Significance filtering induces denominator selection bias in local genetic correlation: closed-form characterisation, a gate-aware correction, and an applicability diagnostic

Technical Summary: Significance Filtering and Denominator Selection Bias in Local Genetic Correlation

Problem Statement
Local genetic correlation ($\rho$) is estimated as the ratio of local genetic covariance to the geometric mean of two local heritabilities. The standard implementation, LAVA, reports this ratio only when both traits pass a univariate heritability significance test ($P < 0.05$). While LAVA's documentation frames this filter as a stabilizer to remove unstable estimates, the authors demonstrate that it acts as a selection event on the denominator of the ratio estimator.

Because the univariate test statistic and the denominator are derived from the same sum of squares, conditioning on the statistic being large (passing the filter) conditions on the denominator being large. This selection preferentially admits loci where sampling noise has inflated the heritability estimate. Consequently, the denominator is systematically inflated, causing the resulting ratio (the correlation estimate) to be attenuated (biased toward zero) rather than inflated. Unlike the "winner's curse," which inflates estimates, this mechanism attenuates them. Furthermore, this selection interacts with sample overlap between GWAS to manufacture non-zero expected correlations even when the true correlation is zero.

Methodology
The authors derive a closed-form characterization of this bias under the noncentral Wishart model assumed by LAVA's own interval routines.

1. Theoretical Derivation:

   • The gate-conditional mean of the estimator is shown to be affine in the true correlation ($\rho$): $E[\hat{\rho} | G] = S_{loc}\rho + I_{loc}$.
   • $S_{loc}$ (Slope): Represents the attenuation factor. It is a function of the local signal-to-noise ratios ($\theta_i$), the number of principal components ($K$), and the gate-induced inflation of the diagonal elements ($\Delta_i$).
   • $I_{loc}$ (Intercept): Represents the "manufactured" correlation arising from sample overlap ($r_e$) and the filter. It is proportional to $r_e$ and the inflation terms, vanishing as power increases.
   • Crucially, these quantities depend only on data already reported by LAVA: the retained principal components ($K$), the univariate $P$-values, the reported correlation, and the sample-overlap correlation.

2. Correction Protocol:

   • Stage 1 (De-biasing the Denominator): The observed gated heritability is recovered from the $P$-value. The authors invert the mapping $g_K$ (from true signal $\theta$ to gated expectation) to recover the underlying signal strength. This inversion is unique but has a "floor" where true signal cannot be distinguished from noise.
   • Stage 2 (De-biasing the Ratio): The corrected correlation is calculated as $\hat{\rho}_{corr} = (\hat{\rho} - I_{loc}) / S_{loc}$.
   • Aggregation: Because the correction inverts a conditional mean, it is applied to aggregates (pooled loci) rather than individual loci to avoid instability, as individual loci sit within the distribution rather than at the mean.

3. Applicability Diagnostic:
   A three-part diagnostic determines if the correction is identified for a given trait pair:

   • C1 (Stability): The aggregate slope must be stable across different stratifications of $K$.
   • C2 (Floor Distance): The observed gated heritability must be statistically distinguishable from the distribution expected under a complete local null ($\theta=0$).
   • C3 (Non-Null Output): The fraction of gate passes attributable to chance must be below a threshold (0.40).

4. Validation:
   The authors employ a three-tier simulation strategy:

   • Tier A: Draws directly from the noncentral Wishart distribution.
   • Tier B: Injects known $\rho$ into real loci and recovers it through LAVA's actual code path.
   • Tier C: A control experiment isolating the filter's effect from LAVA's output capping mechanisms.

Key Results

• Bias Characterization: The bias is strictly affine in $\rho$. Across six psychiatric trait pairs, the measured attenuation slopes ($S_{loc}$) range from 0.579 to 0.923. The closed-form prediction matches the measured slopes within 0.007 without fitting any parameters to the panel.
• Source of Bias: The filter is the primary cause of attenuation. While LAVA's output capping contributes slightly, the filter alone accounts for the vast majority of signal loss (e.g., 37–42% signal destruction in low-power regimes).
• Manufactured Correlation: In the absence of true correlation, the filter combined with sample overlap generates significant positive correlations. For example, in a high-overlap pair (ASD $\times$ ADHD), the manufactured intercept was 0.139, exceeding many published local genetic correlations.
• Correction Performance: Applying the gate-aware correction in aggregate reduces the mean absolute bias from 0.247 to 0.026 (a 9.7-fold improvement). A single pooled correction factor (e.g., dividing by a global slope) performs poorly (bias 0.195) because it fails to capture the heterogeneity of $S_{loc}$ across loci.
• Interval Width: Post-filter confidence intervals are correctly calibrated (94.9% coverage) but are nearly vacuous. The mean width spans 69% of the parameter space $[-1, 1]$, rendering sign determination impossible for the majority of loci (e.g., 68 of 78 loci in one pair had intervals crossing zero).
• Diagnostic Failures: Three of the six trait pairs failed the applicability diagnostic. In these cases (e.g., ASD $\times$ SCZ), the weaker trait's local heritability was indistinguishable from a complete local null. For these pairs, the reported loci are not weak findings but artifacts of the filter acting on noise; correction is not identified.

Significance and Claims
The paper claims to provide the first closed-form characterization of the selection bias induced by LAVA's reporting rules. Its significance lies in:

1. Correcting a Misconception: It demonstrates that the filter is not a benign stabilizer but a selection event that systematically attenuates estimates and manufactures correlation.
2. Actionable Correction: It offers a correction method that requires no re-analysis of raw data, only the standard LAVA output and an overlap estimate.
3. Contextualizing Published Results: The authors argue that most published estimates from well-powered traits are likely fine (bias < 10%), but estimates from underpowered traits or those with high sample overlap are severely biased.
4. Reporting Standards: The paper proposes specific reporting guidelines, including the necessity of reporting the "attrition cascade" (how many blocks were lost to negative variance estimates vs. the filter), checking for "Channel 2" (silent removal of traits with negative variance), and using the proposed diagnostic before applying corrections.

The authors explicitly state that this work does not invalidate the biological findings of previous studies but rather provides the tools to quantify and correct the analytical bias inherent in the current standard pipeline. They caution that for pairs failing the diagnostic, the reported correlations are likely artifacts of the filter and should not be interpreted as biological signals.