GLM Introductory Overview - Analysis of Covariance
In general, between designs that contain both categorical and continuous predictor variables can be called ANCOVA designs. Traditionally, however, ANCOVA designs have referred more specifically to designs in which the first-order effects of one or more continuous predictor variables are taken into account when assessing the effects of one or more categorical predictor variables. A basic introduction to analysis of covariance can also be found in Analysis of Covariance (ANCOVA) - Basic Ideas.
To illustrate, suppose a researcher wants to assess the influences of a categorical predictor variable A with 3 levels on some outcome, and that measurements on a continuous predictor variable P, known to covary with the outcome, are available. If the data for the analysis are
then the sigma-restricted X matrix for the design that includes the separate first-order effects of P and A would be
The b2 and b3 coefficients in the regression equation
Y = b0 + b1X1 + b2X2 + b3X3
represent the influences of group membership on the A categorical predictor variable, controlling for the influence of scores on the P continuous predictor variable. Similarly, the b1 coefficient represents the influence of scores on P controlling for the influences of group membership on A. This traditional ANCOVA analysis gives a more sensitive test of the influence of A to the extent that P reduces the prediction error, that is, the residuals for the outcome variable.
The X matrix for the same design using the overparameterized model would be
The interpretation is unchanged except that the influences of group membership on the A categorical predictor variables are represented by the b2, b3 and b4 coefficients in the regression equation
Y = b0 + b1X1 + b2X2 + b3X3 + b4X4
Between-subject designs
Within-subject (repeated measures) designs
Multivariate designs