The QUANTREG Procedure

Analysis of Fish-Habitat Relationships

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 @@;
 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;

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 $Q(\tau | X=x) = \mb {x}^{\prime }\bbeta (\tau )$ be estimated by solving

\[  \hat\bbeta (\tau ) = {\arg \min }_{\bbeta \in \mb {R}^2}\sum _{i=1}^ n\rho _{\tau } (y_ i- \mb {x}_ i^{\prime } \bbeta ) \]

where $\tau =(0.5, 0.9)$.

By default, the regression coefficients $\hat\bbeta ( \tau )$ 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
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
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:

\[  {y}_{90} = \exp (0.0576 - 0.0215 x) \]

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 $\tau $ in the interval $(0,1)$. The following statements use the QUANTREG procedure to request the estimated quantile processes $\hat\bbeta (\tau )$ for the slope and intercept parameters:

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

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 $\tau $ in the interval $(0,1)$. The table also provides predicted values of the conditional quantile function $Q(\tau )$ at the mean for WDRatio, which can be used to estimate the conditional density function.

Figure 77.8: Objective Function

Objective Function for Quantile
Label Quantile Objective
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

Quantile Processes for Intercept and Slope