HAWASSAUNI

VERSI
TY
BENSADAYECAMPUS
Depar
tmentofAgr
oeconomi
cs
ECONOMETRI
CSASSI
GNMENT|

NAME;
TEDROSTESFAW
I
DNO;
2360/
13

1
TABLEOFCONTENT
Bi
nar
yLogi
tModel
…………………………………3
Tobi
tModel
…………………………………………………4
pr
oitModel
…………………………………………………5
Mul
ti
nomi
alModel
……………………………………6
Or
der
edl
ogi
tmodel
…………………………………9
HeckManModel
………………………………………10
Doubl
ehur
dlemodel
………………………………12
Mul
ti
plel
i
nearr
egr
essi
onmodel
…………13
si
mpl
eli
nearr
egr
essi
onmodel
…………16
PSM model
………………………………………18
REFERENCE…………………………………………21

2
BI
NARYLOGI
TMODEL

i
sacl assifi
cationalgori
thm usedt of i
ndtheprobabil
it
yofeventsuccess
andev entfail
ure.Itisusedwhent hedependentvari
ableisbinary(0/1,
True/
Fal se,Yes/No)i nnature.Itsupport
scategori
zingdataint
odi scret
e
cl
assesbyst udy i
ngtherelationshipfr
om agivensetoflabel
leddat a.It
l
earnsal inearrelati
onshipfrom thegivendatasetandthenintroducesa
non-l
i
near it
yint heform oftheSi gmoidfunct
ion.

Thef
ormul
afort
hebi
nar
ylogi
tmodel
isasf
oll
ows:

P(
Y=1|
X)=1/(
1+exp(
-βX)
)

Wher
e:

-P(
Y=1|X)i
stheprobabi
li
tyofthedependentvar
iabl
eYbei
ngequal
to1
gi
ventheval
uesoftheindependentv
ariabl
esX.

-βi
sav ectorofcoef
fi
cient
sthatrepr
esentst
heeffectofeach
i
ndependentvari
abl
eont hel
og-oddsofthedependentvar
iabl
e.

-Xi
sav
ect
orofi
ndependentv
ari
abl
es.

-expi
stheexponent
ial
funct
ion.

I
tisbasedonsigmoi
df unct
ionwher
eout
puti
spr
obabi
l
ityandi
nputcanbe
f
rom -
infi
nit
yto+i
nfi
nit
y

ASSUMPTI
ON

1.Bi
nar
youtcome:Thedependentv
ari
ablemustbebinary,
taki
ngonl
ytwo
val
uessuchas0and1,yesandno,orsuccessandf
ailur
e.

2.I
ndependenceofobservati
ons:
Theobservat
ionsmustbeindependent
ofeachother
.Thereshouldbenocorr
elat
ionordependencebetweenthe
observ
ati
onsinthedataset.

3.Li
neari
tyi
nthel
og-
odds:
Therel
ati
onshi
pbetweentheindependent
vari
abl
esandthel
og-
oddsoft
hedependentv
ari
ableshouldbeli
near.Thi
s

3
assumptioni
snecessaryf
ort
hel
ogi
sti
cregr
essi
onmodel
toest
imat
ethe
coeff
ici
entsaccur
atel
y.

4.Nomul t
icol
li
nearit
y:Theindependentvari
abl
esshoul
dnotbehi
ghl
y
correl
atedwi t
heachot her
.Mul t
icol
li
near
itycanl
eadtounst
abl
eand
unreli
ablecoeffi
cientest
imates.

5.Noi nfl
uent
ialoutli
ers:Outli
erscanhaveasi gni
fi
cantimpactonthe
esti
mat edcoeffi
cients.I
tisimportanttocheckforinf
luent
ial
outl
ier
sand
considerthei
rremov alortransfor
mationifnecessary
.

6.Adequatesamplesi
ze:Logist
icregr
essi
onmodel srequi
reasuffi
cient
samplesizetopr
oducereli
ableesti
mates.Asageneralrul
e,ther
eshould
beatleast10-
20caseswiththeleastf
requentout
comeperi ndependent
var
iabl
e.

7.Absenceofperfectsepar
ati
on:Perf
ectsepar
ationoccur
swhent herei
sa
combinati
onofindependentvari
abl
esthatcanperfectl
ypr
edictt
he
outcomevari
abl
e.I nsuchcases,t
helogi
sti
cregressionmodelcannotbe
est
imated.

8.Assumpt i
onofindependenceofer
ror
s:Theerr
orsint
hemodelshoul
d
beindependentofeachother,meani
ngther
eshouldbenocor
rel
ati
onor
patt
ernintheresi
dual.

I
ti simport
anttonotet
hatvi
olati
ngtheseassumpti
onsmayaff
ectt
he
validi
tyandrel
i
abil
it
yoftheesti
matedcoeff
ici
ent
sandpredi
cti
onsf
rom t
he
binarylogi
tmodel.

CONDI
TIONS

Binar
ylogitmodeli
suseful
intheanal
ysi
sofmulti
plef
act
orsi
nfl
uenci
nga
negati
ve/posi
ti
veoutcome,oranyot
hercl
assi
fi
cat
ionwher
ether
eareonl
y
twopossibleout
comes
STRENGTH

Itmakesnoassumpti
onsaboutdist
ri
but
ionsofclassesi
nfeat
urespace.
Itcaneasi
l
y
extendt
omulti
plecl
asses(
mult
inomialr
egressi
on)andanatur
alprobabi
l
isti
cvi
ewof

4
cl
asspredict
ions.
Itnotonlyprov
idesameasureofhowappropri
atea
pr
edict
or(coeff
ici
entsize)
is,
butalsoi
tsdi
rect
ionofassoci
ati
on(posi
ti
veornegat
ive)
.

Goodaccuracyf ormanysi mpl

edat aset
sanditper
formswellwhenthedatasetis
l
inearl
yseparable.
Logisti
cregr
essionisl
essincl
i
nedtoover-
fit
ti
ngbuti
tcanov er
fi
tin
highdimensi
onal dataset
s.OnemayconsiderRegul
ari
zat
ion(L1andL2)techniquesto
avoidover
-fi
tt
inginthesescenar
ios.

WEAKNESS

Ift
henumberofobser vat
ionsi
slesserthant
henumberoffeatures,
Logisti
cRegressi
on
shouldnotbeused,other
wise,i
tmayleadtoover
fi
tti
ng.
Themaj orli
mitat
ionofLogist
ic
Regressi
onistheassumpt i
onofli
neari
tybet
weenthedependentvari
ableandthe
i
ndependentvari
abl
es.

Non-l
i
nearproblemscan’tbesol v
edwi t
hlogist
icregressionbecausei thasalinear
deci
si
onsur f
ace.Linear
lyseparabledat
ai srar
elyfoundi nreal-wor
ldscenari
os.InLi
near
Regr
essionindependentanddependentv ari
abl
esar erelatedlinear
ly.ButLogi
stic
Regr
essionneedst hati
ndependentvari
ablesareli
nearlyrelatedtothelogodds
(l
og(
p/(1-p)
).

TOBI
TMODEL

CONDI
TION

Tobitr
egr
essi
onmodelisusedtoest
imat
eli
nearrel
ati
onshipsbet
weendependent
var
iabl
eandexpl
anat
oryvari
abl
eswhendependentvar
iabl
eislef
tcensor
ing.

Standar
dTobitassumpti
onshol
donnormal
it
yandhomoscedast
ici
tyofr
esi
dual
s,and
Tobitmodel
shav eshownpoorr
obust
nesst
othei
rvi
olat
ion.

Tobi
tmodel
shavebeenusedtoaddressseveral
quest
ionsinmanagementresear
ch.
Revi
ewi
ngexi
sti
ngpract
icesandappl
icat
ions,
wediscussthreechal
l
enges:
(a)

TheTobitmodel
isusedwhent hedependentvar
iabl
ehasali
mitedr
ange,
suchaswhen
i
tiscensor
edortruncat
ed.Thefor
mulafortheTobitmodel
isasfol
l
ows:

Y*=βX+ε

Y=max
(Y*
,0)

Wher
e:

-Y*ist
helatentvari
able,
whi
chr
epr
esent
stheunder
lyi
ngcont
inuousv
ari
abl
ebef
ore
censor
ingortruncat
ion.

5
-Yi
stheobser
veddependentv
ari
abl
e,whi
chi
sli
mit
edt
oacer
tai
nrange.

-βisav ect
orofcoef
fi
cient
sthatr
epr
esent
stheef
fectofeachi
ndependentv
ari
abl
eon
thel
atentvari
abl
e.

-Xi
sav
ect
orofi
ndependentv
ari
abl
es.

-εist
heerrorter
m,whi
chi
sassumedt
obenor
mal
l
ydi
str
ibut
edwi
thmeanzer
oand
const
antv
ar i
ance.

assumptionsaboutthenatur
eofdata,
(b)apparenti
nterchangeabi
l
itybet
ween
censori
ngandselectionbi
as,and(
c)potent
ialv
iolat
ionsofkeyassumpt i
onsi
nthe
dist
ri
buti
onofresiduals

ASSUMPTI
ON

Tobi tmodel sassumet hatthev ari

ablesexplai
ningwhet herornott heobser ved
dependentv ar
iabl
eiscensor edmustal soexplaintheleveloft
hev ariabl
ewheni ttakes
posi t
ivevalues.Giv
ent hatthisassumpt ionmaynothol dinsamplesaf fectedby
select i
onbias,orwhent he“y es/no”choiceandt he“howmuch”choi cear eexpl ainedby
diff
er entmechanisms, theuseofTobi tmodelsmayl eadt ounrel
iableest i
mat es.I nour
reviewoft helit
erat
ure,thisissueappear edin7%oft hest udi
esifweonl yinclude
studieswher etheauthor sexplici
tlyst
atethatTobi tmodelsareusedt oaddr ess
select i
onbias.Ifwealsoconsi derstudieswher etheauthorsimplicit
lyarguet hatTobi t
model sar
et hemostsui tabl
echoi ce,theissueisfarmor ecommon.

STRENGTH

Themosti mportantadvantageintheappl
icati
onofTobi tmodel
sisthefactthatt
hey
enableef
fecti
veanaly si
sofeconomicdataev enincaseswherethesampl edatamay
notbefull
yrepresentati
veoftheenti
repopulati
onduet odif
fer
entreasonssuchasdata
avai
labi
li
tyandnat ur
eoft heanaly
sis

WEAKNESS

Themostimportantcri
ti
queofthetobi
tmodeli
sthati
tdoesnotal
l
owf orthesetof
var
iabl
esusedinexplai
ningwhetheryi
sposi
ti
veorzero(say
,x1)t
odiff
erfrom t
heset
ofvari
abl
esusedinexplaini
ngt
hev al
ueofycondi
ti
onalonybei
ngstr
ictl
yposit
ive(
say
,

x2)

PROBI
TMODEL

Probi
tanal
ysi
sexaminestherel
ati
onshipbet
weenabi
naryresponsev
ar i
abl
eanda
conti
nuousst
ressv
ariabl
e.I
thelpstoesti
matet
hepr
obabil
i
t yt
hataninsectwi
l
ldi
e

6
whenexposedt
oacer
tai
namountofpest
ici
deoradi
sinf
est
ati
ont
reat
ment

Probitr
egressi
on,al
socal
ledaprobi
tmodel,i
susedtomodel dichot
omousorbinar
y
outcomev ari
abl
es.Int
heprobi
tmodel,
theinver
sestandardnormaldi
str
ibut
ionoft
he
probabi
li
tyismodeledasali
nearcombinat
ionofthepredi
ctor
s.

CONDI
TIONFORPROBI
TMODEL

Theprobi
tmodeli
susedwhent
hedependentvar
iabl
eisbi
nar
yordichot
omous,
meaningi
tcanonl
ytakeont
woval
ues(e.
g.,0or1).Thef
ormul
afortheprobi
tmodel
is
asfol
l
ows:

P(
Y=1|
X)=Φ(
βX)

Wher
e:

-P(
Y=1|X)ist
heprobabi
li
tyofthedependentv
ari
abl
etaki
ngont
hev
alueof1gi
vent
he
val
uesofthei
ndependentvar
iabl
es.

-Φ(
.)i
sthecumul
ati
vedi
str
ibut
ionf
unct
ionoft
hest
andar
dnor
mal
dist
ri
but
ion.

-βisavect
orofcoef
fi
cient
sthatrepr
esent
stheeff
ectofeachi
ndependentv
ari
abl
eon
thepr
obabi
li
tyoft
hedependentvari
abl
ebeing1.

-Xi
sav
ect
orofi
ndependentv
ari
abl
es.

ASSUMPTI
ON

pr
obi
tmodel'
sident
if
yingassumpti
ons:l
ineari
ndexspeci
fi
cat
ion,
joi
ntnor
mal
i
tyof
er
ror
s,i
nst
rumentexogenei
ty,andr
elev
ancecondit
ions.

STRENGTH

Theprobitanalysi
sensur
esthaty ouresti
mat edpr obabil
i
tiesarebet ween0and1.I t
al
sogivesbet t
eranswerswhent heprobabi l
i
tiesar esmallorlarge.Probitanal
ysiscan
al
sobej usti
fi
edusinguti
li
tyanalysi
s.Probiti
sbet terint
hecaseof" random eff
ects
models"withmoder at
eorlar
gesampl esizes( i
tisequaltologitforsmal lsamplesizes)
.
Forf
ixedeffectsmodels,pr
obitandlogitareequal lygood.

WEAKNESS

Themai nweaknessoftheapproachbasedonr egressionanaly

sisi
st hatitobl
igeseach
vari
abl
et obeeit
herdependentorindependent
,withoutall
owingforthepossibil
it
ythat
thesamev ar
iabl
emaybedependentandi ndependentatthesamet i
me. Theonly
l
imitat
ionofprobi
tmodelsisthattheyrequi
renormal dist
ri
buti
onsforallunobserved

7
componentsofutil
i
t y
.Inmany,
perhapsmostsit
uat
ions,
nor
mal
dist
ri
but
ionspr
ovi
dean
adequat
erepresentati
onoft
herandom component
s.

MULI
TIYNOMI
ALMODEL

Whati
sMul
ti
nomi
alLogi
sti
cRegr
essi
on?

Multi
nomi al
Logist
icRegr essionistheregressi
onanal ysistoconductwhent he
dependentvari
ableisnomi nal wit
hmor ethantwol evels.Simi l
art
omul ti
plelinear
regr
ession,t
hemul t
inomi alregressioni
sapr edi
cti
v eanalysis.Mult
inomialregressi
oni
s
usedtoexplaint
her el
ationshipbet weenonenomi nal dependentvariabl
eandoneor
moreindependentvariables

mul t
inomiall
ogist
icregressi
onisaclassi
fi
cati
onmet hodt hatgeneral
izeslogist
ic
regressi
ontomul ti
cl
asspr obl
ems,i.
e.wit
hmor ethant wopossiblediscrete
outcomes.[1]Thati
s,itisamodel t
hatisusedtopredicttheprobabili
ti
esoft he
dif
ferentpossibl
eoutcomesofacat egori
call
ydistr
ibut
eddependentv ari
able,gi
vena
setofindependentvariabl
es(whichmayber eal
-val
ued, bi
nary-
valued,categori
cal
-
valued,et
c.).

Mult
inomial
logi
sticregr
essionisknownbyav ari
etyofothernames,i
ncl
uding
pol
ytomousLR,[2]
[3]mult
iclassLR,soft
maxregr
ession,multi
nomial
logi
t(mlogi
t)
,the
maximum entr
opy( MaxEnt)classi
fi
er,
andthecondit
ionalmaximum ent
ropymodel.

Mul t
inomial
logist
icregr
essionisusedwhenthedependentvar
iabl
einquest
ioni
s
nomi nal(
equival
entl
ycategori
cal,
meaningthati
tfal
l
sintoanyoneofasetof
categori
esthatcannotbeor der
edinanymeaningful
way)andforwhichther
earemor
e
thant wocategori
es.Someexampl eswouldbe:

Whichmaj
orwi
l
lacol
l
egest
udentchoose,
giv
ent
hei
rgr
ades,
stat
edl
i
kesanddi
sli
kes,
et
c.?

Whi
chbl
oodt
ypedoesaper
sonhav
e,gi
vent
her
esul
tsofv
ari
ousdi
agnost
ict
est
s?

I
nahands-
fr
eemobi
lephonedi
ali
ngappli
cat
ion,
whi
chper
son'
snamewasspoken,
gi
venv
ari
ouspr
oper
ti
esofthespeechsi
gnal
?

Whi
chcandi
dat
ewi
l
laper
sonv
otef
or,
giv
enpar
ti
cul
ardemogr
aphi
cchar
act
eri
sti
cs?

Whi
chcountr
ywil
lafir
mlocateanof
fi
cei
n,gi
vent
hechar
act
eri
sti
csoft
hef
ir
m andof
t
hevar
iouscandi
dat
ecountr
ies?

Mul
ti
nomi
alLogi
sti
cRegr
essi
onModel

MLogi
tregr
essi
oni
sagener
ali
zedl
i
nearmodel
usedt
oest
imat
ethepr
obabi
l
iti
esf
or

8
t
hem cat
egor
iesofaqual
i
tat
ivedependentv
ari
abl
eY,
usi
ngasetofexpl
anat
ory
v
ari
abl
esX:

Pr
Yk=Pr
Yi=K|
Xi,
B1,
B2,
.
.,
Bm

wher
eBki
sther
owv
ect
orofr
egr
essi
oncof
fi
centof

Xf
ort
heKt
hcat
egor
yofY

wher
eβki
sther
owv
ect
orofr
egr
essi
oncoef
fi
cient
sofXf
ort
hekt
hcat
egor
yofY.

Unfort
unat
ely
,thi
smodel
speci
fi
cat
ioni
snoni
dent
if
iabl
e;t
hepr
obabi
l
iti
esar
eident
ical
foreachβ0k+q,

Thecondi
ti
onst
ousemul
ti
nomi
all
oggest
icr
egr
essi
onmodel

Amul t
inomi
al l
ogi
sti
cr egr
ession(ormult
inomial
regr
essi
onforshor
t)i
susedwhent he
outcomevari
ablebei
ngpr edi
ctedisnominalandhasmorethantwocat
egori
esthatdo
nothaveagivenrankororder.Thi
smodel canbeusedwithanynumberofi
ndependent
var
iabl
esthatarecategori
calorconti
nuous.

CONDI
TIONFORMULTI
NOMI
ALREGRESSI
ONMODEL

Mult
inomi all
ogist
icr
egr
essi
oni susef
ulf
orsit
uati
onsinwhi chy ouwanttobeablet o
cl
assif
ysubj ect
sbasedonvaluesofasetofpredi
ctorvar
iables.Thi
stypeofregression
i
ssimilartologist
icr
egr
essi
on,butiti
smoregeneralbecauset hedependentvar
iableis
notr
est r
ictedtotwocat
egor
ies.

Whennott
ousemul
ti
nomi
all
ogi
sti
cregr
essi
on?

Whennott
ousemul
ti
nomi
alr
egr
essi
on

Natural
l
yorderedout
comev ari
abl
e.I
ngener
al,
multi
nomialr
egressi
oni
sint
endedtobe
usedwheny ouhaveamulti
classout
comevari
abl
ethatdoesnothaveanat
uralOr
der
ed
outcomeswithasmallsamples

ASUMPTI
ON

Multicl
assout comeandi nfer
encei sy ourprimar ygoal.Ifyouhaveamul t
icl
ass
outcomev ariableandy ouaremor einterestedini nf
erencet hanpr
edict
ion,y
oushould
almostcertainlyreachforamul t
icl
assr egressionmodel suchasmulti
nomial
regressi
onoror di
nall
ogist
icregression.Likeotherr egressionmodels,t
hesemodels
provideint
erpretablecoeff
ici
entsthatquant if
yt herelati
onshipbetweenyourfeat
ures
andy ouroutcomev ar
iabl
e.

9
Or dinaloutcomet hatdoesnotmeetpr oport
ional oddsassumpt i
on.Evenify ouar e
usi nganout comev ariablethathasanat uralordert oit
,youmaybebet terof fusing
mul t
inomial r
egressi oni nsomecases.Thi sisbecauseor dinall
ogist
icregression
model smakeaf air
lyst rongassumpt ioncall
edt hepr oportionaloddsassumpt i
on.This
assumpt i
onessent iall
yi mpliesthatthediff
erencesassoci atedwithmov i
ngf rom one
cat egoryoft heout comev ari
abletothenexthi ghercategor yarethesameacr ossal l
cat egori
es.Thi sisaf air
lystringentassumptiont hatdoesnotappl yi
nallcasesandi f
youdonott hinkt hi
sassumpt ionappli
esfory ourusecase, youmaybebet t eroff
treat i
ngyourout comev ariableasanunor deredv ariabl
eandusi ngmul ti
nomi allogist
ic
regr essi
oni nstead.

STRENGTH

I
nt erpret
abl ecoef fi
cient s.Oneoft hemai nadv antagesofmul t
inomialregressionist hat
i
tpr ovideshi ghl
yinterpr etabl
ecoef f
icientsthatquant i
fyt her elat
ionshipbetweeny our
featuresandy ourout comev ari
able.Ther earenotmanyot hermodel sthatpr ovidethis
l
ev elofint erpretabi
li
tyf ormul ti
classout comes. Mor eflexiblethanor di
nal l
ogistic
regression.Anot herbenef i
tthatmul t
inomi allogisti
cregr essionhassov erordinal
l
ogi sti
cr egr essi
onist hatmul ti
nomial l
ogisti
cr egressioni samor eflexi
blemodel than
ordinal l
ogi sticr
egressi on.Thatmeanst hatitdoesnotmakeasmanyst r
ong
assumpt ionsaboutt hest ructureofyourdat a.

WEAKNESS

Cannotaddr essor der edv ariables.Onedi sadv antageofmul ti

nomi alregr essioni sthatit
cannotaccountf ormul ti
classout comev ari
ablesthathav eanat uralorder ingt othem.
Youcanst il
l usemul tinomi al regressi onint heset y
pesofscenar ios,buti twi l
l not
accountf oranynat uralor der i
ngbet weent helevelsoft hosev ariables.Mor epar ameters
thanmul ti
nomi al l
ogi sti
cr egr ession.Anot herdisadv antaget hatmul ti
nomi alregression
hascompar edt oor dinal logist i
cregr essioni sthatmul t
inomi alregr essionmodel shav e
manymor epar amet erst hatneedt obeest imatedt hanor dinal l
ogi sticregr ession
model s.Theseaddi tional par amet ersar ewhatal l
owst hemul ti
nomi alregr essionmodel
tohav emor ef l
exibili
tyt hant heor dinallogisticregressionmodel .Notav ailableinsome
l
ibraries.Anot herdi sadv ant ageofmul ti
nomi all
ogisticregressi oni st hati tisar el
ativel
y
nichemodel thatisnotav ailableinal lcommonmachi nelearningl i
braries.Si nce
regressi onmodel slikemul tinomi alregressi onar emor ecommonl yusedf orinf er
ence
thanpr edict ion,itismor ecommonf ormul ti
nomi allogisti
cr egressi ont obeav ail
ablein
packagesl ikeSASandSt atat hathav eaheav i
erfocusoncl assi cal stati
st icalmodel s
andi nf erence

ORDEREDLOGI
TMODEL

10
Standar dint
erpretati
onoft heor deredlogitcoeffi
cientisthatf oraoneunitincreasein
thepr edict
or,t
her esponsev ariableleveli
sexpect edt ochangebyi t
srespective
regressioncoeffici
enti ntheorder edlog-
oddsscal ewhi letheot hervari
ablesinthe
model arehel
dconst ant.
stati
stics,t
heor deredlogitmodel (alsoorderedlogist
ic
regressionorpropor tionaloddsmodel )i
sanor dinalregressionmodel —thatis,a
regressionmodel forordinaldependentv ari
ables—f i
rstconsideredbyPet erMcCul l
agh.

Theorderedlogitmodelshaveadependentv ar
iabl
ethatareorder
edcat
egori
es.
Examplesincluderat
ingsystems(poor,f
air
, goodexcell
ent),
opini
onsur
veysfr
om
str
ongl
ydi sagreet
ost r
onglyagr
ee,grades,andbondr ati
ngs.

WHENTOUSEORDEREDLOGI
TMODEL?

Youhav eanor dinaloutcomeandi nferencei sy ourpri

mar ygoal.Ingeneral,youshould
reachf orregressi onmodel st hathavehi ghlyinterpret
ablecoeffi
cient
swheni nference
i
sy ourpr i
mar ygoal .Thatmeanst hatyoushoul dreachforregressionmodel st
hatcan
handlemul t
iclassout comesl ikeor di
nallogisticregressi
onmodel sormul ti
nomi al
regressionmodel sanytimei nferenceisy ourpr imarygoalProporti
onalodds
assumpt i
onhol ds.Youshoul dspeci f
icallyuseor dinall
ogist
icregressi
onov erasimilar
model l
ikemul tinomi alregr
essi onwheny ourdat ahasanat uralorderi
ngt oitandy ou
beli
ev ethepr opor ti
onal oddsassumpt i
onhol ds.Inthesescenarios,t
heor dinall
ogistic
regressionmodel isthesimpl ermodel wi thfewerpar ametersthatneedtobeest i
mated.

CONDI
TIONFORORDEREDLOGI
TMODEL

Theorderedlogi
tmodel i
susedwhenthedependentv
ari
abl
eisordinal
,meani
ngi
tcan
takeonmultipl
eorderedcat
egori
es(
e.g.
,l
ow,medium,hi
gh).Theformulaf
ort
he
order
edlogitmodelisasfol
lows:

P(
Y≤j
|
X)=Λ(
αj-βX)-Λ(
αj+1-βX)

Wher
e:

-P(
Y≤j
|X)i
sthepr
obabi
l
ityoft
hedependentvari
ablet
aki
ngonav
aluel
esst
hanorequal
toj
giv
entheval
uesoft
heindependentv
ari
ables.

-Λ(
.)i
sthecumul
ati
vedi
str
ibut
ionf
unct
ionoft
hel
ogi
sti
cdi
str
ibut
ion.

-αj
isav
ect
orofi
nter
cept
sforeachcat
egor
yoft
hedependentv
ari
abl
e.

-βisavect
orofcoef
fi
cient
sthatrepr
esent
stheef
fectofeachindependentv
ari
abl
eon
thepr
obabi
li
tyoft
hedependentvari
abl
ebeingi
nacertai
ncategory.

-Xi
sav
ect
orofi
ndependentv
ari
abl
es.

11
ASSUMPTI
ON

1.Orderedoutcome: Thedependentvariablemustbeorderedorordi
nal,
wit
hmulti
ple
categor
iesarrangedinaspecif
icorder.Forexample,aLikertscal
ewit
hopti
onssuchas
str
onglydisagree,di
sagree,
neutr
al,agree,andst
rongl
yagr ee.

2.I
ndependenceofobservat
ions:
Theobservat
ionsmustbei
ndependentofeachother
.
Ther
eshouldbenocor r
elati
onordependencebetweent
heobser
vati
onsinthedataset
.

3.Proport
ionaloddsassumpti
on: Therelati
onshi
pbetweent
hei ndependentv ar
iabl
es
andthelog-oddsofthedependentv ar
iableshoul
dhaveaconsistentpropor
tionalodds
rat
ioacrossalll
evel
softhedependentv ariabl
e.Thi
sassumpt
ionensur esthatthe
coeff
ici
entsestimat
edbytheor deredlogitmodelar
evali
d.

4.Nomult
icol
l
inear
it
y:Theindependentv
ari
abl
esshoul
dnotbehighl
ycorrel
atedwit
h
eachot
her
.Multi
coll
i
neari
tycanleadtounst
abl
eandunrel
i
ablecoeff
ici
entesti
mates.

5.Noinfl
uenti
aloutl
ier
s:Outl
ier
scanhav
easignif
icantimpactont
heest
imat
ed
coef
fi
cient
s.Iti
simpor t
anttocheckf
ori
nfl
uent
ialoutl
i
ersandconsi
dert
hei
rremov
alor
tr
ansfor
mationifnecessar
y.

6.Adequatesampl esize:Orderedlogi
tmodel
salsor
equi
reasuf
fi
cientsamplesi
zeto
producerel
iableesti
mat es.Asageneralr
ule,
ther
eshoul
dbeatl
east10-20casesper
categor
yoft hedependentv ari
able.

7.Absenceofperf
ectseparat
ion:Si
milartothebinar
ylogi
tmodel,per
fectseparat
ion
occur
swhent her
eisacombi nati
onofi ndependentv
ari
abl
esthatcanperfect
lypredi
ct
theoutcomevari
able.I
nsuchcases,theor der
edlogi
tmodelcannotbeestimated.

8.Assumpti
onofindependenceofer
ror
s:Theerr
orsi
nthemodelshoul
dbe
i
ndependentofeachother
,meaningt
hereshoul
dbenocorr
elat
ionorpat
ter
nint
he
resi
dual
s.

Viol
ati
ngtheseassumptionsmayaffectt
hev al
idi
tyandr
eli
abi
l
ityoft
heest
imat
ed
coeff
ici
ent
sandpr edi
cti
onsfr
om theorderedlogi
tmodel
.

STRENGTH

Handl
esor deredout
comes.Or dinal
logisti
cregr
essi
onisoneofthefewcommon
machi
nel earni
ngmodelsthatwasspeci f
ical
l
ydevel
opedtohandlemult
icl
ass
out
comest hathaveanaturalordertothem.Thatmeansthatiti
sinaleagueofit
sown
whenitcomest ohandl
i
ngor dinaloutcomes.Fewerpar
ameter
st hanot
hermulti
class

12
regr
essi
onmodel s.Theordinall
ogist
icr
egressionmodel i
sasimplemodelthathas
fewerpar
amet er
sthatneedt obeestimat
edt hanotherr
egressi
onmodelsthatcan
handlemult
icl
assdata.Gi
v enthattwomodelshav erel
ati
velysi
mil
arper
for
mance, i
tis
almostal
way sbett
ertogowi ththemoresimplemodel .

I
nt erpr
etablecoeff
icient
s.Aswithmanyot herregressi
onmodel s,ordi
nall
ogisti
c
regressi
onmodel spr ovi
dehighl
yinter
pretablecoeffi
cient
sthatexplai
ntherelat
ionship
bet weenyourfeaturesandyouroutcomev ariabl
e.Thesecoeffi
cientsoft
encomeal ong
withconf i
denceinterval
sandstati
sti
caltestsforevenbetteri
nterpr
etabi
li
ty.

WEAKNESS

Proportionaloddsassumpt ion.Oneoft hemai ndisadvantagesofordinallogi

sti
c
regressionisthatitmakesaf air
lystr
ongassumpt ionthatisnotnecessarilyval
i
dinal l
cases.Thi sassumption,call
edt heproport
ionaloddsassumpt ions,essential
l
yimplies
thatthedi f
ferencesassociatedwi t
hmov i
ngf rom onecategoryoftheout comev ar
iable
tothenexthi ghercategoryar ethesameacr ossallcategori
es.Therearemany
exampl esofsi t
uati
onswher et hisi
snottrue,soyoushoul dconsiderthedomai noft he
problem andassessy ourdatat odeter
minewhet herthisassumptionholds.

Notav ail
ableincommonl i
brari
es.Anot herdownsi deofordinallogi
sti
cr egressionis
thati
tisar elat
ivelynichemodel thatisnotavai
lableinallcommonmachi nel earni
ng
l
ibrar
ies.Ordinallogisti
cregression,andr egr
essionmodel singener al
,tendt obemor e
commonl yusedinf i
eldswher einf er
enceandcl assical
stati
sticsareking.Thatmeans
thatordinall
ogisti
cr egressi
onmodel saremor elikelyt
obei mpl ementedi nlanguages
andpr ogramst hatfavorclassical st
ati
sti
cssuchasSASandSt ata.

HECKMANTWOSTAGEMODEL

CONDI
TION

TheHeckmant wo-st
epselecti
onmet hodpr ovidesameansofcor rect
ingfornon-
randomlysel
ectedsampl es.I
tisat wo-stageest i
mat i
onmethod.Thefir
ststage
perfor
msapr obitanalysi
sonasel ecti
onequat ion.Thesecondstageanalyzesan
outcomeequat i
onbasedont hefirst-
stagebinar yprobi
tmodel.
TheHeckmanmodel
i
ncludestwosepar ateequati
ons–onef ocusingonsel ect
ioni
ntothesampl e(outcome
beingobserv
ed–t hesampleselecti
onequat i
on) ,andthemainequati
onl i
nkingthe
covari
atesofinter
esttotheoutcome.

TheHackmant wo-st
agemodel
,devel
opedbyJ.Richar
dHackman,isafr
ameworkthatex
plai
ns
howt odesi
gnandmanageworkteamseffect
ivel
y.I
tconsi
stsoft
wostages:
thet
askdesign
stageandthegroupfor
mat
ionst
age.

13
Thef
ormul
afort
heHeckmant
wo-
stagemodel
isasf
oll
ows:

Fi
rstSt
age:

Y*=Xβ+Zγ+ε

D=ρY*+u

SecondSt
age:

Y=Xβ+Zγ+δD+v

Wher
e:

-Y*i
sthel
atentv
ari
abl
eofi
nter
est
.

-Xi
sav
ect
orofi
ndependentv
ari
abl
es.

-βisav ect
orofcoef
fi
cient
sthatr
epr
esent
stheef
fectofeachi
ndependentv
ari
abl
eon
thel
atentvari
abl
e.

-Zi
sav
ect
orofi
nst
rument
alv
ari
abl
es.

-γisavectorofcoef
fi
ci
ent
sthatr
epr
esent
stheef
fectofeachi
nst
rument
alv
ari
abl
eon
thel
atentv
ariabl
e.

-εanduar
eer
rort
ermsi
nthef
ir
stst
ageequat
ions.

-Di
sabi
nar
yvar
iabl
eindi
cat
ingsel
ect
ioni
ntot
hesampl
e.

-ρisthecoef
fi
ci
entt
hatr
epr
esent
stheef
fectoft
hel
atentv
ari
abl
eont
hesel
ect
ion
equat
ion.

-δisthecoef
fi
ci
entt
hatr
epr
esent
stheef
fectofsel
ect
ionont
heobser
vedout
come
var
iabl
e.

-vi
stheer
rort
ermi
nthesecondst
ageequat
ion.

TheHeckmant wo-stagemodelisusedtoaddresssampleselect
ionbias,wherethe
select
ionint
othesampl eisrel
atedtotheoutcomevari
able.Thefir
ststageesti
mates
theprobabil
i
tyofselecti
on,andthesecondstageesti
matestheeffectofselect
ionon
theoutcomev ari
ablewhil
eaccountingfort
hissel
ecti
onbias

ASSUMPTI
ON.

TheHackmant wo-st
agemodel i
sastat
isti
calmodel
usedineconomet
ri
cstoaddr
ess
endogenei
tyorsi
multanei
tyi
ssuesi
nregressi
onanal
ysi
s.Herearet
hecondi
ti
onsfor

14
t
heHackmant
wo-
stagemodel
:

1.Endogenei
ty:
Thereshoul
dbeendogeneit
yorsi
mult
anei
typresenti
ntheregressi
on
anal
ysis,
meaningthatt
heindependentv
ari
abl
esar
ecorr
elat
edwi t
htheerr
orterm.

2.Inst
rumentalv
ariabl
es:
Theresearchermustident
if
yinst
rumentalvari
ablest
hatare
correl
atedwi
ththeendogenousindependentv
ariabl
esbutarenotcorrel
atedwit
hthe
err
orterm.Theseinstr
umentsar
eusedt oaddressendogenei
tybyprov i
dingasour
ceof
exogenousvari
ati
on.

3.Exclusi
onrestri
cti
on:Theinstr
umentalv ar
iabl
esmustsatisf
ytheexcl
usionrest
ri
cti
on,
whichstatesthattheyshouldonl
yaffectthedependentvari
ablet
hroughthei
rimpacton
theendogenousi ndependentvar
iabl
e.Inotherwords,
theinstr
umentsshouldnothave
adirectef
fectont hedependentvari
abl
e.

4.Fir
st-
stageregr
ession:I
nthefir
ststageoft heHackmant wo-st
agemodel,the
endogenousindependentvari
abl
esarer egr
essedont heinst
rumentalv
ari
ablesto
obtai
npredict
edv al
uesfortheendogenousv ari
abl
es.Thisstepesti
matesthe
rel
ati
onshipbetweentheendogenousv ari
ablesandtheinstr
uments.

5.Second-st
ager egression:Inthesecondst
age,thepr
edict
edval
uesfrom t
hefi
rst
stagearei
ncludedasi ndependentv ari
abl
esinthemainregr
essi
onequati
on.The
dependentvariabl
eisr egressedonbot ht
hepredi
ctedval
uesandtheexogenous
i
ndependentv ari
ables.

6.Consist
encyandeffi
ciency
:TheHackmantwo-st
agemodelprov
idesconsi
stentand
eff
ici
entesti
matesofthecoeffi
cient
sift
hei
nst
rumental
var
iabl
esareval
idandsatisf
y
thenecessar
ycondit
ions.

7.Over
identi
fi
cati
ontest:I
tisimportanttoconductanov eri
denti
fi
cationtestt
oassess
theval
i
dityoftheinst
rumentalvar
iables.Thi
stestcheckswhet hertherearemore
i
nstr
ument sthannecessaryandhelpsidenti
fypotent
ialproblemswi t
hweak
i
nstr
ument s.

Vi
olat
ingt
heseassumpti
onsmayl
eadt
obi
asedandi
nconsi
stentcoef
fi
cientest
imat
es
i
ntheHackmantwo-st
agemodel
.

STRENGTH

Missingnessinhealt
houtcomecanl eadtosubstant
ialbi
as.Heckman-
sel
ect
ionmodel
s
cancor r
ectfort
hisselect
ionbi
asandy iel
dunbiasedesti
mates,ev
enwhenthe
proporti
onofmissingdataissubst
antial.

Onest
rengt
hoft
heHackmant
wo-
stagemodel
ist
hati
tpr
ovi
desacompr
ehensi
vef
ramewor
k

15
fordesi
gningandmanagi ngworkteams.Byconsider
ingmult
ipl
econdi
ti
ons,
suchastask
i
dentit
y,autonomy,andteam composi
ti
on,or
ganizati
onscanaddressv
ari
ousaspect
sthat
contr
ibutetoteam ef
fect
iveness

Anotherstrengthisthatthemodelemphasizestheimportanceofindivi
dualmot
ivat
ionand
sat
isfacti
on.Byf ocusi
ngonf act
orssuchast asksi
gni
ficance,ski
l
lvariet
y,andf
eedback,the
model recognizesthatteam members'
personalexper
iencesandper cepti
onspl
ayacrucialr
ole
i
ntheirengagementandper for
mance.

Addit
ional
ly,
themodelacknowledgest
hedy namicnatureofteamsbyincl
udi
ngcondit
ionssuch
asteam devel
opmentandnormsandv alues.Thishi
ghlight
stheimpor
tanceofongoi
ngsupport
andnurtur
ingoft
eam dynamicstoensurelong-t
erm success.

Overal
l,
theHackmant wo-stagemodel provi
desaholi
sti
capproachtodesigningandmanagi
ng
workteams,taki
ngintoaccountvari
ousf act
orst
hatinf
luenceteam ef
fect
iveness.I
ts
comprehensi
v enat
ureandf ocusonindiv
idualmoti
vati
onmakei taval
uabletoolfor
organi
zati
onsseeki
ngt ocreatehi
gh-perf
ormingteams.

WEAKNESS

Oneweaknessoft heHackmant wo-stagemodel

isthatitmayov er
simpli
fyt
hecompl ex
dynamicsandinter
acti
onswithinteams.Themodelfocusesonspecifi
ccondit
ionsandfactor
s
thatcont
ri
but
et oteam ef
fect
iveness,buti
tmaynotcaptureal
lthenuancesandcompl exi
ti
esof
real
-l
if
eteam dynamics.

Anotherweaknessi sthatt
hemodel doesnotprov
idespecifi
cguidanceonhowt oaddressor
overcomechallengesthatteamsmayf ace.Whi
leiti
denti
fi
esv ar
iouscondi
ti
onsthatcanimpact
team eff
ect
iveness,itdoesnotof
ferdetai
ledst
rat
egiesorint
erventi
onsforaddr
essi
ngt hese
conditi
onsi
npr acti
ce.

Additi
onally,
themodel maynotf
ull
yaccountfortheinfl
uenceofext er
nalfactors,
suchas
organi
zat i
onalcul
tur
eorexter
nalpr
essures,
ont eam effecti
veness.I
tpri
mar i
lyfocuseson
i
nternalteam dynamicsandmaynotadequatelyconsiderthebroadercontextinwhichteams
operate.

Furthermore,themodel assumest
hatteam membersaremot iv
atedandsati
sfi
edbyt hesame
factors,
whichmaynotal waysbet
hecase.Indi
vi
dualmotivat
ionsandpref
erencescanv ar
y
greatl
ywi t
hinat eam,andthemodeldoesnotprovi
deguidanceonhowt oaddressthese
i
ndiv i
dualdif
ferences.

I
nsummar y,
whi l
etheHackmant wo-stagemodel provi
desacompr ehensi
veframeworkfor
desi
gningandmanagi ngworkt eams, ithaslimitationsincapt
uri
ngt hecomplexi
ti
esofteam
dynamics,pr
ovidi
ngspecifi
cgui dancef orovercomi ngchall
enges,account
ingforext
ernal
fact
ors,andaddressi
ngindi
vidual di
fferenceswi thinteams.

DOUBLEHURDLEMODEL

16
Double-hur
dlemodel sareusedwithdependentv ari
ablesthattakeont heendpointsof
aninter
v al
withpositi
veprobabil
it
yandt hatareconti
nuousl ydistr
ibutedov ert
heinter
ior
oftheinter
val.Forexample,youobservetheamountofal cohol i
ndivi
dualsconsume
overafixedper i
odoftime.Thedistri
but
ionoft heamount swi l
lber oughlyconti
nuous
overpositi
vev al
ues,butt
herewillbea“pileup”atzero,whichist hecor nersol
uti
ont o
theconsumpt ionprobl
em theindi
vidual
sf ace;
noindividualcanconsumeanegat i
ve
amountofal cohol

Thef
ormul
afort
hedoubl
ehur
dlemodel
isasf
oll
ows:

Fi
rstHur
dle:

Y*=Xβ+Zγ+ε

D1=ρ1Y*+u1

SecondHur
dle:

Y=Xβ+Zγ+δD2+v

Wher
e:

-Y*i
sthel
atentv
ari
abl
eofi
nter
est
.

-Xi
sav
ect
orofi
ndependentv
ari
abl
es.

-βisav ect
orofcoef
fi
cient
sthatr
epr
esent
stheef
fectofeachi
ndependentv
ari
abl
eon
thel
atentvari
abl
e.

-Zi
sav
ect
orofi
nst
rument
alv
ari
abl
es.

-γisavectorofcoef
fi
ci
ent
sthatr
epr
esent
stheef
fectofeachi
nst
rument
alv
ari
abl
eon
thel
atentv
ariabl
e.

-εandu1ar
eer
rort
ermsi
nthef
ir
sthur
dleequat
ions.

-D1i
sabi
nar
yvar
iabl
eindi
cat
ingsel
ect
ioni
ntot
hef
ir
sthur
dle.

-ρ1isthecoef
fi
ci
entt
hatr
epr
esent
stheef
fectoft
hel
atentv
ari
abl
eont
hef
ir
sthur
dle
equat
ion.

-Yi
stheobser
vedout
comev
ari
abl
e.

-δisthecoeff
ici
entthatr
epr
esent
stheef
fectofsel
ect
ioni
ntot
hesecondhur
dleont
he
obser
v edout
comev ari
abl
e.

-D2i
sabi
nar
yvar
iabl
eindi
cat
ingsel
ect
ioni
ntot
hesecondhur
dle.

17
-vi
stheer
rort
ermi
nthesecondhur
dleequat
ion.

Thedoublehurdlemodel i
susedt oaddresst wotypesofsampl eselect
ionbias:
select
ioni
ntoparti
cipation(f
irsthurdle)andselectioni
ntothelevelofparti
cipat
ion
(secondhurdl
e).Thefirsthurdleestimatesthepr obabi
l
ityofparti
cipati
on,andthe
secondhurdleesti
mat estheef fectofparti
cipat
ionontheout comev ari
ablewhile
accounti
ngforbotht y
pesofsel ecti
onbias.

ASSUMPTI
ONOFDOUBLEHURDLEMODEL

1.Two- st
agedecisionprocess:Thedat ashoul
dexhibi
tat wo-stagedecisi
on-maki
ng
process,whereindivi
dual
sf i
rstdecidewhetherornott
opar ti
cipatei
ntheact i
vi
ty,
and
then,condit
ionalonparti
cipati
on,decidethel
evelofi
ntensit
yorquant i
tyoftheacti
vi
ty.

2.Hurdl
eeffect
:Thereshoul
dbeahur dl
eeffectpresent,
meaningthatsomei ndi
vi
dual
s
wil
lchoosenottoparti
ci
patei
ntheactiv
ityatal
l(zerooutcome),
whileother
swi l
l
part
ici
pateanddeter
minetheint
ensi
tyorquantit
yoft heacti
vi
ty(
positi
veoutcome).

3.Explanatoryvari
abl
es:Theresearchermustident
if
yr el
evantexpl
anat
oryvari
ables
thatmayi nfl
uenceboththeparti
cipati
ondeci
sionandt heint
ensit
yorquanti
tydecision.
Thesev ari
ablescanincl
udeindiv
idualchar
acteri
sti
cs,soci
o-economicf
actors,
orot her
contextualfact
ors.

4.Het er
ogenei
ty:
Thedoubl ehurdl
emodel all
owsforheterogenei
tyi
nbotht he
part
icipati
ondeci
sionandtheintensi
tyorquanti
tydeci
sion.Thismeansthatdiff
erent
i
ndividualsmayhavediff
erentfact
orsinf
luenci
ngthei
rdecision-
makingprocess.

5.First-
stagehur dl
emodel:I
nthefi
rstst
ageofthedoublehurdlemodel,
abinaryprobit
orlogitregressi
onisusedtoesti
matetheprobabi
li
tyofpart
icipati
oni
ntheacti
vity
.This
stepest i
mat estherel
ati
onshi
pbetweentheexpl
anator
yv ar
iablesandt
heparti
cipati
on
decision.

6.Second-staget
runcatedregression:Inthesecondst age,
at runcat
edr egressi
on
model isusedtoesti
mat etherelati
onshipbetweent heexplanatoryvar
iablesandt he
i
nt ensi
tyorquant
it
yoft heactivi
tyforthoseindi
vidualswhochooset opartici
pate.Thi
s
stepaccountsfortheselecti
onbi asint
roducedbyt heparti
cipati
ondecision.

7.Consistencyandeff
ici
ency:Thedoubl
ehur
dlemodelpr
ovi
desconsi
stentand
eff
ici
entest i
matesofthecoeff
ici
ent
sift
heassumpt
ionsoft
hemodelaremetandt
he
necessaryconditi
onsaresat
isfi
ed.

Vi
olat
ingt
heseassumpti
onsmayl
eadt
obi
asedandi
nconsi
stentcoef
fi
cientest
imat
es
i
nthedoublehur
dlemodel
.

18
STRENGTH

Inaddit
iontonatur
allyi
ncorporati
ngthezeroty
pe, t
hedouble-
hur
dlemodelall
ows
esti
mationoftheproport
ionoft hepopul
ati
onthatisofthezerot
ype.Bet
terst
il
l,
it
all
owsthepr obabi
l
ityofasubject'
sbeingzerotypetodependonthesubj
ect'
s
charact
eri
sti
cs.

WEAKNESS

1.I
tassumesi ndependencebet weenthepart
ici
pationdecisi
onandt helevelof
part
icipati
on:Thedoublehur dlemodelassumest hatthedecisi
ont opar t
ici
pateandthe
l
evel ofparti
cipati
onareindependentofeachother.Howev er
, i
nr eal
ity
,theremaybe
i
nterdependenci esbet
weent hesetwoprocesses.Forexampl e,i
ndi v
idualswhochoose
topar t
ici
patemayhav edifferentmoti
vati
onsorchar act
eri
sti
cst hatinfl
uencethei
rlevel
ofpar t
ici
pati
on.

2.Itmaysuff erfr
om sampleselectionbi
as:Thedoubl ehurdl
emodel isbasedont he
assumpt i
ont hatthesampleusedf oranal
ysi
si srepr
esentati
veofthetargetpopulat
ion.
Howev er,
ifthereissampleselecti
onbias,meani ngt
hatcertai
nindi
vidualsaremor e
l
ikelytobeincludedinthesampl ebasedont heirpar
ti
cipat
ionstat
us,theestimates
obtainedfr
om t hemodelmaybebi ased.

3.Itmaybesensi ti
vetothechoi ceofdistr
ibuti
onalassumpti
ons:Thedoublehurdl
e
model assumesspeci f
icdistr
ibuti
onsfortheer r
orter
msi nbothpart
softhemodel.If
theseassumpt i
onsdonothol dinpract
ice,theesti
matesobtai
nedfrom t
hemodel may
beunreli
able.Addit
ional
ly,
t hechoiceofdistri
buti
onalassumpti
onsmayaf f
ectthe
i
nterpr
etati
onoft heresul
ts.

4.I
tmaybecomput at
ional
l
ydemandi ng:Theesti
mati
onofthedoubl
ehurdlemodel
can
becomputati
onall
ydemandi ng,especial
lywhendeal
i
ngwithlar
gedataset
sorcomplex
model
s.Thiscanlimitit
spracti
cal appli
cabi
l
ityi
ncer
tai
nsit
uati
onswhere
comput
ati
onal r
esourcesarelimited.

5.Itmaynotcapt ureallrelevantfactorsi
nfluencingparti
cipati
on: Whil
ethedouble
hurdl
emodel allowsf oracompr ehensiveunder st
andingoft hefactorsi
nfl
uencing
parti
cipat
ion,
itisstil
lli
mi t
edbyt hevari
ablesincludedintheanal ysi
s.Ther
emaybe
otherunobservableoromi ttedvar i
ablesthatalsoplayasigni f
icantrol
eindeter
mining
parti
cipat
iondecisionsandl evels.

MULTI
PLELI
NEARREGRESSI
ONMODEL

Mul
ti
pleLi
nearRegr
essi
onModel
s

Quest
ions

19
–Whatar
emul
ti
pler
egr
essi
onmodel
s?

–Howdoyouthi
nkmul
ti
plel
i
nearr
egr
essi
onsar
edi
ff
erentf
rom si
mpl
eli
near
r
egr
essi
onmodel
?

–Whyaremul
ti
pler
egr
essi
onmodel
sadv
ant
ageousov
ersi
mpl
eli
nearr
egr
essi
on
model
?

–Doyouthi
nktheest
imati
onandi
nfer
encesi
nmul
ti
pler
egr
essi
onsi
mil
arwi
tht
hosei
n
si
mpl
eli
nearregr
essi
on?

•Manyeconomi
cvar
iabl
esar
einf
luencedbysev
eral
fact
orsorv
ari
abl
es.

•Si
mpl
eli
nearmodel
isnotpr
act
ical
excepti
tssi
mpl
i
cit
ytounder
stand.

•Addi
ngmorevar
iabl
estothesi
mpl
elinearr
egr
essi
onmodel
leadsust
othe
di
scussi
onofmul
tipl
eregr
essi
onmodels

•Modelsinwhi
chthedependentvari
abl
e(orr
egr
essand)dependsont
woormor
e
expl
anator
yvar
iabl
es,
orregressor
s.

•Themul t
ipl
elinearr
egressi
on(
popul
ati
onregressi
onf uncti
on)i
nwhichwehav
eon
dependentvar
iableY,andkexpl
anat
oryvar
iabl
es,X₁
,X₂
,.
...
..
..
Xkisgi
venby

Yi
=B0+B1X1+B2X2+.
..
.+BkXk+Ui

•Wher
eB0,
thei
nter
cept=v
alueofwhenal
lX’
sar
ezer
o

•Bi
=ar
epar
ti
alsl
opecoef
fi
cient
s

•Ui
=ther
andom t
erm

Example,
Biistheamountofchangei
nYiwhenX1changesbyoneuni
t,keepi
ngt
he
eff
ectofothervar
iabl
esconst
ant.

•Si
milar
ly,
B2istheamountofchangei
nYi
whenX2changesbyoneuni
t,keepi
ngt
he
ef
fectofother

v
ari
abl
esconst
ant
.•Theot
hersl
opesar
eal
soi
nter
pret
edi
nthesameway
.

Hy
pot
hesi
sTest
ingi
nMul
ti
pleRegr
essi
on

•I
mpor
tantt
odr
awi
nfer
encesaboutt
heest
imat
esand•

Toknowhowr
epr
esent
ati
vet
heest
imat
esar
etot
het
ruepopul
ati
onpar
amet
er.

hy
pot
hesi
stest
ingassumessev
eral
int
erest
ingf
orms.

20
a.Test
inghy
pot
hesi
saboutani
ndi
vi
dual
par
ti
alr
egr
essi
oncoef
fi
cient
.

b.Testi
ngtheover
all
signi
fi
canceoftheest
imatedmult
ipl
eregr
essi
onmodel
(fi
ndi
ng
outifal
lthepar
ti
alsl
opecoeff
ici
ent
sar esi
mult
aneousl
yequalt
ozero)

c.Test
ingi
ftwoormor
ecoef
fi
cient
sar
eequal
tooneanot
her

d.Test
ingt
hatt
hepar
ti
alr
egr
essi
oncoef
fi
cient
ssat
isf
ycer
tai
nrest
ri
cti
ons

e.Testi
ngthestabi
l
ityoft
heest
imat
edr
egr
essi
onmodel
overt
imeori
ndi
ff
erentcr
oss-
sect
ionaluni
ts

f
.Test
ingt
hef
unct
ional
for
m ofr
egr
essi
onmodel
s.

Mult
ipl
elinearr
egressi
oni
sar egr
essi
onmodelt
hatest
imatestherelati
onshipbet
ween
aquanti
tati
vedependentv
ari
ableandtwoormor
eindependentvar
iablesusinga
st
rai
ghtli
ne

Thegoal ofmult
ipl
elinearr
egr essi
onist
omodel theli
nearrel
ati
onshipbet weenthe
expl
anatory(i
ndependent)vari
ablesandresponse( dependent
)vari
ablesInessence,
mult
ipl
er egr
essionistheextensionofor
dinaryleast-
squares(OLS)regressionbecause
i
tinvol
vesmor ethanoneexpl anator
yvar
iable.

Multipl
elinearr
egressi
onr eferstoastati
sti
cal t
echni
quethatisusedtopr edi
ctthe
outcomeofav ari
ablebasedont hev al
ueoftwoormor evari
ables.I
tissometimes
knownsi mplyasmul ti
pleregression,
anditisanextensionofli
nearregressi
on.The
vari
ablethatwewantt opr edicti
sknownast hedependentvari
able,
whilethevariabl
es
weuset opr edi
ctthevalueoft hedependentv ar
iabl
eareknownasi ndependentor
explanatoryvar
iabl
es.

Multi
pleli
nearregressi
onr eferst
oast ati
sti
calt
echniquethatusestwoormor e
i
ndependentv ar
iablestopredictt
heout comeofadependentv ariabl
e.Mult
ipl
elinear
regr
essionmaybeusedt oinvest
igatetherel
ati
onshipbetweenacont i
nuous(inter
val
scal
e)dependentv ari
able,suchasi ncome,bl
oodpr essur
eorexami nati
onscore

Multi
pleregressi
onisabr oaderclassofregressi
onsthatencompasseslinearand
nonl
inearregressi
onswithmul ti
pleexplanat
oryvari
ables.Whereasli
nearregressonl
y
hasonei ndependentvar
iableimpactingtheslopeoftherelat
ionshi
p,mult
iple
regr
essionincorpor
atesmul t
ipl
eindependentv ar
iabl
es.

Whatar
ethef
ourassumpt
ionsofmul
ti
pler
egr
essi
ont
hatr
esear
cher
sshoul
dal
way
s
t
est?

Speci
fi
cal
l
y,wewi
l
ldi
scusst
heassumpt
ionsofl
i
near
it
y,r
eli
abi
l
ityofmeasur
ement
,

21
homoscedast
ici
ty,
andnor
mal
i
ty

Fiv
emai nassumpti
onsunder l
yi
ngmul t
ipl
eregressi
onmodel smustbesat i
sfi
ed:(
1)
l
inear
it
y,(2)homoskedasti
cit
y,(3)independenceoferrors,
(4)normal
i
ty,and(5)
i
ndependenceofindependentvariabl
es.Diagnosti
cplotscanhelpdet
ectwhetherthese
assumptionsar
esatisf
ied.

WEAKNESS

Multi
pleregr
essioncanal
sosufferf
rom overf
it
ti
ng,whichi
swheny ourmodelfi
tst
he
datatoowell
, andl
osesit
sabi
li
tytogeneral
izetoneworunseendata.Over
fit
ti
ngcan
occurwheny ouhavetoomanyindependentvari
abl
es,orwhenyourvari
abl
esarehi
ghly
corr
elat
edwi t
heachother.

STRENGTH

Mul
ti
plel
inearregressi
onall
owstheinv
esti
gatort
oaccountforal
lofthesepotenti
all
y
i
mport
antfactorsinonemodel.Theadvant
agesofthi
sapproacharethatthi
smayl ead
t
oamor eaccurateandpreci
seunderst
andi
ngoftheassociat
ionofeachindi
vidual
f
act
orwiththeoutcome.

Whatar
ethest
rengt
hsofmul
ti
plel
i
nearr
egr
essi
on?

Mul
ti
plel
inearregressi
onall
owstheinv
esti
gatort
oaccountforal
lofthesepotenti
all
y
i
mport
antfactorsinonemodel.Theadvant
agesofthi
sapproacharethatthi
smayl ead
t
oamor eaccurateandpreci
seunderst
andi
ngoftheassociat
ionofeachindi
vidual
f
act
orwiththeoutcome.

Whati
stheweaknessofmul
ti
plel
i
nearr
egr
essi
on?

Multi
pleregr
essioncanal
sosufferf
rom overf
it
ti
ng,whichi
swheny ourmodelfi
tst
he
datatoowell
, andl
osesit
sabi
li
tytogeneral
izetoneworunseendata.Over
fit
ti
ngcan
occurwheny ouhavetoomanyindependentvari
abl
es,orwhenyourvari
abl
esarehi
ghly
corr
elat
edwi t
heachother.

SI
MPLELI
NEARREGRESSI
ONMODEL

CONDI
TION

Simplelinearregr
essi
oni
susedtoesti
matet herel
ati
onshi
pbet
weent woquanti
tat
ive
vari
ables.Youcanusesimpl
eli
nearregr
essionwheny ouwanttoknow:Howst r
ongthe
rel
ati
onshi pisbet
weentwovar
iabl
es(e.
g.,
ther el
ati
onshi
pbet
weenr ai
nfal
landsoi
l
erosi
on).

Youcanusesi
mpl
eli
nearr
egr
essi
onwheny
ouwantt
oknow:

22
Howstrongt
herelat
ionshipi
sbet
weent
wov
ari
abl
es(
e.g.
,ther
elat
ionshi
pbet
ween
r
ainf
all
andsoil
erosi
on) .

Theval
ueofthedependentvar
iabl
eatacert
ainvalueoft
hei
ndependentv
ari
abl
e(e.
g.,
theamountofsoi
ler
osionatacertai
nlev
elofrai
nfal
l)
.

Exsmpl
eOfSi
mpl
eLi
nearRegr
essi
onModel

Youar easocialresearcheri
nter
est
edintherel
ati
onshipbetweenincomeand
happiness.Yousur vey500peoplewhoseincomesrangefrom 15kto75kandaskt
hem
toranktheirhappinessonascalefrom 1to10.

Yourindependentvar
iabl
e(i
ncome)anddependentvar
iabl
e(happi
ness)ar
ebot h
quanti
tati
ve,soyoucandoar egr
essi
onanal
ysistoseeift
hereisal
inearr
elat
ionshi
p
betweenthem.

ASSUMPTI
ONOFSI
MPLELI
NEARREGRESSI
ONMODEL

1Homogeneityofvar
iance(homoscedasti
cit
y):
thesi
zeoft heerr
orinourpr
edi
cti
on
doesn’
tchangesigni
fi
cantl
yacrosstheval
uesoftheindependentvari
abl
e.

2I
ndependenceofobservat
ions:
theobser
vat
ionsi
nthedatasetwerecoll
ect
edusi
ng
stat
ist
ical
lyval
idsampl
i
ngmet hods,
andther
earenohiddenrelat
ionshi
psamong
observati
ons.

3Nor
mal
i
ty:
Thedat
afol
l
owsanor
mal
dist
ri
but
ion.

4Therelat
ionshi
pbetweent
heindependentanddependentv
ari
abl
eisl
i
near
:theli
neof
bestfi
tthr
ought hedat
apoi
ntsisastrai
ghtli
ne(r
athert
hanacurv
eorsomesortof
groupi
ngfactor)
.

Thef
ormul
aforasi
mpl
eli
nearr
egr
essi
oni
s:

y=B0+B1x+€

yi
sthepredict
edv alueoft
hedependentv
ari
abl
e(y
)foranygi
venv
alueoft
he
i
ndependentvar
iable(x)
.

B0i
sthei
nter
cept
,thepr
edi
ctedv
alueofywhent
hexi
s0.

B1i
sther
egr
essi
oncoef
fi
cient–howmuchweexpectyt
ochangeasxi
ncr
eases.

xi
sthei
ndependentv
ari
abl
e(t
hev
ari
abl
eweex
pecti
sinf
luenci
ngy
).

eistheer
roroft
heesti
mat
e,orhowmuchv
ari
ati
ont
her
eisi
nourest
imat
eoft
he
r
egressi
oncoeff
ici
ent
.

23
STRENGTH

Youcanuset hisequati
ontoestimat
etheval
ueofyf oranygivenv alueofx,ortotest
hypothesesaboutthesigni
fi
canceanddi
rect
ionoftherelat
ionship.Youcanalso
vi
sualizethel
inearr
elat
ionshi
pbyplot
ti
ngthedatapointsandt heregressi
onlineona
graph.

Simplelinearregressi
onisusedt omodel t
her elati
onshi
pbet
weentwocont
inuous
vari
ables.Often,theobjecti
veistopredictthev alueofanout
putv
ari
abl
e(orresponse)
basedont hevalueofani nput(
orpredictor)variabl
e.

WEAKNESS

Lineari
ty:
Theassumptionofl
inear
it
ybet
weenv ari
abl
esrest
ri
ctsli
nearregressi
ons.The
premiseofastrai
ght
-l
inerel
ati
onshi
pisusual
l
yf al
seandmaypr ovi
deinaccurat
e
result
s.

PROPENSI
TYSCOREMATCHI
NHMODEL

CONDI
TION

Propensityscor
emat ching(PSM)isaquasi -
experi
mentalmethodi nwhicht he
researcherusesstat
ist
icaltechni
questoconstructanart
if
ici
alcontrolgroupby
mat chi
ngeacht r
eatedunitwi t
hanon-t
reatedunitofsi
milarcharacter
ist
ics.Using
thesemat ches,t
heresearchercanesti
matet heimpactofanintervent
ion.

PSM i sbasedont heassumpt ionthatt het reat

mentassi gnmenti si ndependentoft he
potentialout
comes, gi
v enasetofcov ariatesthatinf
luencebot h.Thi sisknownast he
conditionali
ndependenceassumpt ion( CIA).PSM estimat esthepr opensit
yscor e,
whichi stheprobabil
it
yofr eceivi
ngt het r
eat mentforeachsubj ect ,
usingal ogist
ic
regressionoranothermodel .Then,itmat chessubjectswi thsimilarpropensityscores,
eit
herbyexactmat chi
ng, nearestneighbormat chi
ng,calipermat ching,orother
met hods.Themat chedpai rsarethenusedt oesti
mat etheav eraget r
eatmentef fect
(ATE)ort heaveragetreatmentef f
ectont het r
eated(ATT) .

Thef
ormul
aforpr
opensi
tyscor
emat
chi
ngi
sasf
oll
ows:

PS=Pr
(T=1|
X)

Wher
e:

-PSisthepr
opensi
tyscor
e,whi
chr
epr
esent
sthepr
obabi
l
ityoft
reat
ment(
T=1)gi
ven
theobser
vedcovar
iat
esX.

-Ti
sthet
reat
mentv
ari
abl
e,whi
chi
sabi
nar
yvar
iabl
eindi
cat
ingwhet
herani
ndi
vi
dual

24
r
ecei
vedt
het
reat
mentornot
.

-Xisavectorofobserv
edcovar
iat
es,whi
charei
ndependentv
ari
abl
est
hatmayaf
fect
bot
hthetreatmentassi
gnmentandtheout
come.

Pr opensityscoremat chingisamet hodusedt oest imatethecausal effectofa

treat mentbymat chingt r
eatedindividualswithsimilarunt
reatedindivi
dual sbasedon
theirpr opensi
tyscor es.Thepr opensi tyscoreisestimatedusingal ogisticregression
model ,wherethet reatmentassi gnmenti sregressedont heobser vedcov ari
ates.Aft
er
obt aini
ngt hepropensi tyscores,treatedindivi
dualsaremat chedwi thcont roli
ndivi
duals
whohav esimil
arpr opensityscor es,creati
ngamat chedsampl e.Theef fectoft he
treat mentcant henbeest imatedbycompar i
ngtheout comesoft het r
eat edand
mat chedcontrolindividuals.

Pr opensi t
yscor emat ching( PSM)hassev eraladv ant
agesov erot hermethodsof
adj ustingf orconf oundi ngf actorsinquasi -experi
ment s.Itcanr educet hedimensi onali
ty
oft hecov ariates,maki ngi teasi ert
obal ancet hem acr osst hetreatmentandcont rol
groups.Addi tional
ly,PSM canav oi
dmodel misspecif
ication,asi tdoesnotr elyona
funct ional form oft heout comeequat i
on.Fur thermore, i
tcanhandl eheterogeneous
treat mentef fects,asi tallowsf ordiff
erentef fectsfordifferentsubgr oupsofsubj ect
s.
Mor eov er
, PSM cani ncr easet hepr ecisi
onoft heest i
mat esbyr educi ngthev ari
anceof
theer rort
er m.

HOW TOI
MPLEMENTPSM

PSM canbei mpl ement edusi ngv arioussof twarepackages, suchasR, Stat a,SAS, or
SPSS.Gener ally
, theprocessi nv ol
v esident i
fyi
ngt hetreatmentandout comev ariables,
aswel l ast
hecov ari
atest hatmayaf f
ectbot h.Then, y
oumustest imat ethepr opensi t
y
scoref oreachsubj ectusi ngal ogisticregressionoranot hermodel .Af t
ert hat,you
shouldchooseamat chingmet hod, suchasexactmat chi
ng,nearestnei ghbormat ching,
cali
permat ching, orothermet hods.Subsequent ly,youmustassesst hebal anceoft he
covariatesacrosst het r
eat mentandcont rolgroupsusi ngst
andar dizedmean
dif
ferences,histograms, orothertest s.Finall
y,youshoul desti
mat et heATEorATT
usingt hematchedpai rsbyt akingt hemeandi fferenceorusingar egr essionmodel .

WEAKNESS

PSM canbei mplement edusingvarioussoftwarepackages,suchasR, Stata,

SAS, or
SPSS.Generall
y,theprocessinvol
v esidenti
fyi
ngthetreat
mentandout comev ari
ables,
aswellasthecov ar
iatesthatmayaf f
ectboth.Then,y
oumustest i
matet hepropensit
y
scoreforeachsubjectusingalogisticregr
essionoranothermodel .Af
terthat,y
ou
shouldchooseamat chingmethod, suchasexactmat chi
ng,nearestneighbormatching,

25
cali
permat chi
ng,orot
hermet hods.Subsequentl
y,youmustassesst hebal
anceoft
he
covariatesacr
ossthetr
eatmentandcont rolgr
oupsusingstandar
dizedmean
dif
ferences,hi
stogr
ams,orothertests.Fi
nal
ly,y
oushouldest i
mat
et heATEorATT
usingt hematchedpai
rsbytakingthemeandi ff
erenceorusingaregressi
onmodel.

Regressionscanbewei ghtedbypropensi
t yscor
esinordert
or educebias.However
,
weighti
ngi sli
kel
ytoincreaserandom er
rorintheesti
mates,andtobiastheestimat
ed
st
andar derrorsdownwar d,
evenwhensel ecti
onmechanismsar ewellunderst
ood.

I
fthegroupshaveli
tt
leoverl
apinpropensit
yscores,
theyareinher
entlyi
ncompar abl
e,
andnostati
sti
calt
ri
ckscanov er
comet hi
spr obl
em.Tradi
ti
onal methodsforcontrol
l
ing
forconf
oundi
ngbyindicat
ionmayfailtorevealt
hisi
rr
econcil
ableli
mitat
ioninthedata,
l
eadingtoerr
oneousconclusi
ons.

STRENGTH

Propensit
yscor
emet hodsreducetheeffectsofconf
oundi
ngduetomeasur edbasel
i
ne
covari
atesbycr
eati
ngamat chedorweightedsampleinwhi
chthedistr
ibut
ionof
measuredbasel
inecovari
atesissimi
larintreat
edandcontr
olpar
ti
cipants

ASSUMPTI
ON

Thecondi
ti
onsf
ort
hePr
opensi
tyScor
eMat
chi
ng(
PSM)model
areasf
oll
ows:

1.Treat
mentassignment:
Thereshoul
dbeacleart
reat
mentassi
gnmentmechanism,
whereindi
v i
dual
sareassi
gnedtoei
therat
reat
mentgrouporacont
rol
groupbasedon
cert
aincri
teri
aorrandomi
zati
on.

2.Over
lap:Thereshouldbeov er
lapinthedistr
ibuti
onofthepropensit
yscor
esbetween
thet
reatmentandcont r
olgroups.Thisensuresthatther
eareindiv
idual
sinbot
hgroups
whohav esimil
arprobabi
li
ti
esofr ecei
vingt
het reat
ment,al
lowingformeani
ngful
comparison.

3.Independence:
Thetreatmentassi
gnmentshouldbeindependentofpotenti
al
outcomes, meani
ngthatther
eshouldbenosy st
emati
cdifferencesbetweenthe
tr
eatmentandcont r
olgroupsthatar
enotaccountedf
orbyt heobservedcovari
ates.

4.Commonsuppor t:Thereshouldbecommonsuppor t
,meani
ngthatthereare
i
ndivi
dualsinboththet r
eatmentandcont rolgroupswhohavesimi
larvaluesofthe
observedcovar
iates.Thisensuresthatthereissuff
ici
entov
erl
apinthecov ar
iat
e
di
stri
buti
onsformeani ngfulcomparison.

5.Unconfoundedness:
Thetr
eatmentassi
gnmentshoul
dbeunconf
ounded,
meani
ng
thatt
herearenounobserv
edfactor
sthatsi
mult
aneousl
yaf
fectt
hetreat
ment

26
assi
gnmentandtheout
comevar
iable.Thi
sassumpt
ioni
snecessar
ytoensur
ethatt
he
esti
matedt
reat
menteff
ecti
snotbiased.

6.Balance:Theobserv
edcov ari
atesshoul
dbebal ancedbet weenthetreatmentand
controlgr
oupsaftermatchi
ng.Thismeanst hat,
onav erage,indi
vi
dualsinthetr
eatment
groupshouldhavesimil
arvaluesoftheobservedcov ar
iatesasindi
vidualsint
hecontr
ol
group.

7.Corr
ectspeci
ficati
on:Themodelusedtoesti
matethepr
opensi
tyscor
esshouldbe
corr
ectl
yspecif
ied,meaningt
hatiti
ncl
udesallr
elev
antcov
ari
atesandcorr
ect
ly
captur
estherel
ationshi
pbetweenthecovar
iat
esandthetr
eatmentassi
gnment.

8.Sensiti
vi
tyanaly
sis:Sensit
ivi
tyanal
ysi
sshouldbeconductedtoassessthe
robustnessoftheesti
mat edtreat
menteffect
stopotent
ial
viol
ati
onsofthe
assumpt i
ons.Thi
shelpst oevaluat
ethereli
abi
li
tyoft
heresult
sandidenti
fypot
ent
ial
sourcesofbias.

I
tisi
mpor t
anttonot
ethatviol
ati
ngtheseassumpt
ionsmayl
eadt
obi
asedand
i
nconsi
stentt
reat
menteffectest
imatesi
nthePSM model
.

SOMEEXAMPLEOFPSM

PSM hasbeenut i
lizedineducat i
onr esearchtomeasur et heef fect
sofv ari
ous
i
nterv enti
onsorpol i
ci
esonst udentoutcomes, suchasacademi cachievement,dropout
rat
es, orcollegeenr oll
ment .Forinst
ance, theeffectofchar terschoolsonst udent
achiev ementcanbedet erminedusingdat afrom t heNat i
onal Assessmentof
Educat ionalProgress( NAEP) .Merit
-basedschol ar shi
pscanbeev al
uatedfortheir
i
mpactoncol legeenr oll
mentandcompl etion,basedondat af rom theFlori
daBright
FuturesSchol arshipPr ogram.Addi t
ionall
y,theeff ectofbili
ngual educati
onon
Englishlanguagepr oficiencycanbeassessedusi ngdat afrom theCal i
forni
aPropositi
on
227.A

27
Ref
erence

Cragg,J.G.1971.Somest
ati
sti
calmodel
sf orl
imit
eddependentvar
iabl
eswi
th
appli
cati
ontothedemandfordurabl
egoods.Econometr
ica39:829–844.

Jones,
A.M.1992.Anot
eoncomput
ati
onoft
hedoubl
e-hur
dlemodel
wit
hdependence

wi
thanappl
i
cat
iont
otobaccoexpendi
tur
e.Bul
l
eti
nofEconomi
cResear
ch44:
67–74.

Jones,
A.M.
,andS.T.Yen.2000.ABox-
Coxdoubl
e-hur
dlemodel
.Manchest
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68:
203–221.

Mar
t´
ınez-
Espĩnei
ra,
R.2006.ABox-
CoxDoubl
e-Hur
dlemodel
ofwi
l
dli
fev
aluat
ion:

Theci
ti
zen’
sper
spect
ive.Ecol
ogi
cal
Economi
cs58:
192–208.

O'
Connell
,A.A.(
2006)
.Logi
sti
cregr
essi
onmodel
sforor
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responsev
ari
abl
es.
ThousandOaks,
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28
Rampichi
ni,
C.,
&Schif
ini
,S.(1998).Ahi
erar
chi
calordi
nal
probi
tmodel
fort
heanal
ysi
s
ofl
if
esati
sfact
ioni
nIt
aly.Soci
alIndi
cat
orsResear
ch,44,
5–39

O'
Connell
,A.A.(
2006)
.Logi
sti
cregr
essi
onmodel
sforor
dinal
responsev
ari
abl
es.
ThousandOaks,
CA:Sage.

Long,S.(
1997).Regr
essi
onmodel
sforcat
egor
ical
andl
i
mit
eddependentv
ari
abl
es.
ThousandOaks,CA:Sage.

29