12.3 Testing for Homogeneity of Regression Slopes

Homogeneity of Regression Slopes

Assumption of Homogeneity in ANCOVA

ANCOVA adjusts group comparisons for a covariate, but that adjustment only works properly if the covariate has the same linear relationship with the dependent variable in every group. This is the homogeneity of regression slopes assumption.

Graphically, it means the regression lines for each group should be parallel. If the slope relating age to test scores is 2.1 points per year in the control group and 2.3 in the experimental group, those slopes are roughly parallel and the assumption holds. If the slopes are 1.0 and 5.0, the covariate's effect depends heavily on which group you're in, and standard ANCOVA breaks down.

Why does this matter? When slopes differ across groups, the adjusted group means shift depending on where along the covariate you evaluate them. The "main effect" of the independent variable becomes ambiguous because the group difference isn't constant. You'd get different answers at different covariate values, so a single adjusted comparison is misleading.

Testing the Assumption

You test this by fitting an ANCOVA model that includes an interaction term between the covariate and the independent variable. The interaction captures whether the slopes differ across groups.

The model with the interaction term is:

$Y_{ij} = \mu + \alpha_i + \beta X_{ij} + \gamma_i X_{ij} + \epsilon_{ij}$

where:

• $Y_{ij}$ is the dependent variable for observation $j$ in group $i$
• $\mu$ is the grand mean
• $\alpha_i$ is the effect of the $i$-th level of the independent variable
• $\beta$ is the common (overall) regression slope for the covariate $X$
• $\gamma_i$ is the deviation of group $i$'s slope from the common slope (this is the interaction)
• $\epsilon_{ij}$ is the error term

The key parameter is $\gamma_i$. If all the $\gamma_i$ values are zero, every group shares the same slope $\beta$, and the assumption holds.

Testing for Homogeneity

Conducting the F-Test

To assess whether the interaction is significant, you compare two nested models:

1. Reduced model (standard ANCOVA): includes the independent variable and the covariate, but no interaction
2. Full model: adds the covariate × independent variable interaction

An F-test compares the residual sums of squares from these two models. The test statistic is:

$F = \frac{(SS_{res,\text{reduced}} - SS_{res,\text{full}}) / df_{\text{diff}}}{SS_{res,\text{full}} / df_{\text{full}}}$

where $df_{\text{diff}}$ is the number of additional parameters in the full model (equal to the number of groups minus 1). A likelihood ratio test provides an equivalent comparison in maximum-likelihood frameworks.

Interpreting the Interaction Term

• Significant interaction (p < 0.05): The slopes are not parallel. The relationship between the covariate and the dependent variable differs across groups. For example, the effect of age on test scores might be steep in the experimental group but nearly flat in the control group.
• Non-significant interaction (p > 0.05): No evidence that the slopes differ. You remove the interaction term and proceed with standard ANCOVA.

A few things to keep in mind about statistical power here. The test's ability to detect unequal slopes depends on sample size, the magnitude of the slope differences, and the number of groups. With small samples, you may fail to detect a real violation simply because the test lacks power. Plotting the within-group regression lines is always a good complement to the formal test.

Interpreting Homogeneity Results

Assumption Met

When the interaction term is not significant, you can interpret the standard ANCOVA results:

• Main effect of the independent variable: the difference between group means after adjusting for the covariate. For example, the difference in job performance across age groups after controlling for IQ.
• Main effect of the covariate: the overall within-group relationship between the covariate and the dependent variable, pooled across groups. For example, the relationship between IQ and job performance, holding age group constant.

The adjusted (least-squares) means for each group are now meaningful because the parallel-slopes assumption justifies evaluating all groups at the same covariate value (typically the grand mean of the covariate).

Assumption Violated

When the interaction is significant, the group difference depends on the covariate value, so a single "main effect" of the independent variable is misleading.

• The adjusted means change depending on which covariate value you choose as the reference point.
• The covariate's slope is not a single number anymore; it varies by group.
• Reporting main effects as though they're constant across covariate values misrepresents the data.

The interaction itself becomes the substantively interesting finding. Rather than treating it as a nuisance, consider whether the differing slopes tell you something meaningful about how the groups respond differently to the covariate.

Action Based on Homogeneity Test

Assumption Met

Proceed with standard ANCOVA:

1. Remove the interaction term from the model.
2. Refit the reduced model with only the main effects of the independent variable and the covariate.
3. Interpret the adjusted group means and test for group differences.
4. Report the F-test for the group effect, the covariate effect, and the adjusted means with confidence intervals.

Assumption Violated

Several alternatives are available, and the right choice depends on your research question and sample size:

• Retain the interaction in the model. Rather than discarding ANCOVA entirely, you can keep the full model and interpret the group × covariate interaction directly. This lets you describe how the covariate's effect differs across groups, which is often the most informative approach.
• Johnson-Neyman technique. This identifies the specific range of covariate values where the group difference is statistically significant. For instance, it might show that age groups differ in job performance only for IQ scores above 110. This is particularly useful when you want to know where along the covariate the groups diverge.
• Separate within-group regressions. Fit the covariate-DV regression separately in each group. This allows completely different slopes but reduces power because each analysis uses only a subset of the data. It's less feasible with small group sizes.
• Multiple regression with all terms. Treat the independent variable (dummy-coded) and the covariate as predictors in a regression that includes their interaction. This is algebraically equivalent to the full ANCOVA model but reframes the analysis in a regression context, which some find easier to interpret.

The choice depends on your priorities:

• If you care most about where groups differ, use the Johnson-Neyman technique.
• If you care about describing the pattern of differing slopes, keep the interaction model and interpret it.
• If group sizes are large enough and you want group-specific slope estimates, separate regressions work.
• If the sample is small, the interaction model or Johnson-Neyman approach will preserve more statistical power than splitting the data.