 0.001 to 0.171).
  • Interaction terms ($u$) were constructed using random two-way interactions of SNPs on Chromosome 21.
  • Target SNPs ($g$) were selected from Chromosome 22 (different chromosome) and Chromosome 21 (same chromosome) to test LD effects.
  • Tool: Used REGENIE (a state-of-the-art LMM tool) to generate summary statistics.
• Validation: Compared the algebraic predictions of the $R$ ratio and $p$-value distributions against the empirical results from REGENIE.

3. Key Results

Mathematical Findings

• Anti-Conservative Regime: The derived model shows that under realistic conditions, the test statistic distribution shifts such that the tail is inflated ($R > 1$).
• Threshold Crossing: For the standard GWAS threshold ($|x| = 5.45$), the probability of observing a significant result under the true null ($p_{true}$) can exceed 0.5 (meaning >50% of "significant" hits are spurious) when:
  • Sample size ($n$) is large (e.g., >100,000).
  • Non-additive variance ($\lambda$) is modest (e.g., 0.03–0.1).
  • Correlation ($\rho$) is low but non-zero (e.g., 0.03).

Simulation Findings

• High Concordance: The simulation results using REGENIE matched the algebraic model predictions perfectly.
• Spurious Significance:
  • In simulations with interaction terms, thousands of SNPs were identified as significant ($p < 5 \times 10^{-8}$) despite having no causal effect.
  • In control simulations (without interaction terms), only 2–4 spurious hits were found across 38 million tests, confirming the bias is driven by the omitted interaction.
• Chromosomal Effects:
  • Same Chromosome: High $\rho$ values were observed due to LD, leading to severe inflation.
  • Different Chromosomes: Even when target SNPs and interaction SNPs were on different chromosomes, $\rho$ was non-zero (mean $\approx 0.032$) due to population structure and global LD, still resulting in an anti-conservative regime for large $n$.

4. Key Contributions

1. Algebraic Proof of Bias: The paper provides the first rigorous algebraic derivation showing that omitting epistasis in linear GWAS models shifts the null distribution of test statistics, creating a systematic bias toward false positives.
2. Quantification of the "Spurious" Zone: The authors define the boundary between conservative and anti-conservative regimes, demonstrating that the "safe" zone for linear models shrinks as sample sizes increase.
3. Strict No-Path Null Concept: Introduction of a rigorous null hypothesis that excludes the target SNP from interaction terms, proving that bias arises purely from correlation (LD) rather than direct causal pathways.
4. Empirical Validation: Validation of the theoretical model using real-world genotype data (Estonian Biobank) and modern GWAS software (REGENIE), confirming that the bias is not just theoretical but observable in practice.

5. Significance and Implications

• Re-evaluation of GWAS Catalog: The findings suggest that a non-negligible portion of statistically significant hits in the GWAS Catalog (especially those with very low p-values in large biobank studies) may be spurious artifacts of unmodeled epistasis rather than true additive causal variants.
• Sample Size Paradox: As GWAS sample sizes grow (now exceeding 1 million in some studies), the power to detect true signals increases, but the power to detect these spurious signals also increases dramatically. The bias scales with $n$, making large-scale studies particularly vulnerable if interactions are ignored.
• Methodological Shift: The paper argues that relying solely on additive linear models is insufficient for understanding complex traits. It advocates for:
  • Explicit modeling of interactions (epistasis).
  • Development of assumption-free or minimum-assumption models.
  • Caution in interpreting significant signals from large-scale linear GWAS without considering potential nonlinear architectures.

In conclusion, the paper demonstrates that omitted interaction terms in GWAS create a systematic anti-conservative bias, leading to a high rate of false positives in large-scale studies, necessitating a shift in how genetic associations are modeled and interpreted.