# Analysis of Covariance

Analysis of covariance (ANCOVA) is a statistical procedure used in cases similar to ones in which a one-way or factorial ANOVA is used. ANCOVA has two major purposes: to adjust initial group differences statistically on one or more variables that are related to the dependent variable but uncontrolled, and to increase the likelihood of finding a significant difference between group means.

Example

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For the first purpose, consider the following example. A researcher uses two classes to investigate whether computer-assisted instruction or noncomputer (traditional) instruction is more effective. On the basis of a pretest, the researcher knows that one class has greater knowledge of the dependent variable (achievement in mathematics) than the other group (for example, the computer group pretest mean is 12 and the noncomputer group pretest mean is 10). If a posttest is given and it is found that the computer group mean is 24 and the noncomputer group mean 20, the researcher might be tempted to conclude that the computer group achieved more than the noncomputer group. This would be likely to happen if the pretest scores were ignored. An alternative approach would be to look at pretest–posttest gain scores and use a t test to determine whether the gain scores are significantly different. This approach would result in comparing 12 (24 – 12) to 10 (20 – 10).

While this approach is theoretically better than not using the pretest scores, for reasons beyond the scope of this chapter, there are technical problems with comparing gain scores. The best method of analyzing the data in this circumstance is by using ANCOVA. ANCOVA would statistically adjust the posttest scores by the differences that existed between the groups on the pretest. In this example, the posttest score of the computer group would be lowered by one point, because this group’s mean was higher by one point than the mean of both groups on the pretest. Similarly, because the noncomputer group pretest mean is one point lower than the mean of the two pretests, the posttest score of 20 would be raised by one point to 21. Instead of comparing 20 to 24, ANCOVA would thus compare 21 to 23.