Editing Analysis of covariance (section)

==Uses==
===Increase power===
ANCOVA can be used to increase [[statistical power]] (the probability a [[Statistical significance|significant difference]] is found between groups when one exists) by reducing the within-group error [[variance]].<ref>{{cite book |last1=Tabachnick |first1=B. G. |last2=Fidell |first2=L. S. |year=2007 |title=Using Multivariate Statistics |edition=5th |location=Boston |publisher=Pearson Education }}</ref> In order to understand this, it is necessary to understand the test used to evaluate differences between groups, the [[F-test]]. The ''F''-test is computed by dividing the explained variance between groups (e.g., medical recovery differences) by the unexplained variance within the groups. Thus,
:<math>
       F = \frac{MS_{between} }{MS_{within}}
</math>
If this value is larger than a critical value, we conclude that there is a significant difference between groups. Unexplained variance includes error variance (e.g., individual differences), as well as the influence of other factors. Therefore, the influence of CVs is grouped in the denominator. When we control for the effect of CVs on the DV, we remove it from the denominator making ''F'' larger, thereby increasing our power to find a significant effect if one exists at all.

[[File:ANCOVA - Partitioning Variance.jpg|right|Partitioning variance]]

===Adjusting preexisting differences===
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 categorical IV that removing the variance on the DV associated with the CV would remove considerable variance on the DV, rendering the results meaningless.<ref>{{cite journal |last1=Miller |first1=G. A. |last2=Chapman |first2=J. P. |year=2001 |title=Misunderstanding Analysis of Covariance |journal=Journal of Abnormal Psychology |volume=110 |issue=1 |pages=40–48 |doi=10.1037/0021-843X.110.1.40 |pmid=11261398 }}</ref>