Data Assumption: Homogeneity of regression slopes (test of parallelism)

Very brief description:

The dependent variable and any covariate(s) such as in ANCOVA and MANCOVA, should have the same slopes (b-coefficient) across all levels of the categorical grouping variable (factors). In other words, the covariate(s) must be linearly related to the dependent variable. On the other hand, covariate(s) and factors should not be significantly correlated.
Who cares
Ordinal regression
Probit response models
Why is it important
The fact is: when groups differ significantly on the covariate (thus an interaction) then placing the covariate into the analysis will not “control for” or “balance out” those differences across the groups. This would suggests that the differences in the dependent variable across the groups vary as a function of the covariate and the results from an ANCOVA won’t be meaningful. The more this assumption is violated, the more likely we will fail to reject the null hypotheses (Type II error, or “false negative error”). This means that there is a significant relationship but is indicated as non-significant, so we do not reject the null hypotheses. 

How to Test
A: Dependent variable(s) and the covariate(s) by factor group should have the same slopes:
  1. Conduct a correlation analysis between the dependent variable(s) and the covariate(s). They should be highly correlated.
  2. A scatter plot of the dependent variable(s) and the covariate(s) by factor group should show that all lines have a similar slope.
B: Covariate(s) and factors should not be significantly correlated:
  1. Covariate(s) should have equal mean scores across the factor groups (meaning no relationship). This can be verified by checking the covariance means via the ANOVA procedure. 
The General Linear Model (GLM) provides an option under “Custom Model” to show all the interactions between independent variables and covariates. The applicable interactions should be non-significant. 
How to fix the problem
I’m not aware of any easy fix. However, you may want to try the Wilcox procedure.

Remember also that these interactions may be interesting enough so rather than trying to remove it, make sure you are fully aware of the interactions, interpret what they mean, and be careful with the results.