16.6 Computing Effect Sizes in an Analysis of Covariance
As you might expect, computing effect sizes is a bit more complicated in analysis of
covariance than it was in the analysis of variance. That is because we have choices to make
in terms of the means we compare and the error term we use. You may recall that with fac-
torial designs and repeated measures designs we had a similar problem concerning the
choice of the error term for the effect size.
As before, we can look at effect size in terms of r-family and d-family measures.
Normally I would suggest r-family measures when looking at an omnibus Ftest, and a
d-family measure when looking at specific contrasts. We will start with an r-family exam-
ple, and then move to the d-family. The example we have been using based on the study by
Conti and Musty produced a significant Fon the omnibus null hypothesis. Probably the
most appropriate way to talk about this particular example would make use of the fact that
Group (or Dose) was a metric variable, increasing from 0 to 2 mg.^8 However I am going to
take a “second-best” approach here because the majority of the studies we run do not have
the independent variable distributed as such an ordered variable.r-Family Measure
As our r-family measure of association we will use h^2 , acknowledging that it is positively
biased. You should recall that h^2 is defined as the treatment SSdivided by the total SS. But
which sums of squares for treatments should we use—the ones from an analysis of vari-
ance on the dependent variable, or the ones from the analysis of covariance? Kline (2004)
offers both of those alternatives, though he uses an adjusted SStotalin the second,^9 withoutSection 16.6 Computing Effect Sizes in an Analysis of Covariance 609Exhibit 16.4 (continued)Spread vs. Level Plot of POSTTESTGroups: Treatment group
Level (mean)1.0 1.5 2.0 2.5 3.0 3.5 4.0Spread (variance)2.52.01.51.00.50.0Mean (adj)1.0 1.5 2.0 2.5 3.0 3.5 4.0Spread (variance)2.52.01.51.00.50.0Spread vs. Level Plot of Adjusted POSTTEST(^8) SPSS will test polynomial contrasts on the adjusted means. Just click on the CONTRAST button and ask for
polynomial contrasts. For this example there is a significant quadratic component.
(^9) SPSS uses this same adjustment if you request effect sizes, and it is simply SS
treat^1 SSerror.