This example uses the data set from a university union survey of salaries of professors in 1991. The survey covered departments in US colleges and universities that list programs in statistics. The goal of this example 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 95.5.1 shows that the relationship is not linear and that a quadratic or cubic regression curve is appropriate. Output 95.5.1 shows a cubic curve.

The curve in Output 95.5.1 does not adequately describe the conditional salary distributions and how they change with length of service. Output 95.5.2 shows the 25th, 50th, and 75th percentiles for each number of years, which gives a better picture of the conditional distributions.

data salary; input Salaries Years @@; label Salaries='Salaries (1000s of dollars)'; 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 51.69 2 53.66 2 58.77 2 56.77 2 53.06 2 54.86 2 50.96 2 56.46 2 51.67 2 49.37 2 56.86 2 49.85 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 95.5.1: Salary and Years as Professor: Cubic Fit

Output 95.5.2: Salary and Years as Professor: Sample Quantiles

These descriptive percentiles do not clearly show trends with length of service. The following statements use polynomial quantile regression to obtain a smooth version.

ods graphics on; proc quantreg data=salary ci=sparsity; model salaries = years years*years years*years*years /quantile=0.25 0.5 0.75 plot=fitplot(showlimits); test years/QINTERACT; run;

The results are shown in Output 95.5.3 and Output 95.5.5. Output 95.5.3 displays the regression coefficients for the three quantiles, from which you can see a difference among the estimated parameters
of the variable *years* across the three quantiles. To test whether the difference is significant, you can specify the option QINTERACT in the TEST
statement. Output 95.5.4 indicates that the difference is not significant (the p-value is greater than 0.05).

Output 95.5.3: Regression Coefficients

The QUANTREG Procedure

Parameter Estimates | |||||||
---|---|---|---|---|---|---|---|

Parameter | DF | Estimate | Standard Error |
95% Confidence Limits | t Value | Pr > |t| | |

Intercept | 1 | 48.2509 | 1.3484 | 45.6011 | 50.9007 | 35.78 | <.0001 |

Years | 1 | 2.2234 | 0.5455 | 1.1514 | 3.2953 | 4.08 | <.0001 |

Years*Years | 1 | -0.1292 | 0.0500 | -0.2275 | -0.0308 | -2.58 | 0.0101 |

Years*Years*Years | 1 | 0.0024 | 0.0013 | -0.0001 | 0.0049 | 1.86 | 0.0634 |

Parameter Estimates | |||||||
---|---|---|---|---|---|---|---|

Parameter | DF | Estimate | Standard Error |
95% Confidence Limits | t Value | Pr > |t| | |

Intercept | 1 | 50.2512 | 1.2812 | 47.7334 | 52.7690 | 39.22 | <.0001 |

Years | 1 | 2.7173 | 0.5947 | 1.5485 | 3.8860 | 4.57 | <.0001 |

Years*Years | 1 | -0.1632 | 0.0632 | -0.2873 | -0.0390 | -2.58 | 0.0101 |

Years*Years*Years | 1 | 0.0034 | 0.0018 | -0.0002 | 0.0070 | 1.85 | 0.0647 |

Parameter Estimates | |||||||
---|---|---|---|---|---|---|---|

Parameter | DF | Estimate | Standard Error |
95% Confidence Limits | t Value | Pr > |t| | |

Intercept | 1 | 51.0298 | 1.5886 | 47.9078 | 54.1517 | 32.12 | <.0001 |

Years | 1 | 3.6513 | 0.7594 | 2.1590 | 5.1436 | 4.81 | <.0001 |

Years*Years | 1 | -0.2390 | 0.0764 | -0.3892 | -0.0888 | -3.13 | 0.0019 |

Years*Years*Years | 1 | 0.0055 | 0.0021 | 0.0013 | 0.0096 | 2.60 | 0.0098 |

Output 95.5.4: Tests for Heteroscedasticity

The three fitted quantile curves and their 95% confidence limits in the Output 95.5.5 clearly show that salary dispersion increases gradually with length of service. After 15 years, a salary more than $70,000 is relatively high, whereas a salary less than $60,000 is relatively low. Percentile curves of this type are useful in medical science as reference curves (Yu, Lu, and Stabder 2003).

Output 95.5.5: Salary and Years as Professor: Regression Quantiles