Fstatistic

The F-statistic is a test statistic used in statistics to compare variances or to test hypotheses about model parameters by assessing how much of the observed variability can be explained by a model relative to unexplained variability. In one-way analysis of variance (ANOVA), the F-statistic is the ratio of the mean square between groups to the mean square within groups, reflecting how much group differences exceed random variation. In linear regression, it is the ratio of the explained variance to the unexplained variance, often written as MSR/MSE, where MSR is the mean square regression and MSE is the mean square error.

Formally, in ANOVA, F = (SSB/dfB) / (SSE/dfE), where SSB is the sum of squares between groups, SSE is the sum of squares due to error, dfB = number of groups minus one, and dfE = total observations minus number of groups. In regression, F = (SSR/dfR) / (SSE/dfE), with SSR being the sum of squares due to regression and dfR the regression degrees of freedom. For multiple regression, F can be expressed in terms of R-squared as F = (R^2/k) / ((1 − R^2)/(n − k − 1)).

Under the null hypothesis (e.g., no group differences or all regression coefficients except the intercept are zero), the F-statistic follows an F-distribution with df1 = dfR and df2 = dfE. A large observed F indicates that the model explains a substantial portion of the variance beyond what would be expected by chance, producing a small p-value.

Assumptions include independent observations, normally distributed errors, and homoscedasticity. Violations can affect validity, and nonparametric or robust alternatives may be used in such cases. The F-statistic is an omnibus test, signaling whether any effect is present but not which specific groups or predictors are responsible.