Editing Analysis of covariance (section)

{{Short description|General linear model that blends ANOVA and regression}}
{{redirect|Ancova|the moth genus|Ancova (moth)}}

'''Analysis of covariance''' ('''ANCOVA''') is a [[general linear model]] that blends [[ANOVA]] and [[regression analysis|regression]]. ANCOVA evaluates whether the means of a [[dependent variable]] (DV) are equal across levels of one or more [[Categorical variable|categorical]] [[independent variable]]s (IV) and across one or more continuous variables. For example, the categorical variable(s) might describe treatment and the continuous variable(s) might be [[covariate]]s (CV)'s, typically nuisance variables; or vice versa. Mathematically, ANCOVA decomposes the variance in the DV into variance explained by the CV(s), variance explained by the categorical IV, and residual variance. Intuitively, ANCOVA can be thought of as 'adjusting' the DV by the group means of the CV(s).<ref>Keppel, G. (1991). ''Design and analysis: A researcher's handbook'' (3rd ed.). Englewood Cliffs: Prentice-Hall, Inc.</ref>

The ANCOVA model assumes a linear relationship between the response (DV) and covariate (CV):

<math>y_{ij} = \mu + \tau_i + \Beta(x_{ij} - \overline{x}) + \epsilon_{ij}.</math>

In this equation, the DV, <math>y_{ij}</math> is the jth observation under the ith categorical group; the CV, <math>x_{ij}</math> is the ''j''th observation of the covariate under the ''i''th group. Variables in the model that are derived from the observed data are <math>\mu</math> (the grand mean) and <math>\overline{x}</math> (the global mean for covariate <math>x</math>). The variables to be fitted are <math>\tau_i</math> (the effect of the ''i''th level of the categorical IV), <math>B</math> (the slope of the line) and <math>\epsilon_{ij}</math> (the associated unobserved error term for the ''j''th observation in the ''i''th group).

Under this specification, the categorical treatment effects sum to zero <math>\left(\sum_i^a \tau_i = 0\right).</math> The standard assumptions of the linear regression model are also assumed to hold, as discussed below.<ref name="Montgomery, Douglas C 2012">Montgomery, Douglas C. "Design and analysis of experiments" (8th Ed.). John Wiley & Sons, 2012.</ref>