Example 72.5 Quantile Polynomial Regression for Salary Data

This example uses the data set from a university union survey of salaries of professors in 1991. The survey covered departments in U.S. colleges and universities that list programs in statistics. The goal here is to examine the relationship between faculty salaries and years of service.

The data include salaries and years of service for 459 professors. The scatter plot in Output 72.5.1 shows that the relationship is not linear, and a quadratic or cubic regression curve is appropriate. Output 72.5.1 shows a cubic curve.

The curve in Output 72.5.1 does not adequately describe the conditional salary distributions and how they change with length of service. Output 72.5.2 shows the 25th, 50th, and 75th percentiles for each number of years, which gives a better picture of the conditional distributions.

   data salary;
     label salaries='Salaries (1000s of dollars)'
           years   ='Years';
     input salaries years @@;
   datalines;
   54.94   2  58.24   2  58.11   2  52.23   2  52.98    2  57.62    2
   44.48   2  57.22   2  54.24   2  54.79   2  56.42    2  61.90    2
   63.90   2  64.10   2  47.77   2  54.86   2  49.31    2  53.37    2
   
   ... more lines ...   

   85.72  25  64.87  25  51.76  25  51.11  25  51.31   25  78.28   25
   57.91  25  86.78  25  58.27  25  56.56  25  76.33   25  61.83   25
   69.13  25  63.15  25  66.13  25
   ;

Output 72.5.1 Salary with Years as Professor: Cubit Fit

Output 72.5.2 Salary with Years as Professor: Sample Quantiles

These descriptive percentiles do not clearly show trends with length of service. The following statements use the QUANTREG procedure to obtain a smooth version by using polynomial quantile regression. The results are shown in Output 72.5.3 and Output 72.5.4.

   ods graphics on;
    
   proc quantreg data=salary ci=none;
      model salaries = years years*years years*years*years
                        /quantile=0.25 0.5 0.75;
   run;
    
   ods graphics off;

Output 72.5.3 shows the regression coefficients for the three quantiles.