Collinearity Diagnostics Within a Locus

Why Diagnose Collinearity Within a Locus?

A joint association model estimates the effects of several variants while holding the other included variants fixed. Within a locus, genotype columns can be strongly correlated because variants are in linkage disequilibrium. If one genotype column is nearly predictable from the others, the data contain little information for separating their conditional effects. The regression can still produce estimates, but small changes in the data or model may lead to substantial changes in those estimates.

Collinearity diagnostics describe this feature of the predictor design. They do not determine whether a variant is causal, establish that an association is real, or diagnose omitted-variant bias. The focus here is whether the included genotype columns provide distinct information for estimating their joint effects.

Model and Diagnostic Quantities

Consider a quantitative outcome \(y\), a matrix \(C\) of an intercept and any non-genetic covariates, and \(m\) genotype columns collected in \(G\). The joint linear regression model is

Here \(\beta_j\) is the conditional regression coefficient for genotype column \(j\), given the other included genotype columns and the covariates. For the usual OLS variance formula, assume the model is correctly specified with errors that have mean zero, are independent, and have common variance \(\sigma^2\). The diagnostics below describe the predictor geometry; their calculation does not by itself verify these error assumptions.

To account for covariates, first remove from each genotype column its linear projection on \(C\). Equivalently, use the residualized genotype matrix from the Frisch–Waugh–Lovell result. Center and scale its nonconstant columns, then form their correlation matrix \(R\). Thus \(R_{jk}\) is the correlation between the covariate-adjusted genotype columns \(j\) and \(k\). For a model without additional covariates, this is simply the correlation matrix of the centered genotype columns.

A VIF of 1 means that genotype column \(j\) is uncorrelated with the other columns in this design. A larger VIF means less information for estimating its conditional coefficient: under the stated linear-model assumptions, its coefficient variance is multiplied by that factor, and its standard error by the square root of that factor, compared with the orthogonal-design reference. VIFs are not themselves standard errors or hypothesis tests.

The eigenvalues of \(R\) give a complementary, joint view. A small eigenvalue means there is a linear combination of genotype columns with little variation. If an eigenvalue is zero, the columns are exactly linearly dependent and the joint coefficients cannot all be uniquely identified. For a nonsingular \(R\), define the condition index associated with eigenvalue \(\lambda_k\) as \(\sqrt{\lambda_{\max}/\lambda_k}\), where \(\lambda_{\max}\) is the largest eigenvalue. Larger indices point to directions of weak information. These indices depend on the chosen scaling, so the correlation matrix is a useful standardization when comparing genotype columns measured on different scales.

Why VIF Measures Variance Inflation

The connection follows directly from the OLS variance formula. Let \(Z\) be the matrix of centered, standardized adjusted genotype columns, scaled so that \(Z^\mathsf{T}Z=(n-1)R\). Conditional on the design, the covariance matrix of the OLS genotype coefficient estimates is \(\sigma^2(Z^\mathsf{T}Z)^{-1}\). Therefore,

For an orthogonal standardized design, \(R\) is the identity matrix and the corresponding coefficient variance is \(\sigma^2/(n-1)\). Dividing the variance for column \(j\) by that reference gives \((R^{-1})_{jj}\), its VIF. The regression interpretation follows from the same geometry: the proportion of column \(j\)'s variation left after predicting it from the other columns is \(1-R_j^2\), so the variance inflation factor is its reciprocal.

Pairwise Correlations and Joint Dependence

A pairwise correlation is easy to inspect, but it can miss dependence involving several columns together. VIFs measure how well each genotype column can be predicted from all the others; eigenvalues reveal weakly varying combinations of the entire set. These diagnostics are complementary: a high pairwise correlation can make a particular VIF large, while a near-dependence involving several columns may be difficult to summarize with any single pairwise correlation.

Recognizing Near-Linear Dependence

When genotype columns are nearly linearly dependent, a weighted combination of them has very little variation. The model then has difficulty distinguishing how much of the outcome association should be assigned to each column. In practice, this can produce large standard errors, sensitive coefficient estimates, and unstable signs or magnitudes across closely related model specifications. A finite VIF or a nonsingular correlation matrix does not guarantee that estimates are precise enough for the scientific question.

How to Use Diagnostics Responsibly

VIF cutoffs are rules of thumb, not universal decision boundaries. A VIF that deserves attention depends on the purpose of the analysis, sample size, desired precision, and how much uncertainty is acceptable. Similarly, condition-index thresholds are conventions rather than tests with a fixed error rate. Report the diagnostic values and the design to which they apply, rather than declaring a model “safe” or “invalid” solely because a number falls on one side of a cutoff.

For a locus-wide assessment, calculate the diagnostics on the genotype columns actually included in the joint model, after accounting for its covariates. If a column is constant after adjustment, it cannot supply an estimable conditional effect. If the correlation matrix is singular or nearly singular, identify the implicated columns or combinations and consider whether the model's scientific target can be supported by the available genotype variation. Changing which variants are included changes the diagnostic design and can change the conditional estimates.