how can you have fractional degrees of freedom?

Question

In certain situations such as Greenhouse-Geisser epsilon or Welch ANOVA, it is possible to have fractional degrees of freedom. Assuming that a degree of freedom represents the number of parameters that are free to vary in the estimation of a statistic, how could this ever be anything but a whole number? You either have a parameter or don't...there's no such thing as a fraction of a parameter, at least not that I can comprehend. My question is twofold: why are fractional degrees of freedom even possible, and how would they be interpreted in real life situations? Would they just be rounded to the nearest whole number in practice? Can a critical value for a fractional DF be obtained through linear interpolation, or does it depend on the distribution used?  Thanks for any clarification! Kevin

Answer 1

I some cases yes. They lie in [0, 1]. Consider, as an example, the k-nearest neighbour smoother, which is the average of the k nearest measured values to the given point. Then, at each of the n measured points, the weight of the original value on the linear combination that makes up the predicted value is just 1/k. Thus, the trace of the hat matrix is n/k. Thus the smooth effort costs n/k effective degrees of freedom. The subject requires some research that is more than just one answer.