F-distribution & F-Test

F-distribution (F-ratio)

X = \frac{S_{1} / d_{1}}{S_{2} / d_{2}}

where $S_{1}$ and $S_{2}$ are independent random variables with Chi-square Test#Chi-square distribution with respective degrees of freedom $d_{1}$ and $d_{2}$ .

F-test

Any test that uses the F-distribution can be called a "F-test".
Generally, F-test can compare two variances.

Overall F-test for Regression Analysis

An overall F-test assesses how well the set of independent variables, as a group, explains the variation in the dependent variable. That is, the F-statistic is used to test whether at least one of the independent variables explains a significant portion of the variation of the dependent variable.
F = MSM / MSE = (explained variance) / (unexplained variance)
where MSM stands for Mean Squares for Model, and MSE stands for Mean Squares for Error.

F = \frac{\frac{\sum ({\hat{y}}_{i} - \bar{y})^{2}}{k}}{\frac{\sum ({\hat{y}}_{i} - y_{i})^{2}}{n - k - 1}}

where k is the number of independent variables , and n is the number of observations.

source as regression, degrees of freedom = k

source as error, degrees of freedom = n-k-1

total, degrees of freedom = n−1

F-test for ANOVA

F = MSB / MSE
where MSB stands for Sum of Squares between the groups, and MSE stands for The Error Mean Sum of Squares

F = \frac{\frac{S S_{b e t w e e n}}{m - 1}}{\frac{S S_{w i t h i n}}{n - m}}

where m is the number of groups, and n is the number of observations (data points).

for n total data points, degrees of freedom = n−1

for m compared groups, degrees of freedom = m−1

for n total data points collected and m groups being compared, degrees of freedom = n−m

F-test for compare two models

The F-test can be used to compare two competing regression models in their ability to “explain” the variance in the dependent variable.

F = \frac{\frac{R S S_{1} - R S S_{2}}{k_{2} - k_{1}}}{\frac{R S S_{2}}{n - k_{2}}}

where RSS stands for residual sum of squares of fitted model 1 or 2, and k stands for degree of freedoms.