Aggregate Data Analysis

8/6/2019 Aggregate Data Analysis

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L O G I S T I C R E G R E S S I O N A N A L Y S I S W I T H A G G R E G A T E

D A T A : T A C K L I N G T H E E C O L O G I C A L F A L L A C Y

Dav id G. Steel, Un iversity of W ollongong, Au straliaMark T ranmer, U nivers ity of Southam pton, UK

D. Holt, O ffice for Na tional S tatistics, Londo n, UKDavid G. Steel , Schoo l of M athem atics an d A pplied Statist ics,

University of Wollongong, N SW 2522, Au stralia (david_steel~uow .edu.au)

K e y W o r d s : E c o l o g i ca l a n a l y si s , e c o lo g i ca l f al -l acy, mul t i l eve l mode l s , r andom e ffec t s , g roup ingvar i ab le s .

1 . I n t r o d u c t i o n

An eco log ica l ana lys i s u ses aggrega te g roup l eve ld a t a t o e s t i m a t e i n d i v i d u a l l e v e l r e l a t i o n s h i p s .

The eco log ica l f a l l acy a r i se s when eco log ica l ana l -ys i s p rov id es b i a sed e s t i ma tes o f i nd iv id ua l l evelr e l a t io n s h i p s . T h e g r o u p s i n v o l v e d a re o f t e n s m a llg e o g r a p h i c a r e a s s u c h a s c e n s u s E n u m e r a t i o n D i s -t r i c t s ( E D s ) . S u p p o s e t h a t w e a r e i n t e r e s t e d i n i n -v e s t i g a t i n g t h e r e l a t i o n s h i p b e t w e e n t w o v a r i a b l e sY a n d X . T h e a g g r e g a t e d a t a a v a il a b le f or a r e a g

_

u s u a l l y c o n s i s ts o f t h e g r o u p l ev e l m e a n sYg a n d

The eco log ica l f a l l acy a r i se s because the ind i -v i d u a l s w i t h i n t h e g r o u p s a r e n o t e q u i v a l e n t t or a n d o m l y f o rm e d g ro u p s . P r o g r e s s i n u n d e r s t a n d -

ing aggrega t ion e ff ec t s r equ i re s a l lowance fo r t hep o p u l a t i o n s t r u c t u r e i n t h e m o d e l u n d e r p i n n i n gthe ana lys i s . S t ee l and Hol t (1996) p rop osed am o d e l f o r c a se s w h e r e t h e r e l a t i o n s h i p b e t w e e n t h etwo va r i ab le s o f i n t e re s t , Y and X , i s l i nea r. Th i sm o d e l i n c o r p o r a t e d a u x i l i a r y v a r i a b l e s , Z , w h i c he x p l a i n m u c h o f t h e w i t h i n a r e a h o m o g e n i e t y o fthe va r i ab le s o f i n t e re s t . Ra nd om grou p l eve l coef -f ic ients were a lso included to ref lec t group level ef -f ec t s add i t iona l t o those due to the aux i l i a ry va r i -ab le s . Based on th i s mode l S tee l and Hol t (1996)d e v e l o p e d a m e t h o d t o a d j u s t g r o u p l e v e l c o v a r i -a n c e m a t r i c e s u s i n g l i m i t e d i n d i v i d u a l l e v e l d a t aava i l ab le fo r t he au x i l i a ry va r i ab le s . S t ee l, Ho l ta n d Tr a n m e r ( 1 9 9 6 ) e v a l u a t e d t h i s a p p r o a c h f o res t ima t ing co r re l a t ion coe ff i c i en t s u s ing ED l eve ld a t a f r o m t h e 1 9 9 1 U K p o p u l a t i o n c e n s u s. U s i n g

This research was supported by the UK Economic So-cial Research Cou ncil, grant Ft000 23 6135

a c o m b i n a t i o n o f b a s i c d e m o g r a p h i c a n d h o u s i n gva r i ab le s a s aux i l i a ry va r i ab le s they were ab le toreduce the aggreg a t ion e ffec ts i n a se t o f eco log icalco r re l a t ions by up to 70 pe r cen t .

In soc ia l r e sea rch the va r i ab le s a re usua l ly ca t -egor i ca l a t t he ind iv idua l l eve l and the a rea l eve lm e a n s a r e t h e c o r r e s p o n d i n g p r o p o r t i o n s . Va r i-ous m e th od s o f eco log ica l in fe rence in th i s s i t ua -

t i o n w e r e e v a l u a t e d b y C l e a v e , B r o w n a n d P a y n e(1995) . Th ese inclu ded l inear ecological regres-s i o n a s o r i g i n a l l y s u g g e s t e d b y G o o d m a n ( 1 9 5 9 )a n d a m e t h o d b a s e d o n a n A g g r e g a t ed C o m p o u n dM u l t i n o m i a l ( A C M ) m o d e l . T h e A C M m o d e l as -s u m e d t h a t t h e f r eq u e n c ie s i n e ac h g r o u p h a v e i n -d e p e n d e n t m u l t i n o m i a l d i s t r i b u t i o n s a n d u s e d aD i r i ch l e t c o m p o u n d i n g d i s t r i b u t i o n . T h e i r e va l u -a t i o n f a v o u r e d t h e A C M b a s e d m e t h o d b u t t h e yrecogn ized th a t i t i s no t easy to implem en t . Re -c e n t ly K i n g ( 1 99 7 ) p r o p o s e d a n e w m e t h o d o f e co -log ica l ana lys i s fo r ca t egor i ca l da t a . Th i s me thod

m o d e l s t h e c o n d i t i o n a l p r o p o r t i o n s o f Y g i v e n Xa s r a n d o m e f fe c ts w i t h a j o i n t t r u n c a t e d N o r m a ld i s t r i b u t i o n a n d e x p l o i t s t h e c o n s t r a i n t s i m p l i e db y t h e g r o u p m e a n s .

I n t h i s p a p e r w e d e ve l o p a n d e v a l u a t e s o m e s i m -p le ad jus t ed eco log ica l ana lys i s p rocedures basedo n t h e i d e a o f i n c o r p o r a t i n g a u x i l i a r y v a r i a b le sand us ing ind iv idua l l eve l da t a ava i l ab le fo r t hesevar iables .

0 A d j u s t e d E c o l o g i c a l A n a l y s i s f o rD i c h o t o m o u s Va r i a b l e s

Let Yi and Xi be the va lues o f t he va r i ab le s o fin t e re s t fo r t he i t h ind iv idua l i n the popu la t ion .Supp ose the re i s a s ample , s , o f g roup s and wi th int h e s a m p l e d g r o u p , g , a s am p l e o f n g i n d i v i d u a l s

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f o ll o w in g e s t i m a t o r s o f t h e c o n d i t i o n a l p r o b a b i l i t yP ( Y = I IX = b) = 7rllb, for b = 0, 1.

( a ) SA R re l a t ive f r equenc ie s . Th e r e l a t ive f re -q u e n c y o b t a i n e d f r om t h e S A R ,nlb+/n+b+.

(b ) SAS re l a t ive f r equenc ie s . Fo r pa i r s o f va r i-a b l e s f o r w h i c h t h e S A S c o n t a i n s t h e r e l e v a n tc r o ss t a b u l a t i o n w e c a n c a l c u l a t eNlb+ IN+b+.

( c ) Eco log ica l l i nea r r eg res s ion .a r o u n d t h e r e l a t i o n s h i p

Th i s i s bu i l t

~'g = P11og(1 - X. ) -4- P11 1g)(. (3)

I f P ll 0 g a n d P x ll g a r e r a n d o m v a r i a b l es w i t hE[P~Io, IXg] - 7r~10 a n d E [ ~ I ~ , I X , ] _ - 7r11~the n a l i nea r r eg res s ion o f Yg on Xg g ivesun bia sed es t im ate s of 7r110 an d 7r111. Th is i st h e c l a s s i c a l G o o d m a n r e g r e s s i o n a p p r o a c h( G o o d m a n , 1 9 59 ). T h i s a p p r o a c h c a n p r o-

d u c e e s t i m a t e s o u t s i d e [ 0, 1] a n d s i m p l e d i r e c tu s e o f t h i s m o d e l i s n o t u s u a l l y r e c o m m e n d e d .

(d ) Eco log ica l l og i s ti c r eg res s ion . A com mo nm o d e l u s e d i n a n a l y s i n g a d i c h o t o m o u s r e -sponse va r i ab le i s l og i s t i c r eg res s ion which a s -sum es Y~ i s a B inom ia l va r i ab le b ased o n onet r i a l , B (1 , E [Y~IXi] ) , where

E [r, lx , ] )lo g 1 - E [ Y ~ I X , ] - ~ +DX~

I f g r o u p s w e r e c o m p l e t e l y h o m o g e n e o u s w i t hr e s p e c t t o X t h e n e a c h g r o u p t o t a l Y g w o u l db e a B i n o m i a l v a r i a b l e b a s e d o n N g t r i a l s w i t hp r o b a b i l i t y s u c h t h a t

E [ ~ a l X g ] )log 1 - -~[Ygg~ffg - oL + Z f( g

G r o u p s a r e n o t h o m o g e n e o u s w i t h re s p e c t t oX b u t t h i s m o d e l h a s t h e a d v a n t a g e o f n o tg iv ing p red ic t e d p rob ab i l i t i e s ou t s ide [0,1] .

( e ) Ad jus t e d eco log ica l l i nea r r eg res s ion . Here we

u s e l i n e a r r e g r e s s i o n b a s e d o n a g g r e g a t e d a t at o e s t i m a t e P ( Y I X , Z )a n d P ( X I Z ) . E s t i m a -t i o n o f P ( Y I X ) i s t h e n b a s e d o n e q u a t i o n ( 2 )w i t h t h e i n d i v i d u a l l e v e l d a t a b e i n g u s e d t oe s t i m a t e P ( Z ) .

( f ) Ad jus t e d eco log ica l l og i s t ic r eg res s ion . Herewe use log i s t i c r eg res s ion based on aggrega te

d a t a t o e s t i m a t eP ( Y I X , Z )a n d P ( X I Z ) . E s -t i m a t i o n o fP ( Y [ X ) i s t h e n b a s e d o n e q u a t i o n( 2 ) w i t h t h e i n d i v i d u a l l e v e l d a t a b e i n g u s e dt o e s t i m a t e P ( Z ) .

( g ) A d j u s t e d c o r r e l a t i o n a p p r o a c h . F o r t w o d i -c h o t o m o u s v a r i a b l e s t h e c o r r e l a t i o n c o m b i n e dw i t h t h e m a r g i n a l t o t a l s d e t e r m i n e s t h e p r o -por t ions in the c ross c l a s s if i ca t ion i .e .

Ply+ = PI++ P+~+ +

Ry x v / P I + + ( 1 - P l + + ) P + l + ( 1 - P + l + ) (4 )

w h e r e R y x i s t he co r re l a t ion coe ff ic i en t basedo n t h e t a b l e o f p r o p o r t i o n s P ~b +. T h e m e t h o dp r o p o s e d b y S t e e l a n d H o l t ( 1 9 9 6) i s u s e dt o p r o d u c e a d j u s t e d e s t i m a t e s o f t h e c o r r e l a-t ion R y x ( Z ) w h i c h c a n t h e n b e s u b s t i t u t e di n t o e q u a t i o n ( 4 ). T h i s m e t h o d e n a b l e s u s e o fi n f o r m a t i o n a b o u t s e v e r a l a u x i l i a r y v a r i a b l e se v e n i f o n l y t h e t w o w a y c r o ss t a b u l a t i o n s a r eava i l ab le .

( h ) K i n g ' s e c o lo g i ca l i n f e re n c e m e t h o d . T h i sm e t h o d i s a l so b u i l t a r o u n d e q u a t i o n ( 3 ). T h eg r o u p l e v el c o n d i t i o n a l p r o p o r t i o n s P ll 0g a n dP l [l g a r e a s s u m e d t o h a v e a j o i n t N o r m a l d i s -t r i b u t i o n w h i c h is t r u n c a t e d s o t h a t t h e y e a c hl ie in the [0 ,1 ] i n t e rva l . Th e m e th od incorpo-ra t e s th e f ac t t ha t g iven ] ?g and i f g , equ a t ion( 3 ) i m p l i e s b o u n d s a n d c o n s t r a i n t s f o r P l l l ga n d P l l 0 g . T h i s a p p r o a c h p r o d u c e s e s t i m a t e so f P l l l g a n d P ll 0g w h i c h c a n t h e n b e c o m -b i n e d t o p r o d u c e e s t i m a t e s o f P lf l a n d / 9 11 0 .

T h i s m e t h o d c a n i n c o r p o r a t e g r o u p l e v e l c o -v a r i a t e s t o h e l p m o d e l t h e v a r i a t i o n b e t w e e ng r o u p s .

The va r i ab le s used in the ana lys i s were a s fo l -lows:

Y: e m p l o y e d , u n e m p l o y e d ,X " m a r i t a l st a t u s ,Z : age 45 -59 , age 60+ , l i v ing in ow ner occu p ied

dwel l ing , r en t ing f rom loca l au thor i ty.T h e s e a u x i l i a r y v a r i a b l e s w e r e c h o s e n b e c a u s e o ft h e i r s u c c e s s i n r e m o v i n g a g g r e g a t i o n e f f e c t s i n

c o r r e l a t io n s i n t h e e v a l u a t i o n b y S t e e l, H o l t a n dTr a n m e r ( 1 9 9 6 ). T h e a n a ly s i s w a s c o n f i n e d t ot h o s e a g e d 1 6 o r m o r e .

Ta b l e 1 g i v es t h e e s t i m a t e d p r o b a b i l i t i e s o f b e-i n g e m p l o y e d ( Y = 1 ) g i v e n m a r i t a l s t a t u s ( X ) .T h e e s t i m a t e s a r e o b t a i n e d u s i n g t h e m e t h o d sl i st e d a b o v e . T h e e co l o g ic a l m e t h o d s a n d a d j u s t e dc o r r e la t i o n m e t h o d s w e re i m p l e m e n t e d u s i n g t h e

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S A S p a c k a g e a n d K i n g ' s m e t h o d w a s i m p l e m e n t e dus ing EzI so f tware , deve loped by King and co l -l eagues (King , 1997) . The f i rs t row of the t ab leg ive the " t rue" va lues o f the p robab i l i t ie s a s e s-t ima ted f rom census c ross t abu la t ions , ava i l ab lefo r these pa r t i cu la r va r i ab les f rom the SAS da tabase , wh ich were the same as the co r respond inge s t i m a t e s o b t a i n e d f r o m t h e S A R .

W h e n a d j u s t m e n t v a r i a b l e s a r e n o t i n c l u d e d ,the ecological l inear and ecological logis t ic es t i -ma tes a re cons ide rab ly d i ff e ren t f rom the t rue va l -ues . The inc lus ion o f "owner occup ied housing"as an ad jus tmen t va r i ab le l eads to eco log ica l l i n -ea r and log i s t i c e s t ima tes tha t a re much c lose r tothe t rue va lues . Used as a s ing le ad jus tmen t va r i -ab le "aged 60 and over" does no t improve the e s -t i m a t e s . T h e e s t i m a t e s o b t a i n e d b y t h e a d j u s t e decological l inear method are fa i r ly c lose to the t rueva lues when seve ra l ad ju s tme n t va r i ab les a re used .In pa r t i cu la r, t hose combina t ions o f ad ju s tme n t

va r i ab les tha t inc lude hous ing t enure . The ad -jus ted l inea r r eg ress ion me thod genera l ly worksb e t t e r t h a n t h e a d j u s t e d c o r r e l a t i o n m e t h o d s u g -ges ted by S tee l and Hol t (1996). In th i s exam-p le , K ing ' s me thod wi tho u t covar i a te s g ives r e su l t ss imilar to those for the ecological l inear and lo-g i s ti c me thod s wi tho u t covari a t e s. W hen "owneroccup ied" i s used as a covar i a t e s , t he e s t ima tedprobab i l i t i e s a re fu r the r f rom the t rue va lues thanthose ob ta ined us ing the ad jus ted l inea r r eg res -s ion me thod . However, when the covar i a t e " ren t -ing f rom a loca l au thor i ty" i s added the e s t ima tes

f r o m K i n g ' s m e t h o d a r e s li g h tl y b e t t e r t h a n t h o s ebased on the ad jus ted l inea r r eg ress ion me thod .Use o f King ' s me th od wi th more tha n two covari -a tes proved diff icul t in pract ice and no resul ts forsuch cases are included.

F o r t h e e s t i m a t e s o f th e p r o p o r t i o n u n e m p l o y e dby mar i t a l s t a tus g iven in Tab le 2 , the unad-jus ted eco log ica l l i nea r and log i s t i c me thods p ro -v ide poor e s t ima tes . The l inea r me thod l eads toan ou t o f r ange es t ima te . Inc lud ing "owner occu-p ied" a s an ad jus tmen t va r i ab le in these me th -

ods l eads to some improve ment . The ad jus tedeco log ica l l i nea r r eg ress ion me thod works r eason-ab ly we l l fo r combina t ions o f ad jus tme n t va r i-ab les tha t inc lude hous ing t enure and age . King ' sm e t h o d w i t h o u t c o v a r i a t e s w o r k s s o m e w h a t b e t -t e r in th i s case than in the p rev ious example , be -cause d i ff e rences in the p ropor t ions unemployedand mar r i ed ac ross the EDs l ead to more in fo r-

ma t ive bounds . However, t he e s t ima tes a re s ti llworse than those ob ta ined f rom the adus ted log i s -t i c r eg ress ion me thod . W hen the va r i ab le "owneroccup ied" i s used as a covar i a te , t he e s t ima tes ob-t a ined f rom King ' s me thod a re e ffec t ive ly equa l tothe t rue va lues , wh i l e those based on the ad jus tedl inea r and log i s t i c r eg ress ion me thods a re no t .

Whi le these r e su l t s a re l imi ted to two re l a t ion-sh ips , t hey h igh l igh t the im por ta nce o f us ing aux-i l iary var iables in ecological analysis to obta in rea-sonab le e s t ima tes . The cho ice o f aux i l i a ry va ri -ab les i s imp or ta n t and me thod s o f iden t i fy ing e f-fec t ive ad jus tmen t va r i ab les need to be used , a ssugges ted by S tee l and H ol t (1996) . I f appro pr ia t eaux i l i a ry va r i ab les a re used , qu i t e s imple me thodscan pe r fo rm a lmos t a s we l l a s more soph i s t i ca ted ,com pute r in t ens ive ones . W e expec t tha t the s im-p le ad jus ted eco log ica l me thods can be ex tendedto a l so inc lude random e ffec t s wi th in the so r t o fmul t i l eve l f r amework deve loped , fo r example , byGolds te in (1995). Me thods tha t so le ly use r an -dom effects to account for the var ia t ion in the re-la t ionship between groups wi l l , in general , not bevery success fu l. More in fo rm at ion needs to be in-corpo ra ted in o rde r to ge t use fu l e s t ima tes . Th i sin fo rmat ion ma y be in the fo rm of ind iv idua l l eve lda ta on aux i l i a ry va r i ab les , g roup l eve l covar i a t e so r the cons t ra in t s exp lo i t ed in King ' s me thod .

4 . R e f e r e n c e s

Cleave, N. , Brown, P.J . and C. D. Payne (1995)

Eva lu a t ion o f Me thods fo r Eco log ica l In fe rence .Journal of the Royal Statistical Society, A,158 ,p p 5 5 - 7 2Goldste in , H. (1995) . Multilevel Statistical Mod-els, 2nd Edition, E d w a r d A r n o l d , L o n d o n .Goodman , L .A. (1959) . Some a l t e rna t ives to Eco-logical regression. Am erica n Jou rnal o f Sociologi-cal Review, 18, 663-664.King, G. (1997). A Solution to the Ecological In-ference Problem :Reconstructing Individual Behav-ior f rom Aggregate Data. P r i n c e t o n U n i v P r e s sSteel , D. and Hol t , D. (1996) . Analysing and Ad-

jus t ing A ggrega t ion Effect s : The Eco log ica l Fa l -l acy Rev i s t ed . International Statistical Review,64, pp 39-6OStee l D . , Ho l t D . and Tranmer M. (1996) . Makingun i t l eve l in fe rences f rom aggrega ted da ta ,SurveyMethodology 22, 3-15

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Ta b l e 1: E s t i m a t e d p r o b a b i l i t i e s : Y - e m p l o y e d ; X - m a r r i e d

l (a ) : Eco log ica l l inea r l (b ) : Eco log ica l log i s ti c

Pl lo Pl l l Pl lo Pl l lSAS ' t ru th ' . 41 .50 SAS ' t ru th ' .41 .50

cova r ia te ( s ) : covar ia te ( s ) :

no ne .26 .67 non e .26 .67age2 .23 .71 ag e6 0+ .29 .65

age1, age2 .28 .65 age1, age2 .33 .59

oo .48 .42 oo .49 .41

oo, r la .47 .43 oo, r la .46 .44

oo, r la , age1 ,2 .45 .45 oo, r la , age1,2 .45 .46

l ( c ): C o r r e l a ti o n m e t h o d l ( d ) : K i n g ' s m e t h o d

Pllo Pl l lS A S ' t r u t h ' .4 1 .5 0 S A S ' t r u t h '

covar ia te(s)"

non e .27 .66

age2 .22 .70

age1, age2 + .20 .73

oo .55 .35

oo, rla .51 .39

oo, r la , ag el , 2 .47 .43

covar ia te ( s ) :

n o n e

a g e 6 0 +

a g e 4 5 5 9 , a g e 6 0 +o o

oo, r la

oo , r l a , age l ,2

Pl l0 Pl l !.41 .50

.25 .69

.50 .39

.44 .46

Source : 1991 UK census da ta .P o p u l a t i o n : R e s i d e n t s a g e d 1 6 o r m o r e i n h o u s eh o l d s , M a n c h e s t e r L A D .

Y take s the va lue 1 fo r ' employ ed ' 0 fo r ' no t em ployed ' .

X t a k e s t h e v a l u e 1 f o r 'm a r r i e d ' a n d 0 fo r ' n o t m a r r i e d ' .

P l l o m e a n s P ( Y = 1 I X = 0 ) ; P l l l m e a n sP ( Y = 1 I X = 1)A d j u s t m e n t v a r i ab l e s : o o - o w n e r o c c u p i e d ; rl a - r e n t e d f r o m l o c al a u t h o r i t y ;

age1 = pe rsons aged 45 - 59 ; age2 = pe rsons aged 60 and over.

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T a b l e 2 : E s t i m a t e d p r o b a b i l i t i e s : Y - u n e m p l o y e d ; X - m a r r i e d

l ( a ) : E c o l o g i c a l l i n e a r l (b ) : Eco log ica l l og i s t i c

P i l o P i l lS A S ' t r u t h ' . 14 . 07

covar i a t e ( s ) "

non e .24 - . 06age2 .24 - .05

ag e l , age2 .22 - . 03

oo .18 .01

oo, r la .19 .01

oo , r l a , ag e l ,2 . 16 .04

P ~ 1 0 P ~ l i

S A S ' t r u t h ' . 14 . 07


non e .17 .02a g e 6 0 + . 17 . 02

age l , age2 .11 .09

oo .16 .04

oo, r la .15 .04

oo , r l a , age l ,2 . 11 .09

2 ( c) : C o r r e l a t i o n m e t h o d 2 ( d ): K i n g ' s m e t h o d

Pi l0 P i l lS A S ' t r u t h ' . 14 . 07

covar i a t e ( s ) -

non e .27 - . 09

age2 .27 - .09

ag e l , age2 .28 - . 09

oo .19 - .00

oo, r la .18 .01

oo , r l a , ag e l ,2 . 16 .03

P l lo P i l lS A S ' t r u t h ' . 14 . 07


non e .19 .00

age2

a g e l , a g e 2

oo .14 .07

oo , r l a

oo , r l a , age l ,2

S o u r c e : 1 9 9 1 U K c e n s u s d a t a .P o p u l a t i o n : R e s i d e n t s a g e d 1 6 o r m o r e i n h o u s e h o ld s , M a n c h e s t e r L A D .

Y t a k e s t h e v a l u e 1 f o r ' u n e m p l o y e d ' 0 f o r ' n o t u n e m p l o y e d ' .

X t a k e s t h e v a l u e 1 f o r 'm a r r i e d ' a n d 0 f o r 'n o t m a r r i e d ' .

P i l 0 m e a n s P(Y = l I X = 0 ); P i l i m e a n sP(Y = l I X = 1 )A d j u s t m e n t v a r i a b le s : o o = o w n e r o c c u p i ed ; r l a = r e n t e d f r o m l o c a l a u t h o r i t y ;

a g e l = p e r s o n s a g e d 4 5 - 5 9 ; a g e 2 = p e r s o n s a g e d 6 0 a n d o v e r.

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