### Fractional Degrees of Freedom

Sometimes - either in the output of a statistical package, or in the results section of a published paper - you will find the quoted **degrees of freedom** of a *t*- or *F*-statistic are not whole numbers, *i.e.* they are **fractional**.

The reason for the occurrence of these fractional degrees of freedom is usually because two or more sums-of-squares have been combined in some way to create appropriate approximate numerator and denominator terms for testing the effect that is being investigated.

Such statistics are sometimes called **quasi t- or quasi F-statistics**. Their null-hypothesis distribution does not have exactly a t or F, but

**F E SATTERTHWAITE**in 1946 showed how to calculate and approximation to the null-hypothesis sampling distribution of these quasi

*t/F*statistics. The approximation is a member of the extended family of

*t/F*distributions which interpolates with fractional parameters the more familiar family with whole number degrees of freedom.

An important place where these fractional degrees of freedom occur is in **repeated measures ANOVA** where, in order to overcome problems with **heterogeneity of covariances**, a correction factor (between 0 and 1) called **epsilon** is calculated and used to deflate both the numerator and denominator degrees of freedom in F statistics involving the repeated measures factor. Different versions of *epsilon* are associated with the names of BOX, of GREENHOUSE & GEISSER, and of HUYHN & FELDT.

The other frequently-encountered situation where Satterthwaite's approximation is used is in **two-sample t-tests with unequal variances**.

To be included later: Graphics illustrating the relation of distributions with fractional degrees of freedom to ones with interger degrees of freedom.

**Satterthwaite** (vb.) To spray the person you are talking to with half-chewed breadcrumbs.