F-test

An F-test is any statistical test in which the test statistic has an F-distribution if the null hypothesis is true. A great variety of hypotheses in applied statistics are tested by F-tests. Among these are given below:

• The hypothesis that the means of multiple normally distributed populations, all having the same standard deviation, are equal. This is perhaps the most well-known of hypotheses tested by means of an F-test, and the simplest problem in the analysis of variance (ANOVA).

• The hypothesis that the standard deviations of two normally distributed populations are equal, and thus that they are of comparable origin.

Note that if it is equality of variances (or standard deviations) that is being tested, the F-test is extremely non-robust to non-normality. That is, even if the data display only modest departures from the normal distribution, the test is unreliable and should not be used.

In many cases, the F-test statistic can be calculated through a straightforward process. Two regression models are required, one of which constrains one or more of the regression coefficients according to the null hypothesis. The test statistic is then based on a modified ratio of the sum of squares of residuals of the two models as follows:

Given n observations, where model U has k unrestricted coefficients, and model R restricts m of the coefficients (typically to zero), the F-test statistic can be calculated as

${\frac {\left({\frac {RSS_{R}-RSS_{U}}{m}}\right)}{\left({\frac {RSS_{U}}{n-k}}\right)}}.$

where $RSS_{i}$ is the residual sum of squares of model $i$.

The resulting test statistic value would then be compared to the corresponding entry on a table of F-test critical values, which is included in most statistical texts.