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For a full-column-rank design matrix in a normal linear model, a fitted mean at covariate vector $v$ is $v^T\hat{\beta }$. The covariance matrix of the least-squares estimator immediately gives the displayed variance. Replacing ${\sigma }^2$ by the independent unbiased residual variance gives its squared standard error and a Student t confidence interval on $n-p$ statistical degrees of freedom. A future response instead contributes an additional error variance ${\sigma }^2$ when its error is independent of the fitted data.

// Target: probability-and-statistics.bigb

Wright's path tracing rule expresses a covariance in a standardized linear structural equation model as a sum of products of edge coefficients over admissible unblocked paths.