F-statistic calculator

The F-statistic operates on the principle of comparing two variances, yielding a value that follows an F-distribution under the null hypothesis. Depending on your specific application, the calculator uses one of two primary formulations:

1. Basic Variance Ratio (Two-Sample F-Test): Used for comparing the variances of two independent normally distributed populations. The formula is F_basic = Var1 / Var2, where Var1 and Var2 represent the sample variances.

2. Multiple Regression (Nested Model Comparison): Used to test whether a set of additional predictor variables improves a regression model significantly. The formula is F_multiple = ((SSR_rest - SSR_full) / J) / (SSR_full / (N - K)), where SSR_rest is the sum of squared residuals of the restricted model, SSR_full is the sum of squared residuals of the full model, J is the number of restrictions, N is the sample size, and K is the total number of coefficients in the full model.

Let us walk through a regression example to see how the multiple model F-statistic is computed. Imagine you are testing whether adding two new variables improves your model. You have a sample size (N) of 50, and the full model contains 5 coefficients (K = 5). You impose 2 restrictions (J = 2).

Step 1: Determine the Sum of Squared Residuals. Suppose your restricted model yields an SSR (SSR_rest) of 450, while your full model yields a lower SSR (SSR_full) of 300 due to better fit.

Step 2: Calculate the difference in error: 450 - 300 = 150.

Step 3: Divide by the number of restrictions (J = 2): 150 / 2 = 75.

Step 4: Calculate the error variance of the full model: SSR_full / (N - K) = 300 / (50 - 5) = 300 / 45 = 6.67.

Step 5: Compute the final F-statistic: 75 / 6.67 = 11.25. A higher F-statistic relative to your critical value suggests that the added variables provide significant explanatory power.

Check Your Degrees of Freedom: Always ensure your sample size (N) is substantially larger than your total coefficients (K). A small degrees of freedom denominator can artificially inflate your F-value.

Understand Directionality: The F-test is typically a one-tailed test. You are looking to see if the calculated value exceeds the critical threshold determined by your chosen significance level (alpha, e.g., 0.05) and degrees of freedom.

Verify Assumptions: Ensure your data meets underlying assumptions such as homoscedasticity and normality of residuals, particularly when working with variance ratios and regression models.