Statistical Interaction Example

To help demonstrate the concept of a statistical interaction, an example is provided below. The following represents the relationship of an interaction between sex and age on the response variable, salary. In this model X1 represents age and X2 represents sex. The general model estimated is:
Y= 2 + 6 (X1 ) +1 (X2 ) +1 (X1 X2)

It is possible using the above regression equation to derive separate equations for each group of the sex variable. For instance, the following equations represent the equation for men (group 1) and women (group 0):

E(Y)= 2 + 6(X2) for group 0
E(Y)= 3 + 7(X2) for group 1

The regression lines for male and female are plotted in the figure below. This figure demonstrates that the groups have different intercepts and different slopes. Substantively, this means the relationship between sex and salary varies according to age. A model not controlling for the interaction between age and sex cannot account for the varying relationship between the independent variables and the response variable. The failure to account for this interaction leads to a misinterpretation of the relationship.
Interaction Example
The interaction can also be represented by a three dimensional space, as seen below. The form of the general equation outlined above is plotted here on a warped surface in three dimensional space (see Darlington 1990). There two ways to recognize the interaction plotted in this figure, both the regression line and the color scheme. The color of the distribution indicates that it is skewed. Although, the both males and females have an increasing salary rate across the age categories, the graph indicates that the gap in salary between males and females increases across the age categories. The graph also indicates that the regression line for males and females are not parallel, meaning that the gap between the groups increases across the age categories.
Interaction Example
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Revised: August 17, 1999.