dfresid

dfresid, short for residual degrees of freedom, is a quantity used in regression analysis to quantify the amount of information remaining to estimate the model’s error variance after fitting the model. It reflects how many observations are left to inform the error term once the model parameters have been estimated.

Formally, df_residual = n − rank(X), where n is the number of observations and rank(X) is the rank of the design matrix X. In the common case where the model has p estimated parameters with full column rank (including an intercept), df_residual = n − p. If the design matrix is not of full rank due to linear dependency among predictors, rank(X) is smaller and df_residual > n − p, reflecting more remaining degrees of freedom than the naive count.

Usage and interpretation: df_residual serves as the denominator in the estimate of the error variance, with s^2 = RSS / df_residual, where RSS is the residual sum of squares. It also underpins the mean square error in ANOVA and influences the standard errors of coefficient estimates. In software such as R, df_residual is reported as part of model summaries for linear and generalized linear models and corresponds to the residual degrees of freedom used to gauge the precision of variance estimates.

Relation to model fit: A larger df_residual generally indicates more reliable variance estimation and greater statistical power for testing coefficients, while a small df_residual can inflate the variability of variance estimates and reduce test reliability, particularly in small samples or highly parameterized models.