residualvarianser

Residualvarianser, or residual variances, denote the variance of the residuals—the differences between observed values and those predicted by a statistical model. They quantify the portion of variation in the dependent variable that remains unexplained after fitting the model. In regression analysis, the total variance of the dependent variable can be partitioned into explained variance (due to the regression) and residual variance (the error variance). The residual variance is related to the mean squared error and to the standard errors of the estimated parameters, and it influences the width of confidence and prediction intervals. It is also connected to the coefficient of determination, R^2, which measures the proportion of total variance explained by the model.

Estimation of residual variance typically uses the sum of squared residuals (SSE). If n observations and k parameters are estimated (including the intercept), the unbiased estimator of the error variance is s^2 = SSE/(n - k). In maximum likelihood settings, the residual variance corresponds to the estimated variance of the error term, with the appropriate degrees of freedom determined by the model.

Interpretation and cautions: A smaller residual variance indicates a better fit, but the value depends on the scale of the data, so direct comparisons across different datasets or units are not meaningful without standardization. Residual variance rests on model assumptions; heteroscedasticity, non-normality, or model misspecification can distort inference and the reliability of the estimator.

Applications: Residualvarianser are used in model diagnostics, to assess goodness of fit, and to construct prediction intervals and standard errors. They also play a role in weighted estimation and in comparing alternative models.