Effective Degrees of Freedom: A Flawed Metaphor

To most applied statisticians, a fitting procedure's degrees of freedom is synonymous with its model complexity, or its capacity for overfitting to data. In particular, it is often used to parameterize the bias-variance tradeoff in model selection. We argue that, contrary to folk intuition, model complexity and degrees of freedom are not synonymous and may correspond very poorly. We exhibit and theoretically explore various examples of fitting procedures for which degrees of freedom is not monotonic in the model complexity parameter, and can exceed the total dimension of the response space. Even in very simple settings, the degrees of freedom can exceed the dimension of the ambient space by an arbitrarily large amount. We show the degrees of freedom for any non-convex projection method can be unbounded.