Generalized Least Squares for Fixed Effects

Why Covariance Changes Fixed-Effect Estimation

In a genetic mixed model, individuals are not necessarily independent. Relatives may share polygenic effects, and a genomic relationship matrix can represent the pattern of that sharing. Even when the scientific target is a fixed effect, such as a covariate-adjusted mean or a marker coefficient, the correlation among observations matters: it changes how much information the data contain and how observations should be weighted.

Generalized least squares (GLS) estimates fixed effects using the covariance matrix of the observations. The result is conditional on the covariance matrix used. In a mixed model, that matrix may be built from a residual covariance and one or more genetic covariance components. The focus here is the fixed-effect estimator and its covariance when that observation covariance is specified.

The covariance matrix \(V\) has variances on its diagonal and covariances off the diagonal. For example, a simple genetic mixed-model covariance might be \(V=\tau^2K+\sigma^2I\), where \(K\) is a genomic relationship matrix, \(\tau^2\) is a genetic variance component, and \(\sigma^2\) is a residual variance component. This is an example of the covariance structure, not a claim that its components are known in applications. The GLS calculations below treat \(V\) as specified; estimating variance components is a separate problem.

The GLS Estimator and Its Covariance

Ordinary least squares minimizes the unweighted sum of squared residuals. GLS instead minimizes the residual quadratic form weighted by \(V^{-1}\). Because directions with greater variance or shared covariance contribute differently to the information, this weighting can change both estimates and uncertainty.

The inverse in this expression exists under the stated assumptions. Since \(V\) is positive definite, so is \(V^{-1}\). For any nonzero \(p\)-vector \(a\), full column rank of \(X\) gives \(Xa\ne0\), and therefore \(a^\mathsf{T}X^\mathsf{T}V^{-1}Xa=(Xa)^\mathsf{T}V^{-1}(Xa)>0\). Thus \(X^\mathsf{T}V^{-1}X\) is invertible.

Derivation: Unbiasedness and Minimum Variance

The estimator is linear in \(y\). To see why its form is appropriate, consider any linear estimator \(Ay\), where \(A\) is a \(p\times n\) matrix. Its expectation is \(AX\beta\). It is unbiased for every \(\beta\) precisely when \(AX=I_p\). Among these linear unbiased estimators, GLS has the smallest covariance matrix.

Consequently, \(\operatorname{Cov}(Ay)-\operatorname{Cov}(A_0y)=DVD^\mathsf{T}\) is positive semidefinite: no linear unbiased estimator has smaller variance in any linear combination of the coefficients. Direct substitution gives \(A_0VA_0^\mathsf{T}=(X^\mathsf{T}V^{-1}X)^{-1}\). This establishes the GLS covariance formula and the best linear unbiased property under the specified covariance. If \(e\) is multivariate normal, any linear transformation of \(y\) is normal, so \(\hat{\beta}_{\mathrm{GLS}}\sim N\bigl(\beta,(X^\mathsf{T}V^{-1}X)^{-1}\bigr)\).

Worked Calculations

The examples use hypothetical data. Outcomes and covariates are specified explicitly, and each covariance matrix is treated as known. When a model includes an intercept and a covariate, the intercept is the fitted mean at covariate value zero; that value need not be substantively typical.

Interpretation, Scope, and Common Pitfalls

GLS separates the fixed-effect design from the covariance weighting. The columns of \(X\) specify which fixed effects are being estimated; \(V\) specifies how outcome deviations co-vary across observations. A coefficient is interpreted in the scale and units of its outcome and covariate, conditional on the other columns in \(X\). Its standard error describes sampling variability under the specified \(V\), not uncertainty about whether \(V\) itself is correct.

In genetic applications, relatedness can make a group of individuals less informative than the same number of independent individuals. Covariates can also be correlated with patterns in \(V\), so GLS estimates may differ from ordinary least-squares estimates, not merely have different standard errors. A correctly chosen covariance model is therefore central to calibrated fixed-effect inference. In practice, variance components often need to be estimated; the formulas here do not prescribe how to estimate them.

• Wrong reasoning: “There are \(n\) individuals, so the intercept variance is the residual variance divided by \(n\).” Correction: That shortcut assumes independent observations with a common variance. With general \(V\), the intercept variance is \((1^\mathsf{T}V^{-1}1)^{-1}\).
• Wrong reasoning: “A GLS coefficient is automatically causal because relatedness has been modeled.” Correction: GLS adjusts the weighting and covariance under the stated model. Causal interpretation requires additional design and identification assumptions.
• Wrong reasoning: “A smaller GLS standard error proves the covariance model is better.” Correction: The standard error is conditional on the specified covariance. A misspecified \(V\) can produce misleading uncertainty, even when the formula has been computed correctly.
• Wrong reasoning: “The ordinary mean and GLS mean must differ whenever observations are correlated.” Correction: They can coincide in special designs, as in the two-person equal-variance example, even though their variance calculations differ.