# Introduction to Quantile Regression

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Taupo, Biometrics 2009

Introduction to Quantile
Regression

David Baird

VSN NZ, 40 McMahon Drive,

Christchurch, New Zealand

email: David@vsn.co.nz

Taupo, Biometrics 2009

Reasons to use quantiles rather than means

Analysis of distribution rather than
average

Robustness

Skewed data

Interested in representative value

Interested in tails of distribution

Unequal variation of samples

E.g. Income distribution is highly skewed so
median relates more to typical person that mean.

Taupo, Biometrics 2009

Quantiles

Cumulative Distribution Function

Quantile Function

Discrete step function

)
Prob(
)
(
y
Y
y
F

)
)
(
:
min(
)
(

y
F
y
Q
CDF
1.0
0.6
0.2
2.01.51.00.50.0
0.4
-0.5-1.0
0.0
0.8
-1.5-2.0
Quantile (n=20)
-1.0
-1.5
1.0
0.0
1.00.8
1.5
0.6
0.5
0.40.2
-0.5

Taupo, Biometrics 2009

Optimality Criteria

Linear absolute loss

Mean optimizes

Quantile
τ

optimizes

I
= 0,1 indicator function

i
y
min

i
i
i
i
y
e
e
I
e
)
0
(
min
-1
1
0
-1
1
0

1

Taupo, Biometrics 2009

Regression Quantile

X
y
e
e
I
e
i
i
i
i

)
0
(
min

Optimize

Solution found by Simplex algorithm

split
e
i

into positive and

negative residuals

Solution at vertex of feasible region

May be non
-
unique solution (along edge)

-

so solution passes through
n

data points

0

,
0

i
i
i
i
i
v
u
v
u
e

Taupo, Biometrics 2009

Simple Linear Regression

Food
Expenditure
vs Income

Engel 1857

survey of 235
Belgian households

Range of
Quantiles

Change of
slope at
different
quantiles?

Taupo, Biometrics 2009

Variation of Parameter with Quantile

Taupo, Biometrics 2009

Estimation of Confidence Intervals

Asymptotic approximation of
variation

Bootstrapping

Novel approach to bootstrapping by
reweighting rather than resampling

W
i

~ Exponential(1)

Resampling is a discrete
approximation of

exponential weighting

Avoids changing

design points so

faster and identical

quantiles produced

5
7
0
3
1
6
4
2
32 54 610 7

Taupo, Biometrics 2009

Bootstrap Confidence Limits

Taupo, Biometrics 2009

Polynomials

Support

points

Taupo, Biometrics 2009

Groups and interactions

Taupo, Biometrics 2009

Splines

Generate basis functions

10 30 50200 40 60
Motorcycle Helmet data

Acceleration vs Time
from impact

Taupo, Biometrics 2009

Loess

Generate moving weights using
kernel and

specified

window width

Taupo, Biometrics 2009

Non
-
Linear Quantile Regression

Run Linear
quantile
regression

in non
-
linear
optimizer

Quantiles for
exponential
model

Taupo, Biometrics 2009

Example Melbourne Temperatures

Taupo, Biometrics 2009

Example Melbourne Temperatures

Taupo, Biometrics 2009

Wool Strength Data

5 Farms

Breaking
strength
and cross
-
sectional
area of
individual
wool fibres
measured

Taupo, Biometrics 2009

Fitted Quantiles

Taupo, Biometrics 2009

Fitted Quantiles

Taupo, Biometrics 2009

Fitted Quantiles

Taupo, Biometrics 2009

Fitted Quantiles

Taupo, Biometrics 2009

Fitted Quantiles

Taupo, Biometrics 2009

Wool Strength Data

Taupo, Biometrics 2009

Between Farm Comparisons

Taupo, Biometrics 2009

Software for Quantile Regression

SAS Proc QUANTREG

(experimental v 9.1)

R Package quantreg

GenStat 12 edition procedures:

RQLINEAR & RQSMOOTH

Menu: Stats | Regression | Quantile Regression

Taupo, Biometrics 2009

Reference

Roger Koenker, 2005.

Quantile Regression
,

Cambridge University Press.