Question

Question: Factors identifying urban counties. The Professional Geographer (February 2000) published a study of urban and rural counties in the western United States. Six independent variables—total county population (x₁), population density (x₂), population concentration (x₃), population growth(x₄), proportion of county land in farms (x₅,) and 5-year change in agricultural land base (x₆)—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 (most urban). 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 oncounties.

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?

Short answer

Answer

a. From the correlation matrix, it can be seen that all the values are below 0.20. the negative sign just indicates the inverse relationship between the variables. But the correlation value for x₁ and x₂ is 0.45 and for x₃ and x₂ is 0.43 which is not very alarming but still indicate moderate multicollinearity.

b. The overall model p-value is less than 0.001 indicating the overall adequacy of the model however, the p-values of some coefficients like x₂ x₄ ,x₆ and are very insignificant at 0.230, 0.860, and 0.580 which might indicate that the variables might be correlated and that there might be multicollinearity in the model.

Step-by-step solution

Given Information

There are six independent variables county population (x₁), population density (x₂), population concentration (x₃), population growth(x₄), farm land (x₅), and agricultural change (x₆).

Interpretation of correlation matrix 

From the correlation matrix, it can be seen that all the values are below 0.20. the negative sign just indicates the inverse relationship between the variables. But the correlation value for x₁ and x₂ is 0.45 and for x₃ and x₂ is 0.43 which is not very alarming but still indicate moderate multicollinearity.

Evidence for multicollinearity

The overall model p-value is less than 0.001 indicating the overall adequacy of the model however, the p-values of some coefficients like x₂ x₄ ,x₆ and are very insignificant at 0.230, 0.860, and 0.580 which might indicate that the variables might be correlated and that there might be multicollinearity in the model.