ANCOVA Models
A regression model that contains a mixture of both quantitative as well as qualitative variables is called an Analysis of Covariance (ANCOVA) model. ANCOVA models are extensions of ANOVA models. They are capable of statistically controlling the effects of quantitative explanatory variables (also called covariates or control variables), in a model that involves quantitative as well as qualitative regressors.
Now we will illustrate how qualitative and quantitative regressors are included to form ANCOVA models. Suppose we consider the same example used in the ANOVA model with 1 qualitative variable : average annual salary of public school teachers in 3 geographical regions of Country A. If we include a quantitative variable, State Government expenditure on public schools, to this regression, we get the following model:
- Yi = α1 + α2D2i + α3D3i + α4Xi + Ui
where,
- Yi = average annual salary of public school teachers in state i
- Xi = State expenditure on public schools per pupil
- D2i = 1, if the State i is in the North Region
- D2i = 0, otherwise
- D3i = 1, if the State i is in the South Region
- D3i = 0, otherwise
Say the regression output for this model looks like this:
- Ŷi = 13,269.11 − 1673.514D2i − 1144.157D3i + 3.2889Xi
The result suggests that, for every $1 increase in State expenditure on public schools, a public school teacher's average salary goes up by about $3.29. Meanwhile, for a state in the North region, the mean salary of the teachers is lower than that of West region by about $1673 and for a state in the South region, the mean salary of teachers is lower than that of the West region by about $1144. This means that the mean salary of the teachers in the North is about $11,595 ($13,269.11 − $1673.514) and that of the teachers in the South is about $12,125 ($13,269.11 − $1144.157). The mean salary of the teachers in the West Region (omitted, base category) is about $13,269 (intercept term). Note that all these values have been obtained ceteris paribus. Figure 3 depicts this model diagrammatically. The average salary lines are parallel to each other as the slope is constant (keeping State expenditure constant for all 3 regions) and the trade off in the graph is between 2 quantitative variables: public school teacher's salary (Y) in relation to State expenditure on public schools (X).
Read more about this topic: Dummy Variable (statistics)
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