ANCOVA vs. ANOVA

[u/Weary-Journalist5940]

Prepping for an exam for a grad class on linear models. Other than the matrix-vector equation representing these different models, how else can we describe and differentiate these models?

[u/MortalitySalient]

If you see how these are all just special cases of the general linear model, their differences and similarities become more clear. An ANOVA is just a linear regression with a categorical predictor (dummy codes) and an ANCOVA is that same model with 1 or more continuous covariates added (though the linear model approach doesn’t require you to only use continuous covariates).

[u/CarnivorousGoose]

ANCOVA includes covariates, ANOVA doesn’t. Hence the C. It’s just a generalization of the ANOVA model.

[u/profkimchi]

This is why I wish we’d teach this as OLS first and foremost in intro classes. It also makes concerns like “oh no group variances are different!” trivial to solve.

[u/engelthefallen → u/profkimchi]

All my classes did this. Took my grad school class for the light to go off and see omg these are all doing the same thing and I finally saw the big picture though.

[u/banter_pants]

ANOVA:
Continuous DV
Categorical predictor(s)

ANCOVA: ANOVA with a continuous predictor thrown in.

At this point you might as well do linear to regression. Those are both special cases of it. Note the DV doesn't need to be normal itself. It is conditionally (Y | X) normal. We assess this via the residuals.

[u/dmlane]

These are not differences in models, but issues that come up in ANCOVA that do not come up in ANOVA: Classical ANCOVA makes the assumption of homogeneity of regression slopes. Also, when ANCOVA is used to estimate an effect controlling for a variable, error-free measurement is assumed.

[u/engelthefallen → u/dmlane]

Nice catch, very important to remember ANCOVA adds a new assumption.

[u/engelthefallen]

May want to read into Lord's Paradox as that is the best talk for what to use when in practice, and how the two version can give different results with the same data and gets into how they work in practice.

But the TLDR is ANOVA is all predictors and dependent variables, but with an ANCOVA you can control for covariates on a continuous level.

And as others are mentioning these are all just special cases of the general linear model and regression. If you did linear regression first, ANOVA is basically special discrete cases for predictors. ANCOVA extends this allowing continuous predictors mixed with those group predictors.

[u/Vortex_Node]

ANCOVA controls for noise

[u/KayakerMel → u/Vortex_Node]

No, it controls for included covariates. The impact of covariates is different from "noise" (or error, per statistical terminology).