Multiple degree of freedom tests

Steve Simon


Someone sent me an email describing a situation where an interaction effect in SPSS had a large p-value, but one of the individual components of that interaction had a small and statistically significant p-value.

This can occur in many statistical models where you are testing a factor or interaction that involves multiple degrees of freedom. You might be looking at a variable which indicates race/ethnicity. If there are five possible values for this variable (e.g., White, Black, Hispanic, Asian, and Other), then testing the effect of race in a statistical model involves four degrees of freedom. The hypothesis that you are testing is that the average outcome measure is the same for all five levels for race.

SPSS and other statistical software will then often present individual components of this hypothesis. They will select a reference category and then compare each of the remaining categories to that reference. SPSS choose the last or largest value of the category as the reference category, which may not always be exactly what you want. In the race/ethnicity example, that leads to four single degree of freedom comparisons:

It doesn’t happen too often but sometimes the single degree of freedom comparisons are not consistent with the multiple degree of freedom test. What should you do in this situation?

First, make sure that you are looking at a logical reference category. The reference category might be a control group or it might be the largest group in your sample, or the youngest group. A lot depends on the context of the problem. One thing for sure is that the computer understands nothing about the context of your problem, so it only chooses a good reference category by dumb luck. In the example above, using Other as your reference category is probably the worst possible choice.

You should also apply a Bonferroni correction to the single degree of freedom components. A p-value of 0.03 doesn’t look quite as impressive if it is just one of five degrees of freedom.

If you still get significance after applying Bonferroni, then I think you have to be honest and mention the fact that your analysis produced ambiguous and conflicting results. Report the p-values both for the multiple degrees of freedom test and for each of the individual degrees of freedom test. Then let the reader decide.

Of course, with an interaction, you should also draw a graph, because certain types of interactions, especially ones where the lines cross, are more interesting than others.

The reverse can happen as well. Suppose the test with multiple degrees of freedom has a statistically significant p-value, but the p-values for all of the individual degree of freedom tests are large. Here, you might want to do a bit of investigation. Maybe it is some oddball combination (groups 1 and 4 combined versus groups 2, 3, and 5) that is showing a large effect. This sort of thing might be missed by the particular single degree of freedom tests that the software has chosen for you.

You can find an earlier version of this page on my original website.