ARS Multilevel Models - Methods and Substance

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MULTLV MODS:Methds an Substance

hmas A. Prete ad Jer FrrstaDepaent of Sioloy, Duke University, Box 88 Dura, North Carolina770888

KYORDS: contextual-effects mels, meology micromacro eaccal models

bstractTs pape evews ecet developmets te applcato o multlevel modelsto substnve poblems sology. Thee s o sgle mullevel model socology but ae a et

·o moe o less cloely elated appoaches o

exploig the ln betwee the maco ad mco levels o socal pheomea.Methodologcal developmets o te last te years e discussed ad cotastedwt olde meod. llatve eple o ow mltlevel ly a cotributed to socologcal owledge re povded o seveal aeas o the dsciplne, ncludig demogapy, educati, statcato, ad cology autos e use o ese models o empcal esec e dscussed, alog possble e developmets

NTRDCTN

Mullevel models e used sooloy to spey e eet o soal otexton divdual-level outcomes Te dea that dvduals espod to he socalcontex s a deg clam o e sologcal dscple wc s oud M's wo on politcal economy (1846) urheims studes o te mpacto commuty o oma ad sucde (1897) Webes eseh o owelgous commutes sape ecoomc behavo (1905) Meos wo oncommutes elatve depvato, nd socal compaso teoy (1968) and eelso et a's (195) esec to e eect o sal context o votg.lau (1960) nd avs (1961) publsed mpott papes devoted to cotextual eects d a thvg esec ageda to ese ssues exsted te1960s ad 1970s (oyd & Ivese 1979 laloc 198 Ivese 1991) Moe

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2 DPRT ORRST

reenty, Coeman (1986) assied soioogial theores into three groupsaording to their mutieve ontent. In the rst group, variation in a dependentvaabe is expained through independent variabes that appy to the samesia eve (e.g. ouny organization oa ommunity individua). n thesecond grop, variation in a dendent variabe at one eve is expained byproesses that operate at a higher level In a ird type, variation in outomesat one eve is expained by variation n outomes at a ower eve heoriesin the seond and third groups are mtieve theories, though neary a existingmutieve researh fas into grop two.

Bak (1984) reviewed the theoretia and metodoogia iterature onmutieve modes abot 10 years ago Most of his observations-ptiaryabout the probems of interpreting mutieve effets and about onsisteny of

esimates-are sti vaid However muh has hanged in the past deade tTe most sefu metatheoretia fouation of miro and maro eves ofsoia reaity and te reationship between those eves ontinue to be ativetopis for debate in eoretia soioogy (Aexander et a1 1987 Huber 1991).Multlevel teoies for partular sia phenomena have beome more reinedMoreover, new statistal tehniques have been appied in severa areas ofsoioogy with usefu resuts Tese new methods are not a panaea for teprobems raised by Bak in his 19 review as Mason (1991) notes buthey do represent a signant advane over earer methods and they have

ed to new tys of substantive anayses as we.Beause so muh work is ongoing in so many different areas we annot

attempt an exhaustive review of a soioogia researh that is arguabymutieve. nstead we seetivey review researh in order to desribe tereent methodoogia deveopments to show ho the newer metodoogiaork is being used in substantive reas to disuss ho the new modes reateto eah oter and to identify areas where urher deveopments are needed orin proess.

Te term an be used in different ways There is no singe

mteve moe; there is rater a vaiety of modes that have ben used toanaye sia presses pstuated to operate at more than one eve of anaysis. One an haraterz oes ethods and daa olletion as eitherunieve or mutieve whie they tend to be assoiated the assoiation is notprfet. Researhers sometimes theorie at mutipe eves but use data at onyone eve They sometimes theorie a one eve bt se da at a differenteve. The isses raised by these efforts (aggregation the "coogia faayet) are beyond the so of this review (see Bak 1984 for a summary).n his review we use e tem mutieve mde to designate the siation and testing of mutieve teories with mutieve data

Bak (19) dened ontexua anaysis as foows: "The essentia featureof a ontextuaeffts mes is an aowane for mar presses that are

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MULTVL MODLS 333

presumed o have an impact on e individual actor over and above the effectso ay idividual-lvl vaabl ha may b opraig" (p. 354). w grali i u o dvd o apply o ay ui at i ro rlavto o or maro lvl i h aalyi, his diio i ll qit rviabl

The noion o oex i uite general and can include spaial ontexts(egcountiestaecomuniies)temporal conexs (ie hsory)organizaionalconexs (clarm hls rms) and soiacultura/economic conexs(eni groupssoial claseconomc secors) Analogous mods are employd wh ox i h mhod o daa ollcion or som orcommon proty or a se o daa poits as in meaaalyihe ue ofsttistical models o reconcile results rom dierent studies on the same topicor he analysis o factoral surveys (Hedges & O 198Braum 1989Hox al 1991Bryk & Raudnbuh 12) he thorial picatio o ontx ipl or rar ivolvd (lalk 14Ivr 11; or a a mororal lvl lxdr 197) Mlil ox a alyo a given nit. Context an overlapping or nested. ey can have fuzzy boundries or clear ones No urpisigly more complex ciications canirodue oidable mhologial diculie.

Grally sakingmultilevl modls xplai irolvl ouo i twoways: (i) by howing that parameters o modls pied a th iro lvlwhere cro level covaiaes re used to explain micro level outcomesre a

function of contexnd (ii) by showing that hs mcromacro relationship can be exprd in erm of haaciic o th coxwhch take h orm omaolvl vaabl Two yp o marolvl varabl ar coonly oundi ullvl modls (a) ox-si ma or hghr mos o mrolvl viabl, d (b) global vaabl (Laarfld & Ml 6) a arnot expressible as unction o icrolevel vriables or a leas no the micolevel vable found in the iroleve uaion.

W hav aady nod implicitly ha multlevl odls are also calledconexual models. Readers o the rece lieraure will have noted many oher

ms or thse modelsincluding bu ot lied oherarcal linear modelhirarhical lina rgrsionrandom oeiciet moelherarchal mxedlier modelor bayesian liner model (Kref e a 190). o a certain extenth prolifrao o a o rom h saial propi of vouodlig agi ud o aly lilvl daa T a apply ply o h xio o oxal grio aalyi a hav b udxtivly i iology i 160 (a rviw i oyd & vr 19or Blaloc 1984)

Howevr svral othr yps o ultilvl model ar alo i u or big

dvlod in oiologyiludig (i) xio o otingcy tbl aalyito include mulilvel ts (ii) extios o ve-hisoy aalyis to includmulilvl ffc (iii) ndogous switcing rgrios (iv) xtio of

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334 DPTE FOT

aevaabe mes icude mutieve eecs; v) he devepme micrmcr psed mcrmcr) mdes d vi) sisic mehds muteve ysis We discuss hese eive meds er i hispper

The amiiar me r ceua ecs ad mre rece eesis arehe mdes ms widey assiaed wih he em v d These areregressi mdes a re iear i hei ccies Geery speakig, heyare used aayze daa ha csis mupe mcr uis cets) admutipe micr uis wihi ech macr u These regressi mdes ca epressed i w agebraicay equiva rms: i) as a equai reaig amcreve ucme a se mcreve variabes ag with a se equais i whch he ceciets his creve mde are epressed as

ucis macreve vaiabes, r ii) as a sige equai where he creve dede vaiabe is epressed as a uci bth c ad mcrviabes Tis secd geeray icudes ieracis ewee he micrad macr variabes

The majr advaces i the pas e years cce he ucia rmreaig mcr ad macr varabes, bu rer a mre sphisicaed reame the errr scure r tese modes Wereas he der mdes ca

characerzed as ed eecs regressi mes, the ew mdes speciy heregressi cciets as radm eecs I a ed eecs muieve regressi mde, he micreve cecie is epressed as a eac ci macreve vaabes Rdm eecs mueve mdes, i cras, cier ems i e macr equas Te icusi hese errr erms a emacr eve impies a mre cmpe errr strucure i the sigeequaversi the muteve mde The ue radm cecie mdes awsthe da ays decmpse he varace i he depede varae i hewihce variace ad he wee ce varace, ad study hese

w surces varai r the creve ucme Thus, radm cciemuieve mes are a y varace cmpets mde

The disict bewee he e ees muieve me ad he radmeec muieve me ca see i a simpe eampe, adped rm ryk& audebush (12). Supse ha mah acheveme i eemeary schY ) r idividua "i is uci siecmic saus SES), ad suppsetha he reshp ws e wig simpe equai

= E

where X is he SS sude i As wre he iercep gives a esime the epeced math achieveme r a stude whse scicmc sausscre is zer Sice er is meaigu vaue r scicmic saus,

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MULTEVEL MODE 335

we cn "cener X (e.g Wllms 1986, Iversen 1991 Byk & Raudnbush 1992)as a deviaio fom he man SES i he sampl 1

Y + PI(Xj + £

Hee Po pvde an etmate of the expeted mah ahevement fo a tudentwho SS s at e ma ad PI pode a etmate of efft of a untof SS o ma ahmt Th quato a typal moleel equatowhh pe h ahevment of all tdts wth man SS to hae theame epted math ahemnt P and p the efft of SS (PI) to ame fo all tudent n oe wod e effct n h mel e euafor all dnts an em altav to hs ed efft ml h aalystmht hypoth hat eah tudet ha he ow unque P, whh would

pedt h/ math ahvmet f t tudnt we to ome fom a meanSS faly Futhmor h alyt ht aue at ah tdent ha hihow uqu to aad adanta o dadanta rlatd tofamly SS. Under the asumpton hat each sudnt has hshr own oft th modl ow om:

Y + Pli(X; + Ei 1

wh P h ntept fo tudnt and Pli fft of SES fo udnt" Beaus h paramt of Equaton 1 y by dvdual hs modl

led a vayn paamet mdl f t aumed that paamt aed but uow the e mdel a d fft ayn paamt modl(Jud et al 15 Gene 10) f tudent n te ampl ottut thnt populaton of tudnt te oneptualzaton of thee paat aayn but d plauble on toetal ound Th pamter ouldot be etmated wout fuhe oat te numb of paam tw a at a th populaton Uually howe h tudnt n ttudy wold only a ampl fo o lae populaton and t would mkemoe onptual en to w ndvdl-ll ont n Euaton 1a andom daw fom ome lae populaton of nt ju a testudnt n the ampl ae adom daw from om laer populaton oftudnt

Und t aumpton th cnt an eepd to ndate thattey ae ayn and andom

J=l+U Pli = i I + O

ntn t aue ba to he molevl equato

IS Brk and Raudnbuh (12) v' ) f an ndd dun fh vau t o eter eploe lleel ork

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336 DIP FOT

This model which is someimes called he Swamy regression model (171

is siilr to uation 1 except hat it has a more complex heteroskedasticerror struture that inludes the random omonents from quation 2 multi

plied by mcrolevel covraes. e sasical consequences of hs change refairly sraighforward Under he sandrd (song) assumpions found in le-!en discussions of regression ordinry leas sqres (OLS) esimaes of

� o and �I are consisen, b O esiaes o their standard erors are incon-sistent, nd OLS is ess efcient than teatve estimation rocedures

Model is clerly more general n Model l b i is obviously nsaisfying fm a sbstntve point of view; i provides no explanation fo howhese cfciens vr across individas The random effecs mlilevel modelcan viewed as a way o imposing furher srcure on Equaion 3 and, inthe ress rvidng eanation fo e vaying cefciens (Kre e a10 I does so in he scic case where e analys cn arge ha hesemcrolevel nis re locaed in differen and disinguishable sial conexs,and aso a e propees of ese soca conexs rovde an exanaon forviaion n e micrlevel ceciens

In he sden example above, a naral conex s he scol (oher possi

biiies are e classrm or the sch disrc If we se e eer "j o ndexe shl of eac sden we an eae quatons 2 by a moe substantveymeaningfl se of eqaions

� = Y + � = 'I +

4

In quaons 4 al the indvidua-eve varaion in crolevel coeciens

is aribued o e school, nd ever suden in a given school is presumed oave he same crlevel coeciens n oter words e inerceps and slopes

vy sysemaically by school, bu thee is no further vration wihn schoolsOne an go her model e execed value or each school's coeciensas functions of hat school's characteristics. For eame eting Zj reresentthe mean SES in shool j and ettng Z2 j be a dmmy vrable a ndicaesweer scl j s a Caolc shol or not we ould reeress Equation 4as:

� = YZj Y2Z2j j 5

21e semnal discusson of ths approach o the lnear mel s found n Lndley& Smth's (972)dscusson of bayesan approaches o egesson Bayesan approaches closely elaed tandn some ests equvalen Sen-lke esmos (ron & Morrs 972, o Judge et 985 oa recen exended dscusson)

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o in a single equaion om as:

MULTEVEL MODE 337

Y = � + Zl + + '1OX - X) + 11 Z ( X - + 2 2 (Xi - + {E + U + (Xi - 0}

6

Euaion 6, e macroleve viabes inerac wi e co evel vrials, d o suctu in ackts conains o colvl tms andmaco tems. conin muple mico e osaions witnt sa conx uaon 6 mplis ta th ros wl htosscand coeated acoss micoel units Und he usua assumpons OLS

simaion of uation 6 wil pouc consistn estimates of coecints ut inconsistn estimaes of i sandrd os

th analys nw aanc-coinc ma fo o sucu ofquaon 6 t woud saightowd to stima quaton 6 th gnazdas squres (GLS (s eg. Godsn 98. Bcaus ths matr is gnllynot nown howe i mus e simaed long wi e cciens inEuation 6 s ms o doin ts re iteaie meds such as eM algoim, whch alaly sims e prms o quaion 6 hnth aiancecovarianc mai o he o and so o unil congence is

eached (Mason al 983Goldsein 98 & Raudnush 992 sKf a 990 o a discussion o aleaive iteaive mehos) Thsemhs ha n incoad ino a colcion o standalon saisical pogams such as GNMD HM M3 VAC and outins tat un underSAS, GAUSS and BMDP (se K al 990 o a discussion of mn oes ouines)

Th asc dnc twn th ndom eecs multlvl ml shownn quaon 6 and he old edeffecs mulv mo ound n mos ohe iraure eiewd in od & Iversen 99) or Baloc 98) concs o suctu shown in uations he ed ffcs moel assumshs os a o This assumptio cons Euaion 6 into a standdgssion a contains intaction s and cocn o his gssioncan esmad usng OLS wou any o e sttscal dcues descdaove As B & audenush 99) no e choce of effects oandom effecs is not ity ecaus e hypotsis of o os in maco quatons can st It is ays poss to cont a ndomffcsomulation ino a dfcts olation n whch t eo aiancs in

quation com o y ncuding dummy varas fo ach cone (withappopa constains o ensue dncaon n quaions 4 o hisappoach, howe, has two diicuies i inrduces a lge nume of addiona cociens and lus e disnction etwn sample nd populaion

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338 P FOT

tha s preserved by te radom-effects model The random effects model ssuperor on bot grounds.

qin 6 incpes he ssmpin h he inecep nd sl

cfcens nctns f cntext One culd esc the model s hat onlysme sls vay wle othe are xed, or ne cld spcfy a only hentercept ccens va, wle ll slope cfcents re xed One couldlso scfy th some but not ll of the error varances n the macr eqaonsre ze hese hyptheses e ll testble (k & Rdenbs 199). Fnllyit is nt necessy ht ll f e wihn-cnext regressins identicllyspcied; ll is eqid is t ne mre f he miclevel cfciens e cmparable crss cntexts (Wng Masn 989)

The use of newer statstcal techques to study pacular multlevel

poblem cn lte ne's subsnve cnclsns3 rhemre he newercniqes cn led ineresing new ssnive nyses e impnceof contex in shapng otcomes s illumnaed by n analyss of e vaancecomponens of e multlevel mel or exaple f os of the vaon nsrtcan cmes cs eween fmles, if ms f he varton n w ne raonably conclude that tefmly is mjor deeminn f scn tcomes t e schl is m deeminn f chievemen cnst, ndng h ms ll hevain in sricin cmes ws wihn fmlies o h lms ll he

varatn n achevemen outcomes was wthn schools leds to dfferent substatve cnclusons Tese typs of analyses are natural products of randmeffect multilevel nlyss (Goldsn 197, B & Rudenbush 1992,see ls Huser Mssel 195)

Randomeffects multlevel mels also allow more sophscated attenon te tas of esmatng mcrolevel mels for specc contexs Wthncontextmdels re f parcular nterest o lcy alyss wen te contexs (egschools) can manpulated f ther rformance s wntng (Wllms 1992)Multilevel d pvide dsinct estimts f wiincntext ccientshe nlys cld se wincnx d esime wincntext egesson bu e estmaes wll be mprese f the wncnex sample is smllAletvely, te alyst could use the mco mel shown n Equaons 4 or t btn esmtes f te wtncotext ccents, ether by usng teestmted men f Euions 4 or by pluggng n he pproprte values for emcro vrables n Eqtons Ths cnd estmaor uses dt fo ll conex,nt js the cntext f which cfcent vles e desired, ecse hereresson cciens of the macro equaon are estimate wh d for

3See• for example. & Leuw's 988 demonsaon ow concusons about the etso a schs sex-rato on indvidua eadng aby changed when one used a ndom eects nseadof a effcs muevel

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30 DIT & FOIST AL

as funcions of schl characterisics, which are measured contemporaneouslywith the individual-level measures in the rstlevel equaton. These schoolcharacterstcs at he second level re expressed as deviaons frm the average

effecs for he schl over time The hird level model hen species ecoecients of these timevarying schol vables as functions of schoolcharacterstics averaged over time. This model is designed to distinguishthe eects of shoterm and long-term characteristics of schools on studentachievement

Nonnested mels have also en develod for siological problems Onetype of nonnested model aises when two macrolevel contexts are presumedto affect the same icrolevel unit at the same time By crossing the twodimensions along which hese multipe contexts re dened, one obtins a

larger number of multidimensional contexts such tat each mcro unit is nestedwithn a single multdimensional context Wih this modicaon the presscan sudied wih the usual two-level multilevel model (Braun 1989).

A second type of nonnested model can be used to allow contextual effectsto e heterogeneous wihin contexs The usual multlevel model assumes hathe effec of context e homogeneous for all unis located in he same contextAs Blalk (1984) noed ths assumpion of homogeneity may often be unwarranted. As a remedy, Blalk suggested that actors be divided into groupshat hae hoogeneous "atachent dependenies nd vulneabilities.

DiPrete Grusky (1990ab) employed a simlar strategy when they developeda threelevel nonnested model to overcome the homogeneity problem in theiranalysis of temporal variation in the ccients of he basic satus attainenteuation Their hreelevel model used data from reated crosssectionalsaples of he Amecan population. The rstlevel equaton was a basiccoleel status atanment equation wih indiidual-leel viables measured at he different time points avalable in the repeated cross-sections. Thesecond level expressed the ccients of these individuallevel variables asfuncions of lagged macro factors The third level permtted the stength ofthe macro factors on individual presses to vary across individuals DiPrete Grusky used an individual-level variablea measure of laor force experenceto capure the differentia vunerabilities of workers to macro processes but in prnciple these vulnerabilities could have en specied as afuon of ohe acro varables suc a sze laor rke sector, etc

alteative elaoration of the basic multilevel mel involves relaxinghe assumption hat he dependent vrable is a continuous, normally distuted random vaiable Wong & Mason (1985) developed a hierarchical logistic

regression model for multilevel analysis. Goldstein 1987) developed a hiern a h nx, f xamp Wm (986 uad ha h mpa f h m·

pn n vau um mgh a funn f h na aby f h pup.

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MULEEL MODE 341

archical loglnear moel Other such work s ongong (e.g Kreft et a990,Godsten 99 Bresow & Cayton 99)

OTHE FOMS OF MUTEVE MODESMost of the dscusson to ths nt has conceed the analss of data formutple contexts where several mcroevel daa ponts are avalabe for eachconext This type of daa alows a vance componens anayss ha canseparate the wihin-cotext and tweecotext conribuios to total vaiace some cases, however, reseachers may develop a multilevel theoyand olet multeve data, ut annt use vaance components mutevemodels ecause the researchers do not have measurements for multple cro

unts whn each macro unt A notabe case n pont s found n contemporaryresearch i straticatio ad work A cetral heme i the recet literate ishe nuence o wor organzaon and labor market segment on sracationoutcomes Several schoars (Viemez & Brdges 988, Pace et al 99TomoskovcDevey et a 99 have coected the names of employers fomndvdual surve respondents to generate a sampe of organzatons fomwhch addtonal nfomaton was coected The combnaton of the ndvdualand the empoyer suveys s a mueve daabase on workers and ther workcontext The most recent data of hs fom comes from the Naton Organ

zaton Survey (NOS whch combnes mcroevel data from the eneral SalSurvey wh macro data about he employng organzatons of SS sampemembers and her empoyed spouses Unorunaey, because hese daasetstypically coi ifoaio o oly oe icrolevel uit wii each acroevel unt it s not ssble to estimate he vaace coments mulilevelmodes descred aove, and correcons for e heteroskedastct in adomeffets mulevel modes t to these data must foow the moeg strategesdscussed, for exampe, by addaa (977 or utn & Satorra (989

Anayses wh data such as he NOS can spfy mcrolevel ccents as

funcons of macolevel vabes but hey canot estmate he "toal effecof conex cause ony one mcro obsevao s avalabe r organaoAnother mpoant modeng strategy n stratcaton researchsbng modesn effect ds he reverse Hauser & Mossel (985, see also Hauser &Sewel 986 Hauser & Wong 989 note that her sbng model s a sccaton for the contexual effects of he famy of ogn on sratcatonoutcomesn fact t can concepuaed as a random effects mode for thentercept of a regresson of satus on schng Unlke the varance components modes dscussed earler or he es for he effcts of work organzaton just dscussed Hause et al's sbng model was not desgned to podea subsantve explanaton for varyng ccents of ndvdua-leve varabes(eg schooing motvaton meta abity as a functon of falyeve va-

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MULEVEL MODE 4

Mare Researchers using this sategy specify eparate regessionmels or ech o small numr o tinct contexts (e. the pmar orecody ector o n ecoomy or the diffent eductionl cks i hih

chool), d hey o cy mel ht epl the obered dbuoof micro uits cross the contex Becue the number of contexts i mllmcro variles cannot inuce o povide quantitaive explanation otween-context dierence in rereso ccients

Latent tucure mels re relte n some wy to multlevel model Altet ucture mel c houht o s type of multlevel model whchsiton tween mico level vrible re "eplned by the ucovernof ltent classes which argualy generate the icrolevel asitons Gdmn 4Clo Mcutcheon ) A ltent srctre moel is nl

oous to reon mel which the mesured covrtes re conceptualize s ndictors of ltent context wiout rue drect eects of theiron e ntercepts o e crolevel mels re specied to vry by ltentconext A compur serch is done n order to ident ltet contexts tminize he mesured drect efects o he covtes nd mximze evron explned y he varng intercepts Te latent clsses cannot nteprete pure contextul eecs however becuse ey will eerllynclude individullevel commo ctos s well

Unmesured heteroenety moel i event hitor nyi re lso relate

to multlevel mels Heckmn Sner 94 Maton et l 92) Thepresence o durton dendence in eventhstory mels i ofte interpretedto mean tht uits in the aly der in wys not captured y the mesuredcovar Tese unmeasurd ctor wll bi etts o durtio dendence nd corltion tween these unmesurd factors nd measure covtes will ias the etimte ccies o the meurd covrte Theoluton incresingly use in he analyi of eventhstory model is in eectto specify he intercept o eventhsto models s a random nction ounmeured rson efcts hen to y prametc or ixing disiuon

or thee unmesure efects nd the to ncororte this mxi dirutioninto te lkelih function. However he mxing dirbuton i uully notven substantve nerpreton nd t wll generally e nction o mcroas well as mcro fctors.

Perhaps the use o latent variales t is closest to multilevel analyis is

S No however, hat though a oneptuaizaton, i may ssible to expand the num of contxs n ode ntc mo-eve egss. Fo exampe Goan & M (1989)fod an endogenous swhng ssons analysis of the ets of taking on achevemn

n wih studens om e High Schl a Bo Suey we assed in eite e olege oe nonolege ak Gamor aer 92 lyed Hgh Shol a B data using sha the ontet and he ak stut of e sh as a ma vaae epanng vaaon nwn outcom

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344 DP FOSTA

when scholar spcify parallel covariance stctre models for several distinctconexs (e.g Kohn e al 90). he work by Kohn and assiates, crossnational varation n mel cfcients is given a mulilevel nterpretaon, buthe viaions

no qunaively moeled as a funcon of macro variables.

Of course, when he numr of macro contexs s small-as is the case whena small numbr of counries comparedi is no possible o model vriationn micro parametrs as a funcion of macro variables oher than to est forccient diences. Signican variation in the microlevel coefcents canof course arbuted o various atrbuts of the macro conexts, bu confdence in e validity of such asseions must b gained by idenifying otherfacors (which could be macro variables or cciens of oher icro equations) a vary wth hese abuts in ways prdctd by eory.

Related issues exis in he growing lieraue in compaativehisorical sociology Comparaivehistorcal scholas have paid a grea deal of atenion tohe disincion wn "smllN d "lrgeN sudes (Ragin 1987) buhey have no ye sue in deail he nes from combining "medumszeN macro and largeN micro sudie. One explanation for tis omssion ish compraivehisocal reserchers as a rule prefer "rich complex comprisons hat ruire hem o pu a gre deal of effort in a small number ofcases Tme d money consans lii heir abily o nclude he numbr ofmacro cases requre for sascal analyss In addon ey have no ye

pursu n sufcen deph e possibles a even "smallNmacrolrgeNmcro sudies cn afford of the form usd b Kohn and assiaes Thenumr of such sdies is ginning o grow, ough ey differ from thesaisical models discussed ave in ha crossconex comparisons of wihinconex resuls re ofen done n a qualiive raer hn a quaave way(Janoski 99; se also Hag e al 989 Flora e al 988ab)

Lstl here hve n some nascen efforts o generae models that explainmacro presses or oucomes as a ncion of micolevel presses andoucomes While he mrnce of usng mcrolevel processes o explan

macrolevel oucomes has long been recognizd in he sial sciences (Meron936 Olson 965 chllng 98) reaivly lle rsarch has amp ospcif moels chrcere or eplan proesses he macro leel s olem & 989 pps s med blong a individuallevel exchanges o characerze e exchange relaionshps in a system. This mel diffe from more conventional multilevelreession models however in tha it employs no macro characerisicsiher as indenden or as depnden variablesher than those revealedhough daa anlysis of crolevel relaionships mong acors (eg the

or an example of a network nalysis hat lks insead a how macro network level proiesfec mco bvio,sMrkovsky al1988).

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MLTEVEL MODE 345

"pce of items being exchanged). Models that use micolevel relaionshipsto explain chactestics of he system othe than hose that emege fom themodel iself ae still udedeveloped

SUBSTANTIV SAH WITH MULTVLMODELS

The mehodological developments discussed above have been used to study avety of substatve poblems, may of wich have been alluded to tespace available we can only biey illusate e utlity of multilevel melsfo subsantive reseach

In the aea of demogaphy andom-effects mulilevel mels have, to date,been used pmaly to analyze the addtive and inteactive effects of communitylevel chaactestcs on the epoductive behavio of women in developingoe (Ctele 195) twle on (195) ued multlevel moelsto stuy he connecon between siconomc status and felity They foundat, while sociconomc factos affect e numbe of childen eve bo temagnitude of tee effects dend on a county's ioecomc devpmetand on the stength of its family planning pogams. ounes at a low levelof siconomc development and with no famly planning pogams havepositive mco siconomic felity diffeentials ounties at a high level

of siconomc development have song negative individuallevel socioeconomc diffeentials when famly plning pogams ae we but weaeidvduallevel cooc dffeeal wen fy pang pogme stonge. In a study of contaceptive use ntwisle et al (1986) fnd thatwhile individul siconomc effects on contaceptive e ae tong countylevel effecs ae wek in contast to some elie studies (see alo ntwisleet a1989)

Some et effot hve alo be mde to ee te mn effet ofvaous communitylevel factos on e epoductive and ml behavio of

young adults in he United Sates. Hogan Kagawa (1985) and Hogan et al(198) found hat gowin up in a p highly segegated communty deceases a black females chances of using conaception at st intecouse andsignicantly inceases he sk of having a pematal pegnacy. ane (1991)ecenly foud that te likelihoo of having a bit amog young black adwhite female is invesely eltd to the quality of local neighho asmeasued by he popoion of all wokes in the community who wee employed pofessionals and manages. Finally Billy Mooe (1992) foundthat the popoon of women wokng fulltme the pooion of wte colla

woe d e popoio of femle who e cuet epated o dvocedn e ommuity e all ivesely elat to e of eeecng a bihaong mared nonblack women whle the female uneployment ate the

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346 P FOST

sex rato of he never maed pulatio, and he chlwoman rato for womenaged to 24 are sitvely assiatd with the risk of eiecig a bior unmaed noblack women.

he recent sudy b Lcher et al (2 is e rst to use mutlevel modelsto exae he man eects of maage maket factors on frst matal anstions of oung black and whte women in e Uited States hese authorsfoud hat whle differeces between backs ad whites in maage marketortunities are qute large, such dfferences unabe to accout or raciaderences n the timng o rst maage hree addonal studes whchexamne he imact o factors such as the proportion o female headed falies,te roion o adus wth a coege degree, or the dvorce rate, generalysuppot Lchter et al's concuson (Homan et al , L 2, Forrista

uubshed Studies o communit actors on rates o marrage ad etiityave so ar arge used on e mai eecs o contet; studes hat examinewheher maco vaables aect he vaues o mcoleve ccents other hane mcroeve itecet rema to doe

e iteaue o schl eects s voumious Coem et al (82 oudat SES eecs on test scoes e atteed Caoc scs compad wiubc scs ad & Byk 8 eteded e sec by usgrandom eects muievel mels Their reseach aso suggests that the eectsof race are reduced in ordely schs, while he efects o class and academic

backgoud are reduced i smaer schos, in schoos where te math curcuum s moe omogeus, d i scs wee discipe pcedues ea d ectve Wms & Raudebus 8 e yss o estabty o e ecs o sc comsto ad cy o studet acevemetover tme oud at vaaton in sco eects or a gve sc teds to smal reatve to the vaaton in schl efects acoss scls Oer studiesave okd at he segh of ll a maket efects o decisions to remaini schl ae ilms 8 and sex dferences and schl effects ine growth i mahemacs sks in lower nd mdde schs (Wms

Jacobsen 0 St other studes have examned he eects o lacementwhn educatoal acks or attendance at schoos that use educationa tackso achevement outcomes (eg Gamoran Mare 8, Gamoran 2, Kerckhoff 3 A comlete review of he mullevel educational research wouldreue an entre revew ar (see audenbush & Byk 86, Goldstein 87,Bk audenbush 88 Bk 8 ilms Jacobsen 0, umberger Wlms , or Crane , or examles o reated reseach)

In the area of straticaton Gsky & Hauser (84 examned the imact

of sietal atbutes i elaiig coss-aoal viaio i sial mobilityTey ound reasonab song tough comex eects o economc develoment on sial mobity, and tey ound at the eects o tca organzatonwee euay stog Trema & Y 8 studed how macro actors mgt

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MULlEVEL MODE 347

explain cross-national vaiation in the cfcients o the basic status attanmentequation. ey ound that ndustalize seties tend to be more on tandeveloping sieties, and that sieties which are o indusralied and

which ave mo stas quality show songer eects of ducation, and weakereffects of aily background on status attainment Erikson & Goldthor(1992), in conast, concluded tat cross-natonal vaaton in sial mobilitys lagely unsstematic once dierences in cupational dstributons controlle t a more micro conexal leel, reeach into neighoh eecton poverty sially in h Uit Stts found signiat native ffectsof comng rom a welfaredependent comunity or ro a highly seregatedcommunity, but many eects of neighborhd context on economc outcomesapar weak Datcher 182 Mayer & Jencks 18 Tienda 11, Ger

Raudnbush 1 991 Mass et al 199, Corcor et aI 199Meanwhle oher sholas have exaned how sial chan wihn a iven

sety can cange te press o stratcaon DPete & Gsky (1990b)ound that the radual rowh o bueaucratic sonnel polices in he ntedtates ma have plaed a role n equalng opportunty, but at drect eectso litical intervenon may have been ore iportnt sources of change Athe hird level o heir ultilevel oel, hey showed thatconsistent withprevalent heories about how labor arkets oratee manitude o heseches was stroner for new entrants into te lar maket for exi

ence wores n addition many studies ave analyzed trends n te ccents o satication models wout necessaly intoducing macro vaablesto explain ese rends (Hauser & Featherman 198, Mae 1981 Hout 1988,Grus & DPrete 10, Blosseld & hait 1993

A huge literature examines the ipact o organiatonal structure and labormaet seent on caeer outcomes (Baron 18, Rosenfeld 1992) usuallyw data ro scc rms or wh ata about rms and labor maetscollected from individuals But as noted above soe recent research usesultilevel dat that was collected at th he individual and the level Forexaple, Villeme & ridges (188) have explored he inuence of r siand oter oranzational chaactestcs on eins and how these eects vayby ender and cupaton Publised reseach based on the new NationOrizaton tud s epeced in e nea uture

n te aea o ciminoloy, several scholas have exaned the impact ofcontextual actors on individual rsks of vicaion (apson et al 198th Joura 18, Kennedy Forde 10, Miee McDowall 1;PW Rounee, C Land, Miehe unpublished ms For example, Miehe cDowall (193) sowe tat community measures such as the extent ofamly disption densty of ownership of various consumer gos racialheteroeneity, edian income popoton livin alone uneployment rates,and divorce rats inuence risks o vicimiation Aside m their ndn

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348 DIP O

hat indvidual saey prcauions agans burglay had a greaer impac inmore aluent neighborhoods an in poorer or commercial neighborhoodscounit hacteistics aped to hae only weak effects on the rela

tionship twn microleel iables d ictiization hough Rountree etal ound evidence ha ineactions ween communiy and individual-leelvarabes ae lager han od appe rom esmaon o a ed eecsmoel

Mutilevel research is ound in many other areas o socioogy and relatededs asde om what is covered in ou ilusative discussion above. The mainpurpose o he above dscussion is o present eidence that for some sociologically iman otcomes, contex maers. It is probably a air generalization o sate ha most such evidence involes the main effects o context on

behaior, though impon research ha obviousy aso been done in stdyinghe interaction sucure oo he evidence ceinly ds no jusiy he conclsion ha contet always maers whn heoy sggests i mgh nor a itseecs, ne o oher factors, wil always be substanvey imporant (Mason1991) Furemore the above discussion dos not address the underlyingeoreca quesons conceng he corec iepreon o measred contexal effcts (assumin hese measured effects are, in act, "real and not heprodc o ay estimao)

Wh soogy s me to "me range eog (Meon 968,

ere s no genera heoy o muee neeaonsps Teoreca reseacin different substantie areas howeer is preedig toward he goal ofconsucting plausible explnaions for what context mes and how it ightaffect indiidual haior while empiical reseach tests hese ideas wihmlilevel daa. he amount o multilevel research in sioloy should owrapidy as heoes ae ren, as aready-reseached problems ae revisedwh newer methods and as new daa ao he estng o ne hyeses onte reationship between macro and mcro.

TE NEED O UTON

Despite the adaages o he new techques o multee aalsis, oe stbe caeful in appyng hese tecnques o substanie probems. e newtechniques whe superor o he olde ones rely on assumpons ha oenwill not hold in specic sbsnive contexts Some of he potetial problemsre:

l<ote that the intection effects not necesariy inreased in size when a rndom-efftsme is used. CfKre{t de Leeuw (

I Paay oveapping ists o the na iuies aisd by mueve meng cfound in Bak (4 t a and Maon

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MULLEVEL MEL

[1] Model complxity. Th esimaion o the coeciens o andom-ecs

multilvl modls s not trival, nd h socologcal magnaion can easlyoutun capaciy o th data t compur and cuen opmizaon ch

niques o provde obust estimates et a's observaon n ths rgard srnchan: "Invsgars f h pas s any ndcan) wll end chsemodels hat are too complcated (ve levels wth 10 varables on each lvel.hs leads o impossibly dfficul search problms ver e space models

and mpssbly dfcul lklihood maxmaon poblems Kre al

10 p [2] e assumpon f xd rgrssrs e sascal o underlyng

randmefecs multlevel mdls assumes hat th rgressrs are xed and

estmaes th modls cndnal on er valus Bu n mos practcal appl

cations he regressors are random and uncndinal stmation s desrdKrf al 0

he poblem of mssng daa Whle ncreasingly sphscatd prcedurs are now avalable r hndlng mssng data n statitical nalyss Ltle Rubin 1 isin sotwar or randmcts mulilvl rressn

modls dos not pod o intnal o sowa paca atmnt othe missing data problm Given the growing evidence that naive treatment o

he mssng daa prblm prduces based esmates Wang e al and

gven e amun f cmpuatn rqurd genrae randm-ffecs mult

levl rgresson estimas it s que possible a an analys wo s faced bhwh mssng data problms and w consrains on compung would do br

by combining sophisticatd methods o handling missing data with smpl

d ffcs mullvl egssn mtds an by cmbnng nave mhds

r handling missing daa wit sphistcated randmefects multlevl regesson mehods. fomr strategy would yeld consstn esmates o hgssin paramtrs thouh biasd timates o standard erors but h lattersatgy mht poduc nconsstnt stmats o both the regrsson paramersand t standad eos

Corelain btwen micr varabls and errrs n the macr quatnAs Equation 6 shws te ror n the macro quaton s a componnt o the

or n t mo quaton oo ths ro is corlatd wth mrlevlrgrssors h estimats o the microlvl cocients will be biased n hisstuain he use o dummy varables o convrt a randomefecs multlevel

regressn mdl a xd effcts mullvel rgressn mdl may b advsedJudge t al 8 rene 10

Nmally dsrbud rrs sng sascal packages genrally as

sm normall dsibted ors ttl is nwn about te obustness o hs

1 2S Muthen () for a discussion of r progss in h amn o ssing aa n

onx of a andom ffcs muliv mod.

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P

assumpt. te mcrolevel eors are rmal or can be made nal roga siable ansfonaion, nonnoaliy in e acro eqaion an he addressedas in [], ro e se of d varabes o onver fro a randoeffes a xedeffes mueve reresson problem

[6] Cela betwee mac vrables ad eors te macr equa,r betwee eter macr r mcro varables and er te mcro eqaonis roble bedevils al resear a eloys non exrienal daa o akecasa nferences (e. Holand 98, Berk 988; see also ook apbel979, Bak 98 Bk Radenbus 992 If e err n e arequa re celaed wi e measured mac-level vrables e macrlevel cfcets are based, and a causal tepreat ma inaurateSimilarly if e eor in e crolevel eqaion is correlaed eier wi

ro eve or acroeve vaabes e esaes w be inconsisenTee ae aios easons wy s oreaions ay ase ase (90,

97 discssed e case were indivdallevel vaabes re ory measred,s a macrolevel vaables ncon as prxes for mcrlevel vrables a regression For exale average SES n a scool g be a roxy forindivida SS i f e laer vaabe s easred wi error. Anoer probemocrs wen aracesis of easremen error vary by onex. For exaple f measuremen error for SS was greaer aol a i pblc sclsales, or f e re varane of SS in aolic sools was resriced

reaive o bic sools wile e eor vaiane was e same in e woconexs e easred effec of SS on aeveen n aolic sools gbe saller an in pblic sools wen e e ees are e sae sze (RMauser persoal cmmucao) I all f tese examples wat appears obe a mlileve effec ay be an aifac of e falre coec fr easreen eror in e icro evel odel

Anoer siaion were s oeaions old aise is wen e arolevelvaables are in some sense endogenos e miolevel poess Blalock98 ake an exaple fro e publised cnrvers over oleman e al' s

rept Pub Pub 98, fpares f cdren wfrnmeasured reasnsare ig acievers seek alc scols e corelaion bween e indiidalee eos and e sool ye aabe wod eado bias n e effe o soo ye in e acro eqaions e.g Godberger ain98 As wi e robe o easreen eror seecvy bas is noeiinaed by e se of mre soisaed mueve redures Cobng asscaed reamen f measreme er selecvy bas and relaedrbems wi e se of smpler fixed efets mullevel regressio mdels maofen rodce or arae resls an a saegy wic cobines naivereaen o ese isses wi randoeecs eve oeing

[7) e assion o exangaby: An asson nderying e se oempral Bayes prcedures o irease e relabily of woex es

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maes is at he diffeent contexs "exchngeable (Lindley Smith 972 e absence of explict covaiates in the macro equaion, this amonts toassuming hat he icrolevel cfcients differ across contexts only by chancevaiaion. e psence of explicit covarates in he macro equaion hisounts to ssung that te sidual vaiaton n te maco quaons isunsystematic. the contexts exchngeable, then the esercher cn improvete precision of wihin-context cfcient estimates by "boowng infonaion om oher contxts However, f he contexs ae not exchangeable-noher words if the dsinctveness of a gven context is not fully capted bye macolevel vaiablehen te snkae esmator fo e context-spcc ccients may be biased (McCullagh 8 Bk Raudenbush )

Sce we do not ow how bad the bias s, we do not now (eg n a

meansquae eo sense) wheer e adeoff is wowile for any paiculaset of whncontet estmates Fueoe e basvarace tadoff maydffer for each context in dl

The falue of exchaneability may viewd as a poblem of daa colltionat te maco level, or t may viewed as a poblem of sample size at hemaco level e numbe of factos at me contexts dstnctve is largeelative to he numbr of contexs, hen he abiliy of e nalyst to mel esevaiaons is obviously limd Aguably this situaion is more of a poblemfo e nalysis of complex unique enties such as counies ha it is for

moe snddized oganizaons such as schools. Wen nlyzing a set ofuque contexts he juscaon for rforing randomeffets regessonmodels may be weak and a xedeffects dummyvarable model may moreappropriate (Greene 0). Unforunately ee is no easy answe to thequestion of when a xed o a random effts mel is moe appropate

RC R R RRC

Te rate of development in he mehodology of multlevel mels and n heirsubstanve applcaion to a vae of sioloical poblems has n enor

mous in the past decade s rate s likey to contnue in te ne (e

Hox Keft A vaiety of methological problems awat solutons Ate mundane level hee is e pospect of faste computonal algotmsand mo capable hdwe at wil allow uses to nalyze moe complexproblems We can also ho for a closer lnage twen he lrature onmutlevel estmaon and mullevel study desgn Often a radeoff existstwn e numr of mcrolevel obsevaons at ca collete wineach context d he numr of contexts at c smpled sers awaitusel guidace aut how they should allate ther dat collcton sources

We should aso ext he development of more geneal mulilevel melsand software wih whch to estimat hem One sch development would e extension of he randomeffects mullevel reression mel to the case of

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event-histoy aysis, whle anothe would a simlar exension for loglinearand elated models fo coningency ables Because may common eventhistoy mels ad mels fo the alysis of coningency tbles ae genealized linea mels (McCullagh NeIde 1989), developments of multilevelsatistical theoy fo this mo geneal case (along with appopiate compuesowae) would have wide applicabiliy in siology. Genelizations of egenealize nea me to nclude adom tes n e linea pedicto aecalled genealized mxed linea models (GLMM) ad ecent wok by egGoldsin (99) o Bslow layton (993) on appomat meods ofinfeence in GLs may lead to mpoved pactcal appoaches to te esimaton of mulilevel event stoy mels multilevel logline d logmultiplicative models and othe elatd multilevel mels in the nea fuue

A second imtan development would be the development of mulilevelmdels fo systems of miolevel ad macolevel endogenous vaiables hecurent eciques ca hadle only a single micrlevel dendent vaiableResech is cuntly undeway to ceate a "pat aalysis fo multilevel

eseach (e et a 990 p .A id od fo development would be te exension of latnt vaiable

mes so at ey could applied to multilevel poblems ne wecomeadvce would involve e case whee te micolevel viables e laentwhile the contex d macolevel vaables ae measued exactly As Muhn

atoa noted (989) estimation of such a mel euies both futhesiscal developmen and new ssal softwae (s also Muhn 994).

A moe ambitious laten iable mel would alow e mao onexs o laen as well e use of latent cotexs would obviously equie at emaco vaiables fo these contexs also latent since the boundies ofobseable contxts would not exactl coespond to the boundaes oflatentoexs u a mdel mgt iewed as exension of latent class analysisnsead of seaching fo laent classes that explain vaaion in some outcomevaiable by ducing te paial assation (condtional on te latent class)

tween measued covaiates ad this outcome vaiable to a minimum aextended latent class mel migt employed to explain emaining vationin e effe of oaes as a funon of poes of te laen lassesmodeled as a function of ma indiato ables

o example seveal scolas in te 97s d 9s employed notions o adual eonomy o a segmented la mket in atmpts to explain vaation insaication outcomes (Hdson Kafman 982) Howeve ese stuctueswee no pecisely measued; often it was not obvious which segment any givenwoke belonged to is fuzziness did no pevent scholas fm proposingviables at we indicatos of make segment (eg measues of techologym size oganizational and industry atibutes o cupational caacteisics)n a latnt mullevel mel market segments mgt laent ontexs ta ae

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MULTLEVEL MODEL

explaned by nd indicae by , cupatonal nd indusal chacestics.e laten popees of hese laten conexs would explan coecien vaaioni microleve quaio for aicaio oucome13 Give e formidable

compuaioa dicule ha read ex for cue verio of mulilevelmode i i hrd o ay wha he proc re fo he deveopmen of aiicaleoy ad usel algohm o such mulievel aen conexual modes.Clearly, however, ere is om fo rhe advcemen

The exnce of a mueve mel mh alo ceae a uied frmework fo e ieeo of e ccien of ordir mcolevel modelSupe n al ha coruc a mode fo aiude owrd priculgovemen plicies ha iclude measues of educaon and income and ha

i esimaed wi O egeion upe we e coside he inereive

and iicl dfcue raed b he ccien fo icome ha de ime? From a ubve rcve, ieeo of he icome effec reoen conexal come ha a ec o o becaue oe' iere eread o he moun of money one ha i he ba, bu also cause icomeis aociaed wih paes of inesonal assiaion of upbnging and ofexpoue o mass media Idel one would lie o idef e chaacericof io-economcculual cox d ue is aibue o expain heeffec of individuaevel vriables on oucomes Fm h pspcive, viually all ociological mdel become coexual, wih mico ad maco

vaiables dening and explaining conex ha povide he explaaions fovaying pamees a he individual level. Nedes o a, idenicaion issuesfor uch ae ca me oud quie formdabe The ehd fucio woud complex, ad e robuses of eu woud unceri

n his conex i is helpful o eall he sll-geane obseaons made byBla in h 198 rview of coexual nali The ulimae goal of mulileve anali i o deermine how ocial coex affec and i affeced bci bhavior mprved we o queio quire palel advce

i eor ad daa coco a w a i ical medoo hie heei pen of m fo impovemen in all of ese domans, e ae of pogessis encouagig.

CO

We wish o acnowledge he helpul comens of Robe M Haue ThomasF. Janosi, and ll Pell on vious apecs of is manuscp This wowa uprd b Naioa ciece Foudaion g E-90269 d E05 d by a gr from he Aexde umbod Foundaio

S cens & Lang (98) fo alcaton of endogenous swchng regressons modelwh laen regmes o he oblem of emlong dual lar mares as a cone whou a rordeniion of lar mare undares

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34 D & S

y Annu vw chapter well y arcle aAnnu vw chapter,y b pu the Revws Preprn Repri

1�37; 415-29-S17; em: arr@org

Anr C, sn B, unc R, Smlsr N.9 Mir-ar Axader, B Gin, R Munch, J meler pp.12 Bekeley Univ. Clif Pess

Baron 198 Organizaona sves onsaicaion Ann. Rv. i. 0379

Brlson BP P cP W 195

Vtin Ccago: niv Chicago PrssBk 1988. Causa infeen for siolog d Hk fgy, mese, pp 155-74. ewby Pk, Cifage

Bily JO M DE 1992. A mutievl nay i of mit d nonmaa feii in thUS. S. Frc 70977-101

la H 1984 onxtuaef meseoca d mehoogica issues Av c 1035372

Blau P. 19. Sctura ff Am iv 26:178-93

Blssed PBvt Y 1993.

Prsistnt Inqa: Canging cat Aanmntn n Cts Bo Coo: Wesvew

Bk 199 tiv Asis Edatina Data San Diego Calif: Academic

oyd H, ven GR 979. nxta Ana yss Cncpts and tatstica TcnqesBemont Cif Wadwoh

Ban H 1989 Empiical Bays meh: at for xpoto anaysis. Bk 99pp 1955

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