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Here one can distinguish between regression effective degrees of freedom and residual effective degrees of freedom. The effective degrees of freedom of the fit can be defined in various ways to implement goodness-of-fit tests, cross-validation and other inferential procedures. The distribution is a generalized chi-squared distribution, and the theory associated with this distribution provides an alternative route to the answers provided by an effective degrees of freedom. is not an orthogonal projection), these sums-of-squares no longer have (scaled, non-central) chi-squared distributions, and dimensionally defined degrees-of-freedom are not useful. However, because H does not correspond to an ordinary least-squares fit (i.e. Where is the vector of fitted values at each of the original covariate values from the fitted model, y is the original vector of responses, and H is the hat matrix or, more generally, smoother matrix.įor statistical inference, sums-of-squares can still be formed: the model sum-of-squares is the residual sum-of-squares is. However, these procedures are still linear in the observations, and the fitted values of the regression can be expressed in the form

Many regression methods, including ridge regression, linear smoothers and smoothing splines are not based on ordinary least squares projections, but rather on regularized (generalized and/or penalized) least-squares, and so degrees of freedom defined in terms of dimensionality is generally not useful for these procedures.