Applied Statistics

9Analysis of Covariance (ANCOVA)

9.1 Introduction

Analysis of covariance combines elements of the analysis of variance and of regression analysis. Because the analysis of variance can be seen as a multiple linear regression analysis, the analysis of covariance (ANCOVA) can be defined as a multiple linear regression analysis in which there is at least one categorical explanatory variable and one quantitative variable. Usually the categorical variable is a treatment of primary interest measured at the experimental unit and is called response y. The quantitative variable x, which is also measured in experimental units in anticipation that it is associated linearly with the response of the treatment. This quantitative variable x is called a covariate or covariable or concomitant variable.

Before we explain this in general, we give some examples performed in a completely randomised design.

1. In a completely randomised design treatments were applied on tea bushes where the yields y_ij are the yields in kilograms of the tea bushes. An important source of error is that, by the luck of the draw by randomisation, some treatments will be allotted to a more productive set of bushes than others. Fisher described in 1925 in his first edition of “Statistical Methods for Research Workers” the application of the covariate x_ij, which was the yield in kilograms of the tea bushes in a period before treatments were applied. Since the relative yields of the tea bushes show a good deal of stability from ...