# What is the intuition behind (M)ANCOVA and when/why should one use it?

As per my understanding here's the what/why/when of the following hypotheses tests in a crude sense:

• t-test: Used when comparing means between two samples
• ANOVA (one way): Used when you have one dependent variable and one independent (i.e., categorical) variable and you wish to analyze the 'means' (i.e., effects) across multiple groups. Simply stated, multi-way t-tests in essence.
• ANOVA (two way): Similar to one-way except you have two independent (i.e., categorical) variables
• MANOVA: ANOVA with multiple dependent variables
• ANCOVA: ??
• MANCOVA: ??

Intuitively, the concepts/intuition behind (M)ANOVA makes sense and I understand when/how to apply it and why is it necessary. I've just overly simplified my understanding about them above. However, I lack the similar intuition behind (M)ANCOVA.

• *ANCOVAs allow for both continuous and categorical variables. You can also have $n$-way versions, not just two-way. – Dimitriy V. Masterov Oct 25 '13 at 1:13

## 1 Answer

To complete your scheme:

• ANCOVA: ANOVA conducted to compare multiple (possibly only two) conditions on at least one independent variable while controlling for a set of continuous nuisance variables (possibly only one).
• MANCOVA: MANOVA conducted to compare multiple (possibly only two) conditions on at least one independent variable while controlling for a set of continuous nuisance variables (possibly only one).
• If you need more info than this, clarify your question w/ what more you want to know. – gung Oct 25 '13 at 1:24
• Could you please clarify what this means: ...while controlling for a set of continuous nuisance variables? What does 'controlling' mean w.r.t. the math? – PhD Oct 25 '13 at 1:29
• Let's say you want to assess the mean weight of vegetarians vs. omnivores, but you know that weight will vary in part as a function of height. You don't actually care about the relationship b/t height & weight, you just want to control for that so as to get a clearer picture of how mean weight differs by diet. If you ran a straight ANOVA (in this case, equivalent to a t-test), you could not include what you knew about each individual's height; when you (statistically) control for height, you are running an ANCOVA. – gung Oct 25 '13 at 1:35
• Makes sense. When you say "control for it" does it mean "just add it to the equation" (e.g., as an IV on the RHS of the equation) or is there something more subtle? Sorry, I'm not a statistician by trade. I just happen to keep stumbling into using it and seek clarification of the terms so as to not confuse myself. – PhD Oct 25 '13 at 1:37
• Yes, it means nothing more than "add it to the equation". My answer here: How are regression, the t-test, and the ANOVA all versions of the general linear model? may help as well. – gung Oct 25 '13 at 1:40