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In an ordinary least squares regression with an intercept and a nonconstant response, $R^2=1-\mathrm{RSS}/{\sum }_i(Y_i-\overline{Y}{)}^2$. It measures the fitted proportion of sample variation relative to an intercept-only baseline. It increases when regressors are added, even if those regressors are unhelpful for prediction. A high value does not establish the assumptions of a normal linear model or rule out a systematic regression residual pattern.

For a full-rank ordinary least squares design matrix, the hat matrix $H$ and residual projection $G=I-H$ satisfy $HG=GH=0$. Hence fitted values and regression residuals are perpendicular as observed vectors. Under isotropic errors their cross-covariance matrix is zero, and under a normal linear model the two vectors are independent.

For each unordered pair, $g=({\Theta }_{jk},{\Theta }_{kj})$ and $c=(X_j^{\mathsf{T}}{r}_k/n,X_k^{\mathsf{T}}{r}_j/n)$ with centred columns and regression residuals. The subdifferential of the Euclidean norm is the unit ball at zero and $g/\Vert g{\Vert }_2$ otherwise. These KKT conditions are necessary and sufficient for the convex paired objective and explain joint edge deletion.

In an additive mean model, the partial residual for component $j$ subtracts every fitted term except that component. It equals the full regression residual plus the fitted component. Plotting it against that component's predictor shows the conditional pattern after accounting for other fitted effects, and supplies the response for a backfitting update.

The regression residual vector is $e=Y-X\hat{\beta }=(I-H)Y$, where $H=X(X^{T}X{)}^{-1}{X}^T$ is the hat matrix. Both $H$ and $I-H$ are symmetric idempotent matrices, and $(I-H)X=0$. Therefore $e=(I-H)\epsilon$ and the linear image of a multivariate normal vector givesThis multivariate normal distribution is singular, with rank $n-p$: fitted-residual orthogonality forces $X^{T}e=0$. Specifically, $\mathrm{Var}(e_i)={\sigma }^2(1-h_{ii})$ and $\mathrm{Cov}(e_i,e_j)=-{\sigma }^2{h}_{ij}$ for distinct observations. Thus regression residuals are generally neither independent nor identically distributed, even though the model errors are. Also $\mathrm{RSS}/{\sigma }^2\sim {\chi }_{n-p}^2$ and is independent of $\hat{\beta }$, which justifies the standard errors and exact Student t-tests below.