Polynomial Example
The following equation represents a simple polynomial
regression:
Y= b0 + b
1(age) + b2(age*age)
The inclusion of these types of regressors fits a curve
to the data, indicating that the effect of the independent variable on the dependent
variable depends on the level of that same independent variable.
An Example:
Let the response variable represent individual-level attitudes toward the following
statement: "The United States should provide military support to
friendly nondemocratic
regimes, which are threatened by opposing forces". It may be plausible to expect that
the generation that came of age during the Vietnam era may be generally more opposed to
such views than earlier or later generations. This expectation implies that the
relationship between the response variable and the independent variable may be curvilinear.
In this particular example, the response variable is a 100 point thermometer scale, with
higher vaules representing more favorable and lower values representing less favorable. The
age covariate is a five category ordinal variable. The model is estimated producing the following results:
E(Y)=58.3 + -4.1(age) + .7(age*age)
The figure below represents the plot of the regression line. Substantively, this plot
implies that the relationship between age and an individual's attitude toward military
intervention is decreasingling negative to some age at which attitudes become increasingly
positive. It is possible to derive the inflection point (point where slope shifts
direction) by using the following formula: -b1/2b2. In this case the
inflection
point is as follows: -(-4.1)/2(.7)=2.93. Substantively, this means at
approximately the
third category of the age variable, the slope changes from decreasingly negative to
increasingly positive.
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