Can degrees of freedom be a non-integer number?

When I use GAM, it gives me residual DF is $26.6$ (last line in the code). What does that mean? Going beyond GAM example, In general, can the number of degrees of freedom be a non-integer number?

> library(gam)
> summary(gam(mpg~lo(wt),data=mtcars))

Call: gam(formula = mpg ~ lo(wt), data = mtcars)
Deviance Residuals:
Min      1Q  Median      3Q     Max
-4.1470 -1.6217 -0.8971  1.2445  6.0516

(Dispersion Parameter for gaussian family taken to be 6.6717)

Null Deviance: 1126.047 on 31 degrees of freedom
Residual Deviance: 177.4662 on 26.6 degrees of freedom
AIC: 158.4294

Number of Local Scoring Iterations: 2

Anova for Parametric Effects
Df Sum Sq Mean Sq F value    Pr(>F)
lo(wt)     1.0 847.73  847.73  127.06 1.239e-11 ***
Residuals 26.6 177.47    6.67

• In general, yes, df can be a floating point number. – David Lane May 21 '17 at 17:14
• You probably mean to ask about real number (or a number that's non-integer); a floating point number is a computer concept (a way of approximatiing real numbers) that relates to implementation but you're really asking about the underlying mathematical idea (and so better to ask a mathematical question). One often encounters situations where (for one reason or another, not always good) a quantity that is conceptually an integer is nevertheless in implementation stored as a floating point number. I suggest "Can a model have non-integer degrees of freedom?" for the title. – Glen_b -Reinstate Monica May 22 '17 at 2:34

Degrees of freedom are non-integer in a number of contexts. Indeed in a few circumstances you can establish that the degrees of freedom to fit the data for some particular models must be between some value $$k$$ and $$k+1$$.

We usually think of degrees of freedom as the number of free parameters, but there are situations where the parameters are not completely free and they can then be difficult to count. This can happen when smoothing / regularizing, for example.

The cases of locally weighted regression / kernel methods an smoothing splines are examples of such a situation -- a total number of free parameters is not something you can readily count by adding up predictors, so a more general idea of degrees of freedom is needed.

In Generalized Additive Models on which gam is partly based, Hastie and Tibshirani (1990) [1] (and indeed in numerous other references) for some models where we can write $$\hat y = Ay$$, the degrees of freedom is sometimes taken to be $$\operatorname{tr}(A)$$ (they also discuss $$\operatorname{tr}(AA^T)$$ or $$\operatorname{tr}(2A-AA^T)$$). The first is consistent with the more usual approach where both work (e.g. in regression, where in normal situations $$\operatorname{tr}(A)$$ will be the column dimension of $$X$$), but when $$A$$ is symmetric and idempotent, all three of those formulas are the same.

[I don't have this reference handy to check enough of the details; an alternative by the same authors (plus Friedman) that's easy to get hold of is Elements of Statistical Learning [2]; see for example equation 5.16, which defines the effective degrees of freedom of a smoothing spline as $$\operatorname{tr}(A)$$ (in my notation)]

More generally still, Ye (1998) [3] defined generalized degrees of freedom as $$\sum_i \frac{\partial \hat y_i}{\partial y_i}$$, which is the sum of the sensitivities of fitted values to their corresponding observations. In turn, this is consistent with $$\operatorname{tr}(A)$$ where that definition works. To use Ye's definition you need only be able to compute $$\hat y$$ and to perturb the data by some small amount (in order to compute $$\frac{\partial \hat y_i}{\partial y_i}$$ numerically). This makes it very broadly applicable.

For models like those fitted by gam, those various measures are generally not integer.

(I highly recommend reading these references' discussion on this issue, though the story can get rather more complicated in some situations. See, for example [4])

[1] Hastie, T. and Tibshirani, R. (1990),
London: Chapman and Hall.

[2] Hastie, T., Tibshirani, R. and Friedman, J. (2009),
The Elements of Statistical Learning: Data Mining, Inference, and Prediction, 2ndEd
Springer-Verlag.
https://statweb.stanford.edu/~tibs/ElemStatLearn/

[3] Ye, J. (1998),
"On Measuring and Correcting the Effects of Data Mining and Model Selection"
Journal of the American Statistical Association, Vol. 93, No. 441, pp 120-131

[4] Janson, L., Fithian, W., and Hastie, T. (2013),
"Effective Degrees of Freedom: A Flawed Metaphor"
https://arxiv.org/abs/1312.7851

• It is not relevant to this case but the Welch two sample t test when the variances are unequal can have a non-integer number of degrees of freedom. – Michael R. Chernick May 21 '17 at 18:01
• As can the epsilon-corrected df in repeated measures ANOVA. – David Lane May 21 '17 at 19:49
• Another reference is statweb.stanford.edu/~tibs/ElemStatLearn/printings/… section 5.4.1 Degrees of Freedom and Smoother Matrices – Adrian May 22 '17 at 14:57
• @Adrian thanks; I had been tossing up whether to add just that reference (and in particular whether to mention eqn 5.16 in the section you point to). I've concluded that it's a good idea to add it in. – Glen_b -Reinstate Monica May 23 '17 at 6:33