Deriving Haseman–Elston Regression from Pairwise Covariance

From Genetic Covariance to a Pairwise Regression

Haseman–Elston regression connects two levels of a genetic model. At the individual level, a genomic relationship matrix describes how genetic effects covary across people. At the pair level, the product of two people’s centered phenotypes carries information about that covariance. The regression equation follows by taking the expected value of that product under the model.

The key is to distinguish a model-implied expectation from an observed pairwise product. A particular pair’s product can differ substantially from its expectation because phenotypes contain residual variation. The regression uses many pairs to relate those noisy products to genomic similarity.

Assumptions and the Pairwise Expectation

For the basic derivation, assume that the genetic effects have mean zero and covariance matrix \(\sigma_g^2K\), and that residuals have mean zero, variance \(\sigma_e^2\), and are uncorrelated across individuals. Assume also that genetic effects and residuals are uncorrelated. In matrix notation, these assumptions are

Here \(\sigma_g^2\) is the genetic variance component for the SNP set under the chosen GRM scaling, and \(\sigma_e^2\) is the residual variance. The derivation needs these covariance assumptions and zero means; normality is not required. Since \(r_i=g_i+e_i\), expand the cross-product and take its expectation:

The two mixed terms vanish because the genetic and residual effects are uncorrelated and have mean zero. The final equality uses the specified covariance matrices. For a pair of distinct individuals, \(I_{ij}=0\), so the expression reduces to

This is the central link: under the model, the expected cross-product is proportional to genomic relationship, and the proportionality constant is the genetic variance component. For a single pair, \(r_i r_j\) is not itself an estimate of \(\sigma_g^2K_{ij}\); the equality is about its expectation over the random effects.

Why the Equation Is a Regression

For each distinct pair \(i,j\), the model gives an expected response \(E[r_ir_j]\) and a predictor \(K_{ij}\). Across pairs, these expectations lie on a line through the origin with slope \(\sigma_g^2\). The corresponding moment-regression equation is

The approximation sign matters: the left side is observed and noisy, while the right side is its model-implied expectation. In an analysis, the regression slope is obtained by matching observed pairwise products to genomic relationships across the chosen pairs. This tutorial’s equation is a consequence of the covariance model, not a claim that every observed point lies exactly on its line.

An intercept can be included when the pairwise products are expected to have a shared covariance component even at \(K_{ij}=0\). For example, if the residuals have a common off-diagonal covariance \(\tau\), then for distinct individuals

In that model, the expected intercept is \(\tau\), and the expected slope remains \(\sigma_g^2\). Under the basic independent-residual model, \(\tau=0\), so the model-implied intercept is zero. An estimated intercept can nevertheless differ from zero in a finite sample. It should not automatically be interpreted as residual variance: the residual variance appears on the diagonal of the covariance matrix, not in the off-diagonal expectation for distinct individuals.

From the Slope to SNP Heritability

A variance component’s numerical value depends on the scale of the GRM. To connect the slope to SNP heritability, use a GRM normalized so that \(K_{ii}=1\). The marginal variance of a centered phenotype for an individual is then

For this model and SNP set, SNP heritability is the fraction of that marginal variance attributable to the additive genetic component. Thus, once the regression slope identifies \(\sigma_g^2\), the corresponding variance ratio is

The pairwise regression supplies the genetic variance component through the slope; it does not, by itself, turn that slope into a heritability. The residual variance is also needed for the denominator. The unit-diagonal condition matters: if \(K_{ii}\) is not one, the genetic contribution to individual \(i\)’s marginal variance is \(\sigma_g^2K_{ii}\), so the displayed ratio cannot be used without accounting for the GRM scale.

GRM Scaling and the Meaning of the Slope

The relationship predictor’s scale and the variance component’s scale are linked. Suppose a GRM is multiplied by a positive constant \(c\), producing \(K^*=cK\). The same genetic covariance matrix can be written either as \(\sigma_g^2K\) or as \(\sigma_g^{2*}K^*\), provided

Consequently, a slope expressed per unit of \(K^*\) is \(\sigma_g^2/c\), not \(\sigma_g^2\) on the original scale. The covariance model has not changed; its component has been re-expressed using a differently scaled predictor. A heritability calculation must use a variance component and a marginal variance that correspond to the same GRM convention.

Derivation as a Sequence of Steps

When the Derivation Can Mislead

The regression equation depends on the covariance assumptions, not merely on the fact that a GRM has been computed. If residuals from related individuals share environmental influences, their off-diagonal covariance need not be zero. If that covariance is constant across pairs, it contributes an intercept; if it varies with genomic relationship, it can be difficult to distinguish from the genetic contribution using this equation alone. Population structure, misspecified fixed effects, or a mismatched GRM can also make the assumed relationship between phenotype covariance and \(K_{ij}\) inadequate.

There is also an important distinction between pairwise expectations and independent observations. Pairs that share an individual are statistically dependent because they contain that individual’s phenotype. The covariance derivation explains the regression target; it does not establish that all pairwise products are independent or justify treating them as such for uncertainty calculations.