9.2 - Steps in ANCOVA

First, we need to confirm that for at least one of the treatment groups there is a significant regression relationship with the covariate. Otherwise, including the covariate in the model won’t improve the estimation of treatment means.

Once this is confirmed, we need to examine whether the regression relationship between the response and the covariate has the same slope for each treatment group. Graphically, this means that the regression line at each factor level has the same slope and therefore the lines are all parallel.

Depending on the outcome of the test for equal slopes, we have two alternative ways to finish up the ANCOVA:

  1. Fit a common slope model and adjust the treatment SS for the presence of the covariate.
  2. Evaluate the differences in means for at least three levels of the covariate.

These steps are illustrated in the following two sections and are diagrammed below:

No
No
Yes
Yes
Step 1: Are the regression slopes all equal to 0?
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No
No
Yes
Yes
Step 2: Are the regression slopes all equal?
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Stop!
The analysis is just an ANOVA
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Step 3 (different slopes): If slopes differ significantly, then individual regressions for each level of the treatment are reported. LSmeans are compared at 3 points along the covariate.
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Step 3 (equal slopes): If slopes differ not significantly, generate a common slope and use LSmeans to compare responses at a common (mean) value of the covariate.
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Note! The figure above is presented as a guideline and does require some subjective judgment. Small sample sizes, for example, may result in none of the individual regressions in Step 1 being statistically significant. Yet, the inclusion of the covariate in the model may still be advantageous as pooling the data will increase the number of observations when fitting the joint model. Exploratory data analysis and regression diagnostics also will be useful.