University of South Carolina, Arnold School of Public Health, Dept. of Health Service Policy and Management Curricula and Courses, HSPM J716 Jan. 22, 2002

The assumption is that if you don't use your regression line, you'll ignore what X is and use the mean of your Y values as your prediction.

The least possible improvement is 0. This is if the regression line is no help at all. You might as well use the mean of the Y values for your prediction.

The most possible improvement is 1. This is if the regression line fits the data perfectly.

That is why the R-squared is always between 0 and 1. The regression line is never worse than worthless (0), and it can't be better than perfect (1).

All this is based on the assumptions behind using least squares being true. If those assumptions are not true, then it is possible that using the regression line to predict could be worse than worthless.

The formula for the R-squared, and the assumptions behind using least squares regression, are in the course booklet.

If your regression results show an R-squared of less than 0 or greater than 1, then one of these is true:

1. There is a calculation error, or
2. Your results are reporting an "adjusted" R-squared. (The template you have constructed does not do this, but some commercial statistical software does report an adjusted R-squared.) An "adjusted" R-squared tries to allow for the fact that an X variable that is really completely unrelated to your Y variable will probably have some relationship to Y in your data just by luck. The adjusted R-squared reduces the R-squared by how much fit would probably happen just by luck. Sometimes this reduction is more than the calculated R-squared, so you wind up with a negative adjusted R-squared.