Statistics Review slides

7/23/2019 Statistics Review slides

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PLCY 2305:

Statistics II for Public PolicyCourse Meeting: Tuesdays, 5:30!"

Lab Meeting: Mondays, #2!#

$%ce &ours: T'ursdays, (!)() *eorge Street

P+$-SS$+: ./Y/TI $1-S./Y/TI$1-S4+$1-67

T/: 4+7$ */SP-+II

4+7$*/SP-+II4+$1-67
mailto:[email protected]:[email protected]:[email protected]:[email protected]


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4rief $8er8ie9 of t'e CourseMuc' of social science analysis is about 'o9 to

understand iortant relations'is, often 9it'olicy ilications, by estiating ;uantitati8eagnitudes of causal e


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T'is course is about using

data to easure causaleed test scores?

4ut alost al9ays 9e only 'a8e obser8ational

BnoneAeriental data= returns to education= cigarette rices

= onetary olicy

Most of t'e course deals 9it' di%culties arising fro using

obser8ational to estiate causal e


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Learn et'ods for estiating causal e


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Empirical problem: Class si>e andeducational outut

=Policy ;uestion: 1'at is t'e eeby one student er class? by "studentsHclass?

=1e ust use data to Jnd out Bis t'ere any9ay to ans9er t'is withoutdata?

+e8ie9 of Probability and StatisticsBS1 C'aters 2, 3


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T'e California Test Score

6ata Set/ll K!( and K!" California sc'ool districts Bn20

Nariables:=5t'grade test scores BStanford!Oac'ie8eent test, cobined at' and

reading, district a8erage=Student!teac'er ratio BST+ no ofstudents in t'e district di8ided by no full!tie e;ui8alent teac'ers


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Initial loo@ at t'e data:(You should already know how to interpret this table)

T'is table doesnt tell us anyt'ing about t'e relations'ibet9een test scores and t'e STR


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6o districts 9it' saller

classes 'a8e 'ig'er testscores?Scatterplot of test score 8 student!teac'er ratio

What does this figure show?


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If 9e get soe nuerical e8idence on 9'et'erdistricts 9it' lo9 ST+s 'a8e 'ig'er test scores,t'en 'o9 to analy>e?

# Coare a8erage test scores in districts 9it' lo9

ST+s to t'ose 9it' 'ig' ST+s BDestimationE

2 Test t'e DnullE 'yot'esis t'at t'e ean test

scores in t'e t9o tyes of districts are t'e sae,against t'e Dalternati8eE 'yot'esis t'at t'ey

di


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Initial data analysis: Coare districts9it' DsallE class si>es BST+ Q 20 8s

DlargeE BST+ R 20 class si>es:

1. Estimationof di


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# -stiation

G Bt'isis ust notation for

t'e

eans (5) G (500

)

Is t'is a large di


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2 &yot'esis testing

6i


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Coute t'e di


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3 ConJdence inter8al

/ O5W conJdence inter8al for t'e di


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1'at coes neAtZ

T'e ec'anics of estiation, 'yot'esis testing,and conJdence inter8als s'ould be failiar

T'ese concets eAtend directly to regression and its8ariants

4efore turning to regression, 'o9e8er, 9e 9ill

re8ie9 soe of t'e underlying t'eory of estiation,'yot'esis testing, and conJdence inter8als:=1'y do t'ese rocedures 9or@, and 9'y use t'ese rat'er

t'an ot'ers?

=1e 9ill re8ie9 t'e intellectual foundations of statistics and

econoetrics


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+e8ie9 of Statistical T'eory# The probability framework for statistical inference

2 -stiation

3 Testing

ConJdence Inter8als

The probability framework for statistical inference

a Poulation, rando 8ariable, and distribution

b Moents of a distribution Bean, 8ariance, standard

de8iation, co8ariance, correlationc Conditional distributions and conditional eans

d 6istribution of a sale of data dra9n randoly fro aoulation: Y#, Z, Yn


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Ba Poulation, rando

8ariable, and distributionPopulationT'e grou or collection of all ossible entities of interestBsc'ool districts

1e 9ill t'in@ of oulations as inJnitely large B[ is an

aroAiation to D8ery bigE

Random variable Y

uerical suary of a rando outcoe Bdistrict a8erage

test score, district ST+


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Population distribution of Y

T'e robabilities of di


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Bb Moents of a oulation distribution: ean,8ariance, standard de8iation, co8ariance, correlation

mean eAected 8alue BeAectation of Y

EBY

Y

long!run a8erage 8alue of Yo8er reeatedreali>ations of Y

variance EBYGY2

easure of t'e s;uared sread oft'e distribution

standard deviation !Y

Y2

variance


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"oments# $td%

skeness easure of asyetry of a distribution

skewness 0: distribution is syetric

skewnessV BQ 0: distribution 'as long rig't Bleft tail

kurtosis

easure of ass in tails

easure of robability of large 8alues

kurtosis 3: noral distribution

skewnessV 3: 'ea8y tails BDleptokurtotic)

E Y Y( )

3

Y3

E Y Y( )

4

Y

4


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2 rando 8ariables: oint

distributions andco8ariance+ando 8ariables&and''a8e a!oint distributionT'e covariancebet9een&and'is= co8B&,' E\B&G&B'G'] !&'

T'e co8ariance is a easure of t'e linear association bet9een

&and'F

Its units are units of&units of'

co8B&,' V 0 eans a ositi8e relation bet9een&and'

If&and'are indeendently distributed, t'en co8B&,' 0

Bbut not 8ice 8ersaT'e co8ariance of a r8 9it' itself is its 8ariance:

co8B&,& E\B&G&B&G&] E\B&G&2]

X

2


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If t'e co8ariance bet9een Test S$oreand STRisnegati8e:

So is t'e correlationZ


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T'e $orrelation $oe$ientis deJned in tersof t'e co8ariance:

corrB&,' r&'

# G# ^ corrB&,' ^ #

# corrB&,' # ean erfect ositi8e linear association# corrB&,' G# eans erfect negati8e linear association

# corrB&,' 0 eans no linear association

cov(X,Z)var(X)var(Z)

= XZ

X

Z


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The $orrelation $oe$ient measures linear asso$iation


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Bc Conditional distributions andconditional eans"onditional distributions

T'e distribution of Y, gi8en 8alueBs of soe ot'er rando8ariable,&

-A: t'e distribution of test scores, gi8en t'at ST+ Q 20

"onditional e#pectations and conditional moments

$onditional mean ean of conditional distribution= EBYU& Bimportant concept and notation

$onditional varian$e 8ariance of conditional distribution

Eample: EBTest s$oresUSTRQ 20 t'e ean of test scoresaong districts 9it' sall class si>es

The di$erence in means is the di$erence beteen themeans o% to conditional distributions&

EBTest s$oresUSTRQ 20 G EBTest s$oresUSTRR 20


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*onditional mean# $td%

$t'er eAales of conditional eans:

1ages of all feale 9or@ers BY 9ages,& gender

Mortality rate of t'ose gi8en aneAeriental treatent BY li8eHdieF&treatedHnot treated

If EB&U' EB&), t'en corrB&,' 0 Bnotnecessarily 8ice 8ersa 'o9e8er

The conditional mean is a 'possiblyne( term %or the %amiliar idea o% the


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Bd 6istribution of a sale ofdata dra9n randoly fro a

oulation: Y#,Z, Yn)e ill assume simple random samplingC'oose and indi8idual Bdistrict, entity at rando fro t'eoulation

Randomness and data

Prior to sale selection, t'e 8alue ofYis rando becauset'e indi8idual selected is rando

$nce t'e indi8idual is selected and t'e 8alue of Yis obser8ed,t'en Yis ust a nuber G not rando

T'e data set is BY#, Y2,Z, Yn, 9'ere Yi 8alue of Yfor t'e it'

indi8idual Bdistrict, entity saled


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+istribution of Y#,Z,Ynunder simplerandom samplin,

4ecause indi8iduals _# and _2 are selected atrando, t'e 8alue of Y#'as no inforation contentfor Y2

T'us:

=Y#and Y2are independently distributed

=Y#and Y2coe fro t'e sae distribution, t'at is,

Y#, Y2are identically distributed=T'at is, under sile rando saling, Y#and Y2

are indeendently and identically distributedBi.i.d.

=More generally, under sile rando saling,Y i # Z n are iid


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This framework allows ri,orous statisti$al inferen$esabout moments of population distributions usin, asample of data from that populationZ# T'e robability frae9or@ for statistical inference

2 Estimation

3 Testing

ConJdence Inter8als

Estimation

is t'e natural estiator of t'e ean 4ut:

a 1'at are t'e roerties of ?

b 1'y s'ould 9e use rat'er t'an soe ot'er estiator? oreAale:

= Y#Bt'e Jrst obser8ation

= aybe une;ual 9eig'ts G not sile a8erage

= edianBY#,Z, Yn

T'e starting oint is t'e saling distribution of Z

Y

Y

Y

Y


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Ba T'e saling

distribution ofis a rando 8ariable, and its roerties aredeterined byt'e sampling distributionof=T'e indi8iduals in t'e sale are dra9n at rando

=T'us t'e 8alues of BY#, Z, Yn are rando

=T'us functions of BY#, Z, Yn, suc' as , are rando: 'ad adi


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The samplin, distribution of

# $td%E#ample: Suose Yta@es on 0 or # Ba *ernoullirando

8ariable 9it' t'e robability distribution,Pr\Y 0] 022, PrBY# 0)"

T'en

EBY p# B# Gp 0 p 0)"

/6 E\YG EBY]2pB# Gp -remember this./

0)" B#G0)" 0#)

T'e saling distribution of deends on n

Consider n 2 T'e saling distribution of is,

= PrB 0 0222 005

= PrB 20220)" 03

= PrB # 0)"2 0(#

Y

Y

2

Y

Y

Y

Y

Y


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T'e saling distribution of 9'en Yis 4ernoulli Bp )":

Y


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T'ings 9e 9ant to @no9 about t'esaling distribution:

1'at is t'e ean of ?=If EB true )", t'en is an unbiasedestiatorof

1'at is t'e 8ariance of ?=&o9 does 8arB deend on nBfaous #Hnforula

6oes becoe close to9'en nis large?=La9 of large nubers: is a consistentestiator of

Gaears bell s'aed for nlargeZis t'isgenerally true?=In fact, Gis aroAiately norally distributed for n

large

BCentral Liit T'eore

Y Y

Y

YY

Y

Y

Y


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T'e ean and 8ariance of

t'e saling distribution of*eneral case G t'at is, for Yiiid fro any distribution, notust 4ernoulli:ean: EB EB Y

Nariance: 8arB E\ G EB ]2

E\ GY]2

E

E

Y

Y

1

nYi

i=1

n

1

nE(Y

i)

i=1

n

1

n

Yi=1

n

Y Y Y

1

n

Yi

i=1

n

Y

2

1

n(Y

i

Y)

i=1

n

2

Y


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So 8arB E

1

n(Y

i

Y)

i=1

n

2

Y

E1

n(Y

i

Y)

i=1

n

1

n(Y

j

Y)

j=1

n

1

n2

E (Yi

Y)(Y

j

Y)

j=1

n

i=1

n

1

n2cov(Y

i,Y

j)

j=1

n

i=1

n

1

n2 Y

2

i=1

n

2

Y

n


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"ean and varian$e of samplin,distribution of # $td%

EB Y

8arB

0mpli$ations:# is an unbiasedestiator ofYBt'at is, EB Y

2 8arB is in8ersely roortional to n

# t'e sread of t'e saling distribution isroortional to #H

2T'us t'e saling uncertainty associated 9it'is roortional to #H Blarger sales, lessuncertainty, but s;uare!root la9

Y

Y

Y

Y

2

n

YY

Y

n

nY


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T'e saling distribution of

9'en nis largeor sall sale si>es, t'e distribution of iscolicated, but if nis large, t'e salingdistribution is sile

# /s nincreases, t'e distribution of becoesore tig'tly centered aroundYBthe +a o%

+arge ,umbers

2 Moreo8er, t'e distribution of GYbecoesnoral

Bthe "entral +imit Theorem

Y

Y

Y

Y


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T'e 1aw of 1ar,e 2umbers:

/n estiator is consistentif t'e robability t'at its falls 9it'inan inter8al of t'e true oulation 8alue tends to one as t'esale si>e increases

If BY#,Z,Yn are iid and Q [, t'en is a consistent estiator

ofY, t'at is,Pr\U GYU Q]# as n[

9'ic' can be 9ritten, Y

BD YE eans D con8erges in robability toYE

Bthe math: as n[, 8arB 0, 9'ic' ilies t'at

Pr\U GYU Q 3] #

Y2 Y

Y

p

Y

Y Yp

Y

Y

2

n

Y


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T'e *entral 1imit Theorem

BCLT:If BY#,Z,Yn are iid and 0 Q Q [ , t'en 9'en nislarge t'e distribution of is 9ell aroAiated by anoral distribution

= is aroAiately distributed 2BY, BDnoraldistribution 9it' eanYand 8ariance HnE

= B GYH!Yis aroAiately distributed 2B0,# Bstandard

noral

=That is, standardized = =

is approximately distribted as ,!",#)

=The lar$er is n, the better is the approximation%

n Y

Y2

Y

Y

Y

2

n

Y

2

YYE(Y)

var(Y)

Y Y

Y/ n

Y E(Y )


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Same eample: saling distribution of :

YE(Y)

var(Y)


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Suary: T'e Saling 6istributionof

or Y#,Z,Yniid 9it' 0 Q Q [,

T'e eAact BJnite sale saling distribution of 'as eanYBD is an unbiased estiator ofYE and 8ariance Hn

$t'er t'an its ean and 8ariance, t'e eAact distribution of is

colicated and deends on t'e distribution of YBt'e oulationdistribution

1'en nis large, t'e saling distribution siliJes:

("a$ o% large n&'ers)

is aroiatel* N(0+1) (C",)

Y

Y2

Y

Y

2

Y

p

Y

YE(Y)

var(Y)

Y


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Bb 1'y 7se To -stiate

Y? isunbiased: EB Y is consistent: Y is t'e Dleast s;uaresE estiator ofYF sol8es,

so, inii>es t'e su of s;uared DresidualsEoptional derivation (also see 4pp% 5%6)

Set deri8ati8e to >ero and denote otial 8alue of mby :

or 7

Y

Y

Y

Y

Y

Yp

Y

Y

minm

(Yi

m)2

i=1

n

d

dm (Yi m)2

i=1

n

d

dm(Y

i m)2

i=1

n

2 (Yi m)i=1

n

Y

i=1

n

1

n

i

m= nm

m

1

nYi

i=1

n

Ym


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1'y 7se To -stiateY,

ctd 'as a saller 8ariance t'an all ot'er linear unbiased

estiators: consider t'e estiato r, , 9'ere

`ai are suc' t'at is unbiasedF t'en 8arB ^ 8arB

Broof: S1, C' #)

isnt t'e only estiator ofYG can you t'in@ of a tie you

ig't 9ant to use t'e edian instead?

-T ST-PS:#T'e robability frae9or@ for statistical inference

2-stiation

&%'ypothesis Testin$

ConJdence inter8als

Y

Y

1

1

n

Y i i

i

aYn

=

= Y

Y Y

Y


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&yot'esis Testing

T'e hypothesis testing robleBfor t'eean:

Ma@e a ro8isional decision based on t'ee8idence at 'and 9'et'er a null 'yot'esisis true, or instead t'at soe alternati8e'yot'esis is true

T'at is, test

=80: EBY Y,08s 8#: EBY VY,0B#!sided,V

=80: EBY Y,08s 8#: EBY QY,0B#!sided,


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Some terminolo,y for

testin, statisti$alhypotheses:p(value robability of dra9ing a statistic Beg at least asad8erse to t'e null as t'e 8alue actually couted 9it' yourdata, assuing t'at t'e null 'yot'esis is true

T'e signifcance levelof a test is a re!seciJed robability

of incorrectly reecting t'e null, 9'en t'e null is true"alculating the p-valuebased on :

p!8alue

1'ere is t'e 8alue of actually obser8ed Bnonrando

PrH

0

[Y Y,0

>Yact Y,0

!

Y

Y

Yact Y


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*al$ulatin, the p9value#

$td%To coute t'ep!8alue, you need t'e to @no9 t'e salingdistribution of , 9'ic' is colicated if nis sallIf nis large, you can use t'e noral aroAiation BCLT:

p!8alue ,

robability under leftrig't 2B0,# tails

9'ere std de8of t'e distribution of !YH .

Y

PrH

0

[Y Y,0

>Yact Y,0

!

PrH

0

[Y

Y,0

Y/ n

>Yact

Y,0

Y/ n

!

PrH

0

[Y

Y,0

Y

>Y

act Y,0

Y

!

Y Y n


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*al$ulatin, the p9value with!

Yknown:

or large n,p!8alue t'e robability t'at a 2B0,#rando 8ariable falls outside UB GY,0H U

In ractice, is un@no9n G it ust be estiated

Yact

Y

Y


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Estimator of the varian$e of

Y: Dsale 8ariance of YEact:

If BY#,Z,Yn are iid and EBY Q [ , t'en

1'y does t'e la9 of large nubers aly?=4ecause is a sale a8erageF see /endiA 33

=Tec'nical note: 9e assue EBY Q [ because 'ere t'e

a8erage is not of Yi, but of its s;uareF see / 33

sY2

1

n 1 (Yi Y)2

i=1

n

sY2p

Y2

sY2

2


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*omputin, the p9value with estimated:

p!8alue ,

Blarge n

so

robability under noral tails t'at is outside Uta$tU

9'ere t Bt'e usual t!statistic

.

p-val&e . ( estimated)

.

Y2

PrH0 [Y

Y,0 >Yact Y,0 !

PrH

0

[Y

Y,0

Y/ n

>Yact

Y,0

Y/ n

!

PrH0[Y

Y,0

sY/ n

>Yact

Y,0

sY/ n

!

Y Y,0

sY/ n

Pr

H0

[ t> tact ! Y2


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1'at is t'e lin@ bet9een t'ep!8alueand t'e signiJcance le8el?

T'e signiJcance le8el is reseciJed oreAale, if t'e reseciJed signiJcancele8el is 5W,

=You reect t'e null 'yot'esis if UtU R #O(

=-;ui8alently, you reect ifp^ 005

=T'ep!8alue is soeties called t'e marginalsignifcance level

=$ften, it is better to counicate t'ep!8alue t'an sily9'et'er a test reects or not G t'ep!8alue contains ore

inforation t'an t'e DyesHnoE stateent about 9'et'ert'e test re ects


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/t t'is oint, you ig't be 9ondering,1'at 'aened to t'e t!table and t'e degreesof freedo?

The Stdent t distribtion

If Yi, i #,Z, nis iid, 2BY, , t'en t'e t!statistic

'as t'e Student t!distribution 9it' nG # degrees of

freedoT'e critical 8alues of t'e Student t!distribution istabulated in t'e bac@ of all statistics boo@s+eeber t'e recie?

# Coute t'e t!statistic

2 Coute t'e degrees of freedo, 9'ic' is nG#

3 Loo@ u t'e 5W critical 8alue

If t'e t!statistic eAceeds in absolute 8alue

Y2


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Coents on t'is recie

and t'e Studentt!distribution# T'e t'eory of t'e t!distribution 9as one of t'e earlytriu's of at'eatical statistics

It is astounding, really: if Yis iid noral, t'en you can@no9 t'e ea$t, nite9sampledistribution of t'e t!statistic Git is t'e Student t

So, you can construct conJdence inter8als Busing t'eStudent tcritical 8alue t'at 'a8e ea$tlyt'e rig'tco8erage rate, no atter 9'at t'e sale si>e

T'is result 9as really useful before couters 9it' sallsale si>es

4utZ


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*omments on Student tdistribution# $td%2 If t'e sale si>e is oderate Bse8eral do>en or large

B'undreds or ore, t'e di


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d di ib i


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*omments on Student t distribution#$td%3 So, t'e Student!tdistribution is only rele8ant

9'en t'e sale si>e is 8ery sallF4ut e8en so, you ust be sure t'at t'e

oulation distribution of Yis noral

In ost data, t'e norality assution is rarely

credible

or eAale:= /re earnings are norally distributed?

* S d di ib i


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*omments on Student t distribution#$td% Pooled Nariance: Consider t'e t!statistic testing t'e

'yot'esis t'at t9o eans Bgrous s, l are e;ual:

-8en if t'e oulation distribution of Yin t'e t9o grous isnoral, t'e ooled 8ariance statistic doesnt 'a8e aStudent tdistribution

T'e ooled 8ariance t!statistic is only 8alid if t'e 8ariancesof t'e noral distributions are t'e sae in t'e t9o grous

1ould you eAect t'is to be true, say, for ens 8

t=Ys Y

l

ss2

ns

+ sl2

nl

=Ys Y

l

SE(Ys Y

l)

h d di ib i


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The Student9t distribution ;SummaryT'e assution t'at Yis distributed 2BY, is rarely lausible

in ractice BIncoe? uber of c'ildren?

or nV 30, t'e t!distribution and 2B0,# are 8ery close Bas ngro9s large, t'e tnG#distribution con8erges to 2B0,#

T'et!distribution is an artifact fro days 9'en sale si>es 9eresall and t'ere 9ere noHfe9 couters

or 'istorical reasons, statistical soft9are tyically uses t'e t!

distribution to coutep!8alues G but no die is oderate or large

or t'ese reasons, in t'is class 9e 9ill focus on t'e large!naroAiation gi8en by t'e CLT

-T ST-PS:

#T'e robability frae9or@ for statisticalinference

2-stiation

3Testing

Y2


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ConJdence Inter8als

/ O5W confdence intervalforYis an inter8alt'at contains t'e true 8alue ofYin O5W ofreeated sales

2ote: 1'at is rando 'ere? T'e 8alues of Y#,,Ynand t'us any functions of t'e G including t'econJdence inter8al

T'e conJdence inter8al 9ill di


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*onden$e intervals# $td%/ O5W conJdence inter8al can al9ays be constructed as t'e

set of 8alues ofYnot reected by a 'yot'esis test 9it' a 5WsigniJcance le8el

Y: ^ #O( Y: G#O( ^ ^ #O(

Y: G#O( ^ GY^ #O(

Y B G #O( , #O(

This $onden$e interval relies on the lar,e9n results that is

approimately normally distributed and

Lin@ to Table

Y Y

sY/ n

Y Y

sY/ n

sY

n

sY

n

sY

nY

Y

sY2p

Y2

Y
http://www.had2know.com/academics/normal-distribution-table-z-scores.htmlhttp://www.had2know.com/academics/normal-distribution-table-z-scores.html


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Suary:

ro t'e t9o assutions of:# sile rando saling of a oulation, t'at is,

`Yi, i#,Z,n are iid

2 0 Q EBY Q [

Statisticians de8eloed, for large sales Blarge n:= T'eory of estiation Bsaling distribution of

= T'eory of 'yot'esis testing Blarge!n distribution of t!statistic and coutation of t'ep!8alue

= T'eory of conJdence inter8als Bconstructed by in8ertingt'e test statistic

/re assutions B# h B2 lausible in ractice? es

Y

L t b @ t t' i i l li


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Lets go bac@ to t'e original olicy;uestion:

1'at is t'e e


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