Fu Ch10 Categorical

7/25/2019 Fu Ch10 Categorical

1/54

EPI809/Spring 2008EPI809/Spring 2008 11

Chapter 10Chapter 10

Hypothesis testing: CategoricalHypothesis testing: Categorical

Data AnalysisData Analysis


2/54

EPI809/Spring 2008 2

Learning !"ecti#esLearning !"ecti#es

1$1$ Co%parison o& !ino%ial proportion 'sing ( an)Co%parison o& !ino%ial proportion 'sing ( an)

22*est$*est$

2$2$ E+plainE+plain 22*est &or In)epen)ence o& 2 #aria!les*est &or In)epen)ence o& 2 #aria!les,$,$ E+plain *he -isher.s test &or in)epen)enceE+plain *he -isher.s test &or in)epen)ence

$$ ce%ar.s tests &or correlate) )atace%ar.s tests &or correlate) )ata

$$ 3appa Statistic3appa Statistic

4$4$ 5se o& SAS Proc -6E75se o& SAS Proc -6E7


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Data *ypesData *ypes

Data

Quantitative Qualitative

Discrete Continuous

Data

Quantitative Qualitative

Discrete Continuous


4/54


7'alitati#e Data7'alitati#e Data

1$1$ 7'alitati#e 6an)o% aria!les iel)7'alitati#e 6an)o% aria!les iel)6esponses *hat Can e P't In Categories$6esponses *hat Can e P't In Categories$E+a%ple: ;en)er E+a%ple: ;en)er

2$2$ eas're%ent or Co'nt 6e&lect ? in Categoryeas're%ent or Co'nt 6e&lect ? in Category

,$,$ o%inal or r)inal Scale o%inal or r)inal Scale

$$ Data can !e collecte) as contin'o's !'tData can !e collecte) as contin'o's !'t

reco)e) to categorical )ata$ E+a%plereco)e) to categorical )ata$ E+a%pleor%al tension= hypertension >


5/54


Hypothesis *estsHypothesis *ests

7'alitati#e Data7'alitati#e DataQualitative

Data

Z Test Z Test 2 Test

Proportion Independence1 pop.

2 Test

2 or morepop.

2 pop.

Qualitative

Data



2 Test

2 or morepop.

2 pop.


6/54

EPI809/Spring 2008EPI809/Spring 2008 44

( *est &or Di&&erences in( *est &or Di&&erences in

*o Proportions*o Proportions


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Hypotheses &orHypotheses &or

*o*oProportionsProportions


8/54




Research Questions

Hypothesis

No Dierence

!ny Dierence

Pop 1

Pop 2

Pop 1 " Pop 2

Pop 1 Pop 2

Pop 1 # Pop 2

H$

Ha

Research Questions

Hypothesis

No Dierence

!ny Dierence

Pop 1

Pop 2

Pop 1 " Pop 2

Pop 1 Pop 2

Pop 1 # Pop 2

H$

Ha


9/54




Research Questions

Hypothesis

No Dierence

!ny Dierence

Pop 1

Pop 2

Pop 1 " Pop 2

Pop 1 Pop 2

Pop 1 # Pop 2

H$ p1%p2 & $

Ha p1%p2 $

Research Questions

Hypothesis

No Dierence

!ny Dierence

Pop 1

Pop 2

Pop 1 " Pop 2

Pop 1 Pop 2

Pop 1 # Pop 2

H$ p1%p2 & $

Ha p1%p2 $


10/54




Research Questions

Hypothesis

No Dierence

!ny Dierence

Pop 1

Pop 2

Pop 1 " Pop 2

Pop 1 Pop 2

Pop 1 # Pop 2

H$ p1%p2 & $ p1%p2 $

Ha p1%p2 $ p1%p2 " $

Research Questions

Hypothesis

No Dierence

!ny Dierence

Pop 1

Pop 2

Pop 1 " Pop 2

Pop 1 Pop 2

Pop 1 # Pop 2

H$ p1%p2 & $ p1%p2 $

Ha p1%p2 $ p1%p2 " $


11/54




Research Questions

Hypothesis

No Dierence

!ny Dierence

Pop 1

Pop 2

Pop 1 " Pop 2

Pop 1 Pop 2

Pop 1 # Pop 2

H$ p1%p2 & $ p1%p2 $ p1%p2 $

Ha p1%p2 $ p1%p2 " $ p1%p2# $

Research Questions

Hypothesis

No Dierence

!ny Dierence

Pop 1

Pop 2

Pop 1 " Pop 2

Pop 1 Pop 2

Pop 1 # Pop 2

H$ p1%p2 & $ p1%p2 $ p1%p2 $

Ha p1%p2 $ p1%p2 " $ p1%p2# $


12/54




Research Questions

Hypothesis

No Dierence

!ny Dierence

Pop 1

Pop 2

Pop 1 " Pop 2

Pop 1 Pop 2

Pop 1 # Pop 2

H$ p1%p2 & $ p1%p2 $ p1%p2 $

Ha p1%p2 $ p1%p2 " $ p1%p2# $

Research Questions

Hypothesis

No Dierence

!ny Dierence

Pop 1

Pop 2

Pop 1 " Pop 2

Pop 1 Pop 2

Pop 1 # Pop 2

H$ p1%p2 & $ p1%p2 $ p1%p2 $

Ha p1%p2 $ p1%p2 " $ p1%p2# $


13/54


( *est &or Di&&erence in *o( *est &or Di&&erence in *o

ProportionsProportions1$1$ Ass'%ptionsAss'%ptions Pop'lations Are In)epen)entPop'lations Are In)epen)ent

Pop'lations -ollo ino%ial Distri!'tionPop'lations -ollo ino%ial Distri!'tion or%al Appro+i%ation Can e 5se) &oror%al Appro+i%ation Can e 5se) &or

large sa%pleslarge sa%ples >

2$2$ (@*est Statistic &or *o Proportions(@*est Statistic &or *o Proportions( ) ( )

( ) 21

21

21

2121 where

111

nn

XXp

nnpp

ppppZ

++

=

+

( ) ( )

( ) 21

21

21

2121 where

111

nn

XXp

nnpp

ppppZ

++

=

+


14/54


Sa%ple Distri!'tion &or Di&&erenceSa%ple Distri!'tion &or Di&&erence

eteen Proportionseteen Proportions

( ) ( )1 1 2 21 2 1 2

1 2

0 1 2

1 2

1 2

1 2

1 1 N ;

1 1N 0; :

,

p p p pp p p p

n n

pq under H p pn n

x xp

n n

+

+ =

+=

+

2 2

1 2

1 2 1 2

1 2

~ N ;X Xn n

+


15/54


( *est &or *o Proportions( *est &or *o Proportions

*hinBing Challenge*hinBing Challenge

o'.re an epi)e%iologist &or the 5So'.re an epi)e%iologist &or the 5S

Depart%ent o& Health an) H'%anDepart%ent o& Health an) H'%an

Ser#ices$ o'.re st')ying theSer#ices$ o'.re st')ying thepre#alence o& )isease in topre#alence o& )isease in to

states $ Instates $ In '!'!== ()()o&o&

1*$$1*$$people s'r#eye) erepeople s'r#eye) ere

)isease) an) in)isease) an) in C!C!== 12+12+ o&o& 1*$$1*$$

ere )isease)$ Atere )isease)$ At .$*.$*le#el= )oesle#el= )oes

'!'!ha#e aha#e a lo,erlo,erpre#alence ratepre#alence rate

'!

C!


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Sol'tionSol'tion


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Test -tatisticTest -tatistic

DecisionDecision

ConclusionConclusion


Sol'tionSol'tion

HH$$

HHaa

&&

nn'!'!&& nnC!C!&&

Critical /alue0sCritical /alue0s


18/54



DecisionDecision



Sol'tionSol'tion

HH$$pp'!'!%%ppC!C!& $& $

HHaapp'!'!%%ppC!C!" $" $

&&nn'!'!&& nnC!C!&&



19/54



DecisionDecision



Sol'tionSol'tion

HH$$pp'!'!%%ppC!C!& $& $


&&.$*.$*nn'!'!&& 1*$$1*$$ nnC!C!&& 1*$$1*$$



20/54



DecisionDecision



Sol'tionSol'tion

HH$$pp'!'!%%ppC!C!& $& $


&&.$*.$*nn'!'!&&1*$$1*$$ nnC!C!&&1*$$1*$$


(0@1$4/2

$02

6e"ect

(0@1$4/2

$02

6e"ect


21/54



Sol'tionSol'tion

( ) ( )

( ) ( )

00.4

1500

1

1500

10677.10677.

00860.0493.

0677.15001500

12974

0860.1500

1290493.

1500

74

=

+

=++=

++=

======

Z

nn

XXp

n

Xp

n

Xp

CAMA

CAMA

CA

CACA

MA

MAMA

( ) ( )

( ) ( )

00.4

1500

1

1500

10677.10677.

00860.0493.

0677.15001500

12974

0860.1500

1290493.

1500

74

=

+

=++=

++=

======

Z

nn

XXp

n

Xp

n

Xp

CAMA

CAMA

CA

CACA

MA

MAMA


22/54


Z & %).$$Z & %).$$


Sol'tionSol'tion

HH$$pp'!'!%%ppC!C!& $& $


&&

.$*.$*

nn'!'!&& 1*$$1*$$ nnC!C!&&1*$$1*$$



DecisionDecision


(0@1$4/2

$02

6e"ect

(0@1$4/2

$02

6e"ect


23/54


Z & %).$$Z & %).$$


Sol'tionSol'tion

HH$$pp'!'!%%ppC!C!& $& $


&&

.$*

.$*

nn'!'!&& 1*$$1*$$ nnC!C!&&1*$$1*$$



DecisionDecision


(0@1$4/2

$02

6e"ect

(0@1$4/2

$02

6e"ect Reect atReect at & .$*& .$*


24/54


Z & %).$$Z & %).$$


Sol'tionSol'tion

HH$$pp'!'!%%ppC!C!& $& $


&&

.$*.$*

nn'!'!&&1*$$1*$$ nnC!C!&&1*$$1*$$



DecisionDecision


(0@1$4/2

$02

6e"ect

(0@1$4/2

$02

6e"ect Reect atReect at & .$*& .$*

There is evidence '!There is evidence '!

is less than C!is less than C!


25/54


26/54


Hypothesis *estsHypothesis *ests

7'alitati#e Data7'alitati#e DataQualitative

Data



2 Test

2 or morepop.

2 pop.

Qualitative

Data



2 Test

2 or morepop.

2 pop.


27/54


22*est o& In)epen)ence*est o& In)epen)ence

1$1$Shos I& a 6elationship E+ists eteen 2Shos I& a 6elationship E+ists eteen 2

7'alitati#e aria!les= !'t )oes7'alitati#e aria!les= !'t )oes NotNotShoSho

Ca'salityCa'sality

2$2$Ass'%ptionsAss'%ptions

'ltino%ial E+peri%ent'ltino%ial E+peri%ent

All E+pecte) Co'ntsAll E+pecte) Co'nts

,$,$5ses *o@Fay Contingency *a!le5ses *o@Fay Contingency *a!le


28/54


22*est o& In)epen)ence*est o& In)epen)enceContingency *a!leContingency *a!le

1$1$ Shos ? !ser#ations -ro% 1 Sa%pleShos ? !ser#ations -ro% 1 Sa%ple

Gointly in 2 7'alitati#e aria!lesGointly in 2 7'alitati#e aria!les


29/54


22*est o& In)epen)ence*est o& In)epen)enceContingency *a!leContingency *a!le

1$1$Shos ? !ser#ations -ro% 1 Sa%pleShos ? !ser#ations -ro% 1 Sa%ple

Gointly in 2 7'alitati#e aria!lesGointly in 2 7'alitati#e aria!les

3evels o varia4le 23evels o varia4le 2

3evels o varia4le 13evels o varia4le 1


30/54


22*est o& In)epen)ence*est o& In)epen)enceHypotheses StatisticHypotheses Statistic

1$1$HypothesesHypotheses HH00: aria!les Are In)epen)ent: aria!les Are In)epen)ent

HHaa: aria!les Are 6elate) : aria!les Are 6elate)


31/54



1$1$HypothesesHypotheses

HH00: aria!les Are In)epen)ent: aria!les Are In)epen)ent


2$2$*est Statistic*est Statistic 54served count54served count

67pected67pected

countcount

2

2

=

n E n

E n

ij ij

ij

c h

c hall cells

2

2

=

n E n

E n

ij ij

ij

c h

c hall cells


32/54



1$1$HypothesesHypotheses

HH00: aria!les Are In)epen)ent: aria!les Are In)epen)ent


2$2$*est Statistic*est Statistic

Degrees o& -ree)o%: Ro,sRo,s ColumnsColumns

54served count54served count

67pected67pected

countcount

2

2

=

n E n

E n

ij ij

ij

c h

c hall cells

2

2

=

n E n

E n

ij ij

ij

c h

c hall cells


33/54



E+pecte) Co'ntsE+pecte) Co'nts1$1$Statistical In)epen)ence eans GointStatistical In)epen)ence eans GointPro!a!ility E'als Pro)'ct o& arginalPro!a!ility E'als Pro)'ct o& arginal

Pro!a!ilitiesPro!a!ilities

2$2$Co%p'te arginal Pro!a!ilities 'ltiplyCo%p'te arginal Pro!a!ilities 'ltiply

&or Goint Pro!a!ility&or Goint Pro!a!ility

,$,$E+pecte) Co'nt Is Sa%ple SiJe *i%esE+pecte) Co'nt Is Sa%ple SiJe *i%es

Goint Pro!a!ilityGoint Pro!a!ility


34/54


E+pecte) Co'nt E+a%pleE+pecte) Co'nt E+a%ple


35/54



112112

18$18$'ar9inal pro4a4ility &'ar9inal pro4a4ility &


36/54



112112

18$18$

(:(:

18$18$

'ar9inal pro4a4ility &'ar9inal pro4a4ility &



37/54



112112

18$18$

(:(:

18$18$



;oint pro4a4ility &;oint pro4a4ility &112112

18$18$(:(:

18$18$


38/54



112112

18$18$

(:(:

18$18$



;oint pro4a4ility &;oint pro4a4ility &112112

18$18$(:(:

18$18$

67pected count & 18$


39/54


E+pecte) Co'nt Calc'lationE+pecte) Co'nt Calc'lation


40/54



67pected count &Ro, total Column total

-ample si=e


-ample si=e


41/54



1127:21127:2

18$18$

):7(:):7(:

18$18$

):7:2):7:2

18$18$

1127(:1127(:

18$18$


-ample si=e


-ample si=e


42/54


o' ran)o%ly sa%pleo' ran)o%ly sa%ple 2:82:8se+'ally acti#ese+'ally acti#e

in)i#i)'als an) collect in&or%ation on their HIin)i#i)'als an) collect in&or%ation on their HI

stat's an) History o& S*Ds$ At thestat's an) History o& S*Ds$ At the .$*.$*le#el= isle#el= is

there e#i)ence o& athere e#i)ence o& a relationshiprelationship

22*est o& In)epen)ence*est o& In)epen)enceE+a%ple on HIE+a%ple on HI


43/54


22*est o& In)epen)ence*est o& In)epen)enceSol'tionSol'tion


44/54



Sol'tionSol'tion

HH$$

HHaa

&&

d &d &



DecisionDecision


2$

Reect

2$

Reect

2 * t & I ) )


45/54



HH$$ No RelationshipNo Relationship

HHaa RelationshipRelationship

&&

d &d &



DecisionDecision


2$

Reect

2$

Reect

22 &* t & I ) )


46/54





&& .$*.$*

d &d & 02 % 102 % 1 & 102 % 102 % 1 & 1



DecisionDecision


2$

Reect

2$

Reect

22 * & I ) )* t & I ) )


47/54





&& .$*.$*

d &d & 02 % 102 % 1 & 102 % 102 % 1 & 1



DecisionDecision


2$ >.:)1

Reect

2$ >.:)1

Reect & .$*& .$*

22 * t & I ) )* t & I ) )


48/54


EE00nnijij* in all* in all

cellscells

1($71>21($71>2

2:82:81($71*)1($71*)

2:82:8

11871>211871>2

2:82:81*)71181*)7118

2:82:8


22 * t & I ) )* t & I ) )


49/54


2

2

11 11

2

11

12 12

2

12

22 22

2

22

2 2 2:) *> *

*> *

>2 82 *

82 *

122 +1*

+1**) 2+

=

=

=

=

n E n

E n

n E n

E n

n E n

E n

n E n

E n

ij ij

ij

.

.

.

.

.

..

c hc h

a fa f

a fa f

a fa f

all cells

2

2

11 11

2

11

12 12

2

12

22 22

2

22

2 2 2:) *> *

*> *

>2 82 *

82 *

122 +1*

+1**) 2+

=

=

=

=

n E n

E n

n E n

E n

n E n

E n

n E n

E n

ij ij

ij

.

.

.

.

.

..

c hc h

a fa f

a fa f

a fa f

all cells


22 * t & I ) )* t & I ) )


50/54





& .$*& .$*

dd& 02 % 102 % 1 & 1& 02 % 102 % 1 & 1



DecisionDecision


2$ >.:)1

Reect

2$ >.:)1

Reect & .$*& .$*

22& *).2+& *).2+

22 * t & I ) )* t & I ) )


51/54





& .$*& .$*

dd& 02 % 102 % 1 & 1& 02 % 102 % 1 & 1



DecisionDecision

ConclusionConclusionReect atReect at & .$*& .$*

2$ >.:)1

Reect

2$ >.:)1

Reect & .$*& .$*

22& *).2+& *).2+

22 * t & I ) )* t & I ) )


52/54





& .$*& .$*

dd& 02 % 102 % 1 & 1& 02 % 102 % 1 & 1



DecisionDecision

ConclusionConclusionReect atReect at & .$*& .$*

There is evidence o aThere is evidence o a

relationshiprelationship

2$ >.:)1

Reect

2$ >.:)1

Reect & .$*& .$*

22& *).2+& *).2+


53/54


22*est o& In)epen)ence*est o& In)epen)enceSAS CDESSAS CDES

DataData)isK)isK

inp't S*Ds HI co'ntKinp't S*Ds HI co'ntK

car)sKcar)sK

1 1 81 1 8

1 2 ,21 2 ,2

2 1 82 1 82 2 1222 2 122

KK

runrunKK

ProcProcre?re?)ata)is or)er)ataK)ata)is or)er)ataK eight Co'ntKeight Co'ntK

ta!les S*DsHI/ta!les S*DsHI/chischisKK

runrunKK


54/54

22*est o& In)epen)ence*est o& In)epen)enceSAS 5*P5*SAS 5*P5*

Statistics for Table of STDs by HIV

Statistic DF Value Prob

-------------------------------------------------------Chi-Square 154.150 !.0001 "i#elihoo$ %atio Chi-Square 1 55.&'( !.0001 Co)ti)uity *$+. Chi-Square 1 5.,'&1 !.0001 a)tel-Hae)sel Chi-Square 1 5,./(0/ !.0001 Phi Coefficie)t 0.4,51

Co)ti)e)cy Coefficie)t 0.,//0 Craer2s V 0.4,51

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