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### 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.

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