Best Linear Unbiased Prediction of Polygenic Effects

Why Predict Polygenic Effects?

A genetic mixed model separates a phenotype into systematic effects, such as an intercept or measured covariates, and latent polygenic effects that vary among individuals. The fixed effects describe the modeled mean; the polygenic effects describe genetic deviations around that mean. Because those deviations are not observed directly, the model predicts them from phenotypes and the covariance implied by genetic relatedness.

A best linear unbiased predictor, or BLUP, is a linear prediction of a random effect that is unbiased under the model and has minimum prediction-error variance among linear unbiased predictors. It is not an estimate of a fixed effect. The distinction matters: a fixed-effect estimate targets a parameter in the mean, while a BLUP targets a realized latent effect for an individual.

Model and Assumptions

Let \(y\) be a vector of \(n\) observed phenotypes, \(X\) a known design matrix for \(p\) fixed effects, and \(\beta\) the unknown fixed-effect vector. Let \(g\) be a vector of \(m\) individual polygenic effects, and \(Z\) the design matrix connecting those effects to observations. The Gaussian linear mixed model is

Assume that \(g\sim N(0,G)\), \(\varepsilon\sim N(0,R)\), and \(g\) and \(\varepsilon\) are independent. In a single-component additive genetic model, \(G=\sigma_g^2K\), where \(K\) is the genetic relationship matrix under the course's chosen scaling, and \(R=\sigma_e^2 I\) for independent residuals with common variance. More generally, \(R\) can be a specified positive-definite residual covariance matrix. For the standard BLUP formulas below, treat \(G\) and \(R\) as known and positive definite, and assume \(X\) has full column rank.

The marginal distribution of the observed data is \(y\sim N(X\beta,V)\), where

Thus, phenotypes are correlated when their polygenic effects are correlated. The off-diagonal entries of \(K\) encode that genetic covariance; they do not mean that one individual's phenotype causes another's.

Known Fixed Effects: Conditional Gaussian Prediction

First suppose \(\beta\) is known. The joint distribution of \(g\) and \(y\) is Gaussian, with \(E(g)=0\), \(E(y)=X\beta\), \(\operatorname{Var}(g)=G\), and \(\operatorname{Cov}(g,y)=GZ'\). The conditional mean of one jointly Gaussian vector given another is its mean plus the cross-covariance times the inverse covariance of the observed vector, applied to the observed deviation. Therefore,

This conditional mean is the best predictor under squared-error loss, and it is linear in the phenotype residuals \(y-X\beta\). Its conditional prediction-error covariance is

For the familiar case in which each individual contributes one phenotype and has one corresponding polygenic effect, \(Z=I\). Then the predictor is \(GV^{-1}(y-X\beta)\). The matrix multiplication combines each individual's residual phenotype with the covariance pattern across all individuals; it is generally not a separate calculation for each person.

Unknown Fixed Effects: The BLUP Formula

In practice, fixed effects are usually unknown. Estimate them by generalized least squares (GLS), using the marginal covariance \(V\):

Define

This matrix removes the contribution of the fitted fixed effects in the \(V^{-1}\)-weighted geometry. In particular, \(PX=0\), and \(Py=V^{-1}(y-X\widehat{\beta}_{\mathrm{GLS}})\). The BLUP of the polygenic effects is

The formula predicts random effects from the part of the phenotype vector remaining after fixed effects are accounted for. It does not turn the individual polygenic effects into fixed regression coefficients. The fixed effects are estimated from the mean structure; the random effects are predicted using both the residual phenotypes and their covariance.

Why This Predictor Is Best Among Linear Unbiased Predictors

A short derivation establishes both unbiasedness and optimality. Consider a linear predictor \(Ay\) of \(g\), where \(A\) is any fixed matrix. Its expectation is \(AX\beta\), so it is unbiased for \(g\), whose expectation is zero, for every \(\beta\) only if \(AX=0\). The proposed predictor has this property because \(PX=0\), so \(GZ'PX=0\).

To check optimality, let \(A_0=GZ'P\) and compare it with any other linear unbiased predictor \(Ay\). Set \(D=A-A_0\). Both predictors are unbiased, so \(DX=0\). The prediction-error vector for \(A_0y\) is \(g-A_0y\). Its covariance with \(y\) is

The expression on the right ends in \(X'\), so its covariance with \(Dy\) is zero: multiplying by \(D'\) gives a factor \(X'D'=0\), which follows from \(DX=0\). Since \(Ay=A_0y+Dy\), the cross terms in the prediction-error covariance vanish. Consequently,

Because \(V\) is positive definite, \(DVD'\) is positive semidefinite. Thus no other linear unbiased predictor has smaller prediction-error covariance than \(A_0y\). This proves the BLUP property under the stated model and known covariance matrices.

For unknown fixed effects, the prediction-error covariance of the BLUP is

This expression accounts for estimating the fixed effects. It differs from the conditional covariance with known \(\beta\), which uses \(V^{-1}\) in place of \(P\).

How Relatedness Changes Shrinkage

With multiple individuals, genetic relatedness determines how much information a phenotype provides about another individual's polygenic effect. In a simple two-individual illustration, suppose the target individual has no observed phenotype, but one relative has a phenotype residual \(r_y\) after subtracting a known mean. Let \(G=\sigma_g^2K\), with \(K_{11}=K_{22}=1\) and \(K_{12}=K_{21}=k\), and let residual variance be \(\sigma_e^2\). The relative's phenotype residual has variance \(\sigma_g^2+\sigma_e^2\), and its covariance with the target's polygenic effect is \(k\sigma_g^2\). The conditional mean is therefore

The coefficient depends on both the genetic relationship and the reliability of the relative's phenotype as a measure of their own polygenic effect. Greater relatedness increases the information about the target's effect; greater residual variance reduces it.

Known Versus Estimated Variance Components

The BLUP formulas above assume \(G\) and \(R\) are known. In an analysis, their variance components are commonly estimated, for example by REML, and then substituted into the prediction formula. This produces an empirical BLUP (EBLUP). It is a plug-in prediction, because the covariance matrices used to calculate it depend on estimated rather than known variance components. Its uncertainty is not fully represented by the known-variance prediction-error formula alone; that formula treats the covariance parameters as fixed at their specified values.

A BLUP can be zero even when an individual's phenotype differs from the fitted mean, for example when the model assigns little genetic variance relative to residual variance. Conversely, information from relatives can yield a nonzero prediction even when the target individual's own phenotype is unavailable. In both cases, the prediction reflects the assumed covariance structure as well as the observed data.

Common Mistake: Calling a BLUP a Fixed-Effect Estimate

• Incorrect reasoning: “The predicted polygenic effect is a regression coefficient, so it estimates the fixed genetic effect for that individual.”
• Correction: A BLUP predicts a realized random effect with model expectation zero. Fixed-effect estimates describe systematic mean differences. The BLUP is a covariance-weighted prediction, usually shrunk toward zero, not a separate fixed coefficient estimated without shrinkage.
• Incorrect reasoning: “A prediction of 0.5 trait units means the individual's true effect is known to be 0.5.”
• Correction: The value is a point prediction conditional on the model and data. Prediction error remains, summarized under known covariance parameters by the prediction-error covariance.