fstatistics

F-statistics are test statistics used to compare model fits in analysis of variance (ANOVA) and in regression analysis. They assess whether the amount of variance explained by a model is significantly greater than would be expected by chance relative to the unexplained variance. Under the null hypothesis, the F-statistic follows an F-distribution with two degrees of freedom determined by the model: df1 for the model and df2 for the error.

In one-way ANOVA, the F-statistic is computed as F = MS_between / MS_within, where MS_between = SS_between/df_between and MS_within = SS_within/df_within. Here df_between = k−1 (k groups) and df_within = n−k (n observations). In regression settings, F = MS_model / MS_error, with MS_model = SSR/df_model and MS_error = SSE/df_error. Typically df_model equals the number of predictors (or parameters estimated), and df_error = n − p − 1 for a model with p predictors (excluding the intercept).

Applications and interpretation: the F-statistic tests the null that all group means are equal in ANOVA, or that the regression model does not explain more variance than a simple baseline model. In simple linear regression, the F-statistic for the overall model is mathematically equivalent to the square of the t-statistic for the slope. A large F value yields a small p-value, indicating significant effects, while a small F value suggests no evidence against the null hypothesis.

Assumptions and notes: valid F-tests rely on independent observations, normally distributed residuals, and homoscedastic (equal) variances. They can be sensitive to outliers and departures from normality, especially with small samples. When multiple F-tests are performed, adjustments may be needed to control error rates.