presented by dr. neil w. polhemus
TRANSCRIPT
Quantile Regression
• Constructs linear models for predicting specified quantiles.
• Useful when:
– primary interest concerns a percentile of the distribution rather than the mean.
– the distribution of the data at a specified combination of the predictor variables is not Gaussian.
– the variance of Y depends on X.
– there are outliers present.
Applications
• Growth curves
• Ecology
• Epidemiology
• Health services utilization
• CEO pay
• Household income
• Home prices
• Sea ice extent
• Astrophysics
• Chemistry
• Genomics
• Waiting times
• Product reliability
Basic Model Structure
Qt(Y): conditional t-th quantile of dependent
variable Y
X1, X2, … Xp: predictor variables
Note that the coefficients depend on t.
𝑄𝜏 𝑌 = 𝛽0 𝜏 + 𝛽1 𝜏 𝑋1 + 𝛽2 𝜏 𝑋2 +⋯+ 𝛽𝑝 𝜏 𝑋𝑝 + 𝜖
Statgraphics
• Statgraphics uses the quantreg program in R to fit
models.
• Quantreg was written by R. Koenker.
• You should download the latest build (19.2.02)
which includes a few tweaks to the Quantile
Regression procedure.
Example #1
• First example is taken from the Journal of Statistics
Education Data Archive.
• Information about 247 men and 267 women
sampled at fitness centers in California:
– 21 body dimension measurements
– age
– height
– weight
– gender
Comparison of Regression Lines
Genderfemalemale
Plot of Fitted Model
58 62 66 70 74 78
Height
90
120
150
180
210
240
270
Weig
ht
Methods
• Barrodale and Roberts – for up to several
thousand observations.
• Frisch-Newton – useful for larger problems.
• Frisch-Newton with preprocessing – useful for
even larger problems. Best for large n and small p.
• Frisch-Newton with sparse algebra – useful for
large n and large p.
Effect of Gender
Estimated Quantiles
Height=68.0,Age=40.0
female male
Gender
90
120
150
180
210
240
270
Weig
ht
QuantileQ95Q90Q75Q50Q25Q10Q5
Effect of Height
Estimated Quantiles
Gender=1,Age=40.0
58 62 66 70 74 78
Height
90
120
150
180
210
240
270
Weig
ht
QuantileQ95Q90Q75Q50Q25Q10Q5
Effect of Age
Estimated Quantiles
Gender=1,Height=68.0
18 28 38 48 58 68
Age
90
120
150
180
210
240
270
Weig
ht
QuantileQ95Q90Q75Q50Q25Q10Q5
Comparing Quantile Coefficients
Estimated Coefficients for Age
with 95% confidence intervals
0 0.2 0.4 0.6 0.8 1
Quantile
-0.2
0.1
0.4
0.7
1
1.3C
oeff
icie
nt
Example #2
• Second example is also taken from the Journal of
Statistics Education Data Archive.
• Information about 93 makes and models of
automobiles manufactured in 1993
– Price of automobile
– Weight
Scatterplot
1600 2100 2600 3100 3600 4100 4600
Weight
0
20
40
60
80
Mid
Pri
ce
Plot of Mid Price vs Weight
Estimated Quantiles
Estimated Quantiles
1600 2100 2600 3100 3600 4100 4600
Weight
0
20
40
60
80
Mid
Pri
ce
QuantileQ95Q90Q75Q50Q25Q10Q5
Comparing Quantile Coefficients
Estimated Coefficients for Weight
with 95% confidence intervals
0 0.2 0.4 0.6 0.8 1
Quantile
-4
6
16
26
36(X 0.001)
Co
eff
icie
nt
References
• StatFolios and data files are at: www.statgraphics.com/webinars
• R Package “quantreg” (2021) https://cran.r-project.org/web/packages/quantreg/quantreg.pdf
• Quantile Regression (2005) Roger Koenker. (Econometric Society Monographs, Series Number 38)
• Body and car data from: http://jse.amstat.org/jse_data_archive.htm