# PROC QUANTREG – Modeling a Quantile not the Average

Proc QuantReg helps build quantile regression models.

The median regression is one such example. Often it may be necessary to build model for a specific percentile. This is where PROC QUANTREG becomes handy. Look at the picture that provides scatter plot of the measures of trout density in 71 test places in a stream where the WD Ratio provides the width and depth ratio. The hypothesis being that less the ratio (means at higher depths if the width is small or lower depth if the width is higher) we are likely to see more trout habitats (due to less likely disturbances).

I also plot the 90th percentile; we want to find out the relationship between density and WDRatio where 90% of the trout population habitats densities could be estimated. That is for a given WD Ratio, we want to find how much density of trout habitats will be found at 90th percentile. Note that if we were building linear regression for example, it will be the red line. But we are interested in building a quantile regression, a function, which best fits the red dots.

If you plot Median or average where will the red dots be? That will give indications as to which curve we are modeling and how it is different from the 90% curve.

If we want to model this data for 90% regression then the SAS codes are as follows. An attempt is made to take key points from the reference mentioned at the bottom of this notes.

proc quantreg data=trout alpha=0.01 ci=resampling;
model LnDensity = WDRatio / quantile=0.9
CovB CorrB
seed=12345;
test WDRatio;
run;

The output is
The QUANTREG Procedure
Model Information
Data Set WORK.TROUT
Dependent Variable LnDensity LOG(Density)

Number of Independent Variables 1
Number of Observations 71
Optimization Algorithm Simplex
Method for Confidence Limits Resampling
Summary Statistics
Standard

Variable Q1 Median Q3 Mean 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

The QUANTREG Procedure
Quantile and Objective Function
Quantile 0.9
Objective Function 7.2303
Predicted Value at Mean -0.5709
Parameter Estimates
Standard 99% Confidence
Parameter DF Estimate Error Limits t Value Pr > t
Intercept 1 0.0576 0.2727 -0.6648 0.7801 0.21 0.8333
WDRatio 1 -0.0215 0.0073 -0.0408 -0.0022 -2.96 0.0042

Testing does not come out as a standard output; one has to mention that in the command list, as in above. The PROC uses SIMPLEX method for minimizing the error and Monte Carlo Marginal Bootstrap method for confidence interval and testing.

Another Example:
proc quantreg data=salary ci=none;
model salaries = years years*years years*years*years
/quantile=.25 .5 .75;
run;

The data plot and the quantile regression fit is shown here.

For many other examples, and some good amount of theory, see the pdf article, from where most of the material is summarized here.
The QUANTREG Procedure – Power Point Presentation (Experimental)
PROC QUANTREG – Experimental – SAS document – Material contributing to the above PPT.
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