Quantile regression is used extensively in ecological studies (Cade and Noon, 2003). Recently, Dunham, Cade, and Terrell (2002) applied quantile regression to analyze fishhabitat 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 widthtodepth 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 widthtodepth 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 widthtodepth 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 pvalues 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
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 pvalues 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 widthtodepth 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 widthtodepth 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  ChiSquare  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).
Figure 77.10: Quantile Processes for Intercept and Slope