Validity of ANCOVA for Linear Models with Different Slopes

Question

Does ANCOVA require homogeneity of regression slopes? In other words, do the slopes of the lines need to be the same in order to use this method?

Per this blog post ANCOVA can be used to discriminate between models, with different slopes, but this blog post says the slopes must be parallel for ANCOVA to be used. Please explain the discrepancy in plain language if possible. I have some simulated data with Python code below as an example for context.

Example Data:

## Module Imports
import numpy as np
import matplotlib.pyplot as plt
import pandas as pd
import statsmodels.api as sm
import statsmodels.formula.api as smf
Generate sample data for Z statistic method
g1_count = 100  # group 1 number of datapoints
g2_count = 100  # group 2 number of datapoints
x1 = np.arange(5, 15, 10/g1_count)  # array of x values for group 1
y1 = x1 + 3 + np.random.normal(0, 1, g1_count)  # array of x values for group 2
x2 = np.arange(0, 10, 10/g2_count)  # array of y values for group 1
y2 = 2*x2 + 3 + np.random.normal(0, 1, g2_count)  # array of y values for group 2
df = pd.DataFrame({"x1": x1, "x2": x2, "y1": y1, "y2": y2})  # create dataframe
Plot Data
fig, ax = plt.subplots()
ax.scatter(x1,y1)
ax.scatter(x2,y2)

## Sample data for ANCOVA, combines x and y values to a single column each and adds a categorical column
x_c = np.concatenate((x1, x2))
y_c = np.concatenate((y1, y2))
group_list = np.concatenate((np.array((len(x1)*["A"])),np.array((len(x2)*["B"]))))
df_c = pd.DataFrame({"x": x_c, "y": y_c, "group": group_list})
df_c

Fit Model and See Summary Statistics

lm_ancova = smf.ols('y ~ group + x', data=df_c).fit()
lm_ancova.summary()

Null hypothesis is rejected. The two linear models are likely not the same given the low p-values.

However, the assumption that there is no interaction between the group and covariate (x) fails according to an ANOVA test as described here.

inter_lm = smf.ols('y ~ group * x', data=df_c).fit()  # fit linear interaction model
sm.stats.anova_lm(inter_lm, typ=3)  # ANOVA test on interaction model

If you have any suggestions for a more appropriate test implemented in Python it would be much appreciated.

Answer 1

The formal answer to this question is "yes", the ANCOVA test requires homogeneity of slopes (equal slopes for the groups). But with regards to the underlying motivation for the question, the answer is "no", equal slope is NOT a requirement...if you are asking if the groups are different (in a broad sense).

Briefly, the question driving the null hypothesis statistical test (NHST) for the ANCOVA (analysis of covariance) is this: ¿Are my groups different on the dependent variable AFTER having controlled for a covariate that comparably impacts that dependent variable? And the power of this test is that it allows you to "correct" for variation across the groups if the covariates are not the same from group to group. For this analysis, you do indeed need a common slope (particularly if you intend to calculate the adjusted means for the groups).

However, if the core of your question is this: ¿are my groups different from each other? ...then the requirement for equal slope is less pressing. If the slopes are not equal, then you have a situation of moderation (where the covariate moderates the relationship between the groups and the dependent variable). Thus, your groups are REALLY different from each other...it is not just the intercepts that are (may be) different, but the slopes ALSO are different.

Hope this helps clarify.

Answer 2

The "ANCOVA model" usually refers to a model in which the slopes for the covariates are assumed equal across groups. The first post is using an unusual form of ANCOVA. Allowing the sloeps to vary across groups is sometimes called "the full model in every cell". See the following quote from Muller and Fetterman (2003, Ch 16):

The full model in every cell is a generalization of ANCOVA that allows interactions between continuous and categorical predictors. [...] The traditional analysis of covariance (ANCOVA) model is a special case of the full model in every cell. Adding the restriction of equal slopes reduces the full model to ANCOVA.

The blog also describes an invalid methodology of testing whether the slopes are different and then fitting a model assuming they are the same if nonsignificant. The absence of statistical significance doesn't mean an interaction is absent, and building a model by backward selection invalidates p-values computed in the final step.

What happens if the true data-generating model has different slopes for the two groups and you fit a model that assumest hey have the same slope? This is described in detail by Słoczyński (2022) and Chattopadhyay and Zubizirreta (2022). Basically, the effect of the grouping variable will be estimated for a population other than the one under study, and the papers describe the exact nature of that population, which tends to resemble the smaller group. In contrast, allowing the slopes to differ ensures the estimated effect generalizes to the population from which the sample was drawn. Schafer and Kang (2008) also describe why ANCOVA is problematic and why you should allow the slopes to vary. In the context of experiments, Lin (2013) explains why you not only need to allow the slopes to vary, you need to use a special "robust" standard error to correctly calculate the p-values.

I disagree with Gregg H that the choice should depend on what hypothesis you are testing. Even if you are testing the simple hypothesis of whether the groups differ after adjusting for covariates and have no substantive interest in the interaction, you should still include the interaction between the grouping variable and the centered covariates. You must center the covariates at their mean in the sample in order to interpret the coefficient on the grouping variable as the adjusted difference in means. You should not use a hypothesis test of whether an interaction is present to decide how to fit the model; just fit a single model and interpret it. There is very little cost to including an interaction when there is none in the data-generating model.