Variant Inclusion and LD-Proxy Sensitivity

Why Variant Inclusion Is a Sensitivity Question

A conditional association estimate answers a question about a particular model: how strongly is a variant associated with the phenotype given the other included variants and covariates? In a locus with correlated genotypes, changing the model can change the estimate, its standard error, or both. A signal that appears stable under reasonable alternatives is less dependent on one modeling choice; that stability does not, by itself, establish a causal effect.

Variant-inclusion sensitivity asks whether the conclusion changes when a correlated variant is added to or removed from the locus model. LD-proxy sensitivity asks whether the conclusion depends on which correlated variant is used to represent a signal. These checks are especially informative when candidate variants have different imputation quality or when a preferred variant is unavailable in some data.

Definition: A variant-inclusion sensitivity analysis compares association results from models that differ in the inclusion of one or more locus variants, while keeping the phenotype, analysis sample, covariates, and other modeling choices as comparable as possible. An LD-proxy sensitivity analysis repeats that comparison using alternative correlated variants as proxies for a signal. The quantities being compared are model-specific conditional estimates, not interchangeable measurements of a single guaranteed biological effect.

The Model and the Meaning of a Change

For a continuous phenotype, a joint linear model for individual \(i\) can be written as

Y_i=\alpha+\sum_{k=1}^{q}\gamma_k C_{ik}+\sum_{j=1}^{m}\beta_j G_{ij}+\varepsilon_i.

Here \(Y_i\) is the phenotype, \(C_{ik}\) are covariates, and \(G_{ij}\) is the genotype dosage for variant \(j\), commonly on a 0-to-2 scale for a biallelic variant. The coefficient \(\beta_j\) is the expected phenotype difference per one-unit increase in dosage, conditional on the other variables in this model. A model-based standard error and test additionally require appropriate assumptions for the chosen regression: for ordinary least squares inference, the conditional mean is correctly specified and errors are independent with constant variance, or a suitable robust variance estimator is used. The genotype and phenotype data must also be aligned to the same individuals and alleles.

A sensitivity comparison is only interpretable when differences in the models are clear. If the sample changes because one variant has more missing data, an estimate may move because of sample composition rather than variant inclusion alone. Likewise, changing covariates, phenotype transformations, or genotype coding at the same time prevents a clean attribution of the change. These choices should be held fixed when possible, and unavoidable differences should be reported.

Why Correlated Variants Can Change an Estimate

A short calculation shows why adding a correlated variant can alter a coefficient even when no data errors are present. Consider two centered, standardized genotype predictors \(X_1\) and \(X_2\), after accounting for the same covariates, with correlation \(r\). Let \(Y^*\) be the phenotype after accounting for those covariates. Write \(b_{1m}=\operatorname{Cov}(X_1,Y^*)\) and \(b_{2m}=\operatorname{Cov}(X_2,Y^*)\); these are the coefficients from separate one-predictor regressions under this standardization. The two-predictor least-squares equations are

\begin{pmatrix}1&r\\r&1\end{pmatrix} \begin{pmatrix}\hat{\beta}_1\\\hat{\beta}_2\end{pmatrix} = \begin{pmatrix}b_{1m}\\b_{2m}\end{pmatrix}.

When \(|r|<1\), the matrix is invertible. Solving the two equations gives

\hat{\beta}_1=\frac{b_{1m}-r b_{2m}}{1-r^2}, \qquad \hat{\beta}_2=\frac{b_{2m}-r b_{1m}}{1-r^2}.

For example, if the phenotype has a linear association with both predictors, the one-predictor coefficient for \(X_1\) contains association shared with \(X_2\). In the two-predictor model, the coefficient is based on the variation in \(X_1\) not explained by \(X_2\). As \(|r|\) approaches 1, that remaining variation becomes small, so estimates can become imprecise and sensitive to modest changes in the data or model. This calculation is a regression identity, not evidence that either variant is causal.

Worked Example: Adding a Correlated Variant

Suppose a hypothetical analysis uses the same individuals and covariates in each model. The outcome and both genotype predictors are standardized after covariate adjustment. The separate marginal coefficients are \(b_{1m}=0.120\) and \(b_{2m}=0.100\), and the genotype correlation is \(r=0.800\). Compare the coefficient for variant 1 when variant 2 is omitted and included.

Variant 2 omitted. In the one-predictor model, the coefficient for variant 1 is \(0.120\) outcome standard deviations per genotype standard deviation.

Both variants included. The conditional coefficient for variant 1 is

\hat{\beta}_1 =\frac{0.120-(0.800)(0.100)}{1-(0.800)^2} =\frac{0.120-0.080}{0.360} =\frac{0.040}{0.360} =0.1111.

The conditional coefficient for variant 2 is

\hat{\beta}_2 =\frac{0.100-(0.800)(0.120)}{0.360} =\frac{0.100-0.096}{0.360} =0.01111.

Variant 1's estimate decreases by \(0.00889\) outcome standard deviations per genotype standard deviation, or \(0.00889/0.120=7.41\%\) of its one-predictor estimate. This change is consistent with the two variants sharing predictive information. It does not show that variant 2 is responsible for the association, and the example provides no standard errors with which to assess statistical uncertainty.

Checking Sensitivity to Imputation Quality

A dosage is an expected allele count based on genotype uncertainty; it is not necessarily a directly observed genotype. Poorly imputed dosages can contain less reliable information about a variant. In a joint model, this may affect the variant's own estimate and the conditional estimates of correlated variants. The direction and size of the change are not guaranteed: the result depends on the uncertainty pattern, LD, sample, and phenotype associations.

A useful diagnostic is to distinguish two questions. First, does the result change when the uncertain variant's dosage is added to the model? Second, does it change when that dosage is replaced by a more reliable measurement, such as a high-quality genotype in a validation sample? The second comparison is informative only if the sample and other analysis choices are held fixed or their differences are made explicit. Imputation-quality measures help describe the concern, but no single quality threshold is universal across methods and settings.

Worked Example: A Poorly Imputed Variant in a Joint Model

In a hypothetical matched sample of 8,000 individuals, researchers fit the same covariate-adjusted linear model for a standardized phenotype. Variant T is the target signal. Variant U is correlated with T and has an imputation-quality estimate of \(0.42\). The following estimates are illustrative regression outputs, in phenotype standard deviations per one-unit allele dosage:

Model for the same 8,000 individuals	Estimate for T	Standard error
T included; U excluded	0.082	0.020
T and imputed dosage for U included	0.055	0.023
T and high-quality genotype for U included	0.078	0.024

Adding the imputed dosage changes T's estimate by \(0.055-0.082=-0.027\), a decrease of \(0.027/0.082=32.9\%\) in magnitude relative to the T-only estimate. Replacing that dosage with the higher-quality genotype gives a change of \(0.078-0.082=-0.004\), or \(4.88\%\) in magnitude. For context, the estimate divided by its standard error is \(0.055/0.023=2.39\) in the imputed-dosage model and \(0.078/0.024=3.25\) in the high-quality-genotype model.

The contrast suggests that the result is sensitive to how U is represented in this hypothetical analysis. It does not prove that imputation error caused the difference: the genotype measurement, model conditioning, and their interaction with LD all matter. The same-sample comparison removes one important source of variation, while the quality information motivates checking the dosage. The coefficient remains a conditional association estimate, not a causal effect.

Comparing Alternative LD Proxies

Sometimes the variant used to represent a signal is not available or is poorly measured. An alternative proxy may be strongly correlated with it, but “strongly correlated” does not mean identical. Different proxies can have different allele frequencies, LD with other locus variants, and measurement quality. Their conditional coefficients therefore need not agree, even if both track related genetic variation.

When comparing proxies, record which variant is the outcome-model predictor and which variants are conditioned on. Verify allele alignment and effect-allele coding before comparing signs. Also compare the same sample and covariates, and report the proxy correlation in the relevant ancestry and dataset when available. A change in estimate can reflect a different conditional predictor, not necessarily a disappearance or creation of the underlying locus association.

Worked Example: Two Proxies Give Different Conditional Estimates

Suppose a hypothetical standardized analysis compares target variant T with two proxies, P1 and P2. After the same covariate adjustment, the marginal coefficients are \(b_{Tm}=0.100\), \(b_{P1m}=0.096\), and \(b_{P2m}=0.075\). The correlations with T are \(r(T,P1)=0.900\) and \(r(T,P2)=0.700\). For each comparison, fit a two-predictor regression of the phenotype on T and that proxy.

For the pair T and P1, the conditional estimate for T is

\hat{\beta}_{T\mid P1} =\frac{0.100-(0.900)(0.096)}{1-(0.900)^2} =\frac{0.0136}{0.190} =0.07158.

For the pair T and P2, it is

\hat{\beta}_{T\mid P2} =\frac{0.100-(0.700)(0.075)}{1-(0.700)^2} =\frac{0.0475}{0.510} =0.09314.

Thus, the conditional estimate for T differs across the two fitted models: \(0.07158\) versus \(0.09314\) outcome standard deviations per genotype standard deviation. P1 is more correlated with T and shares more of its association in these inputs. The comparison does not license choosing whichever estimate is preferred; it shows that the conditional result depends on the proxy and model. Standard errors and the full LD structure would be needed for a more complete assessment.

A Practical Sensitivity Workflow

Specify the signal and baseline model.
Name the target variant, phenotype, covariates, genotype representation, analysis sample, and other conditioned variants.

Choose plausible alternatives.
Consider adding or excluding a correlated variant, using a higher-quality measurement, or substituting a justified LD proxy. Avoid arbitrary changes that do not address a real modeling concern.

Keep comparisons aligned.
Use the same individuals, covariates, phenotype scale, and allele coding when possible. If sample sizes differ, report them and do not attribute the entire change to variant inclusion.

Compare estimates, not only significance labels.
Report coefficients with units, standard errors or intervals, sample sizes, and variant-quality information. Note changes in sign, magnitude, precision, and which predictors were conditioned on.

State the scope of the conclusion.
Describe the signal as robust or sensitive to the tested model choices. A difference in estimates is not automatically statistically significant; inference about that difference requires accounting for their dependence.

The comparison of two estimates from the same individuals is generally dependent because both use the same phenotype and correlated genotypes. It is therefore not valid to treat their standard errors as if the estimates came from independent samples when testing whether they differ. If formal inference about a difference is needed, its standard error must account for the covariance between estimates, for example through an appropriate joint-model calculation or resampling procedure.

Common Pitfalls and Interpretation

“The p-value crossed a threshold, so the signal changed.” A threshold crossing can result from a small change in standard error as well as a change in effect estimate. Compare coefficients, uncertainty, and sample sizes rather than reducing the result to significant versus not significant.
“The proxy with the larger estimate is the better variant.” Different proxies are different predictors with different conditional meanings. Larger magnitude is not evidence of better measurement or greater biological importance.
“The poorly imputed variant caused the shift.” Low imputation quality is a reason to investigate, not proof of the mechanism. Check matched-sample results and, where possible, compare with more reliable genotype information.
“Stable estimates prove a causal signal.” Stability across the tested models supports robustness to those particular choices. It does not rule out untested model errors, population structure, or other explanations for association.

Key takeaway: Conditional results belong to specified models. Assess variant-inclusion and proxy sensitivity by making controlled comparisons, reporting estimates and uncertainty in context, and describing stability only across the alternatives actually examined.

Check Your Understanding

Use the model-specific meaning of conditional estimates in answering these questions.

Why can a target variant's coefficient change when a correlated variant is added, even if the analysis sample is unchanged?
In a sensitivity analysis, the sample size falls after adding a variant with missing genotypes. What comparison would help separate sample-composition changes from variant-inclusion effects?
Why does a proxy with a stronger correlation to the target not necessarily produce the same conditional coefficient?
A target estimate changes little across three tested models. What can be concluded, and what cannot be concluded, from that stability?
Why is it generally incorrect to compare two same-sample conditional estimates as independent when testing whether they differ?