Calculation Procedure for the Randomization F Test for Regression Coefficients

Given: n paired X,Y data values. These are the observed (or real) data.

Step 1: Generate a random number data set consisting of n paired X’,Y’ values that are ‘modeled’ after the observed sample data (X,Y). The modeling is such that the X’,Y’ data have the same distributional properties as X,Y.

Step 2: Fit the 1^st, 2^nd and 3^rd degree polynomials to the X’,Y’ data exactly as done in the Parametric F Test. This includes finding the ‘Sum of Squares due to Regression’ for the linear, quadratic and cubic terms (SSR_L, SSR_Q and SSR_C), the ‘Mean Sum of Squares due to Deviations’ for the 3^rd degree polynomial (MSD₃), and the F statistics for each term (F_L, F_Q and F_C).

Note that F_L/s, F_Q/s and F_C/s are spurious F values because they are based on a random number data set (X’,Y’).

Step 3: Repeat Steps 1 and 2 a large number of times (q) using different X’,Y’ data sets (q should be at least 100).

Step 4: Plot the ‘probability distributions’ of the q spurious F_L/s, F_Q/s and F_C/s values (one plot for each F statistic).

Step 5: Calculate the F statistics for the linear, quadratic and cubic terms for the observed X,Y data as is normally done in the Parametric F Test (F_L/Obs, F_Q/Obs and F_C/Obs).

Step 6: For each F statistic, find the percent of spurious F values (F_k/s) that are larger than the single observed F value (F_k/Obs). This is the statistical significance of the k^th regression term.