F = msRegression ÷ msError
You need four things to calculate an ANOR (analalysis of regression): the XY correlation coefficient (Pearson r), the number of columns (k), the number of people in the study (N), and the SS of Y (assuming X is predicting Y). From these, you can calculate an F value which you compare to a value you look up in a table.
Here is a set of data to play with:
X Y
11 1
4 2
8 8
2 12
7 11
16 2
When we are done, we have filled in a summary table like this one:
SS df ms
Regression _____ ____ ____
Error _____ ____ ____
Total _____ ____ ____
There are six people in the study; 2 measures on the same people. The total degrees of freedom are N-1, so 6 minus 1 equals 5.
The df for regression is k-1 (columns minus 1. Two minus one equals 1.
The df for error is total minus regression, which is 4.
Here is where we are so far:
SS df ms
Regression ____ 1 ____
Error _____ 4 ____
Total ____ 5 ____
The SStotal equals the SS of Y. In this case it is 122.
The SSregression is calculated bu multiplying SSregression by the coefficient of determination (the square of Pearson’s r). The correlation between X and Y in this case is .56. Consequently, r-squared is .34, and the SSregression is 41.14.
The Sserror is the correlation coefficient multiplied by the coefficient of non-determinism (1 – r-squared). You check your work by subtracting the SSregression from the SStotal.
Here is where we are:
SS df ms
Regression 41.14 1 ____
Error 80.86 4 ____
Total 122.00 5 ____
To complete the summary table, divide each SS by its appropriate df.
SS df ms
Regression 41.14 1 41.14
Error 80.86 4 20.21
Total 122.00 5 25.20
The F statistic is the mean squares of Regression divided by the mean squares of Error. So F = 41.14 / 20.21. When divided through, you get: F = 2.04.
We test the significance of this F by comparing it to the critical value in the F Table. We enter the table by going across to the dfregression (1) and down the dferror (in this case it’s 4). So the critical value = 7.71. In order to be significant, the F we calculated would have to be larger than 7.71. Since it isn’t, the pattern we see is likely to be due to chance.
