Another use of ANCOVA is to adjust for preexisting differences in nonequivalent (intact) groups. This controversial application aims at correcting for initial group differences (prior to group assignment) that exists on DV among several intact groups. In this situation, participants cannot be made equal through random assignment, so CVs are used to adjust scores and make participants more similar than without the CV. However, even with the use of covariates, there are no statistical techniques that can equate unequal groups. Furthermore, the CV may be so intimately related to the IV that removing the variance on the DV associated with the CV would remove considerable variance on the DV, rendering the results meaningless. There are several key assumptions that underlie the use of ANCOVA and affect interpretation of the results. The standard linear regression assumptions hold; further we assume that the slope of the covariate is equal across all treatment groups (homogeneity of regression slopes). The regression relationship between the dependent variable and concomitant variables must be linear. The error is a random variable with conditional zero mean and equal variances for different treatment classes and observations. The errors are uncorrelated. That is, the error covariance matrix is diagonal. The residuals (error terms) should be normally distributed formula_9 ~ formula_13. The slopes of the different regression lines should be equivalent, i.e., regression lines should be parallel among groups. The fifth issue, concerning the homogeneity of different treatment regression slopes is particularly important in evaluating the appropriateness of ANCOVA model. Also note that we only need the error terms to be normally distributed. In fact both the independent variable and the concomitant variables will not be normally distributed in most cases. If a CV is highly related to another CV (at a correlation of 0.5 or more), then it will not adjust the DV over and above the other CV. One or the other should be removed since they are statistically redundant. Tested by Levene's test of equality of error variances. This is most important after adjustments have been made, but if you have it before adjustment you are likely to have it afterwards. To see if the CV significantly interacts with the IV, run an ANCOVA model including both the IV and the CVxIV interaction term. If the CVxIV interaction is significant, ANCOVA should not be performed. Instead, Green & Salkind suggest assessing group differences on the DV at particular levels of the CV. Also consider using a moderated regression analysis, treating the CV and its interaction as another IV.