CHAPTER 11 Comparing Average Values between Groups 153

»

» ^{The numerator degrees of freedom: This number is often designated as df}N

or df1, which is one less than the number of groups.»

» ^{The denominator degrees of freedom: This number is designated as df}D ^or

df2, which is the total number of observations minus the number of groups.

The p value can be calculated from the values of F, df1, and df2, and the software

performs this calculation for you. If the p value from the ANOVA is statistically

significant — less than 0.05 or your chosen α level — then you can conclude that

the group means are not all equal and you can reject the null hypothesis. Techni-

cally, what that means is that at least one mean was so far away from another

mean that it made the F test result come out far away from 1, causing the p value

to be statistically significant.

Picking through post-hoc tests

Suppose that the ANOVA is not statistically significant (meaning F was larger than

0.05). It means that there is no point in doing any t tests, because all the means

are close to each other. But if the ANOVA is statistically significant, we are left

with the question: Which group means are higher or lower than others? Answering

that question requires us to do post-hoc tests, which are t tests done after an ANOVA

(post hoc is Latin for “after this”).

Although using post-hoc tests can be helpful, controlling Type I error is not that

easy in reality. There can be issues with the data that may make you not trust the

results of your post-hoc tests, such having too many levels to the group you are

testing in your ANOVA, or having one or more of the levels with very few partici-

pants (so the results are unstable). Still, if you have a statistically significant

ANOVA, you should do post-hoc t tests, just so you know the answer to the ques-

tion stated earlier.

It’s okay to do these post-hoc tests; you just have to take a penalty. A penalty is

where you deliberately make something harder for yourself in statistics. In this

case, we take a penalty by making it deliberately harder to conclude a p value on a

t test is statistically significant. We do that by adjusting the α to be lower than

0.05. How much we adjust it depends on the post-hoc test we choose.»

» ^{The Bonferroni adjustment uses this calculation to determine the new,}

lower alpha: α/N, where N is the number of groups. As you can tell, the

Bonferroni adjustment is easy to do manually! In the case of our three marital

groups (M, NM, and OTH), our adjusted Bonferroni α would be 0.05/3, which is

0.016. This means that for a post-hoc t test of average fasting glucose

between two of the three marital groups, the p value would not be inter-

preted as significant unless the it was less than 0.016 (which is a tougher