Following a significant ANCOVA, we can use the usual approach to Tukey’s HSD and the Tukey-Kramer follow up test, as described in Unplanned Comparisons, except that we need to modify the value of MSW as follows:
where SSB(x) and SSW(x) are the sum of squares between and within the covariance groups (based on one-way Anova on the covariate data), i.e.
The standard error can then be calculated as
if all the group sizes are equal (to m). This is the Tukey HSD test. If the group sizes are unequal, the standard error is (i.e. the Tukey-Kramer test)
Example 1: Repeat Example 1 of Contrasts for ANCOVA using the Tukey-Kramer test.
The results are shown in Figure 1.
Figure 1 – Tukey-Kramer for ANCOVA
See ANCOVA Analysis Tools for information about how to conduct Tukey’s HSD or Tukey-Kramer using the Real Statistics ANCOVA data analysis tool.