Exploratory Data Analysis

Hal Varian20 March 2006

What is EDA? Goals

Examine and summarize data Look for patterns and suggest hypotheses Provide guidance for more systematic analysis

Methods of analysis Primarily graphics and tables Online reference

http://www.itl.nist.gov/div898/handbook/eda/eda.htm http://www.math.yorku.ca/SCS/Courses/eda/

Tools for EDA We will use R = open source S

Very widely used by statisticians Libraries for all sorts of things are

available Download from

cran.stat.ucla.edu http://www.r-project.org/

Recommend ESS (=Emacs Speaks Statistics) for interactive use

Windows interface is not bad

Interactive R session

> library("foreign")

> dat <- read.spss("GSS93 subset.sav")

> attach(dat)

> summary(AGE)

Min. 1st Qu. Median Mean 3rd Qu. Max.

18.0 33.0 43.0 46.4 59.0 99.0 > hist(AGE)

Histogram of ageHistogram of AGE

AGE

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Recode missing data AGE[AGE>90] <- NA plot(density(AGE,na.rm=T))

#plot both together hist(AGE,freq=F) lines(density(AGE,na.rm=T))

Density and density + hist

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density(x = AGE, na.rm = T)

N = 1495 Bandwidth = 3.633

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Histogram of AGE

AGE

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Boxplot Boxplot

Outlier 1.5 interquartile range 3rd quartile Median 1st quartile Smallest value 20

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Boxplot enhancements Notches: confidence interval for

median Varwidth=T: width of box is sqrt(n) Useful for

comparisons2

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Comparing distributions boxplot(AGE~RACE) boxplot(AGE~RACE,notch=T,varwidth=T)

Doesn’t seem to be big diff in age distn

white black other

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EDUC v RACEboxplot(EDUC[EDUC<90]~RACE[EDUC<90],notch=T,varwidth=T)

other black white

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Violin plot Combines density plot and boxplot Good for weird shaped

distributions…

Back to Back Histogram library("Hmisc") histbackback(EDUC[RACE=="black"],EDUC[RACE=="white"],probability=T)

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EDUC[RACE == "black"] EDUC[RACE == "white"]

Two-way table GT12 <- EDUC>12 temp <-table(GT12,RACE)

GT12 white black other FALSE 614 100 37 TRUE 640 67 38

prop.table(temp,2) GT12 white black other FALSE 0.4896332 0.5988024 0.4933333 TRUE 0.5103668 0.4011976 0.5066667

Comparing distributions qqplot = quantile-quantile plot

Fraction of data less than k in x Fraction of data less than k in y

Shapes Straight line: same distribution Vertical intercepts differ: different mean Slopes differ: different variance

Reference distribution can be theoretical distn qnorm – compare to standardized normal Skew to right: both tails below straight line Heavy tails: lower tail above, upper tail below line

qqplot(x,y) examples

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1=12=2

identical

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Normal Q-Q Plot

Theoretical Quantiles

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Sample vN(0,1),with refline

More qqnorm examples

Skewed to right Heavy tails

www.maths.murdoch.edu.au/units/statsnotes/samplestats/qqplot.html

Pairs of variables Is one variable related to another? Scatterplot

Basic: plot(x,y) Enhanced from library(“car”):

scatterplot(x,y) Scatterplot matrix

Basic: pairs(data.frame(x,y,z)) Enhanced:

scatterplot.matrix(data.frame(x,y,z))

Basic and enhanced scatterplot

Scatterplot matrix

Labeling points in scatterplots identify(x,y,labels=“foo”) Color is also useful

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Cigarettes and taxes Discussant on paper by Austan

Goolsbee, “Playing with Fire” Question: did Internet purchases of

cigarettes affect state tobacco tax revenues?

Cigarette Prices in 1990s

1990 1992 1994 1996 1998 2000

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Price of cigarettes

Internet usage

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Internet usage

Price elasticity of use/sales Across all states and years

Taxable sales elasticity: -0.802 Use elasticity: -0.440

Sales are much more responsive to price than usage suggesting that there is some cross border trade (aka “buttlegging”)

Use vs Sales in 2000

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q.p[year == 2000]

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Reduced form dp = log(p2001) – log(p1995) dq = log(q2001) – log(q1995) Regress dq/dp on internet

penetration in 2000 See next slide for result

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Elasticity v Internet penetration

What is Internet providing? It was always a good deal for some to buy

cigarettes out-of-state (in high tax states) Mail order has been around for a long time

and is certainly cost-effective Internet makes it easier to find merchants

– just type into search engine Internet is great at matching buyers and

sellers

Price of a match Google doesn’t accept cigarette

advertisements, but Overture does Price for top listing: $1.20 per click

Avg price for click on Overture is 40 cents

Conversion rates might be 5%, so advertiser is paying $24 for introduction

But think of lifetime value…

Value of a match Google doesn’t accept cigarette

advertisements, but Overture does Price for top listing: $1.20 per click

Avg price for click on Overture is 40 cents

Conversion rates might be 5%, so advertiser is paying $24 for introduction

But think of lifetime value…

Straightening out and scaling data Find transform so that data looks

linear, or normal, or fits on same scale Log10 (easier to interpret than log) Square root Reciprocal Box-Cox transform (xr – 1)/r which

combines many of above; r=0 is log

City sizes: regular & log10

Histogram of log10(pop1980)

log10(pop1980)

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