Tutorials › Statistical Genetics › Reference LD Quality in Summary Analysis

Home  /  Tutorials  /  Conditional and Joint Association

Conditional and Joint Association · Tutorial 611 of 989

Reference LD Quality in Summary Analysis

Use reference-LD comparisons and allele-harmonization checks to assess whether summary-statistic conditional estimates are stable and interpretable.

Advanced 10 min read

What You'll Learn

  • Express two-variant conditional estimates in terms of marginal effects and reference-LD correlation
  • Explain why ancestry mismatch can change conditional estimates even when marginal summaries are fixed
  • Assess how small reference samples make estimated LD—and conditional results—unstable
  • Detect allele-orientation errors by checking effect alleles and signed LD together
  • Use comparison and sensitivity checks to distinguish robust results from LD-dependent ones

Why Reference LD Quality Matters

Summary-statistic conditional analysis combines marginal association estimates with linkage disequilibrium (LD) information to estimate associations conditional on other variants. The marginal estimates describe each variant one at a time; the LD information describes how their genotypes co-vary. If that LD information does not represent the association sample, the conditional estimates can shift even though the marginal summary statistics have not changed.

The main concern is not simply whether a reference panel is large or comes from a similar population. Its signed LD values must be appropriate for the variants, ancestry composition, and allele coding used in the analysis. A small reference sample can estimate those values imprecisely, while an allele-orientation mismatch can reverse their signs. These problems can produce unstable estimates, misleading apparent secondary signals, or even implausibly large conditional effects.

Two-Variant Conditional Estimates and Their LD Input

For a precise statement, consider two variants with genotypes standardized to mean zero and variance one. Let \(R\) be their genotype correlation matrix in the association sample, and let \(\theta_1\) and \(\theta_2\) be their joint regression coefficients for an outcome on a fixed scale. Let \(b_1\) and \(b_2\) be the corresponding population marginal regression coefficients, each obtained from a separate regression of the outcome on one standardized genotype. In a common linear-regression setting, the marginal and joint coefficients satisfy \(b=R\theta\). Summary-statistic conditional analysis replaces the association-sample LD matrix with an estimated matrix from a reference sample.

Definition: For two standardized genotypes, write their signed LD correlation as \(r\). If the marginal effect vector is \(b=(b_1,b_2)^\mathsf{T}\) and the LD matrix used for conditioning is \(R=\begin{pmatrix}1&r\\r&1\end{pmatrix}\), the two-variant conditional estimates are obtained from \(\hat{\theta}=R^{-1}b\). Thus, \(\hat{\theta}_1=(b_1-rb_2)/(1-r^2)\) and \(\hat{\theta}_2=(b_2-rb_1)/(1-r^2)\), provided \(|r|<1\).

These equations make the quality issue visible. The conditional estimate depends on both the marginal effects and the signed correlation supplied by the reference data. It also divides by \(1-r^2\), which becomes small when the variants are highly correlated. The formula is exact for the stated standardized population-regression setup. In summary-data applications, it is an approximation when covariate adjustment, sample composition, phenotype scaling, or other details prevent the marginal summaries from matching that setup. The analyst should therefore regard agreement between the analysis and reference samples as a substantive assumption, not as a detail of computation.

How a Mismatch Changes the Estimate

A reference panel is useful only to the extent that its LD represents the association sample at the variants being conditioned. LD can differ across ancestry groups, and the amount or pattern of difference can vary across a locus. If the association sample has correlation \(r_A\) but the reference panel supplies \(r_R\), the computed first conditional estimate is \((b_1-r_Rb_2)/(1-r_R^2)\), not the value based on \(r_A\). A reference panel that is broadly described as ancestry-matched may still be a poor local match for particular variants.

Worked Example: Ancestry Mismatch Changes a Conditional Estimate

Hypothetical data. Suppose the two joint effects in the association population are \(\theta_1=0.080\) and \(\theta_2=0.025\), in outcome units per one-standard-deviation increase in genotype. In that population, let the genotype correlation be \(r_A=0.80\). The expected marginal effects are

$$ b_1=\theta_1+r_A\theta_2=0.080+0.80(0.025)=0.100, \qquad b_2=r_A\theta_1+\theta_2=0.80(0.080)+0.025=0.089. $$

Using the association-population LD recovers the first joint effect:

$$ \hat{\theta}_1 =\frac{0.100-0.80(0.089)}{1-0.80^2} =\frac{0.0288}{0.36} =0.0800. $$

Now suppose a reference sample from a different ancestry has \(r_R=0.50\). Holding the same marginal summaries fixed, the estimate becomes

$$ \hat{\theta}_1^{(R)} =\frac{0.100-0.50(0.089)}{1-0.50^2} =\frac{0.0555}{0.75} =0.0740. $$

The reference-based estimate is \(0.0060\) outcome units per genotype standard deviation below the value recovered using the association-population LD, a 7.50% difference relative to \(0.0800\). This is a hypothetical illustration of the formula, not evidence that one ancestry is intrinsically preferable. The relevant issue is whether the reference LD represents the association sample for these variants.

Small Reference Samples and Sensitivity

Even when a reference sample has suitable ancestry and allele coding, a small sample can yield a noisy estimate of \(r\). Because the conditional estimate is a nonlinear function of \(r\), plugging in one noisy correlation can conceal the resulting uncertainty. The risk is especially important when \(|r|\) is large: the denominator \(1-r^2\) is then small, so changes in LD can produce substantial changes in the conditional estimates. Conversely, a small change in \(r\) does not always have a large effect; the sensitivity also depends on the marginal effects.

Key relationship: For fixed marginal effects, the local sensitivity of the first conditional estimate to the LD correlation is
$$ \frac{\partial\hat{\theta}_1}{\partial r} = \frac{2rb_1-b_2(1+r^2)}{(1-r^2)^2}. $$
The denominator shows why estimates can be sensitive near \(|r|=1\), although the numerator determines the actual local change at a given set of effects.

A useful diagnostic is to compare estimates across plausible values of \(r\), or across suitable reference panels, while holding the marginal summaries and allele coding fixed. This is a sensitivity analysis, not a substitute for identifying the best-matched reference data. If conclusions change materially across credible LD inputs, report that dependence rather than presenting a single conditional estimate as if the LD were known exactly.

Worked Example: Quantifying Sensitivity to a Noisy Reference Correlation

Hypothetical data. Retain \(b_1=0.100\) and \(b_2=0.089\). Suppose a reference sample of \(n=80\) independent individuals gives \(\hat r=0.70\). The point conditional estimate is

$$ \hat{\theta}_1 =\frac{0.100-0.70(0.089)}{1-0.70^2} =\frac{0.0377}{0.51} =0.0739. $$

For illustration, under a bivariate-normal approximation for independent reference observations, the Fisher-transformed correlation \(z_r=\operatorname{atanh}(\hat r)\) has approximate standard error \(1/\sqrt{n-3}\). Here that standard error is \(1/\sqrt{77}=0.1140\). An approximate 95% range on the Fisher scale is

$$ \operatorname{atanh}(0.70)\pm1.96(0.1140) =0.8673\pm0.2234 =(0.6439,1.0907). $$

Transforming the endpoints back to the correlation scale gives approximately \(r=0.567\) to \(r=0.797\). This is a rough reference-LD uncertainty range under the stated approximation, not a confidence interval for the conditional effect that accounts for all sources of uncertainty. To see the implication for the point estimate, evaluate the conditional formula at the two endpoints:

$$ \hat{\theta}_1(0.567) =\frac{0.100-0.567(0.089)}{1-0.567^2} \approx\frac{0.04954}{0.6785} =0.0730, \qquad \hat{\theta}_1(0.797) =\frac{0.100-0.797(0.089)}{1-0.797^2} \approx\frac{0.02907}{0.3648} =0.0797. $$

Thus, this approximate range of plausible LD values maps to conditional point estimates from about \(0.0730\) to \(0.0797\), with the marginal effects held fixed. The example shows why the sample size and precision of the reference LD matter. In real analyses, genotype data may not satisfy the bivariate-normal approximation exactly, and uncertainty in marginal effects and their covariance also matters; this calculation is a diagnostic illustration, not a complete inferential interval.

Allele Orientation Must Match the LD Sign

An LD correlation is signed: its sign depends on how each allele is coded. A marginal effect is also signed relative to its stated effect allele. Therefore, the effect allele and the LD coding must be harmonized together. If one variant is recoded from one allele to the other, its marginal effect changes sign and the correlation between that genotype and the other variant changes sign. The conditional estimate should not change when the recoding is done consistently.

Worked Example: Detecting an Allele-Orientation Error

Hypothetical data. Let \(b_1=0.100\), \(b_2=0.089\), and \(r=0.80\), with all values initially coded consistently. The first conditional estimate is

$$ \hat{\theta}_1 =\frac{0.100-0.80(0.089)}{1-0.80^2} =\frac{0.0288}{0.36} =0.0800. $$

Now reverse the effect-allele coding for variant 2. Its marginal effect becomes \(-0.089\), and its correlation with variant 1 becomes \(-0.80\). Using both recoded values gives

$$ \hat{\theta}_1^{(\mathrm{recoded})} =\frac{0.100-(-0.80)(-0.089)}{1-(-0.80)^2} =\frac{0.0288}{0.36} =0.0800. $$

The estimate is unchanged, as it should be. If the effect were flipped but the LD correlation were mistakenly left at \(+0.80\), the result would instead be

$$ \hat{\theta}_1^{(\mathrm{mismatched})} =\frac{0.100-0.80(-0.089)}{1-0.80^2} =\frac{0.1712}{0.36} =0.476. $$

This large change is created by inconsistent coding, not by evidence for a genuinely large conditional effect. A practical check is to align effect alleles and allele frequencies between the summary statistics and reference data, and then verify that the signed LD matrix uses the same allele orientation. Strand-ambiguous alleles require particular care because allele labels alone may not resolve orientation.

A Practical Quality-Check Sequence

Reference-LD diagnostics should be performed before interpreting a conditional estimate as evidence that a signal is distinct from another. The following sequence separates three different sources of trouble: population mismatch, sampling imprecision, and inconsistent allele coding.

1
Check variant and allele alignment.
Confirm that the variants in the marginal summaries and reference data correspond to the same alleles and that the signed LD matrix uses those same effect-allele orientations.
2
Assess reference relevance.
Compare the reference sample’s ancestry composition with that of the association sample, and consider whether local LD differences could matter for the variants being conditioned.
3
Assess LD precision.
Record the reference sample size and examine whether the estimated correlations are sufficiently precise for the conditional calculation, particularly when correlations are near \(+1\) or \(-1\).
4
Compare plausible LD inputs.
Recompute estimates using suitable alternative reference panels or a justified range of correlation values. Keep the marginal summary statistics and allele coding fixed so the comparison isolates LD dependence.
5
Report instability plainly.
If a conditional estimate, its sign, or the interpretation of a secondary signal changes across credible LD inputs, describe that sensitivity and avoid treating one reference-based result as definitive.

Common Pitfall and Interpretation

Wrong reasoning: “The marginal effect estimates are precise, so the conditional estimates must also be reliable, even if the LD panel is small or from a different population.” This does not follow. The conditional estimates use the marginal effects and the LD matrix, and error in that matrix can alter the result, especially with highly correlated variants.

Correction: Precision of marginal estimates does not establish accuracy or precision of the reference LD. Check allele orientation, ancestry relevance, reference sample size, and sensitivity to plausible LD values. Also distinguish a change in a conditional estimate from evidence of causality: conditional associations describe statistical relationships under the fitted model and do not by themselves identify causal variants.

Key takeaway: A summary-statistic conditional estimate is only as credible as the LD information and allele coding used to calculate it. Compare reference LD with the association setting and test whether the estimate is stable across defensible alternatives.

Check Your Understanding

Use the equations and diagnostics above to answer these questions.

  1. For \(b_1=0.12\), \(b_2=0.08\), and \(r=0.60\), calculate the first two-variant conditional estimate.
  2. Why can the same marginal effects yield different conditional estimates when computed with reference panels from different ancestry groups?
  3. What two quantities must change together when the effect-allele coding for one variant is reversed?
  4. Why can uncertainty in a reference correlation be especially consequential when the absolute correlation is close to one?
  5. If a conditional estimate changes substantially across plausible reference panels, what should be reported about its interpretation?

Previous

Next

Back to Tutorials
Clinical Biostats