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 *MS _{W}* as follows:

where *SS _{B}*

_{(x)}and

*SS*

_{W}_{(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**

Note that some of the formulas in column Q reference cells in Figure 1 of ANOVA Approach to ANCOVA, although the formula for *MS _{W}* in cell Q9 references Figure 3 of ANOVA Approach to 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.