Haseman–Elston Regression as a Moment Estimator

Why Regress Phenotype Cross-Products?

Haseman–Elston (HE) regression estimates a genetic variance component from pairs of individuals. Its basic idea is that individuals with greater genomic similarity, as measured by a genomic relationship matrix, are expected to have more similar phenotypes when the modeled genetic component contributes to phenotypic variation. The method turns that idea into a regression of pairwise phenotype cross-products on pairwise genomic relationships.

HE regression is a moment estimator: it uses relationships between expected products and model parameters, rather than maximizing a likelihood. Its slope is initially a coefficient on the chosen scale of the relationship matrix. It is not automatically a heritability. To interpret it as SNP heritability, one must account for the matrix’s scale and the residual variance.

Model and Pairwise Moment

Let \(y\) be the \(n\)-vector of phenotypes, \(X\) the fixed-effect design matrix, and \(\beta\) its coefficient vector. Use the linear mixed model

where \(K\) is a specified, symmetric genomic relationship matrix. Assume

Here \(\sigma_g^2\) and \(\sigma_e^2\) are nonnegative variance-component coefficients, and \(I\) is the identity matrix. The moment argument requires these means and covariances; normality of \(g\) and \(e\) is not necessary for the pairwise expectation itself. A Gaussian mixed model is often used for likelihood-based analysis, but the HE slope is motivated by the covariance structure.

For exposition, suppose the fixed effects are known and define the fixed-effect-adjusted phenotype \(r=y-X\beta=g+e\). For two distinct individuals \(i\) and \(j\), the model gives

The independent residual term contributes no covariance between distinct individuals. In practice, fixed effects are estimated or otherwise adjusted for; the adjustment must be compatible with the model, since simply projecting out covariates can itself induce dependence among adjusted residuals.

With \(m=n(n-1)/2\) distinct pairs, a common form is an ordinary least-squares regression with an intercept:

where \(q_{ij}\) is an appropriately adjusted phenotype cross-product. In the ideal model with correctly adjusted phenotypes and independent residuals, the expected pairwise cross-product has no intercept, and \(b\) targets \(\sigma_g^2\). An intercept may be included as a nuisance term to accommodate a common offset or to avoid forcing the fitted line through zero. The intercept does not make an incorrectly specified covariance model correct.

With an intercept, the ordinary least-squares slope is

where \(\bar K\) and \(\bar q\) are the means over the pairs included. Without an intercept, the slope is \(\sum_{i<j}K_{ij}q_{ij}/\sum_{i<j}K_{ij}^2\). The two procedures are not generally identical; the intercept choice and pair construction should be specified.

From the Slope to SNP Heritability

The scale of \(K\) is central to interpretation. If \(K\) is multiplied by a positive constant, the same covariance model can be represented by dividing its coefficient by that constant. Thus the regression slope alone does not identify a genetic variance on a scale-independent basis.

Let \(\bar K_{\mathrm{diag}}=n^{-1}\sum_i K_{ii}\), the average diagonal element of the relationship matrix. Conditional on the fixed effects, the average modeled genetic variance across individuals is \(\sigma_g^2\bar K_{\mathrm{diag}}\), while the average modeled residual variance is \(\sigma_e^2\). Under this model, the corresponding average-variance ratio is

If \(K\) is normalized so that its average diagonal is 1, this reduces to \(\sigma_g^2/(\sigma_g^2+\sigma_e^2)\). Unit diagonal is one way to achieve that normalization. The ratio is conditional on, or after adjustment for, the fixed effects in the model; it is not a claim that the genetic component explains the same fraction of every individual’s phenotype.

If the HE slope estimates the coefficient \(\sigma_g^2\), the conversion requires both that coefficient and a residual-variance estimate. If the matrix is not normalized to average diagonal 1, use its actual diagonal scale in the modeled genetic variance. The phenotypic variance used in the denominator must refer to the same adjusted model and scale.

A Short Justification of the Moment Relation

The pairwise expectation follows directly from the covariance assumptions. For \(i\ne j\), write \(r_i=g_i+e_i\) and \(r_j=g_j+e_j\), then expand:

The cross-component terms are zero because \(g\) and \(e\) are uncorrelated and have conditional mean zero. The genetic term is the \((i,j)\) entry of \(\sigma_g^2K\). The residual term is the \((i,j)\) entry of \(\sigma_e^2I\), which equals zero for distinct individuals. Hence \(E(r_ir_j\mid X,K)=\sigma_g^2K_{ij}\). This is the moment relationship HE regression uses; it does not establish that all the pairwise products are independent.

Assumptions and Practical Interpretation

The pairwise regression has a clear interpretation only when its covariance model is credible. Its main assumptions are:

• Correct fixed-effect adjustment. The mean structure, such as an intercept and relevant covariates, is adequately represented. Unadjusted differences in group means can create phenotype cross-products associated with \(K_{ij}\).
• Appropriate genetic covariance. The modeled genetic effects have covariance \(\sigma_g^2K\), where \(K\) is constructed from the SNP set relevant to the estimand. A matrix with the wrong marker set or an inappropriate relationship structure may not represent the intended genetic component.
• Residual covariance as specified. The simple model assumes residuals are uncorrelated across individuals, with common variance \(\sigma_e^2\). Shared environments, related non-genetic effects, or other unmodeled covariance can make phenotype products vary with genomic relationship for reasons not represented by the SNP component.
• Explicit matrix normalization. The diagonal scale of \(K\) must be stated when translating the slope into a genetic variance and heritability. The matrix should also be a valid symmetric relationship matrix for the covariance model.

A further practical point is that pairwise regression rows are not independent observations: many pairs share an individual, and all pairs are formed from the same phenotype vector. Ordinary least-squares arithmetic can define a slope, but treating the \(n(n-1)/2\) pairs as independent can give misleading standard errors and tests. Appropriate uncertainty calculations must account for the dependence induced by shared individuals and by the model.

Common Mistakes

• Calling the slope heritability. The slope estimates a coefficient multiplying \(K\), not a dimensionless variance proportion. Convert it using the residual variance and the matrix’s diagonal scale.
• Ignoring normalization. If \(K\) is rescaled, its coefficient changes inversely. The modeled genetic variance, not the coefficient in isolation, belongs in the heritability ratio.
• Assuming an intercept fixes misspecification. An intercept can represent a constant pairwise offset in the fitted regression. It does not correct covariate confounding or an unmodeled covariance pattern aligned with \(K\).
• Counting all pairs as independent. Pairs sharing an individual are dependent. The large number of pair rows does not equal the number of independent phenotype observations.