Quantile regression is used extensively in ecological studies (Cade and Noon, 2003). Recently, Dunham, Cade, and Terrell (2002) applied quantile regression to analyze fish-habitat relationships for Lahontan cutthroat trout in 13 streams of the eastern Lahontan basin, which covers most of northern Nevada and parts of southern Oregon. The density of trout (number of trout per meter) was measured by sampling stream sites from 1993 to 1999. The width-to-depth ratio of the stream site was determined as a measure of stream habitat.

The goal of this study was to explore the relationship between the conditional quantiles of trout density and the width-to-depth ratio. The scatter plot of the data in Figure 77.1 indicates a nonlinear relationship, and so it is reasonable to fit regression models for the conditional quantiles of the log of density. Since regression quantiles are equivariant under any monotonic (linear or nonlinear) transformation (Koenker and Hallock, 2001), the exponential transformation converts the conditional quantiles to the original density scale.

The data set `trout`

, which follows, includes the average numbers of Lahontan cutthroat trout per meter of stream (`Density`

), the logarithm of Density (`LnDensity`

), and the width-to-depth ratios (`WDRatio`

) for 71 samples.

data trout; input Density WDRatio LnDensity @@; datalines; 0.38732 8.6819 -0.94850 1.16956 10.5102 0.15662 0.42025 10.7636 -0.86690 0.50059 12.7884 -0.69197 0.74235 12.9266 -0.29793 0.40385 14.4884 -0.90672 0.35245 15.2476 -1.04284 0.11499 16.6495 -2.16289 0.18290 16.7188 -1.69881 0.06619 16.7859 -2.71523 0.70330 19.0141 -0.35197 0.50845 19.0548 -0.67639 ... more lines ... 0.25125 54.6916 -1.38129 ;

The following statements use the QUANTREG procedure to fit a simple linear model for the 50th and 90th percentiles of `LnDensity`

:

ods graphics on; proc quantreg data=trout alpha=0.1 ci=resampling; model LnDensity = WDRatio / quantile=0.5 0.9 CovB seed=1268; test WDRatio / wald lr; run;

The MODEL statement specifies a simple linear regression model with `LnDensity`

as the response variable Y and `WDRatio`

as the covariate X. The QUANTILE= option requests that the regression quantile function be estimated by solving

where .

By default, the regression coefficients are estimated with the simplex algorithm, which is explained in the section Simplex Algorithm. The ALPHA= option requests 90% confidence limits for the regression parameters, and the option CI=RESAMPLING specifies that
the intervals are to be computed with the MCMB resampling method of He and Hu (2002). By specifying the CI=RESAMPLING option, the QUANTREG procedure also computes standard errors, t values, and p-values of regression parameters with the MCMB resampling method. The SEED= option specifies a seed for the resampling method.
The COVB option requests covariance matrices for the estimated regression coefficients, and the TEST statement requests tests
for the hypothesis that the slope parameter (the coefficient of `WDRatio`

) is zero.

Figure 77.3 displays model information and summary statistics for the variables in the model. The summary statistics include the median and the standardized median absolute deviation (MAD), which are robust measures of univariate location and scale, respectively. See Huber (1981, p. 108) for more details about the standardized MAD.

Figure 77.3: Model Fitting Information and Summary Statistics

The QUANTREG Procedure

Model Information | |
---|---|

Data Set | WORK.TROUT |

Dependent Variable | LnDensity |

Number of Independent Variables | 1 |

Number of Observations | 71 |

Optimization Algorithm | Simplex |

Method for Confidence Limits | Resampling |

Summary Statistics | ||||||
---|---|---|---|---|---|---|

Variable | Q1 | Median | Q3 | Mean | Standard Deviation |
MAD |

WDRatio | 22.0917 | 29.4083 | 35.9382 | 29.1752 | 9.9859 | 10.4970 |

LnDensity | -2.0511 | -1.3813 | -0.8669 | -1.4973 | 0.7682 | 0.8214 |

Figure 77.4 and Figure 77.5 display the parameter estimates, standard errors, 95% confidence limits, t values, and p-values that are computed by the resampling method.

Figure 77.4: Parameter Estimates at QUANTILE=0.5

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

Parameter | DF | Estimate | Standard Error | 90% Confidence Limits | t Value | Pr > |t| | |

Intercept | 1 | -0.9811 | 0.3952 | -1.6400 | -0.3222 | -2.48 | 0.0155 |

WDRatio | 1 | -0.0136 | 0.0123 | -0.0341 | 0.0068 | -1.11 | 0.2705 |

Figure 77.5: Parameter Estimates at QUANTILE=0.9

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

Parameter | DF | Estimate | Standard Error | 90% Confidence Limits | t Value | Pr > |t| | |

Intercept | 1 | 0.0576 | 0.2606 | -0.3769 | 0.4921 | 0.22 | 0.8257 |

WDRatio | 1 | -0.0215 | 0.0075 | -0.0340 | -0.0091 | -2.88 | 0.0053 |

The 90th percentile of trout density can be predicted from the width-to-depth ratio as follows:

This is the upper dashed curve plotted in Figure 77.1. The lower dashed curve for the median can be obtained in a similar fashion.

The covariance matrices for the estimated parameters are shown in Figure 77.6. The resampling method used for the confidence intervals is used to compute these matrices.

Figure 77.6: Covariance Matrices of the Estimated Parameters

Estimated Covariance Matrix for Quantile = 0.5 |
||
---|---|---|

Intercept | WDRatio | |

Intercept | 0.156191 | -.004653 |

WDRatio | -.004653 | 0.000151 |

Estimated Covariance Matrix for Quantile = 0.9 |
||
---|---|---|

Intercept | WDRatio | |

Intercept | 0.067914 | -.001877 |

WDRatio | -.001877 | 0.000056 |

The tests requested with the TEST statement are shown in Figure 77.7. Both the Wald test and the likelihood ratio test indicate that the coefficient of width-to-depth ratio is significantly different from zero at the 90th percentile, but the deference is not significant at the median.

Figure 77.7: Tests of Significance

Test Results | |||||
---|---|---|---|---|---|

Quantile | Test | Test Statistic | DF | Chi-Square | Pr > ChiSq |

0.5 | Wald | 1.2339 | 1 | 1.23 | 0.2666 |

0.5 | Likelihood Ratio | 1.1467 | 1 | 1.15 | 0.2842 |

0.9 | Wald | 8.3031 | 1 | 8.30 | 0.0040 |

0.9 | Likelihood Ratio | 9.0529 | 1 | 9.05 | 0.0026 |

In many quantile regression problems it is useful to examine how the estimated regression parameters for each covariate change as a function of in the interval . The following statements use the QUANTREG procedure to request the estimated quantile processes for the slope and intercept parameters:

proc quantreg data=trout alpha=0.1 ci=resampling; model LnDensity = WDRatio / quantile=process seed=1268 plot=quantplot; run;

The QUANTILE=PROCESS option requests an estimate of the quantile process for each regression parameter. The options ALPHA=0.1 and CI=RESAMPLING specify that 90% confidence bands for the quantile processes are to be computed with the resampling method.

Figure 77.8 displays a portion of the objective function table for the entire quantile process. The objective function is evaluated at
77 values of in the interval . The table also provides predicted values of the conditional quantile function at the mean for `WDRatio`

, which can be used to estimate the conditional density function.

Figure 77.8: Objective Function

Objective Function for Quantile Process |
|||
---|---|---|---|

Label | Quantile | Objective Function |
Predicted at Mean |

t0 | 0.005634 | 0.7044 | -3.2582 |

t1 | 0.020260 | 2.5331 | -3.0331 |

t2 | 0.031348 | 3.7421 | -2.9376 |

t3 | 0.046131 | 5.2538 | -2.7013 |

. | . | . | . |

. | . | . | . |

. | . | . | . |

t73 | 0.945705 | 4.1433 | -0.4361 |

t74 | 0.966377 | 2.5858 | -0.4287 |

t75 | 0.976060 | 1.8512 | -0.4082 |

t76 | 0.994366 | 0.4356 | -0.4082 |

Figure 77.9 displays a portion of the table of the quantile processes for the estimated parameters and confidence limits.

Figure 77.9: Objective Function

Parameter Estimates for Quantile Process | |||
---|---|---|---|

Label | Quantile | Intercept | WDRatio |

. | . | . | . |

. | . | . | . |

. | . | . | . |

t57 | 0.765705 | -0.42205 | -0.01335 |

lower90 | 0.765705 | -0.91952 | -0.02682 |

upper90 | 0.765705 | 0.07541 | 0.00012 |

t58 | 0.786206 | -0.32688 | -0.01592 |

lower90 | 0.786206 | -0.80883 | -0.02895 |

upper90 | 0.786206 | 0.15507 | -0.00289 |

. | . | . | . |

. | . | . | . |

. | . | . | . |

The PLOT=QUANTPLOT option in the MODEL statement, together with ODS Graphics, requests a plot of the estimated quantile processes.
The left side of Figure 77.10 displays the process for the intercept, and the right side displays the process for the coefficient of `WDRatio`

.

The process plot for `WDRatio`

shows that the slope parameter changes from positive to negative as the quantile increases, and it changes sign with a sharp
drop at the 40th percentile. The 90% confidence bands show that the relationship between `LnDensity`

and `WDRatio`

(expressed by the slope) is not significant below the 78th percentile. This situation can also be seen in Figure 77.9, which shows that 0 falls between the lower and upper confidence limits of the slope parameter for quantiles below 0.78.
Since the confidence intervals for the extreme quantiles are not stable due to insufficient data, the confidence band is not
displayed outside the interval (0.05, 0.95).