dfresidual

dfresidual, often written df.residual in statistical software, refers to the residual degrees of freedom of a fitted model, most commonly a linear model. It measures the amount of information left for estimating the random error after the model has been fitted.

In a dataset with n observations and a design matrix X of rank p, the residual degrees of freedom is n minus p. When an intercept is included and the model is of full rank, p equals the number of estimated parameters. The same idea extends to generalized linear models: df_residual equals n minus the rank of X. If some observations are omitted due to missing values, df_residual is computed from the actual n and p after fitting.

Residual degrees of freedom are used to estimate the error variance: sigma_hat^2 equals the residual sum of squares (RSS) divided by df_residual. They also determine the reference distributions for statistical tests of model terms and for constructing confidence intervals, reflecting how much information remains for error estimation after accounting for model complexity.

Software note: in R, df.residual(object) returns the residual degrees of freedom for a fitted model object such as an lm or glm. It is typically equal to length(residuals(object)) minus the rank of the coefficients, and may be stored as object$df.residual. Analysts examine df_residual to assess the precision of variance estimates and to guide inference and diagnostic procedures.

See also: degrees of freedom, residuals, RSS, model diagnostics.