LD-Induced Sign Reversals

Why a Coefficient Can Change Sign

In a locus-wide association analysis, correlated variants can carry overlapping information about an outcome. A marginal regression describes the association of one variant with the outcome on its own; a joint regression estimates each variant’s association while holding the other included variants fixed. These are different statistical quantities. When variants are in linkage disequilibrium (LD), a coefficient can therefore have a different magnitude, or even the opposite sign, in the joint model than in the marginal model.

A sign change is not automatically an error, nor does it by itself establish that one variant has a biological effect in the opposite direction. It is a feature of the fitted associations given the variants, coding, sample, and model. The key is to understand the algebra behind the change and then check whether the data and allele coding are consistent.

The Two-Variant Regression Relationship

Consider two centered, standardized genotype variables \(X_1\) and \(X_2\), and a centered, standardized outcome \(Y\). Standardizing expresses each variable in standard-deviation units. Let \(r_{12}\) be the correlation between the two genotypes, and let \(r_{y1}\) and \(r_{y2}\) be their respective correlations with the outcome. The marginal regression coefficient for \(X_1\) is \(r_{y1}\); for \(X_2\), it is \(r_{y2}\). In the joint linear regression of \(Y\) on both predictors, the coefficients are:

These equations apply to sample correlations calculated on the same observations, with an intercept in the regressions. They require that neither predictor is an exact linear copy of the other: \(|r_{12}|<1\). If \(|r_{12}|=1\), the two predictor columns are perfectly redundant, and their separate joint coefficients cannot be identified. With standardization, each joint coefficient is measured in outcome standard deviations per predictor standard deviation. The sign comparison is unchanged if a predictor is instead expressed in raw dosage units using a positive rescaling.

The numerator for \(X_1\) subtracts the part of its outcome association that is aligned with \(X_2\). A reversal occurs when this subtraction crosses zero. In particular, if \(r_{y1}>0\) and \(r_{12}r_{y2}>r_{y1}\), then \(\hat{\beta}_1<0\), despite a positive marginal association. This is possible because the joint coefficient asks about the association with \(X_1\) after accounting for the component it shares with \(X_2\), not the association of \(X_1\) alone.

Deriving the Coefficients

The derivation follows from the normal equations for the least-squares fit. For standardized predictors and outcome, the predictor correlation matrix is \(\begin{pmatrix}1&r_{12}\\r_{12}&1\end{pmatrix}\), and the correlations between the predictors and outcome form the vector \(\begin{pmatrix}r_{y1}\\r_{y2}\end{pmatrix}\). The fitted coefficients solve this linear system:

The determinant of the predictor correlation matrix is \(1-r_{12}^2\), which is positive when \(|r_{12}|<1\). Its inverse is \(\frac{1}{1-r_{12}^2}\begin{pmatrix}1&-r_{12}\\-r_{12}&1\end{pmatrix}\). Multiplying the inverse by the right-hand side gives the two formulas above. Thus, each joint coefficient depends both on that variant’s marginal association and on the other variant’s association with the outcome, weighted by the LD correlation.

Suppression: A Conditional Association Hidden Marginally

The same relationship explains suppression. Suppose a genotype has no marginal correlation with the outcome, but is correlated with a second genotype that does have an outcome association. The first genotype can nevertheless have a nonzero joint coefficient. In the formula, if \(r_{y1}=0\), then \(\hat{\beta}_1=-r_{12}r_{y2}/(1-r_{12}^2)\). Unless \(r_{12}=0\) or \(r_{y2}=0\), that coefficient is nonzero. In words, the marginal association can conceal a conditional association when the correlated predictor’s shared component offsets it.

In association analyses, “conditional” means conditional on the other predictors included in the regression model. A joint coefficient is not automatically a direct causal effect. Causal interpretation requires additional assumptions about the study design, confounding, and the relation between measured variants and causal variation; a sign reversal alone supplies none of those assumptions.

Distinguishing a Statistical Reversal from a Coding Error

Allele orientation matters because reversing the dosage scale reverses the coefficient’s sign. For a biallelic variant coded as the count of one allele, changing to the count of the other allele transforms the dosage \(G\) to \(2-G\). After centering and standardizing, this is a sign change of the predictor. The variant’s marginal coefficient and its joint coefficient both change sign, while fitted values and the model’s substantive predictions do not. This differs from an LD-induced reversal, in which the same consistently coded variant has one sign marginally and the opposite sign jointly.

When a sign looks unexpected, check the dosage definition and effect allele for each variant, the direction in which the phenotype is coded, and whether the marginal and joint analyses used the same individuals and compatible data. If the outcome is recoded as its negative, all regression coefficient signs reverse. If a genotype is recoded to the opposite allele, that genotype’s coefficient and correlations with other variables reverse. By contrast, a genuine change between marginal and joint estimates can occur with consistent coding when the variants are correlated.

Common Mistake and Practical Interpretation

• Wrong reasoning: “The coefficient changed from positive to negative, so the allele must have been miscoded.” Correction: LD can produce a change between marginal and joint coefficients even when coding is consistent. Check whether the change follows from the observed correlations, and separately audit allele and phenotype coding.
• Wrong reasoning: “A negative conditional coefficient proves that the variant has a harmful causal effect.” Correction: The coefficient describes a conditional linear association in the specified regression. Causal claims need assumptions beyond the coefficient sign.
• Wrong reasoning: “A null marginal coefficient means the variant cannot contribute information in a joint model.” Correction: As the suppression example shows, shared variation can obscure a conditional association in a marginal analysis.
• Wrong reasoning: “The sign of the marginal and joint coefficients must agree because they use the same individuals.” Correction: The marginal model does not hold the other correlated variant fixed. The joint model does, so the coefficient can change direction.