• Confounder: A third variable that is related to x in a non-causal manner and is related to y either causally or correlationally. The third variable (z) is related to y even when x is not present. Three possible confounding situations can be represented by the diagrams below (a bi-directional arrow indicates a correlation while a unidirectional arrow represents a causal relationship):

• Mediator: In this situation, the nature of the causal relationship between x and y is being obscured by z. In other words, the third variable may be the cause of any relationship between the original two. Two possibilities of this are diagrammed below:

• Moderator: A moderator is a variable (z) whereby x and y have a different relationship between each other at the various levels of z. Note that this is essentially what is entailed in an interaction. This situation can be corrected by doing subgroup analyses or moderated regression.

• Suppressor (Classical): This is a situation where the value of multiple R will increase due the presence of a third variable because the extraneous variable is removing variance from x. In the case of a suppressor, the correlation between y and z equals zero, while the correlation between x and zis greater than zero.

• Covariate: The covariate is essentially the opposite of the suppressor, where the correlation between y and z is greater than zero, while the correlation between x and z equals zero. The covariate relationship becomes an interesting phenomenon and will lead into the next topic: the analysis of covariance.

ANCOVA allows you to remove from a dependent variable (y) irrelevant or error variance that can not be predicted from your independent variable (x). Hence, by accounting for the third variable, you are more able to obtain a accurate picture of the proportion of variance in y that x is capable of accounting for; in other words, your power is increased.

Two general applications exist for ANCOVA:

• Remove Error Variance in the Randomized Experiment: Participants are assigned to treatment and control groups in any ANOVA-type design. ANCOVA is then used as the statistical technique to eliminate irrelevant y variance.

• Equating Non-Equivalent (Intact) Groups: A very controversial use of ANCOVA is to correct for initial group differences (prior to assigned to x) that exists on y among several intact, state variable groups.

• The Covariate: When you reference back to the previous page on extraneous variables, you will note that the definition of a covariate assumes that the third variable (z) is unrelated to x, and that z is related to, and in a sense, acts as a suppressor of y.
• Linearity: Since ANCOVA is a general linear model procedure with much in common with multiple regression, it is also assumed that the covariate has a linear relationship with the dependent variable.
• Homogeneity of Variance: Like previous techniques, ANCOVA assumes homogeneity of variance. In other words, the variance of group one is equal to the variance of group 2 and so on.
• Homogeneity of Regression: ANCOVA assumes that homogeneity of regression exists--that the correlation between y and z is equal for all levels of x. In other words, for each level of the independent variable, the slope of the prediction of the dependent variable from the covariate must be equal.