**Critical F**: The value of the

The *F*-statistic is
the test statistic used in ANOVA and GLM, named in honour of R. A. Fisher, who
first described the distribution and developed the method of analysis of
variance in the 1920s. The continuous *F*
distribution for a given set of test and error degrees of freedom is used to determine
the probability of obtaining at least as large a value of the observed *F*-ratio of explained to unexplained
variation, given a true null hypothesis. The associated *P*-value reports the significance of the test effect on the response
in terms of the probability of mistakenly rejecting a true null hypothesis,
which is deemed acceptably small if *P*
< *α*, where *α* often takes a value of 0.05.

The *P*-value is the
area under the *F* distribution to the right of the corresponding *F* value:

where *p* and *q* are the model test and error degrees of freedom respectively, and the beta
function If
the integral limit *F _{Q}*
exceeds the critical

Fig. 1. In
the absence of a treatment effect, the observed *F* = MS[T]/MS[ε]
follows the *F* distribution, here showing with *α* = 0.05 given by the red-shaded area under its right-hand
tail above *F*_{[α]}. (a) *F*_{[0.05],2,12} = 3.89; (b) *F*_{[0.05],16,100} = 1.75.

The program Ftable.exe provides
critical values of the *F* distribution
for a chosen *α*, and also the
Type-I error probability associated with an observed value of *F*, given test and error degrees of
freedom. For example, it will show critial *F*_{[0.05],2,12}
= 3.89; and at an observed *F*_{2,12}
= 3.98, *P* = 0.047.

Doncaster, C. P. & Davey, A. J. H. (2007) *Analysis of Variance and Covariance: How to
Choose and Construct Models for the Life Sciences*. Cambridge: Cambridge
University Press.

http://www.southampton.ac.uk/~cpd/anovas/datasets/