How do I understand ANCOVA in basic layman's terms?

Question

I need assistance understanding the concept of ANCOVA in very basic layman terms. Also, can someone explain how ANOVA is different from ANCOVA: it seems both these techniques are used to compare means of two different groups?

Answer 1

To answer your question, I would like to invite you to think of a broader picture for, then, take you back to your original question.

First, I would like to introduce a comparison between ANOVA and linear regression with one categorical independent variable; second, I would like to introduce a comparison between ANCOVA and linear regression with one categorical independent variable and one quantitative independent variable; then, I would like to introduce the comparison between ANCOVA and ANOVA.

Let's say, for illustration's sake, that you have three groups of individual (the categorical independent variable with three levels) and one given quantitative additional information about each individual (the quantitative independent variable), beyond their individual responses, of course.

You can use ANOVA to evaluate whether the three groups have the same average response or not, given a set of assumptions with which we will not be concerned here just for pedagogical reasons.

You could take that same data to run a linear regression with 2 dummy variables, one for indicating whether individuals belong to the second group and one for indicating whether individuals belong to the third group. Individual belonging to neither the second nor the third group would, by elimination, belong to the first group, which could be thought of as a "reference group".

You would use that regression to assess by how much the second and third groups differ in average response from the first group (given a set of assumptions, etc., etc., etc.).

One can show that both formulations are mathematically equivalent, so the difference lays in the semantic field.

Statistical technicalities apart, what would be the difference between those two analyses in terms of interpretation?

In the first, the main concern is to detect whether all groups have the same mean or not. In the second, the main concern is to detect by how much the second and the third group differ from the first, sweeping under the rug the discussion about the difference between the second and the third groups.

In terms of interpretation alone: potato/potahto? Perhaps.

Now look at ANCOVA. Beyond the average response of the group, you consider also that quantitative additional information that might or might not influence as well the expected response of the individuals.

ANCOVA, like ANOVA, will be focused in detecting whether the groups might have or not the same average response, considering the additional influence of that quantitative information. A linear regression with the two dummy variables above plus a quantitative regressor would be focused in evaluating also the effect of that quantitative regressor on the expected individual response.

Likewise, one can show that there is mathematical equivalence between those two formulations.

Now, what could be the difference in terms of interpretation? In the regression, you are actually interested in evaluating the effect of the quantitative regressor, while in the ANCOVA all you want is to discount the effect of the quantitative regressor before comparing the average responses among the groups, because you couldn't care less about that quantitative regressor were it not associated with the individual responses.

So, your interest is to diminish the non explained variability on the comparison by accounting some of that variability to a source (that dang quantitative regressor).

In regression, if the effect of the regressor is not statistically significant, you exclude it from your model and adjust it again only with your dummies. In ANCOVA, you don't even test the quantitative regressor for significance, because, significant or not, it has already played its role of controlling for unaccountable variability in the individual responses, external to the group influences and to the attributable systematic variability due to the effect of that quantitative regressor.

Finally, heading to your question, ANCOVA controls for systematic variability that you can attribute to specific sources, reducing the unexplained asystematic variability that is captured in the residual sum of squares, while the ANOVA doesn't.

Okey-dokey? =)

Answer 2

ANOVA looks at the influence of one or more grouping variables (factors) on some continuous dependent measure.

ANCOVA includes at least one grouping variable, but also includes interval-or-ratio-scaled variables on the IV side that are assumed to relate to the DV in linear fashion as in a regression.

ANOVA will let me assess whether extroversion scores (DV) are different for the combinations of sex (grouping variable) and country-of-birth (grouping variable).

ANCOVA will let me assess whether extroversion scores (DV) are different for people of different sex (groups) and different ages (ratio-scaled variable) - though the interaction of these two variables is often left unexamined.