The analysis of covariance or “ANCOVA”: comparing estimation lines

When there is a straight-line statistical relationship between a predictor variate X and a predicted variate Y, the estimation line Ŷ= a + bX is equivalent to a conditional mean of Y given X,that is to a measure of central tendency of Y which changes according to the value of X (section 20.1). The analysis of variance, which has been used in chapter 12 to compare the mean values of several groups of data, can therefore be adapted to the comparison of several estimation lines, since the latter are essentially modified means. The version of the analysis of variance which applies to estimation lines is currently known as the analysis of covariance (acronym: ANCOVA), a name of which the meaning is regrettably liable to be misinterpreted. While the ordinary analysis of variance (chapter 12) corresponds basically to the decomposition of the sum of squared deviations between individual observations Y and the mean $$\bar{Y}$$ $$SS = \Sigma {{\left( {Y - \bar{Y}} \right)}^{2}} = \Sigma {{Y}^{2}} - {{\left( {\Sigma Y} \right)}^{2}}/N$$ the analysis of covariance corresponds to the decomposition of the sum of squares of residual deviations (SSresidual or RSS, section 20.3) between individual observations Y and the estimation line Ŷ= a + bX $$ RSS = \sum {\left( {Y - \hat Y} \right)^2 } = \sum {\left[ {Y - \left( {a + bX} \right)} \right]^2 } = \sum {\left( {Y - \bar Y} \right)^2 } - \left[ {\sum {\left( {X - \bar X} \right)\left( {Y - \bar Y} \right)} } \right]^2 /\sum {\left( {X - \bar X} \right)^2 } . $$ When there is a marked straight-line statistical relationship between variates X and Y, the residual variances are perceptibly smaller that the total variances, and the analysis of covariance is more sensitive than the analysis of variance.