Math 141 - Quantile-Quantile Plots - Reed Collegepeople.reed.edu/~jones/Courses/P14.pdfOrder...

Math 141Quantile-Quantile Plots

Albyn Jones1

1Library [email protected]

www.people.reed.edu/∼jones/courses/141

Albyn Jones Math 141

Outline: Quantile-Quantile Plots

Quantiles and Order Statistics

Quantile-Quantile Plots

Normal Quantile Plots


Quantiles and Order Statistics

Definition: p-th quantile qpEarlier we defined the p-th quantile of the distribution of a RV Xas any number qp satisfying

P(X ≤ qp) ≥ p

andP(X ≥ qp) ≥ (1− p).


Order Statistics

Sample quantiles are based on Order Statistics:Let X1,X2, . . . ,Xn be a sample of size n. The order statisticsX(1),X(2), . . . ,X(n) are just the observations sorted intoascending order.

# The data> x[1] 76 92 83 105 102 109 106 91 110 89

# The order statistics> sort(x)[1] 76 83 89 91 92 102 105 106 109 110


Order Statistics and Sample Quantiles

There are numerous definitions of sample quantiles chosen toperform well under various conditions. All involve interpolationbetween neighboring order statistics.

Suppose that we want the pth quantile, where p liesbetween k/n and (k + 1)/n. There are variations, but allare, for some choice of a ∈ [0,1]:

q̂p = aX(k) + (1− a)X(k+1)

Example, when n is even, the sample median q.5 is usuallytaken to be the average of the two middle observations.


Order Statistics as Sample Quantiles

Let’s turn it around, and ask what sample quantiles correspondto the order statistics?

Consider 4 observations from a population. On the average, weexpect them to at least approximately divide the population intoequal chunks corresponding to equally spaced percentiles:

15,25,35,45

In other words, they correspond to the sample quantiles

q.2,q.4,q.6,q.8


Comparing Two Samples

Suppose we have two samples of size n, X1,X2, . . .Xn andY1,Y2, . . .Yn.

If they were samples from the same distribution, then theorder statistics X(1),X(2), . . .X(n) and Y(1),Y(2), . . .Y(n)would be estimates of the same quantiles.Thus we expect that X(1) ≈ Y(1), X(2) ≈ Y(2), etc.




If they were samples from the same distribution, then theorder statistics X(1),X(2), . . .X(n) and Y(1),Y(2), . . .Y(n)would be estimates of the same quantiles.

Thus we expect that X(1) ≈ Y(1), X(2) ≈ Y(2), etc.




If they were samples from the same distribution, then theorder statistics X(1),X(2), . . .X(n) and Y(1),Y(2), . . .Y(n)would be estimates of the same quantiles.Thus we expect that X(1) ≈ Y(1), X(2) ≈ Y(2), etc.


The QQ plot

The quantile–quantile plot, or QQplot, is a simple graphicalmethod for comparing two sets of sample quantiles. Plot thepairs of order statistics

(X(k),Y(k)).

If the two datasets come from the same distribution, the pointsshould lie roughly on a line through the origin with slope 1.


QQ plot example

● ●

●●

●●

●●●

●●●●●●●●

●●●●●●●●●●

●●●●●●●●●● ●●

●●●●●●●●●●●●●●

●●●●●●●●

●●●●●●●●●●●●●

●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●

●●●●●●●●●●

●●●●●●●●●●●

●●●●●●●●●●

●●●●●●●●●●●●●●●●●

●●●●●●●●●●

●● ●

●●

●

●

−3 −2 −1 0 1 2

−3

−2

−1

01

23

X

Y

QQplot of two N(0,1) samples of size 200


QQ plot example, small sample!

Alert!! With small samples, expect variation!

●

●

●

●

●

●●

●

●

●

● ●● ●

●

●

●

●

●

●

−3 −2 −1 0 1

−2.

0−

1.5

−1.

0−

0.5

0.0

0.5

1.0

X

Y

QQplot of two N(0,1) samples of size 20


QQ plot example: location shift

Two samples from similar distributions which differ only inlocation: the green reference line is y = x .

●

●

●

● ●●●●

●●●●●

●●●●●

●●●●●●●●●●●●●●

●●●●●●●●●

●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●

●●●●●●●●●●●

●●●●●●●●●●●

●●●●● ●●●●●●

●●●●●

●●● ● ● ●

●●

−2 −1 0 1 2

01

23

45

6

X

Y

QQplot of two samples of size 200


QQ plot example: different spread

Two samples from similar distributions which differ only inspread, with reference line.

●

●

●

● ●●●●

●●●●●

●●●●●

●●●●●●●●●●●●●●

●●●●●●●●●

●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●

●●●●●●●●●●●

●●●●●●●●●●●

●●●●● ●●●●●●

●●●●●

●●● ● ● ●

●●

−2 −1 0 1 2

−6

−4

−2

02

46

X

Y



QQ plot example: different shape

Two samples from distributions which differ in shape, as well aslocation and spread, with reference line.

●

●●●●● ●

●●●●●

●●●● ●●●●●●

●●●●●●●

●●●●●●●●●●●

●●● ●●●●●●●●●

●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●

●●●●●●●●●●●●●●●

●●●●●●●

●●●●●●

●●●●●

●●●●●●●●

●●●●●●●●

●

●

●

●

●

●

0.0 0.2 0.4 0.6 0.8 1.0

−2

−1

01

23

X

Y



QQ plot example: Anorexia data

The Family Therapy group had 17 subjects, the Control Therapy26. qqplot() uses estimated quantiles for the larger dataset.

●

●

●

●

●

● ●

●

●●

●●

●

●● ●

●

−10 −5 0 5 10 15

−5

05

1015

20

Control

Fam

ily

QQplot of Family Therapy vs Control


Normal Quantile Plots

Often we wish to compare a dataset to the Normal distribution,a theoretical population, rather than to a second dataset. R hasa function that plots the order statistics of a sample against thecorresponding quaintiles of the standard normal distribution:

qqnorm(X )

If the plot is roughly linear, then our data are approximatelynormally distributed.


Normal Quantile Plot: Normal data

Don’t expect a perfectly straight plot even with normal data!

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

−2 −1 0 1 2

05

1015

20

Normal Q−Q Plot

Theoretical Quantiles

Sam

ple

Qua

ntile

s


Normal Quantile Plot: Mean and SD

We can estimate the mean and SD from a Normal quantile plot:the mean is roughly equal to the median (plotted above 0), andthe slope is roughly the SD.

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

−2 −1 0 1 2

05

1015

20

Normal Q−Q Plot


Sam

ple

Qua

ntile

s

rise: 4.8


Normal Quantile Plot: Short Tails

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

−2 −1 0 1 2

−5

05

1015

2025

Normal Q−Q Plot


Sam

ple

Qua

ntile

s


Normal Quantile Plot: Short Tails

Short tails are hard to diagnose from a density plot!

−5 0 5 10 15 20 25

0.00

0.01

0.02

0.03

0.04

0.05

0.06

Short Tails

N = 50 Bandwidth = 2.436

Den

sity


Normal Quantile Plot: Long Tails

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

−2 −1 0 1 2

−2

−1

01

2

Normal Q−Q Plot


Sam

ple

Qua

ntile

s


Density Plot: Long Tails

−2 0 2 4

0.0

0.1

0.2

0.3

0.4

0.5

Long Tails


Den

sity


Normal Quantile Plot: Positive Skewness

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●

●

●

●

●

●

●

●

●

−2 −1 0 1 2

02

46

810

12

Normal Q−Q Plot


Sam

ple

Qua

ntile

s


Density Plot: Positive Skewness

0 5 10 15

0.00

0.05

0.10

0.15

Positive Skewness


Den

sity


Normal Quantile Plot: Negative Skewness

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

−2 −1 0 1 2

68

1012

14

Normal Q−Q Plot


Sam

ple

Qua

ntile

s


Normal Quantile Plot: Bimodal Data

●

●●

●●

●

●

●

●●

●

●

●

●

●● ●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●

●

●

●

●

●

●

●●

−2 −1 0 1 2

24

68

1012

Normal Q−Q Plot


Sam

ple

Qua

ntile

s


Density Plot: Bimodal Data

0 5 10 15

0.00

0.05

0.10

0.15

Bimodal data


Den

sity


Other distributions

Suppose we would like to make a theoretical quantile plot for adataset X to compare to some other distribution, say aChisquared distribution with 5 degrees of freedom. Easy:

Sort your dataset:SampleQuantiles <- sort(X)

Compute the theoretical quantiles:ChiSqQuantiles <- qchisq(ppoints(X),5)

Plot:plot(ChiSqQuantiles,SampleQuantiles,

pch=19)


Gamma Quantile Plot

●

● ●

● ●

● ●

●

● ● ●

●●

● ●●

● ●●

●

●

●

●

●

●

2 4 6 8 10

24

68

10

Gq

sort

(X)

Gamma Quantile Plot


Summary

QQplots are an excellent graphical tool for comparing twosamples to each other, or one sample to a theoreticaldistribution like the Normal. They reveal differences in location,spread and shape more clearly than do density plots orhistograms.


Math 141 - Quantile-Quantile Plots - Reed Collegepeople.reed.edu/~jones/Courses/P14.pdfOrder...

Documents

Transcript of Math 141 - Quantile-Quantile Plots - Reed Collegepeople.reed.edu/~jones/Courses/P14.pdfOrder...