Variance component ($V={\sum }_k{\tau }_k^2{Z}_k{Z}_k^T+R$)

A variance component is a nonnegative parameter multiplying a specified covariance contribution. In a Gaussian linear mixed model with independent standardized group effects, ${\tau }_k^2{Z}_k{Z}_k^T$ describes the covariance induced by one set of random effects. Testing whether a component is zero is a variance-component likelihood-ratio test at a boundary; ordinary regular chi-squared likelihood-ratio calibration need not apply.

A method-of-moments variance component estimate equates residual mean squares in ANOVA to their model expectations and solves for the variance components. An unconstrained estimate can be negative even when the model component must be nonnegative; boundary treatment then requires a stated estimation convention. The method gives point estimates, not certainty about future experiments.