Question

According to "Assessing the Validity of the PostMaterialism Index" (American Political Science Review [1999]: \(649-664\) ), one may be able to predict an individual's level of support for ecology based on demographic and ideological characteristics. The multiple regression model proposed by the authors was $$ \begin{aligned} &y=3.60-.01 x_{1}+.01 x_{2}-.07 x_{3}+.12 x_{4}+.02 x_{5} \\ &\quad-.04 x_{6}-.01 x_{7}-.04 x_{8}-.02 x_{9}+e \end{aligned} $$ where the variables are defined as follows \(y=\) ecology score (higher values indicate a greater con- $$ \begin{aligned} & \text { cern for ecology) } \\ x_{1}=& \text { age times } 10 \end{aligned} $$ \(x_{2}=\) income (in thousands of dollars) \(x_{3}=\) gender \((1=\) male, \(0=\) female \()\) \(x_{4}=\) race \((1=\) white, \(0=\) nonwhite \()\) \(x_{5}=\) education (in years) \(x_{6}=\) ideology \((4=\) conservative, \(3=\) right of center, \(2=\) middle of the road, \(1=\) left of center, and \(0=\) liberal \()\) \(x_{7}=\) social class \((4=\) upper, \(3=\) upper middle, \(2=\) middle, \(1=\) lower middle, \(0=\) lower \()\) \(x_{8}=\) postmaterialist ( 1 if postmaterialist, 0 otherwise) \(x_{9}=\) materialist \((1\) if materialist, 0 otherwise) a. Suppose you knew a person with the following characteristics: a 25-year- old, white female with a college degree (16 years of education), who has a \(\$ 32,000\) -per-year job, is from the upper middle class and considers herself left of center, but who is neither a materialist nor a postmaterialist. Predict her ecology score. b. If the woman described in Part (a) were Hispanic rather than white, how would the prediction change? c. Given that the other variables are the same, what is the estimated mean difference in ecology score for men and women? d. How would you interpret the coefficient of \(x_{2}\) ? e. Comment on the numerical coding of the ideology and social class variables. Can you suggest a better way of incorporating these two variables into the model?

Short answer

a. The predicted ecology score is calculated using the coefficients given in the model and the characteristics of the person. b. If the woman were Hispanic, her predicted ecology score would change due to the difference in value for the race coefficient. c. The estimated difference in ecology score for men and women is -0.07. d. The coefficient of \(x_{2}\) indicates that for each increase of 1 in the \(x_{2}\) variable (income), the ecology score increases by 0.01. e. One way to improve the coding of the variables for ideology and social class would be to measure and include the magnitude of differences between each category.

Step-by-step solution

Part a - Predicting Ecology Score

Given:- \(x_{1}\) (age): 25*10 = 250- \(x_{2}\) (income): $32,000 = 32- \(x_{3}\) (gender):0 (female)- \(x_{4}\) (race):1 (white)- \(x_{5}\) (education): 16 years- \(x_{6}\) (ideology): 1 (left of center)- \(x_{7}\) (social class):3 (upper middle)- \(x_{8}\) (postmaterialist):0 (not postmaterialist)- \(x_{9}\) (materialist):0 (not materialist)Substitute these values into the given linear equation and solve for y.

Part b - Predicting Adjusted Ecology Score

For this part, the woman is Hispanic, so \(x_{4}\) (race) changes to 0. Substitute this new value into the model and resolve it.

Part c - Estimated Mean Difference in Ecology Score

This question asks for the difference in ecology score between men and women. This is simply the coefficient for the gender variable, which is \(x_{3}\). Thus, the difference is -0.07.

Part d - Interpreting Coefficient of \(x_{2}\)

The coefficient of \(x_{2}\) is 0.01. This indicates that for each increase of 1 in the \(x_{2}\) variable (income in thousands of dollars), the ecology score increases by 0.01.

Part e - Commenting on Variable Coding and Suggesting Improvements

The variables for social class and ideology are arbitrarily coded without considering the distance between categories. One possible method to incorporate these variables into the model in a better way might be to measure and include the magnitude of differences between each category.