Factors identifying urban counties. The

QUESTION:

Factors identifying urban counties. The ProfessionalGeographer (Feb. 2000) published a study of urban and rural counties in the western United States. Six independent variables—total county population (\(x_1\)), population density (\(x_2\)), population concentration (\(x_3\)), population growth (\(x_4\)), proportion of county land in farms (\(x_5\)), and 5-year change in agricultural land base (\(x_6\))—were used to model the urban/rural rating (y) of a county, where rating was recorded on a scale of 1 (most rural) to 10 (mosturban). Prior to running the multiple regression analysis,the researchers were concerned about possible multicollinearity in the data. Below is a correlation matrix for data collected on n = 256 counties.

\(\begin{array}{lrrrrr}
\hline \text { Independent Variable } & x_{1} & x_{2} & x_{3} & x_{4} & x_{5} \\
\hline x_{1}: \text { Total population } & & & & & \\
x_{2}: \text { Population density } & .20 & & & & \\
x_{3}: \text { Population } & .45 & .43 & & & \\
\quad \text { concentration } & & & & & \\
x_{4}: \text { Population growth } & -.05 & -.14 & -.01 & & \\
x_{5}: \text { Farm land } & -.16 & -.15 & -.07 & -.20 & \\
x_{6}: \text { Agricultural change } & -.12 & -.12 & -.22 & -.06 & -.06 \\
\hline
\end{array}\)

\(\text{Source: Based on Berry, K. A., et al. "Interpreting what is rural and urban for}\text{ western U.S. counties," Professional Geographer, Vol. 52, No. 1, Feb. 2000.}\text{ pp. 93-105 (Table 2).}\)

a. Based on the correlation matrix, is there any evidence of extreme multicollinearity?

b. The first-order model with all six independent variables was fit, and the results are shown in the next table. Based on the reported tests, is there any evidence of extreme multicollinearity?

\(\begin{array}{lrc}
\hline \text { Independent Variable } & \beta \text { Estimate } & p \text {-value } \\
\hline x_{1}: \text { Total population } & 0.110 & 0.045 \\
x_{2}: \text { Population density } & 0.065 & 0.230 \\
x_{3}: \text { Population } & & \\
\quad \text { concentration } & 0.540 & 0.000 \\
x_{4}: \text { Population growth } & -0.009 & 0.860 \\
x_{5}: \text { Farm land } & -0.150 & 0.003 \\
x_{6}: \text { Agricultural change } & -0.027 & 0.580 \\
\hline \text { Overall model: } & & \\
R^{2}=.44 & R_{\mathrm{a}}^{2}=.43 F=32.47 & p \text {-value }<.001 \\\hline
\end{array}\)

\(\text{Source: Berry, K. A., et al. "Interpreting what is rural and urban for western U.S.}\text{ counties," Professional Geographer, Vol. 52, No. 1, Feb. 2000 (Table 2).}\)