Analysis of dependent interval variables Y on independent variables X

ANCOVA

ANCOVA (ANalysis of COVAriance) uses features from both ANOVA and multiple regression. The usual one-way classification model in analysis of variance is:

Y_ij= µ_i + e_ij

where
Y_ij - jth observation in the ith group
µ_i - the true mean of the ith group
e_ij - are the residuals or errors in the above model (usually assumed to be normally distributed).

Suppose you have measured a second variable (covariate) with values XX_ij that is linearly related to Y. Further suppose that the slope of the relationship between Y and X is constant from group to group. You could then write the analysis of covariance model:

Y_ij= µ_i + β(X_ij-X) + e^'_ij

where X represents the overall mean of X. If X and Y are closely related, you would expect that the errors, e^'_ij, would be much smaller than the errors, e_ij, giving you more precise results.

The analysis of covariance is useful for many reasons, but it does have the (highly) restrictive assumption that the slope is constant over all the groups. This assumption is often violated, which limits the technique's usefulness.

Example:
Researcher wishes to test for differences in IQ among the three states while controlling for age (the covariate).