Simultaneously Modeling Joint and Marginal Distributions of Multivariate Categorical
Responses
Author(s): Joseph B. Lang and Alan Agresti
Source: Journal of the American Statistical Association, Vol. 89, No. 426 (Jun., 1994), pp. 625632
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SimultaneouslyModeling Jointand Marginal
Distributionsof MultivariateCategorical Responses
Joseph B. LANGand Alan AGRESTI*
We discuss model-fitting
methodsforanalyzingsimultaneously
the joint and marginaldistributions
of multivariatecategorical
responses.The modelsare membersofa broadclassofgeneralizedlogitand loglinearmodels.We fitthembyimproving
a maximum
likelihoodalgorithmthatuses Lagrange'smethodof undetermined
and a Newton-Raphsoniterativescheme.We also
multipliers
testsand adjustedresiduals,and giveasymptoticdistributions
discussgoodness-of-fit
of model parameterestimators.
For thisclass
areequivalentforPoissonand multinomialsamplingassumptions.Simultaneousmodelsforjointand marginal
ofmodels,inferences
distributions
maybe usefulin a varietyofapplications,includingstudiesdealingwithlongitudinal
data,multipleindicatorsin opinion
research,cross-overdesigns,social mobility,and inter-rater
agreement.The modelsare illustrated
forone such application,using
data froma recentGeneralSocial Surveyregarding
opinionsabout varioustypesof government
spending.
KEY WORDS:
maximumlikelihood;Lagrangemultiplier;
Adjustedresiduals;Constrained
Marginalmodels;Ordinaldata;Repeated
measurement.
1. INTRODUCTION
logitmodelsthatone can applysimultaneously
to thejoint
and
marginal
distributions.
The
marginal
distributions
ConsiderTable 1, taken fromthe 1989 General Social
modeled may be of any order.For instance,the modeling
SurveyconductedbytheNationalOpinionResearchCenter
of
pairwiseassociationswithoutcontrollingforothervariat the Universityof Chicago. Subjectsin the sample were
ables
refersto second-order
marginaltablesofthejoint disasked theiropinion regardinggovernmentspendingon (1)
Section
tribution.
3
expresses
the models usingconstraint
(2) health,(3) assistanceto big cities,and
theenvironment,
specifications.
We
then
propose
a model-fitting
approachthat
The common responsescale was (too
(4) law enforcement.
improves
on
approaches
discussed
by
Aitchison
and Silvey
little,about right,too much).
(1958)
and
Haber
(1985a),
using
the
method
of
Lagrange's
types
Two typesofquestionsaboutTable I lead to distinct
multipliers.
The improvedalgorithmapplies
of models.One questionrelatesto how the responsedistri- undetermined
a
Newton-Raphson
iterative
schemein whichthematrixto
butionsdiffer
forthefouritems.For instance,one mightask
be
inverted
has
much
simpler
formthanin previouslyprowhethersubjectsregardedspendingas relativelyhigheron
posed
algorithms.
one itemthanon theothers.Answersto thisquestionrefer
Our analysesassumea multinomialsamplingschemefor
ofTable
to modelingthefourone-waymarginaldistributions
cell counts.Section4 providesasymptoticdistributions
the
1. A second questionpertainsto the dependencestructure
of
the
parameterestimatorsand generalizesresultsof Birch
oftheresponses.For instance,one mightstudythestrength
(1963)
and Palmgren(1981) relatingthesedistributions
to
of associationbetweenresponseson variouspairsof items,
conthose
obtained
assuming
Poisson
sampling.
Section
5
associated
analyzingwhethersome pairsare more strongly
sidersmodel goodnessof fitand showsthatit can be partithan othersor whetherthe pairwiseassociationvariesacintogoodnessof fitforthe marginaland joint comtioned
cordingto responseson theotherindicators.Consideration
This sectionalso generalizesHaberman's (1973)
ponents.
in
ofthesemattersrelatesto modelingthejoint distribution
adjusted
residuals
forinspectingthe fitboth in cells of the
Table 1.
in marginaltotals.Section6 illustrates
and
table
thesimulof categoricalreStandardmodels forjoint distributions
7 distaneous
Table
and
Section
fitting
approach
using
1,
among marginal
sponsesdo not implysimplerelationships
of
simultaneous
cusses
potential
compatibility
problems
joint
ofcatdistributions.
Hence modelingmarginaldistributions
discussesothertypes
and
marginal
models.
Finally,
Section
8
egoricalresponsesis normallyconductedseparatelyfrom
ofdata setsforwhichsimultaneous
jointand marginalmodIn thisarticlewe showhow
modelingofjoint distributions.
is
relevant.
eling
This
modelsof the two typescan be fittedsimultaneously.
processprovidesimprovedmodel parsimony.One also ob- 2. SIMULTANEOUS
JOINTAND MARGINALMODELS
summarizesgoodness
tainsa singletestthatsimultaneously
Let T denotethe numberof componentvariablesin the
valuesand residuals.Estimators
offitand a singlesetoffitted
multivariate
response,letI, denotethenumberofcategories
of model parametersand cell expectedfrequenciesare also
in the response scale for response variable t, and let r
than withseparatefitting
procepotentiallymore efficient
=
ofpossibleresponse
profiles.
t=I It denotethenumber
dures.
thatobservationson the responsesare obtainedat
Suppose
We considerthe modelingof multivariate
categoricalres fixedlevelsofa setofexplanatoryvariables.The observed
responsescale is allowed for
sponses in which a different
data can be displayedin an s X r contingencytable. For
each response.The responsescales may be nominalor ormostofournotationrefers
to a singlepopulation
simplicity,
dinal.Section2 specifiesa generalizedclassoflog-linearand
1).
(s=
* JosephB. Lang is AssistantProfessor,Departmentof Statisticsand
ActuarialScience,University
of Iowa, Iowa City,IA 52242. Alan Agrestiis
of Florida,Gainesville,FL
University
Departmentof Statistics,
Professor,
32611.
? 1994 AmericanStatisticalAssociation
Journalof the AmericanStatisticalAssociation
June4994,Vol. 89, No. 426, Theoryand Methods
625
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Journal of the American Statistical Association, June 1994
626
Table 1. OpinionsAboutGovernment
Spending
1
Cities
Law Enforcement
Environment Health
2
3
1
2
3
1
2
3
1
2
3
1
1
2
3
62
11
2
17
7
3
5
0
1
90
22
2
42
18
0
3
1
1
74
19
1
31
14
3
11
3
1
2
1
2
3
11
1
1
3
4
0
0
0
1
21
6
2
13
9
1
2
0
1
20
6
4
8
5
3
3
2
1
3
1
2
3
3
1
1
0
0
0
0
0
0
2
2
0
1
1
0
0
0
0
9
4
1
2
2
2
1
0
3
NOTE: These data are fromthe 1989 General Social Survey,withcategories 1 = too little,2
= about right,
and 3 = too much.
have the same formas the matricesin the specification
of
themarginalmodel.
Permissiblemodelsforthejoint distribution
includenot
onlysimplelog-linearand logitmodelsbut also modelsfor
log odds ratiosusinggroupingsof cells,such as models for
globalodds ratios(Dale 1986). The marginalmodelcan also
be a log-linearor a corresponding
logitmodel,or some other
formof multinomialresponsemodel,such as a cumulative
logitmodel.Haber (1985a,b) gaveexamplesofnonstandard
modelsthatfallin thisclass of marginalmodels.
3. MAXIMUMLIKELIHOODFITTING
OF
MODELS
SIMULTANEOUS
The standardapproachto maximumlikelihood(ML) fittingof marginalor simultaneousmodels involvessolving
the score equations using the Newton-Raphsonmethod,
Fisher scoring,or some other iterativereweightedleast
squaresalgorithm.
The mostcommonapproach-used, for
instance,by Dale (1986), McCullaghand Nelder(1989, p.
219), Lipsitz,Laird,and Harrington
(1990), and Beckerand
in the
Balagtas(199 1)-reparameterizesthecellprobabilities
in termsofthejointand marginal
multinomial
log-likelihood
distribution
model parameters.A severelimitationof this
is typicallydifficult
approachis thatthereparameterization
and veryawkwardforthegeneralT-variatecase.
An alternative
method,discussedbyAitchisonand Silvey
(1958), Haber (1985a), and Haber and Brown (1986), is
Lagrange'smethodofundetermined
multipliers.
One views
the modelsas inducingconstraints
on the cell probabilities
and maximizesthe likelihoodsubjectto theseconstraints.
Because the algorithmthatwe use is a modificationof Hadescribehis algorithm.
ber's,we firstbriefly
The simultaneousmodelscan be specifiedas
Let Yi denote the numberof subjectshavingresponse
profilei, wherei = (i,, .. , 0i). Let ir = (,xi, ..., 7X'),
where7ridenotesthe probabilitythata randomlyselected
subjecthas responseprofilei. We assumethatY = (YI,
,
withprobabilities
ir. We
Yr)'has a multinomialdistribution
in thecellsofthe
denotecorresponding
expectedfrequencies
tableby,u.For marginaldistribution
t, let k(t)
contingency
ofresponsek. Whenresponsevariable
denotetheprobability
j
t is ordinal,let yj(t) =
k=i M(t) denotethejth cumulative
marginalprobability.
and
Let J(*) denote a model forthejoint distribution,
let M(*) denote a model forfirst-order
marginaldistributions.Let J(A) n M(B) denotethemodel thatspecifiessiand model
multaneouslymodelA forthejoint distribution
B forthe marginaldistributions.
Let S denotethesaturated
model. For instance,themodel J(S) n M(B) assignsstrucC log AIu= Xf,,
ident(,u)= ?,
(1)
and permitsarbitrary
tureonlyto themarginaldistributions
higher-order
interactionsamong the respotises.Fittinga where C = C1 E C2, A' = (A'1, A'), X = X1 E X2, A
the ;si- = (23', fl2)',and ident(It)= 0 denotesmultinomialidentimarginalmodel M(B) alone is equivalentto fitting
multaneousmodel J(S) n M(B). The simplestmodel of fiabilityconstraints.The symbolE denotesa directsum.
I C1 is theblockdiagonalmatrix
marginalhomogeneity,
de- (For example,C1 E C2 = E i2=
interestforM(*) is first-order
C1
with
and
as
the
C2
blocks.)For thecase ofs independent
notedby M(H).
of
multinomial
samples sizesn = (nI, n2, ... , ns)', themulFittinga joint model J(A) alone is equivalentto fitting
tinomial
identifiability
constraintshave the form(ED l r),,
thesimultaneous
modelJ(A) n M(S), whichfocuseson
=
n
a
0.
Let
U
denote
fullcolumnrankmatrixsuchthat
withoutmakingassumptionsabout
theinteraction
structure
the
columns of U is the orthogonal
This class includesordinarylog- the space spannedby
themarginaldistributions.
of
the
space spannedby the columnsof X, so
linearmodelsforexpectedcell countsin contingency
tables. complement
=
U'X
that
0.
Model
(1) is equivalentlyexpressedas the
in marginal
Thesemodelsare notdesignedto estimateeffects
model
constraint
because theirparametershave a conditional
distributions,
interpretation.
The simplestmodel of interestfor J(* ) is
U'C log A, = 0,
ident(u) = 0.
(2)
mutualindependenceof theresponses,denotedby J(I).
This articlefocuseson a moreparsimoniousclassofmod- Haber (1985a) outlineda methodforcomputingU.
The objectiveis to maximizethekernelofthemultinomial
els thatuse unsaturatedcomponentsforboththejoint and
log-likelihood,
1(,; y) = y' log ,u,subjectto model (1) or,
we considermodelsfor
marginaldistributions.
Specifically,
equivalently,
(2),
holding.We expressthemodelparameter
each distribution
thathave formC log A,u= X,8.We denote
as
space
the model forthejoint distribution
by C1 log A1g = XI#,
and themodel forthemarginaldistributions
by C2 log A2,I {,u : U'C log A,u= 0, ident(,u = 0 }
= X2f32.
For each component,we take C to be eitheran
= {,:f(,u)
= 0,
ident(,u) - 0} .
identity,contrast(rows sum to 0), or zero matrixand we
assume thatthemodel matrixX has fullcolumnrank.The Haber (1985a) used a Newton-Raphsonalgorithmto solve
matricesin the specificationof the joint model need not theLagrangianlikelihoodequations
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Lang and Agresti: Simultaneous Modeling forCategorical Responses
[al& Y) +a ident( A), Af (^)1
+
T~+
whereD(e ) denotesa diagonalmatrixwiththeelementsof
et on the main diagonal.We set () = log y and x (?) = 0,
makingthe slightadjustment ? = log(y + E) forsome
small e > 0 whenthereare samplingzeros.
Lang (1992) proved that G(6) is the dominantpart of
A
O,
f0()
g(0+)
627
ident(,)
in thesensethat
where0+ = vec(,u,X, r), withX and r beingvectorsof un- dg(0)/aO',
determinedmultipliers.For the models he considered,T
&g(O)= nG(f) + [(1)
could be solvedforexplicitly
and was nonstochastic.
Hence
he consideredthe solution0 = vec(,, A) to the simplerset
of equations
whereo(l) represents
a sequence thatconvergesto 0 as n*
= min{ ni, . . ., n,} -} oo. The utilityof usingG(O) is due
(9l(ji;y) U + (9f()'1)X
Ay;)
to thesimplicity
ofitsinverse,whichis
'
g(f) =(.
1=0,
0]:
G-1 (0)
whereu = rsdenotesthelengthofthevectorsy and ,u.The
Newton-Raphsoniterativeschemeis
0(t+1)= o(t)-
(90' )
t = 0, 1, 2, .
-Ig(0t)I
wherethederivativematrixcan be calculatedas
'g; + (9 fCO (X?)
(
/
(9g(0)9/.L'(/L
9
[-D-1 + D-1H(H'D-1H)- H'D-'
L
(Hl)-'H)-'H'D-'
D-1H(H'D-1H)-'
(H,D-'H)-l
whereD = D(et) and H = H(t). To invertG, we needonly
inverta diagonalmatrix,D, and a symmetric
positivedefinite
matrix,H'D 1H. Afterobtainingthefitted
values,u= e on
ofthealgorithm,
convergence
we calculatemodelparameter
estimatesusing,B= (X'X ) 1X'C log Aji.
)1
4. ASYMPTOTICBEHAVIOROF ESTIMATORS
One can use thedeltamethodto describetheasymptotic
behaviorof 0 = vec(4, X) and severalcontinuousfunctions
dy'
?
of@,suchas , = et and 13In thissection, 1,,, and ,Bdenote
A drawbackto thisalgorithmis thatthematrixag(0)/490' thetrue(unknown)parametervalues.Assumingthatmodel
is typically
verylarge(withthenumbersofrowsand columns ( 1) holds,Lang ( 1993) showedthatundercertainnonrestricexceedingthenumberofcellsin thecontingency
table)and tiveassumptions,theasymptoticnormaldistributions
does nothavea simpleform,makinginversiondifficult.
Our
X
AN(O, AM),
proposedmodificationsof this algorithmuses an iterative
schemeinvolvinga matrixthatis much simplerto invert.
AN(, If'),
Also, we use a reparameterization
from,uto t = log , for
ii
2(^A
~N(;,g,
boththelog-likelihood
and modelparameterspace,to avoid
and
interimout-of-range
values duringtheiterativeprocess.
Forthereparameterized
kernelofthelog-likelihood,
l(t;y)
,
(5)
N(O, 7, ^
af
L
=
~~~0
y'C, we express the model parameter space as
{1:U'ClogAe
=O,
hold,where
ident(0)=0}
= {t: h(t) = 0,
ident(t) = 0}.
We solve for4 by solvingfor0 in thelikelihoodequations
[
g(b) =
a/t Y)et
(M) = D-
+ H(t)1
= 0,
(3)
whereH(t) = clh()'/(9and 0 = vec( , X). To do this,we
use the modifiedNewton-Raphsoniterativescheme
-
0(t) -(G(0
(t)))-'g(0
(t)),
t=
with
[-D(et)
E) sk
E)s
Irtr -
AIAk
D-'H(HfD-'H)-
H(HfD-'H)
H
D-'
H,
and
(M)
2,6
=
0(t+1)
-'lH)-'
25-M) = D-'l-
49h()
h(s)
=[y-e
2(M) =
()
0, 1, 2,
. .
.,
(4)
(X'X
(-M)A'D(Au)
-1XfCD(AIf1AX
-1C'X (X'X )-1.
The assumptionsare analogousto requiringthata standard
log-linearmodel containall "fixed-by-design"
parameters.
Moreover,X and t areasymptotically
independent.The limiting distributions in (5) hold as n* = min{ n1, ...,
n,}
convergesto infinity
such thateach multinomialindex nk
growsat the same rate.
Under theseassumptions,it followsfromBirch (1963)
thatthesame pointestimatesoccurwhenwe treatcellcounts
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628
Journal of the American Statistical Association, June 1994
as independent
Poissonrandomvariables.
The asymptotictimated
standard
errors
asymptotic
neededto formtheadcovariance
matrices
forPoissonsampling
are
justedresiduals.
Fromthismatrix,
onecanalsoeasilyobtain
theestimated
covariance
matrix
fordifferences
between
obX(P)= (H=
D-1H)servedand fitted
marginal
counts,whichare sumsofcorX P) = D-1-D-'H(H'D-'H
-'HD
responding
differences
forthecells.Usingstandard
errors
of
such
marginal
differences,
one
can
construct
adjusted
residxf) = D -H(H1D-'H)-1H,
ualsformarginal
countsormarginal
cumulative
counts.
and
Atthestart
ofthemodel-fitting
process,
itis often
unclear
z (P)
whichsimultaneous
modelsshouldbe considered.
To help
determine
whichmodelsmayfitwell,onecanfirst
investigate
= (X'X ) -X'CD(Ag)-'A,
(P)A'D(Au) -LC'X (X'X)-.
jointandmarginal
modelsseparately.
Ifa separate
modelof
The limitingdistributions
so willthesimultaneous
hold as i,u= min{ /.i } converges eachtypefitswell,thengenerally
ofthosetwocomponents.
to infinity,
suchthateach Iii growsat thesamerate.The modelconsisting
In fact,formodestimated
in thisarticle,
covariance
foreither
asymptotic
matrices
thesimultaneous
modelJ(A)
sampling elsconsidered
schemeare easilycomputedon convergence
of iterative nM(B) hasa likelihood-ratio
statistic
asymptotically
equivscheme(4), becausetheyinvolve
thatareinput alentto thesum of thestatistical
onlymatrices
valuesfortheseparate
orcomputed
theinversion
ofG(0).
during
models J(A) n M(S) and J(S) n M(B). We now outline
The asymptotic
covariance
ofmodelparameterwhythishappens.
matrices
estimators
undermultinomial
andPoissonsampling
arereConsidertwohypotheses,
[wl and [M2L, suchthatboth
latedbyh:(M) = X(P) - A, where
arespecialcasesofa hypothesis
is a special
[wo]butneither
case
of
the
other.
Aitchison
Following
(1962),
and ['o2 I
[
I]
A = (X'X) -X'CD(AIu)f
arecalledasymptotically
ifthetestsof['oili n['O2I
separable
against[2I(['l ]l [o2]) andof[o1,]
[coI
in ['2] against
X A(Dsk=l-Lk kADD(At) -CXX(X X)
- ([oil
critical
regions
['2]) use thesamelarge-sample
as do theunrestricted
testsof[ oI] against[oo] - [coI] and
is a nonnegative
definite
matrix.
Lang(1992, 1993)generof ['o2] against['wol]- [W21. In ourcontext,
let[Xls] refer
to
alizeda result
ofPalmgren
(1981)forsimplelog-linear
modJ(A) n M(S) and let ['o2] refer
to
n
J(S)
M(B).
From
elsregarding
standard
errors
ofelements
in, beingthesame
results
forG2 withnestedmodels,G2 fortesting
forthetwosampling
schemes(i.e.,thecorresponding
com- standard
the
fit
of
n
J(A) M(B) can be partitioned
into(1) G2 for
ponentsofA are0). Forthegeneralized
log-linear
models
this
model
testing
the
against
alternative
of
J(S) n M(B),
(1), inferences
forall parameters
exceptthetwointercept
plus
(2)
G2
for
testing
n
f(S)
M(B)
against
J(S) n M(S).
parameters
(oneforthejointandoneforthemarginal
model)
If
and
are
then
the
first
separable,
statistic
is
[coI]
['C2]
arethesamewhether
oneassumesfullmultinomial
sampling
G2
to
for
asymptotically
equivalent
the
fit
of
the
testing
seporPoissonsampling.
Whencategorical
covariates
arepresent
G2 for
separability,
and one assumesproductmultinomial
theinfer- aratejointmodel,J(A). Thus,assuming
sampling,
the
simultaneous
model
is
sum
the
G2
of
the
components
encesarethesameas withPoissonsampling
forall thepafortheseparate
models.
jointandmarginal
rameters
exceptthosefixedbythesampling
design.
Aitchison
provided
a sufficient
for[oIl]and['o21
condition
5. GOODNESS OF FITAND RESIDUALS
to be asymptotically
lethi = 0
separable.In ourcontext,
=
0
denote
the
constraints
from
the
set
that
to
pertain
h(t)
To assessmodelgoodness
offit,
onecancompare
observed
=
0
the
model
and
let
joint
the
denote
constraints
from
h2
and fitted
cellcountsusingthelikelihood-ratio
statistic
G2
tothemarginal
model.LetHi = Ohi(t)/
orthePearsonstatistic
X2. Fornonsparse
tables,assuming thissetthatpertain
=
i
B
d9,
1,
2,
and
let
denote
the
information
matrix
under
thatthemodelholds,thesestatistics
haveapproximate
chi= 0
.
n
Aitchison's
condition
['oil
[o21
states
that
H'
B-1
H2
withdegreesoffreedom
squareddistributions
equalto the
for
all
n
He
this
satisfying
['oi
used
condition
to
['o21.
t
i
number
ofconstraints
impliedbyC logA,u= Xf,.From(3),
show
of
['oil
asymptotic
the
equivalence
Wald
statistic
for
X2 can be rewritten
as x2 = X'H''-lHf. Thisstatistic
is
['oil
fortesting
identical
totheLagrange
multiplier
statistic
(Aitchison
and n ['2] and thesumoftheWaldstatistics
and
The
of
['2I
Wald
separately.
asymptotic
equivalence
Silvey1958,1960;Silvey1959),L2 =
statistics
thecorresponding
result
implies
To analyzelackoffitfora simultaneous
modelin a lo- andlikelihood-ratio
for
G2
statistics.
calizedmanner,
wegeneralized
Haberman's
(1973)adjusted
modelthatincludes
log-linear
residuals
forlog-linear
models.Cell i has adjustedresidual WhenJ(A) is an ordinary
allthefixed-by-design
andM(B) constrains
parameters,
only
definedby ei = (yi - 4i)/ASE(yi ).
- 0)
conmarginal
expectedcounts,Aitchison's
From (3), (y-u)
=--H()(
-H(t)(X - 0). thefirst-order
ifthelog-linear
modelhas conBy thedeltamethod,usingtheasymptotic
of ditionholds.Specifically,
distribution
straintsh1 - 0 of formU'1t = 0, and ifthemarginalmodel
X,theestimated
asymptotic
covariance
matrix
of(y -,) is
hasconstraints
h2 = 0 offormU'2C2log
A2et= 0, thenH1
s -,= H(t)(H(t)'D(4)-'
H(t))'1H(t)'
= U1 andH2 = D(,u)A'2D(A2et)-1C'2U2.If therangespace
Forthegeneralized
log-linear
models( 1),thismatrix
isavail- ofX1(forthejointmodel)containstherangespaceofA'2
ableas a byproduct
ofiterative
scheme(4). It yieldsthees- (forthemarginal
model),straightforward
calculation
shows
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Lang and Agresti: Simultaneous Modeling forCategorical Responses
629
thatAitchison's
condition
is satisfied.
Thisholdsforall si- allfourresponses
aremeasured
onthesameordinalresponse
multaneous
modelsconsidered
in thenextsection.
scale.
Aitchison's
results
also implythatfora largeclassofsiTable2 contains
goodness-of-fit
statistics
forseveral
modmultaneousmodels,(1) the goodness-of-fit
testforJ(S) els.Linear-by-linear
terms
usedequallyspacedscoresforthe
n M(B) and thetestforJ(A) n M(B) versusJ(A) n M(S) threecategories,
whichseemsreasonablefortheresponse
are asymptotically
equivalent(providedJ(A) n M(B) scale(toolittle,
aboutright,
toomuch).Themajority
ofcells
holds),and(2) thegoodness-of-fit
testforJ(A) n M(S) and containsmallcounts,andthefitstatistics
aremainlyuseful
the testforJ(A) n M(B) versusJ(S) n M(B) are asymp- forcomparative
purposes.
Separate
fitting
suggested
thatthe
thatJ(A)n M(B) holds).This linear-by-linear
totically
equivalent
(provided
modelprovides
a reasonable
fitforthejoint
resulthas practical
implications
forcasesthatare compu- distribution
andthatthecumulative
logitmodelprovides
a
tationally
complex.Forinstance,
supposethatthejointdis- reasonable
fitforthemarginal
distributions.
Thuswe contribution
structure
is ofsecondary
interest
and thatone is sidereda simultaneous
modelcontaining
boththeseforms.
primarily
interested
intesting
thegoodness
offitofa marginal Table2 showsthatthismodelprovides
a substantially
immodel,M(B). Thejointtablemaycontainmanysampling provedfitovermodelsthatassumeindependence
of rezeros,andsaturated
(or nearlysaturated)
jointdistributionsponsesorhomogeneous
margins.
modelsmaybe difficult
tofit.ThusthemodelJ(S) n M(B)
Becausethedataaresparse,we also computedadjusted
to fit.Providedthata simpler
maybe difficult
jointdistri- residuals
fora cell-by-cell
ofobserved
comparison
andfitted
butionmodelholds,one has an asymptotically
equivalent counts.The fitandtheresiduals
areshownin Table 3. We
waytotestthegoodness
offitofmarginal
modelM(B) with- cannottaketheseresiduals
tooliterally,
becausethesampling
outhavingto fitthesaturated
jointdistribution
model.
design
wassomewhat
morecomplicated
thansimplerandom
sampling.
Nevertheless,
the
model
seems
to fitreasonably
6. EXAMPLEOF SIMULTANEOUS
MODELING
well,withonly5 of 81 adjustedresidualshavingabsolute
Wenowconsider
simultaneous
jointandmarginal
of2 orgreater.
models valuesintheneighborhood
Themorecomplex
forTable 1.Wedenotetheresponses
byE forenvironment,modelcontaining
two-factor
associationtermsof general
H forhealth,C forassistance
to bigcities,and L forlaw form
provides
someimprovement
butwiththelossofmodel
enforcement.
LetPhijk denotetheexpected
forthe parsimony,
frequency
fourparameters
requiring
to describeeach ascell in level h ofE, i ofH, j of C, and k ofL.
sociation.
Simplemodelsforthejointdistribution
are thosethat
Becauselackoffitmayalso resultfrominadequacies
of
assumeno three-factor
interaction.
Theseinclude:
themarginal
model,we studiedadjustedresiduals
formarginalproportions.
Thesearedisplayed
in Table4. The lack
J(I):
log1 hijk =a + aE + aH + aC + aL
offitinthemargins
seemstoreflect
slightly
observed
greater
J(2-factor): log hijk= a + a E + af + af + aL
dispersion
thanwas predicted
forthecitiesresponseand
lessdispersion
thanwas predicted
slightly
forthelaw en+ aEhi + aEh + aEh + (C
+ a4 L + afiJL,
forcement
response.
theobserved
to estimated
Comparing
and
marginal
weseethatthelackoffitis notsevere
proportions,
in substantive
terms.
J(L X L): log1 hijk= a + aE++ aH + ajC + AL
Table 5 showstheassociationand marginal
parameter
+ fEHUhVi
+ fECUhWj + jELUhXk
formodelJ(L X L) n M(PO). The association
estimates
parameter
estimates
revealquitestrong
+
positivepartialasHCV,Wj + fHLV,Xk +
CLWjXk.
sociations
between
onhealthandresponses
responses
onthe
The second model permitsall two-factor
associations,environment
and on lawenforcement.
Forinstance,
given
whereasthethirdmodelis a simplerone assuming
linear- responses
on C andL, theestimated
oddsthattheresponse
formsforthoseassociations,
by-linear
forfixedmonotone onE is "toomuch"
rather
than"toolittle"is exp(4 X .499)
scores{ uh}, { vi }, { wj}, and { xk} assignedto levelsofthe =
7.4 timesas greatwhenH is "toomuch"thanwhenitis
Thelatter
responses.
typeofmodelfits
wellwhenunderlying"toolittle."The
marginal
parameter
estimates
indicate
subcontinuous
variables
havejointnormaldistributions
(Becker stantially
lesssupport
forspending
on cities,particularly
in
1989;Goodman1981;Hollandand Wang1987).Possible
relationto healthand theenvironment.
For instance,
the
modelsforthemarginal
includethecumulative
distributions
estimated
oddsthattheresponse
is aboveanyfixedlevelis
logitmodels
exp(2.34)= 10.4timesas highforcitiesas forenvironment.
M(H): logit Yh(t) = Wh,
M(PO):
logit'Yh(t)
= Wh+ a,
and
M(S):
logitYh(t)= =h15
whereP0 denotesproportional
odds.The linear-by-linear
jointmodelandtheproportional
oddsmarginal
modelare
parsimonious
forms
thatreflect
theordering
oftheresponse
categories.
Theseparticular
modelsarereasonable,
because
Table2. GoodnessofFitforModelsFitted
to Table1
Model
J(S)nM(PO)
J(LX L) fl)M(S)
J(LX L) flM(PO)
J(LXL)fnlM(H)
J(l)fnlM(PO)
df
3
66
69
72
75
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G2
6.2
65.9
71.5
519.2
129.9
X2
6.0
61.5
64.3
455.1
260.1
630
Journal of the American Statistical Association, June 1994
Table 3. Fitand AdjustedCell Residuals forSimultaneousModel
Cities
1
Law Enforcement
Environment
1
Health
1
2
3
2
1
2
3
1
3
2
3
1
2
62
(58.3)
.79
17
(18.0)
-.28
11
(11.9)
-.30
2
(1.6)
.35
2
3
3
1
2
3
1
2
5
(3.2)
1.13
90
(99.1)
-1.26
42
(37.3)
.94
3
(8.1)
-2.00
74
(70.6)
.74
31
(32.4)
-.31
11
(8.6)
1.00
7
(5.8)
.55
3
(1.2)
1.70
0
(1.6)
-1.35
1
(.5)
.65
22
(21.3)
.18
2
(3.0)
-.62
18
(12.6)
1.69
0
(2.8)
-1.74
1
(4.3)
-1.68
1
(1.5)
-.44
19
(16.0)
.93
1
(2.4)
-.99
14
(11.5)
.81
3
(2.7)
.20
3
(4.8)
-.92
1
(1.8)
-.64
11
(9.9)
.43
1
(3.3)
-1.35
1
(-7)
.33
3
(3.0)
.02
4
(1.6)
1.96
0
(.6)
-.76
0
(.5)
-.76
0
(0.4)
-.68
1
(.2)
1.55
21
(22.9)
-.47
6
(8.1)
-.80
2
(1.9)
.09
13
(8.6)
1.64
9
(4.8)
2.04
1
(1.8)
-.58
2
(1.9)
.10
0
(1.6)
-1.31
1
(-9)
.06
20
(22.4)
.63
6
(8.3)
-.89
4
(2.0)
1.47
8
(10.2)
-.77
3
(2.7)
.19
2
(2.5)
-.33
1
(1.5)
-.45
3
(1.2)
1.75
1
(.7)
.42
1
(.2)
1.58
0
(.4)
-.63
0
(.1).
-.26
0
(-1)
-.30
0
(.1)
-.29
1
(1.4)
-.38
1
(1.3)
-.29
0
(.3)
-.57
0
(.3)
-.58
0
(.2)
-.44
2
(3.9)
-1.03
2
(2.2)
-.17
0
(.5)
-.69
0
(.9)
-.97
0
(.8)
-.92
0
(A4)
-.68
9
(5.2)
1.99
4
(3.2)
.51
1
(1.3)
-.26
2
(2.3)
-.24
2
(2.3)
-.18
2
(1.4)
.49
3
5
(6.0)
-.43
3
(2.3)
.47
1
(.6).
.51
0
(.9)
-1.02
3
(.9)
2.28
7. MINIMALLY
SPECIFIEDMODELS
We represent
a modelwithparameter
spacew by [cw].
Consider
a
or
formC logA,
joint
marginal
model
[w]
of
modeling
marginal
andjointdisWhensimultaneously
=
=
(or,
equivalently,
U'C
log
A,u
0)
and
an alternative
X,B
withpossibledependence
onemustbeconcerned
tributions,
=
X*B*
model
C*
[w*]
specified
by
log
A*,
(or U*'C*
in thetwocomponents
betweenconstraints
ofthemodel
log A *,u= 0). We saythat[ * ] is at least as simpleas [ ],
simultaneous
models.This
andconsequent
poorlyspecified
<
forensuring
that denotedby [w*] [cw],ifA* = A and iftherangespace of
thatarehelpful
section
introduces
concepts
is
C'U
a
subset
of
(possiblyequal to) therangespaceof
defined."
simultaneous
modelsare"properly
C* 'U*. ForthespecialcaseC = C*, therangespaceofC'U
is a subsetoftherangespaceofC*'U* ifand onlyifthe
rangespaceofX * is a subsetoftherangespaceofX.
Model
Table4. Marginal
AdjustedResidualsforSimultaneous
Observed
Estimated
A setof constraints
5: f(M) = (fi(M),
...
*, f (A))'
=
0 is
redundant
ifthereexistsa subsetof theconstraints
i* :
f
=
and
also
another
constraint
=
in
0
0
S
*
*(,g)
f(;z)
Adjusted
forall M 3 f *(,g) = O,f,(M) = 0.
satisfying
Marginal
Marginal
Proportions Proportions Residuals
Environment
1
2
3
.732
.211
.058
.730
.215
.055
.58
-.48
.43
Health
1
2
3
.715
.227
.058
.714
.227
.059
.42
.001
-.17
Cities
1
2
3
.221
.395
.386
.207
.419
.374
1.62
-1.65
1.68
1
LawEnforcement 2
3
.623
.311
.066
.630
.286
.084
-2.00
2.20
-2.26
Table 5. ParameterEstimatesforSimultaneousModel
MarginalParameters
AssociationParameters
Parameter Estimate Std. Error Parameter Estimate Std. Error
EH
EC
EL
HC
HL
CL
.499
.314
-.003
.052
.455
.199
.112
.104
.112
.100
.103
.090
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All use subject to JSTOR Terms and Conditions
E
H
C
L
.0
-.081
-2.337
-.462
.115
.117
.120
Lang and Agresti: Simultaneous Modeling forCategorical Responses
631
We call a model [w] minimallyspecifiediftheconstraints As a by-product
ofthesetwomainpoints,we also want
in [w] are functionally
specified
independent,
in thesense toemphasize
thatmaximum
likelihood
methods
havegreater
thatthoseconstraints
arenotredundant.
Forinstance,
the feasibility
thanis commonly
recognized.
In particular,
for
simultaneous
modelJ(SYM) n M(H) specifying
complete marginalmodelingof multivariate
categorical
responses,
symmetry
forthejointdistribution
andhomogeneity
forthe maximum
likelihood
is often
regarded
as having
limited
use
is notminimally
marginal
distributions
specified,
because becauseofitscomplexity.
Currently,
thesemarginal
models
theconstraints
thatspecify
J(SYM) implytheconstraintsareroutinely
fitted
usingweighted
leastsquaresmethods
(e.g.,
usedto specify
M(H). Minimally
specified
modelscan be Kochetal.,1977,Landisetal., 1988).However,
sparsedata,
fitted
usingthealgorithm
in Section3,haveresid- whicharecommonplace
described
whentherearemultiple
responses,
ual degrees
offreedom
equaltothenumber
ofindependentcreateproblems
forthismethod.This,alongwiththepernotincluding
constraints,
themodelidentifiability
constraintceivedcomplexity
of maximumlikelihoodmethods,
has
(i.e.,number
ofrowsminusthenumber
ofcolumnsin X), partly
ledtoalternative
waysoffitting
suchmodels,suchas
andhaveasymptotic
behavior
forML estimators
as derived methods
usinggeneralized
estimating
equations(see,forexin Section4.
ample,Liang,Zeger,and Qaqish 1992). Of course,such
Therearebroadclassesofsimultaneous
modelsthatare methodology
can be usedwithlargetablesforwhichour
necessarily
minimally
specified,
as shownbythefollowingalgorithm
is impractical,
or whenone prefers
to specify
a
resultfromLang (1992), whichis easilygeneralizable
to modelwithout
assuming
a distribution
forthemultivariate
variables
and morethanonecovariate:
response.
multiresponse
Therearemanysituations
inwhichsimultaneous
models
Let J(AP,BP) denotethelog-linear
jointmodelwhereby
the
responses
A andB areconditionally
independent,
given
a covariate
forjointandmarginal
distributions
maybeuseful.
Onebroad
factor
P. LetM(R, OP) denotethelog-linear
marginal
model
application
typeis longitudinal
in whichone'sinstudies,
whereby
theresponse
(R) isjointly
oftheoccasion
independent
terest
focuses
onhowthedistribution
oftheresponse
changes
andthecovariates
thisissimply
(fornocovariate,
M(H)). Then,
if J 2 J(AP, BP), M 2 M(R, OP) and both J and M are minovertime.Modeling
themarginal
distributions
givesa "popimallyspecified,
thesimultaneous
modelJ n M is minimally
ulation-averaged"
description
ofsuchchange.
In longitudinal
specified.
studies,
is oftenofsecondary
modeling
dependence
imporWe outlinetheargument
usedto showthisresult.The tanceto modeling
marginal
changes,
butsometimes
both
modelJ(AP, BP) onlyconstrains
themarginal
expected arerelevant.
Anexampleisthemodeling
ofsocialmobility.
countstosatisfy
themultinomial
constraints.
Therefore,
any We might
consider
howthedistribution
across(upper,midmodelJ ? J(AP,BP) willnotimplyanymarginal
model dle,lower)socialclasseschangesforsuccessive
generations.
On theotherhand,corresponding
constraints.
to anysetof We mightalso considerthepotential
forchange,in terms
countsandanyassociation
andinteraction
marginal
pattern ofthedegree
towhichsocialclassingeneration
t+ 1 depends
as measured
thereexistsa setofcellcounts on socialclassin generation
usingoddsratios,
t.
withthosemargins
andthatpattern
(see,forexample,
Bishop, Modelsforjointand marginal
distributions
arealso relFienberg,
and Holland 1975,p. 375). Becauselog-linearevantin studiesofinter-observer
reliability.
Supposethata
modelconstraints
oftheexpected
arefunctions
countsonly setof observers
ratethesame sampleof subjectsusinga
oddsratios,
itfollows
thatthemarginal
modelcon- categorical
through
scale.Eachmargin
ofthetablerefers
toa different
willnotimplyanyjointmodelconstraints.
straints
Also,the observer,
andthecellspresent
thepossiblecombinations
of
marginal
modelM(R, OP) isminimally
specified
(assuming responses
bytheobservers.
Goodagreement
between
ratings
becausetheOP terms
allowa per- by different
parameter
identifiability),
observers
requiresstrongassociation
between
fectfittothemarginal
totalsthatarefixedbydesign.
theirratings,
andsimilar
distributions.
marginal
BothcomIn thebivariate
casewithresponses
A andB and ponents
response
areneeded.Forinstance,
themarginal
distributions
no covariate,
thesimultaneous
modelJ(A,B) n M(H) is (summarizing
relative
ofthepossibleratings
for
frequencies
becausetheconstraints
thatspecify
the eachobserver)
minimally
specified,
couldbe identical,
could
yetthejointratings
modelJ(A,B) leavethemarginal
jointdistribution
expected revealthattheobservers'
judgments
are statistically
indecountsarbitrary
constraints.pendent.
up to themodelidentifiability
Or therecouldbe strong
butone obassociation,
Allmodelsdiscussed
intheprevious
section
wereminimally servermightsystematically
ratesubjectsa category
higher
specified.
thananother
observer.
Thustodescribe
itis relagreement,
evanttomodelboththejointandthemarginal
distributions.
8. DISCUSSION
Simultaneous
modelsmayalso be usefulforanalyzing
Therearetwomainpointsthatwewishtomakewiththis data fromcrossover
twodesigns.Considera two-period,
article.
wehaveprovided
a moreefficient
crossover
A and B. Each
First,
wayoffittingtreatment
designwithtreatments
andchecking
thefitofa generalized
classoflog-linear
assignedto one oftwosemodels, subjectin thestudyis randomly
A followedby
C log A,u = Xfl. In particular,this methodologyprovides quencegroups,eitherreceiving
treatment
B orreceiving
B followed
nonstandard
improved
waysoffitting
models,suchas joint treatment
byA. We thenhavea
modelsforglobaloddsratiosandmodelsforfirstorsecond- bivariateresponsewiththenumberof covariatelevelsat
ordermarginal
distributions.
Second,we haveshownthe two,thenumberofsequencegroups.Often,interest
refers
tocomparing
utility
ofmembers
ofthisgeneralized
themarginal
classthatcorrespondprimarily
distributions
ofthetwo
to modeling
simultaneously
bothjointand marginal
responses
distri- treatment
to determine
whichtreatment
is most
beneficial.
Butit mayalso be important
butions.
to describe
theas-
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All use subject to JSTOR Terms and Conditions
Journal of the American Statistical Association, June 1994
632
in the
sociationbetweenthe two responses.A difference
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