Analysis of Covariance (ANCOVA)

Definition

Analysis of covariance is a multivariate statistical method in which the dependent variable is a quantitative variable and the independent variables are a mixture of nominal variables and quantitative variables.

Explanation

Analysis of variance is used for studying the relationship between a quantitative dependent variable and one or more nominal independent variables. Regression analysis is used for studying the relationship between a quantitative dependent variable and one or more quantitative independent variables. This immediately raises the question of what to do when, in a common situation, the research hypothesis calls for some of the independent variables to be nominal and some to be quantitative variables.

With the arrival of statistical computer packages, it soon become clear that such a situation can easily be covered by introducing one or more dummy variables for the nominal variables and doing an ordinary multiple regression analysis. This can be done with or without the construction of INTERACTION variables, depending on the nature of the available data and the problem at hand. The use of a separate name for analysis of covariance has been decreasing because regression analysis can handle this analysis without any difficulties.

Historical Account

A good deal of work on the development of analysis of covariance took place in the 1930s. As with so much of analysis of variance, Sir Ronald A. Fisher had a leading role in its development, which was highlighted in a special issue of the Biometrics journal, with an introductory paper by William G. Cochran (1957) and six other papers by leading statisticians of the time.

Analysis of covariance was a natural outgrowth of both analysis of variance and regression analysis. In analysis of variance, it may be reasonable to control for a quantitative variable and compare adjusted means instead of the raw, group means when comparing groups. Similarly, in regression analysis, it may be reasonable to control for a nominal variable. Both of these desires resulted in developments of analysis of covariance and shortcut methods for computations, using desk calculators.

Applications

In its simplest form, analysis of covariance makes use of one quantitative variable and one nominal variable as explanatory variables. In regression notation, the model can be written as

where Y and X are quantitative variables; D is a dummy variable with values 0 and 1, representing two groups; and α, β1, and β2 are the regression parameters.

It is now possible to do analysis of variance comparing the means for the two groups while controlling for the X variable. Instead of comparing the means of Y for the two groups directly, we compare adjusted means of Y. By substituting 0 and 1 for the dummy variable, we get the two parallel lines with the following

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