w4 holland 1986

8/13/2019 W4 Holland 1986

1/9

Statistics nd Causal InferencePAULW. HOLLAND*

Problemsnvolvingausal nference avedogged t theheels f tatisticssince ts arliest ays. orrelation oesnot mply ausation,ndyet ausalconclusionsrawn rom carefully esignedxperimentre often alid.What an a statistical odel ay bout ausation? his question s ad-dressed y using particular odel or ausal nferenceHolland ndRubin 983;Rubin 974) o critique hediscussionsf otherwriters ncausationnd ausalnference. hese nclude elected hilosophers, ed-icalresearchers,tatisticians,conometricians,ndproponents f ausalmodeling.KEYWORDS:Causalmodel;Philosophy; ssociation; xperiments;Mill'smethods; ausal ffect; och's ostulates; ill's ine actors; ran-ger ausality; athdiagrams; robabilisticausality.

1. INTRODUCTIONThe reaction f many tatisticians hen onfronted ith

the possibilityhat heir rofession ight ontribute o adiscussionf causations immediatelyo deny hat hereis any uch possibility. That correlations not causationis perhaps he irst hing hatmust e said" Barnard 982,p. 387).Possibly his vasive ction s nresponse o all ofthose needling ittle eadlines hat pop up in the mostunexpected laces,for xample, If the tatistics annotrelate ause ndeffect, hey an certainly ddto the rhet-oric" Smith 980, . 998).

One need only ecall hat well-designedandomizedexperiment an be a powerful id in nvestigatingausalrelations oquestion he needfor uch defensive ostureby statisticians. andomized xperiments ave trans-formed any ranches f cience, nd he arly roponentsof such tudieswere he sanle statisticians hofoundedthe modern ra of our field.This rticle akes heview hat tatistics asa great ealto ay bout ertain roblems f ausal nference nd ughtto play more ignificantole n philosophicalnalyses fcausation han t has heretofore. n addition, will ry oshowwhy he tatistical odels sed o draw ausal nfer-ences re distinctly ifferent rom hose sed to draw s-sociationalnferences.

The article sorganizeds follows. irst, tatistical odelsappropriate or ssociationalndcausal nferences ill ediscussednd ompared. hen hey ill e applied ovari-ous ideas about causation hat have been expressed yseveralwriters n this ubject. ne difficultyhat rises ntalking bout ausationsthe variety f questions hat resubsumed nder he heading. omeauthors ocus n theultimate eaningfulnessf he notion f ausation. thersare concerned ith educing he auses f a given ffect.Still thers re nterested n understandinghedetails fcausalmechanisms.heemphasis ere will e on measur-ing he ffects f causesbecause his eems o be a place

*PaulW.Holland s Director, esearch tatistics roup, ducationalTesting ervice, rinceton, J 8541.A preliminary raft f his rticlewas he asis f n nvited eneralMethodologyecture or he Amer-ican tatistical ssociation, ugust 985. he ommentsyGlymourndGrangerncluded ere were iven t that ession n

responseo hat raft

of this rticle.

where tatistics, hichs concerned ithmeasurement, ascontributionsomake. t is my pinion hat n emphasison the ffects f auses ather han n the auses f ffectsis, n tself, n mportant onsequencefbringingtatisticalreasoning obearon the nalysis f causation nddirectlyopposesmore raditionalnalyses f causation.

2. MODELFORASSOCIATIONALNFERENCEThemodel ppropriate or ssociationalnferences im-

ply the standard tatistical odel hat relates wo vari-ables over a population. or clarity nd for omparisonwith hemodel or ausal nference escribedn the nextsection, owever, will riefly eview ssociation ere. fI seemoverly xplicit n describing hemodel t s onlybecause wish o be absolutely lear n the fundamentalelements f the heory resented ere.

The modelbeginswith population r universe of"units."A unit n U willbe denoted yu. Units re thebasic bjects f tudy n n nvestigation.xamples f unitsare human ubjects, aboratory quipment, ouseholds,andplots f and.A variablessimply real-valued unc-tion that s defined n every nit n U. The value of avariable or given nit is the number ssigned y omemeasurement rocess ou. A population funits ndvari-ablesdefined n these nits re the basic lements f themodels or oth ssociationndcausation resented ere.They orrespondo the mathematicaloncepts f set ndreal-valued unctions efined n the elements f the et.They re he rimitivesf he heory ndwill ot e furtherdefined.

Suppose hat or ach unit in Uthere s associatedvalue Y(u)of a variableY Supposefurther hat Y is avariable f scientificnterest n the ense hat ne wishesto understand hy hevalues f Y vary ver he units nU. Y is the response ariable ecauseof its status s a"variable o be explained."n making ssociationalnfer-encesone is satisfied ith iscovering owthe values fYare associated ith he values f other ariables efinedon the units f U. Let A be a second ariable efined nU. Distinguish from by calling an attribute f theunits n u. Logically, owever, and Y are on an equal

footing, ince hey reboth imply ariables efined n U.Allprobabilities, istributions,nd expected alues n-volving ariables re computed ver U.A probability illmeannothing orenor essthan proportion f units nU. The expected alueof a variable smerely ts veragevalueover ll of U. Conditional xpected alues re av-erages ver ubsets f unitswhere he ubsets re definedby onditioningnthe alues f variables.t s n his ensethat he models escribed ere re population odels.

The role of timeneeds o be mentioned ere. Popula-

? 1986American tatistical ssociation

Journal f heAmerican tatistical ssociationDecember 986, ol. 1,No.396,Theorynd Methods945

This content downloaded from 14 7.8.31.43 on Mon, 19 Aug 201 3 03:19:41 AMAll use subject to JSTOR Terms and Conditions
http://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsp

8/13/2019 W4 Holland 1986

2/9

946 Journal of the American Statistical Association, December 1986

tions f units xistwithin time rame f ome ort, ndthe measurementsf haracteristics funits hat ariablesrepresent ust lso be made at particular imes. or as-sociationalnference, owever, he roleof time s simplyto affect he definition f the population f units r tospecify heoperationalmeaning f a particular ariable.As we will ee, in causal nference heroleof time as agreater ignificance.Themost etailed nformationne an have n he modeljust described s the values f Y(u) andA u) are all u inU.The oint istribution f Y andA overU is specified yPr(Y = y, A = a) = proportion f u in U for whichY(u)- y and A(u) = a.

The associational arameters re determined y thisjoint distribution. or example, heconditional istribu-tion of Y givenA is specified y Pr(Y = y IA = a) =Pr(Y = y, A = a)/Pr(A = a). This conditional istribu-tion describes ow the distribution f Y valueschangesoverUasAvaries. typical ssociationalarameterstheregression f Y on A, that s, the onditional xpectationE(Y IA = a).

Associationalnference onsists f making tatisticaln-ferencesestimates,ests, osterior istributions,tc.) boutthe ssociationalarameters elating and A on the basisof data gathered bout Yand A from nits n U. In thissense, ssociationalnference s simply escriptivetatis-tics.

3. RUBIN'SMODEL FOR CAUSAL NFERENCEBecauseexperimentations such a powerful cientific

and statistical ool and one that ften ntroduces larityinto discussions f specific ases of causation, una-

bashedly raw n the anguage nd framework f experi-ments or hemodel or ausal nference. t is not hatbelieve n experiments the only roper etting ordis-cussing ausality, ut do feel hat n experiment s thesimplestuch etting. hepurpose sto construct modelthat s complex nough o allowus to formalize asic n-tuitions oncerning ause and effect. he point f depar-ture s the nalysis f causal ffects iven n Rubin 1974,1977,1978,1980). t will be sufficient or ur purposes,however, odealwith simplified,opulation-levelersionofRubin'smodel.This implified odelwasused n Hol-land nd Rubin 1980) oanalyze ausal nference nretro-spective, ase-controltudies sed nmedical esearch ndin Holland nd Rubin 1983)to analyze ord's analysisof ovariance" aradox. refer o this s "Rubin's model"even hough ubinwould rgue hat he deas behind hemodelhavebeenaround ince isher. think hatRubin(1974)wasthe placewhere hese deas were irst ppliedto the tudy f causation.

Thismodel lso beginswith population f units, U.Units n the model or ausal nference re the bjects fstudy n which auses or treatments ay ct. The termscause nd treatment ill e used nterchangeably,ndthenotion hat hese erms onvey s an mportant art f hemodel. t is important o realize hat y using he terms

cause and treatment nterchangeably do not ntend olimit he discussion o the activities ithin controlled

randomized tudy. do it to emphasize n idea that be-lievereceives nsufficientttention n general iscussionsof causation. his s the fact hat heeffect f a cause salways elative o another ause. Forexample, he phrase"A causesB" almost lwaysmeans hat causes relativeto some other ause that ncludes he condition notA." The terminology ecomes ather ortured fwe try ostickwith he usualcausal anguage, ut t s straightfor-ward f we use the anguage f experiments-treatment(i.e., one ause)versus ontrol i.e., another ause). nSec-tion 1 will iscuss he undamental uestion fwhat indsof things an be causes. The keynotion, owever, s thepotential regardless fwhether t an beachievednprac-ticeor not) for xposing r not xposing ach unit o theaction f cause.For ausalnference,t scritical hat achunit be potentially xposable o any one of the causes.As an example, he chooling student eceives an be acause, nour ense, f he tudent's erformancena test,whereas he tudent's aceor gender annot.

For implicityt hall e assumed n his rticle hat hereare ust two causes or levels f treatment, enoted y t(the treatment) nd c (the control). et S be a variablethat ndicates he ause owhich ach unit n U s exposed;that s,S = t ndicates hat heunit s exposed o t and S= c indicates xposure o c. In a controlled tudy, isconstructed y he xperimenter. n an uncontrolledtudy,S is determined osome xtent y factors eyond he x-perimenter's ontrol. n either ase,the ritical eature fthe notion f cause n thismodel s that he valueofS(u)for ach unit ouldhavebeendifferent.

Thevariable is analogous o the variable in Section2, but with he ssential iffkrence hat (u) indicates x-posure f u to a specific ause,whereas u) can ndicatea property r characteristic f u. In this ase the valueofA(u) couldnot havebeendifferent.

Therole f time owbecomes mportant ecause f hefact hat when unit s exposed o a cause hismust ccurat some pecificime r within specificime eriod.Vari-ables now divide nto two classes:pre-exposure-thosewhose alues redeterminedrior oexposureothe ause;post-exposure-those hosevalues re determined fterexposure othe ause.

Therole f response ariable s tomeasure he ffectof he ause, nd hus esponse ariablesmust all nto hepost-exposurelass.Thisgives ise o another ritical le-ment f the model.The values f post-exposureariablesare potentially ffected ythe particular ause, or c, towhich he unit s exposed.This s nothing ess than hestatement hat auses have ffects, hich sthevery eartof the notion f causation. or the model o representfaithfullyhis tate f ffairs e need not single ariable,Y, to represent response ut twovariables, , and Yc,to represent wopotential esponses. he interpretationof these wo values,Y,(u)and Yc(u)for given nit , sthat Y,(u)is the valueof the response hatwouldbe ob-served f he unitwere xposed o t and_(u) is the valuethatwould e observed n he ame nit f t were xposedto c.The notation t(u) and Yc(u)is sometimes onfusing


8/13/2019 W4 Holland 1986

3/9

Holland:Statistics nd Causal Inference 947

because variable sually epresents measurement fsome ort nd a measurementsusually hought f s theresult f process hat sapplied o unit. his s not eallycorrect. or post-exposure ariables hemeasurementsapplied o the pairing u, t) i.e., u after xposure ot) orto u, c) (i.e., u after xposure oc).A notation hatmorenearly xpresses his oint dependence f Y on u andtheexposed cause is Yt(u) = Y(u, t) and Yc(u) = Y(u, c). Ishalluse the Yt, Ycnotation, owever, ince t eads tosimpler xpressions.

The effect f the cause t on u as measured y Y andrelative o cause c is the difference etweenYt(u)andYc(u).n the model hiswill e represented y he lgebraicdifference

Yt(u) - YC(u). (1)I shall allthe difference1) the ausal ffect f t relativeto c) on u (as measured y Y). Expression1) is the waythat hemodel or ausalnferencexpresseshemost asicof ll causal tatements.t says hat reatment causes heeffect ,(u) - Yc(u)on unit U (relative o treatment )or more imply hat

t causes the effect ,(u) - YC(u). (2)Causal nferencesultimatelyoncerned ith he ffects

of causeson specific nits, hat s, with scertaining hevalue of the causaleffect n (1). It is frustrated y aninherent act f observationalife hat call the Funda-mental roblem f Causal nference.

Fundamental roblem f CausalInference. t is im-possible o observe hevalue of Y,(u) and Yc(u)on thesameunit nd, therefore, t s impossibleoobserve heeffect f t on u.

Theemphasisson the word bserve. he mpossibilityof observing othY,(u)and Yc(u)s self-evidentn someexamplesnd essclear nothers. or example, f heunitu is a specific ourth rader, represents novelyear-longprogram f study f arithmetic, represents standardarithmetic rogram, nd Yis a score n a test t the ndof heyear, hen t sevident hatwecould bserve itherY,(u)or Yc(u)but not both. We willnever bservewhatthe ffect f t wason u. Onthe ther and, f u is a roomin a house, means hat flick n the ight witch n thatroom, means hat do not, nd Yindicates hether helight son or not short ime fter pplying ither or c,then might e inclined o believe hat can know hevalues fboth ,(u) ndYc(u)by imply lickinghe witch.It s clear, owever,hat t sonly ecause f he lausibilityof certain ssumptionsbout he ituation hat his eliefofmine anbeshared y nyone lse. f, for xample, helight as beenflicking ff nd on for no apparent easonwhile amcontemplatingeginninghis xperiment,mightdoubt hat wouldknow he valuesof Y,(u) and Yc(u)after licking n the switch-at east until was cleverenough o figure ut a new xperiment

The mplicit hreat f he undamentalroblem fCausal

Inference s that ausal nference s impossible. ut weshould ot ump othat onclusionooquickly. y assert-

ing hat he imultaneous bservation f Y,(u)and Y,(u)is impossible do not mean hatknowledge elevant othesevalues s completely bsent. t will depend n thesituation onsidered. here re two general olutions othe Fundamental roblem, hich or he ake of conven-ience will abelthe cientificolution nd the tatisticalsolution.

Thescientificolutionsto exploit arious omogeneityor nvariance ssumptions. or example, y tudying hebehavior f a pieceof aboratory quipment arefullyscientist ay ome o believe hat hevalueof Yj(u)mea-sured t an earlier ime s equal to the valueof Yj(u)forthe urrent xperiment. llheneeds o donow s toexposeu to t and measure ,(u)and he has overcome heFun-damental roblem f Causal nference. ote, however,that his ypotheticalcientist asmade nuntestable om-ogeneity ssumption. y carefulworkhe may onvincehimself ndothers hat his ssumptionsright, uthe cannever e absolutely ertain. ciencehas progressed eryfar

yusing his pproach. hescientificolutionsa com-monplace spect f our everyday ife s well.We all useit to make the causal nferences hat rise n our ives.These deasare amplifiedn Sections .1 and4.2.

Thestatisticalolutionsdifferent ndmakes seof hepopulation in a typically tatistical ay. The averagecausal ffect, , of t relative oc) over Uis the xpectedvalue of the difference t(u) - Yj(u)over heu's in U;that s,

E( Yt - YC)= T. (3)Tdefined n 3) is the verage ausal ffect. y the usualrules f probability3) may lsobe expressed s

T = E(Y) - E(Yc). (4)Although hisdoesnot ook ike much, 4) reveals hatinformation n different nits hat an be observed anbeused to gainknowledge bout T. For example, f someunits re xposed ot theymay eused ogive nformationabout (Yt)(because his s the mean alue f Yt verU),and f other nits re exposed o c theymaybe used togive nformationbout (YC).Formula4) isthen sed ogainknowledgebout T. Theexactway hat nits wouldbe selected or xposure o t or c is very mportant ndinvolves ll of the usual onsiderationsf good tatisticaldesign f

experiments. he important oint s that hestatisticalolution eplaceshe mpossible-to-observeausaleffect f t on a specific nitwith hepossible-to-estimateaverage ausal ffect f over population funits. heseideaswillbe developed urther n Sections .3 and 4.4.

The usefulness f either he cientific r the tatisticalsolution othe Fundamental roblem f Causal nferencedepends n the truth f different ets of untestable s-sumptions.n Section I willdiscuss ome of the ypicalassumptionshat re often sedto overcome heFunda-mental roblem f Causal nference.

It is useful o have a notation oexpress hefact hatthe ausal ndicator ariable determines hich alue,Ytor Yc, sobserved or given nit. f S(u) = t, then Yt(u)is observed, nd f (u) = c, then Yc(u)s observed. hus


8/13/2019 W4 Holland 1986

4/9

948 Journal f he American Statistical ssociation, December 1986

the observed esponse n unit is Ys(u)(u).The observedresponse ariables, herefore, s.Hence, ven hough hemodel ontains hree ariables, , Yt, nd Yc, heprocessof observation nvolves nly wo, hat s, S, Ys.The dis-tinction etweena) themeasurementrocess, , that ro-duces he response ariable; b) the twoversions f theresponse ariable , ndY,that orrespondsowhich ausethe unit s exposed and n terms f which ausal effectsaredefined); nd c) the bserved esponse ariable s, svery mportant nd, often, s not made n discussionsfcausation. hese distinctions ever rise n the study fsimple ssociation, ut hey re crucial o the nalysis fcausation.

It is useful oreview he model or ssociational nfer-ence and Rubin's model ide by side to emphasize heirdifferences. oth nvolve population f units, U, andboth nvolve woobservable ariables: A, Y) for ssoci-ation nd S, Ys)for ausation. his s all, however, hatthey ave n common. Whereas and Y are simply ari-

ables defined n the units f U, S and Yspresupposemore omplicatedtructure n order or hem oapply oreal ituations. wo or more auses or treatments) ustbe exposable oallof the units, nd the response mustbe a post-exposure ariable n order or he observed e-sponseYsto be defined. ssociational nferencenvolvesthe oint r conditional istributionsfvalues f YandA,and ausal nference oncerns he aluesY,(u) - Y,(u)onindividual nits. ausal nferences roceed rom he ob-served alues f S and Ysand from ssumptionshat d-dress he Fundamental roblem f Causal nference utthat re usually ntestable. ausal nferences onot nec-essarilynvolve tatisticalnferences, ut ssociational n-ferences lmost lways o.

4. SOMESPECIALCASES OF CAUSAL NFERENCEThis ection onsiders ome imple pecial ases fRub-

in's modelfor ausal nference. he purpose s to showhow pecific ssumptionsdded othe model llow ausalinferences f particular ypes.

4.1 Temporal Stability nd Causal TransienceOnewayof applying he cientific olution o the Fun-

damental roblem f Causal nference s to assume hat(a) the value of

Y,(u)doesnot depend n when he e-

quence apply to u then measure on u" occurs nd b)the valueof Y,(u) snot ffected y heprior xposure fu to the equencen a). When hese wo ssumptionsreplausiblet s a simplematter o measure ,(u)and Y,(u)by equential xposure f u to c then , measuring aftereachexposure. he first ssumptions temporal tability,becauset sserts he onstancy f esponse ver ime. hesecond ssumptionscausaltransience, ecause t assertsthat he ffect f he ause and hemeasurement rocessthat esults n Yc(u)is transient nd does not changeenough o affect_(u) measuredater. hese wo ssump-tions ften pply o physical evices nd reroutinely ade

by all of us in everyday ife-for xample, n the "lightswitch" xample mentioned arlier.

4.2 UnitHomogeneityA secondwayof applying he cientific olution othe

Fundamental roblem s to assume hatY,(ul) = YI(u2)and Y,(ul) = Y,(u2) for wounits l and u2.This s theassumptionfunit omogeneity.t, oo, s often pplicableto work one n scientific aboratoriesnd s also a causal

workhorse f everydayife. he causal ffect f t s takentobe the alue fY,(ul) Y,(u2).One way hat aboratoryscientists onvince hemselves hat heunits re homoge-neous s to prepare hem arefully o that hey look"identical n all relevant spects. his, of course, annotprove hat he unit homogeneityssumptions valid, utit can make his ssumption lausible.

4.3 IndependenceIn my discussion f the tatistical olution othe Fun-

damental roblem, didnot give ny pecification o theway hat nitsmight e selected or bservation f Y,or

Y,. I only ndicated hat t wasvery mportant. f course,the mostwell-known ay hat his ccurs nexperimentalwork s by randomization, nd this ection s concernedwith hat opic.

The supposition n using he tatistical olution s thatthe population does not onsist f oneor two units utis "large" n some ense. The observed atafor achunitare values f the pairof variablesS, Ys).

The average ausal effect is the difference etweenthe wo expected aluesE(Yt)and E(Y,). The observeddata S, Ys),however, anonly ive s nformation bout

E(YsS|S t) =E(Yt S = t) (S)

andE(Ys S =c) E(Yc S = c). (6)

It is important orecognize hat (Yt)andE(YtI = t)arenot he ame hing ndneednothave he ame valuesingeneral similarlyor (Yc)andE(YCI = c)].Tostatethis ifference nwords, (Yt) sthe verage alue f Yt(u)over ll u in U,where (YtI = t) is the verage alueof Yt(u)over nly hose n u in Uthat were xposed o t.There s no reasonwhy, n general, hese wo averagesshould eequal.For example, f (u) = t for ll units orwhich ,(u) is small, henE(Y, I = t) willbe smallerthan (Y,).There s,however,n assumptionhat, f lausible, akesthese wo expected alues qual. It is the assumption findependence. henunits re assigned t random itherto cause t or to cause c, certain hysical andomizationprocessesre arried ut o that he etermination fwhichcause t or c) u is exposed o is regarded s statisticallyindependent f all other ariables, ncluding t and Yc.This means hat f the physical andomizations carriedout correctly, hen t splausible hat is independent fY, and Y, and all other ariables ver U. This s the n-dependencessumption.f this ssumption olds, henwehavethe basic quations


8/13/2019 W4 Holland 1986

5/9

954 Journal f he American Statistical ssociation,December 1986

search ection f he Royal tatistical ociety nMarch f1935)Neyman ave an explicit tatement f the dea ofmultiple ersions f he esponsewhichs for Neyman heyield rom n experimental lot f and n an agriculturalexperiment). nfortunately, eyman's iscussion lso n-troduced henotion f a stochastic lement hat s addedto Yto allow for technical rrors" hat re due to inac-curacies f xperimentalechnique.fwe gnore his rob-lemof measurementrror nd assume ero "technical r-rors," hen Neyman's efinition f "true ield" xplicitlyrefers o multiple ersions f the response. Thus X1j(k)willmean he true' ield f hekth bject btainable romthe plot i, )" (p. 110;by object" Neymanmeans reat-ment).Hisnotation svery imilar o that sed byKemp-thorne. o put t nto he notation f Section , the unitsare the plots, ij, ndX1j(k)= Yk(uij), where k(u) isde-fined s in the previous iscussionf Kempthorne.

Neyman lso had an explicit xpression or he veragevalueofXij(k)over ll of the units, ij. t s X..(k). n the

notation f Section this sX..(k) = E(Yk). Hence t sclear that by the time Neymanwaswriting he dea ofmultiple ersions f he esponse, ne for ach reatment,was established. t seems o have been used by writersconcerned bout he etails f he ffects f andomizationin specific xperimental lans e.g., Cox 1958; Kemp-thorne 952)but s generally ot a part f the tandardstatistical otation f many therwriters anexceptionsHamilton 1979)].

The Neyman 1935) reference s also relevant o themodel n Section becauseof the controversy etweenFisher nd Neyman hat t engendered. he controversyrevolves round hechoiceof null hypothesisn experi-ments uch s randomized lock esigns. isherwasquiteclear hat he nullhypothesis hat e proposed s that hecausal ffect aswe have defined t) s0for achunit. orexample, n the famous iscussion t the nd of Neyman(1935)Fisher irst uotedNeyman, s follows:... this bias vanishes when .. the objects compared are reacting odifferencesn soil fertility n exactly he ame manner. . . This s notalways rue. p. 153)

ThenFisherwrote:However,twas lways rue n he ase forwhichtwas equired, amely,when he hypothesisobe tested as rue, hat ifferences f reatmentmadenodifference o the yields. p. 157)

ThenNeyman, nrespondingoFisher's emarks, mpha-sizedhis nterest n what would all the verage ausaleffect.'Ourpurpose n he ield xperimentonsistsn omparing umbersuchasX.(k), or the verage rueyieldswhich ur objects re able to givewhen pplied o hewhole ield.' t s een hat his roblemsessentiallydifferent rom hat rofessor isher uggested. o long s the verageyields f ny reatmentsre dentical, he uestion s to whether hesetreatments ffect eparate ields n ingle lots eems obeuninterestingand cademic. p. 173)

Fisher's ardonic eply ndicates hat, t least,he agreedthat Neyman tated heir ifferences learly. It may befoolish, ut hat s what he test was designed or, ndthe nly urpose or which t has been used" p. 173).Evidently, would onclude hatNeyman's ull hypoth-

esis s one of zero average ausal effect, hat s, E(Y, -YJ)= 0, whereas isher's s one of zero causal ffect orallunits, hat s, Y,(u) - Y,(u) = 0 for ll u E U.

7. WHATCAN BE A CAUSE?It may eem very xtreme osome o limit he notion

of cause to the ense used n Section . Aristotle et thestage or his, owever, y distinguishing ore han nemeaning otheword ause. t might e better oask,whatcan be an "efficient ause" n his sense?Evidently venthis estriction id not imit he notion f cause for uchthinkers s Hume and Mill. Anything an be a cause forthem-or, t east, potential ause.

Put as bluntly nd as contentiously s possible, n thisarticle take he position hat auses re only hose hingsthat ould, n principle, e treatments nexperiments. hequalificationinprinciple"s mportant ecause ractical,ethical, ndother onsiderations ight ake ome xper-iments nfeasible, hat s, imit s to contemplatingypo-

theticalxperiments.or xample,n hemedical nd ocialworldwemight e able to conceive f n experiment, utno one would ver ry o carry t out. nstead,we mighthave to wait for "natural xperiment" o occur. Ob-servational tudy" s the term sedby statisticianse.g.,Cochran 983) orefer o studies or which The objectiveis to study hecausaleffects f certain gents" ut "Forone reason or another he nvestigator an not .. imposeon . . . or withhold rom he subject, a treatment hoseeffects e desires o discover" p. 1).

I believe hat henotion f cause that perates n anexperimentnd n an observationaltudy s the ame.Thedifferences in the degree f control n experimenterasover the phenomena nder nvestigationompared iththat which n observer as. In Rubin'smodel his s ex-pressed y he oint istributionf S with , nd Yc.Totalcontrol an make independent f Y,and Yc.

It maybother ome readers hat have beenusing heterm experiment"n a very estrictedense-though nethat s commonnthe tudy f the design f experiments.For example, xperiments n chemistryn which sub-stance s analyzed nto ts component ngredients r inwhich ngredients re combined ith ach other o syn-thesize new ubstance ften maynot haveclearly den-tifiable nits, reatments, ndresponse ariables. yview

is that n such xperimentsheAristotelian otion f ma-terial ause s often more relevant han hat f efficientcause, ndhence uch xperimentsre not oncerned iththe notion f causethat sdiscussedn this rticle.

To return o the uestion f what an be a cause et meconsiderhree xamplesf tatementshat nvolvehewordcausebut hat ary n ts xact sage.

(A) She did wellon the xam because he s a woman.(B) She did well on the exambecause he studied or

it.(C)Shedidwellon the xam because he wascoached

byher eacher.

I think hat hese tatements, ven hough hey re per-fectly nderstandablenglish entences, ary nthemean-


8/13/2019 W4 Holland 1986

6/9


ing f he because" n each. n each, he ffect, sing heterm oosely, s the ame-doing well on an exam. Thecauses, gain sing he erm oosely, re different. n (A)the "cause" is ascribed o an attribute he possesses. n(B)the cause" s ascribed o some voluntary ctivity heperformed, nd n C) it s ascribed o an activity hatwasimposed n her.

An attribute annot e a cause n n experiment, ecausethe notion f potential xposabilityoes not apply o it.Theonlyway or n attribute ochange ts value s for heunit o change n someway and no longer e the sameunit. tatements f "causation" hat nvolve ttributes s''causes'' re always tatements f ssociation etween hevalues f an attribute nd a response ariable cross heunits n a population. n (A) all that s meant s that heperformancefwomen n the xam xceeds,n ome ense,that f men.

Examples f he onfusion etween ttributes nd ausesfill he ocial cienceiterature. aris nd Stronkhorst1984)

gave he ollowingxample f causal hypothesis:Scho-lastic chievement ffects he hoice f econdarychool"(p. 13). These uthors learly ntended or his ypothesisto state hat n attribute f student i.e., scores n tests,performancenprimary chool) ancause i.e., affect) hestudent's hoice f a particular ype f secondary chool.It is difficult o conceive f how scholastic chievementcouldbe a treatment n an experiment nd, herefore, ea "cause" in the senseused n this rticle. A somewhatstronger tatement f my oint wasgiven y Kempthorne(1978, . 15): "It s epistemological onsenseotalk boutone trait f an individual ausing r determiningnothertrait f the ndividual."

At the other xtreme sExample C). This s easily n-terpreted nterms f hemodel.The nterpretationsthathad she not been coachedby her teacher he would nothavedoneas well s she did. t mplies comparison e-tween he responses otwo auses, ven hough his om-parison s not xplicitlytated.

Example B) is ust one of many ypes f examplesnwhich he pplicability f he model s not bsolutely lear,and t hows ne reasonwhy rguments verwhat onsti-tutes proper ausal nference anragewithout ny efin-itive esolution.

In B)the roblem rises ecause f he oluntaryspect

of the supposed ause-studying or he exam. t is notclear hat wecould xpose person ostudying r not nany erifiableense.Wemight e abletoprevent er romstudying, ut hat would hange he ense f B) to some-thingmuchmore ike C). We could perationally efinestudying s so many ours f nose nbook,"but hat ustdefines n attribute ecouldmeasure n a subject. n myopinion he pplication f the model ostatement B) isproblematicalnd not asily esolved. he voluntary a-ture f much f human ctivity akes ausal tatementsabout hese ctivities ifficultnmany ases.

The voluntary spect f the "cause" in (B) is not theonly ource f difficultyn deciding n the pplicability fRubin'smodel o specific roblems.t s, howevrer,com-mon ource f difficulty.

The general roblem, think, s n deciding hen ome-thing sanattribute funits ndwhen t s a cause hat anact on units. n the former ase all that an be discussedis association, hereas n the atter ase it s possible, tleast, o contemplate easuringausal ffects.

Onemay iew isher's 1957) ttack n thosewhousedthe ssociation etween moking nd ung ancer s evi-denceof a "causal ink"between hem s an example fthe difficulty n deciding hether r not smoking s anattribute r a cause. Certainly he data that beganthisdebate re purely ssociational.oll andHill's tudies 1950,1952, 956) scertained nly moking tatus nd ung an-cer tatus n sets f ubjects. isher rgued hat mokingmight nly e indicative f certain enetic ifferences e-tween mokers ndnonsmokersnd hat hese enetic if-ferences ould e related o he evelopment r not f ungcancer. isher 1957)didfeel hat a good prima acie asehadbeenmadefor urther nvestigation."

Theresponse o Fisher's riticism analsobe viewed s

attempting o show hat moking hould e thought f ncausal erms ather han s ndicative f genetic ttributeof subjects. or example, mong isresponses o Fisher,McCurdy1957)pointed ut hat ung ancer ates ncreasewith he mount f mokingnd hat ubjects ho toppedsmoking ad ower ung ancer ates han hose whodidnot.Both f hese rguments anbe viewed semphasizingthe ausal spects f moking-one ando more r essofit and one might topdoing t. A discussion f the ntiredebatewasgiven yCook 1980).

8. COMMENTSON CAUSAL INFERENCESINVARIOUSDISCIPLINES

This ection ill riefly onsider iscussions f ausationin three disciplines-medicine, conomics, nd "causalmodeling."n eachcasean attempt ill e made o relatethe discussionoRubin'smodel or ausal nference, utnoattempt s made obeexhaustiver even epresentativein the election f topics onsidered.8.1 Causation and Medicine

Webeginwith simple, et basic, xample rom medi-cine-the establishment f pecific acteria s the ause fspecificnfectious iseases. erushalmyndPalmer 1959)described he ituation nthe followingerms:Almost rom he ery eginning, hen acteria ere irst ound o causedisease, acteriologistselt heneed or set f ules o ct sguidepostsin nvestigationf bacteria s possible ausal gents ndisease.p. 28)

These wo uthors escribedhree ostulates ormulatedby hegreat acteriologist, obert och,whodiscovered,among ther hings,he uberculosisacillusn1882.Koch'spostulates also called heKoch-Henle ostulates, vans(1978)] re imple, o-nonsenseriteria or eciding hena microscopicrganisms mplicatedn disease.Accord-ing oYerushalmy nd Palmer 1959), while here s nosingle ormulation fKoch's ostulates-they anbe statedas consisting ssentially f he following:

I. The organism ust e found n all cases of he dis-easeinquestion.


8/13/2019 W4 Holland 1986

7/9


II. It must e solated rom atients ndgrown npureculture.

III. When he ure ulture s noculatednto usceptibleanimals r man, tmust eproduce he isease." p.30)

Rubin'smodel pplies ather learly oPostulates andIII. Postulate is simply ill's method f agreement p-plied o this roblem. t ensures hat here re no data tosupport null ausaleffect n this ase-that is, f therewerebonafide asesof he disease nwhich he rganismwasnot present, longwith ther ases of the disease nwhich t was, then ssuming nithomogeneity e wouldhave an estimate f ero causal ffect or hepresence fthe organism elative o its bsence. Postulate II is likethe ight witch xample-put n the organism nd thediseaseoccurs. he validity f this ostulate tems romthe nstated ssumptionhat ad he nimal r human otbeen noculated ith he ulture he iseasewould ot avebeen xpected ooccur.Note hat he word susceptible"hascrept n, presumablyo dealwith he nevitable non-constant onjunction"f real aboratory ork-in his ase,the mmune ystem.

Koch'ssecondpostulate elates more o good experi-mental echniqueshan o causalnference.f he rganismis isolated rom atients nd grown n pure ulture, henwhen t comes ime o noculate nimals r peoplewith tthe xperimenter nows hat he noculants nfairly xactterms. n a sense, ostulate I is a way f minimizing ea-surement rror n the reatmentt) that s exposed o theunits.

Medicinesmore ifficult hen hebiological heory slesswelldeveloped. s an example now onsider everalsuggestions adeby irAustin radford ill to thosewhomight ish o separate ssociation rom ausation n thestudy f the environment nd disease. He had spentlifetimenpublic ealth nd wasamong he first o argue,quantitatively, or he causal ink between moking ndlung ancer Dolland Hill 1950, 952, 956).Hill 1965)namedninefactors hathe feltwereuseful n suchworkfor deciding hat he most ikely nterpretation f an ob-served ssociations causation. will onsider hese n anorder hat iffers rom ill's.

Temporality. Whichs the art nd which he horse?"(Hill1965, . 297).Hillfelt hatwhile he ime equenceofevents, ausepreceding ffect, ight otbe difficult oestablishn many ases, it certainly eeds o be remem-bered, articularly ith elective actors t work n ndus-try" p. 298). Clearly, emporal uccessions a given orHill.

Experiment. n this ategory illplaced he ccasional"natural xperiment" hat ives trong vidence or aus-ation.He had in mind he effect f preventative ctionstaken oreduce he ncidencef he isease. o they ork?If a person tops moking oes he lowerhis risk f ungcancer? ill clearly iews uch experiments" nthe amewayMillviewed heproduction f an effect y rtificiallyintroducinghepresumed ausal gent-strong ausal vi-dencewhenyou an find t.

BiologicalGradient. By this Hill referred o evidencethat howed n ncreasing isease ate s exposure o theagent nquestionntensified. oth xperiment ndbiolog-icalgradient ay e viewed s emphasizinghe ausalna-ture f heproposed ausal gent, s discussed nthepre-vious ection.

Plausibility, oherence, nalogy. I havegrouped hesethree ogether ecause hey llrefer othe rior nowledgethat he pidemiologist ould eed o consider. s the us-pected ausation iologicallylausible?s t oherent n hesense f not being eriouslyn conflict ith nown acts?Is it nalogous o known ausal elations or imilar gentsand diseases? hesefactors, lthough mportant n somecases, ll reflect he tate f relevant cientific nowledgeand do not directly ranslatento spects f the model fSection . In particular illfelt hat twasunwise oplaceundue mphasis n these becauseof the relatively oorstate f relevant iological nowledgen many ases ofinterest.

Although ill felt hat he ix factors isted bove wereimportant rom ime otime, hey were he ix east ig-nificant actors n his ist. He felt hat he three mostimportant actors re the trength, onsistency,nd peci-ficity f the association n question.

Strength. his sHill'sfirst actor- First pon my istI would ut he trength f the ssociation"p. 295). Thismaybe viewed s simple cceptance f Mill's method fconcomitantariationnpractical erms r of he cientificutility f the prima acie ausaleffect. lthough here sno guarantee or his, t s often more ikely hat largerprima acie ausal effect ill hold up when controlledstudy s performed hanwill smaller rima acie ausaleffect. relevant esult n this regard s the nequalitygiven n Cornfield t al. (1959)that ounds he nfluenceofunmeasured actors n the elative isk a form fprimafacie ausal ffect).

Consistency. Hill's second ignificant actor oncernsthe eneralityf he ssociation cross opulations f nits.This might e viewed s a weakened orm f constantconjunction. t he ery east, nassociationhat spresentin one population nd absent nanother uggests ariablecausal ffects. think hat heres clear ias gainst allingvariable ausal ffects causal"by cientists, ven houghthosewho must ealwithheterogeneous nits, uchashumans, ill enerally gree hat t susually oo much oexpect onstant ffects nthe ealworld.

Specificity. ill's third actor efers ospecific auseshaving pecific ffects.If .. the association s limited ospecificworkers nd to particular itesand types of disease and there s no association between he work andother modes of dying, hen learly hat s a strong rgument n favor fcausation. p. 297)

I think hat pecificitys related o the believabilityfthe ndependence ssumption. he ack of an associationbetween he xposure f person o particular ork laceand the causes of that erson's eath upports he nde-pendence ssumptionn relevant ay but oes not rove


8/13/2019 W4 Holland 1986

8/9


my pinion. or a similar iew ee Kempthorne1978).Oneexpects hat he pplicationfRubin'smodelwill elpclarify hemeaning f complex ausal models nd theirpath diagrams.

9. SUMMARYThis rticle as covered variety f topics hat nvolve

causation, ut here re a few eneral oints hat, think,are mportant nough o emphasizensummary.First f ll, believe t s very elpful otry oseewhat

experimentsas the term s used by statisticians) ellusabout ausation. have mphasizedhree deas bout au-sation nwhich tatisticalxperiments ocus ur ttention.

1. The analysis f causation hould eginwith tudyingthe ffects f causesrather han he raditional pproachof trying odefine hat he auseof given ffect s.

2. Effects f causes re always elative o other auses(i.e., it takes wo auses odefine n effect).

3.Not verythinganbe a cause; n particular, ttributesofunits re never auses.

Let me make a few brief omments n each of theseimportant deas.

Traditional nalyses f ausation tart y ooking or hecauseof n effect. think hat ooking or auses f ffectsisa worthwhilecientificndeavor, ut t s not he properperspectiven a theoretical nalysis f causation.More-over, would hold that he "cause" of a given ffect salways ubject orevision s our knowledgebout he he-nomenon ncreases. or example, o bacteria ause dis-ease? Well, yes .. until we dig deeper and find hat t sthe oxins he bacteria roduce hat eally ause the dis-ease;and this s really ot t either. ertain hemical e-actions are the real causes . . . and so on, ad infinitum.The effect f a causemay edifficult omeasure nsomecircumstances,ut t s, at least,precisely efinable-asdone n Section . It is for his eason hat believe hatformal heories f ausationmust eginwith he ffects fgiven auses ather han ice versa.

That an effect equires wocausesfor ts definition sobvious n the ontext f an experiment utnever eemsto getmuch ecognitiony hosewhodiscuss ausationngeneral erms. his sprobably n mportant ontributionof tatistical hinking odiscussionsf causation. xperi-ments ithout ontrol

omparisonsre simply ot xper-iments. hosewho hink n terms f physical cience x-periments ayhave some difficulty ith his dea,butbelieve hat t strue f ny xperiment.

That verything as a cause ssometimesalled he awof ausality, ut t doesnot mply hat verythingan be acause. The experimental odel eliminatesmany hingsfrom eing auses, nd this sprobably ery ood, ince tgivesmore pecificityothe meaning f the word ause.Donald Rubin nd once madeup the motto

NO CAUSATIONWITHOUTMANIPULATIONto emphasize he mportance f his estriction. lthoughmany eoplebalk t the dea that auses might e imitedin omeway, his dea s a simple onsequence f he truc-

ture f he model nSection .Unless oth ,(u) and Y,(u)can be defined, n principle, t s mpossibleodefine hecausal effect ,(u) - Yj(u). For an attribute (u) we candefine Ya(u) for all u for which A(u) = a, and we candefine Yb(u) for ll u for whichA(u) = b. Attributes refunctions, owever, ndA(u) is either or b (or neither)but not both and b for ny unit, u. Hence Ya(u) - Yb(u)cannot e defined or nyunit, , and attributes re notcauses n the ensethat ausal effects annot e definedfor hem.

The second et of mportant eneral oints wish osummarize oncern he mmediate onsequencesf Rub-in'smodel.There re two onsequenceswish o empha-size.

1.The difference etween he model S, Yt, Y,) andtheprocess f observationS, Ys).

2.TheFundamental roblem f Causal nference-onlyY,or Y, butnotboth an be observed n any nit .

Thesetwo onsequences re really he ame hing aidin different ays. t s a great mistake o confuse ,or Y,with s, ndyet his s done ll the ime. t s also mistaketo conclude rom he Fundamental roblem f Causal n-ference hat ausal nference s impossible.What s im-possible s causal nference ithout making ntested s-sumptions.his oesnot ender ausalnferencempossible,but t does give t an air of uncertainty. t is the sameuncertainty iscussed y Hume. The strength f a modellikeRubin's s that t llows s to make hese ssumptionsmore xplicit han hey sually re. When hey re explic-itly tated he nalyst an then egin o ook for ways oevaluate r to partially est hem.

ACKNOWLEDGMENTSI first earned bout he ausal model nSection from

theperson consider ts riginator, onaldRubin.Don'swork n this rea salways source f nspiration orme.Lindsey hurchill ead an early raft f this rticle ndmadenumerous uggestionshathave mproved nd fo-cused othmy hinking nd he rticle n ubstantial ays.PaulRosenbaum as,very enerously, ivenme he ene-fit f his nsight nto ausal nference n many ccasions.Ben King ncouragedme to put the deas n this rticletogether s a GeneralMethodology ecture or he 1985

meetingsf

heASA.My ther olleaguestETS-HenryBraun, onald Rock, Dorothy hayer, ndHowardWai-

ner-are always source f ntelligencendkeen riticism.Lynne teinberg, s an ETS postdoctoral ellow uring1984-1985,pentmany ours xplainingomehow aus-ation works n experimental sychology. inally, athyFairall'sgoodnature nd many kills nsured hetimelyproduction f the manuscript or he 1985meeting f theASA.

[Received ctober 985. Revised anuary986.]

REFERENCESBarnard, . A. (1982), Causation," n Encyclopedia f Statisticalci-

ences Vol. 1), eds. S. Kotz, N. Johnson, nd C. Read, New York:JohnWiley, p. 387-389.


8/13/2019 W4 Holland 1986

9/9


Blalock,H. M., Jr. ed.) (1971), CausalModels n the ocial Sciences,Chicago: ldine-Atherton.

Bunge,M. (1959),Causalitynd Modern cience 3rd d.), New York:DoverPublications.

Cochran,W. G. (1983), lanning ndAnalysis f Observationaltudies,NewYork:JohnWiley.

Cook,R. D. (1980), Smoking nd Lung Cancer," n R. A. Fisher: nAppreciation,ds. S. Fienberg ndD. Hinkley, ewYork: pringer-Verlag.

Cornfield, .,Haenszel,W.,Hammond, . C., Lilienfeld, . M., Shimkin,M.B., andWynder, . L. (1959), Smoking ndLung ancer: ecentEvidence nd Discussion f ome Questions," ournalfthe ationalCancernstitute, 2,173-203.

Cox,D. R. (1958),The lanning f xperiments, ewYork:JohnWiley.Doll, R., andHill,B. (1950), Smokingnd Carcinoma f theLung,"

British edical ournal, , September 0,739-748.(1952), A Study f the Aetiology f Carcinoma f he Lung,"

British edical ournal, , December 3, 1272-1286.(1956), LungCancer nd Other auses fDeath n Relation o

Smoking," ritish edical ournal, , November 0,1071-1081.Duncan, . D. (1975),ntroductiono tructural quationModels,New

York:Academic ress.Evans,A. S. (1978),CausationndDisease:AChronologicalourney,"

American ournal fEpidemiology,08, 49-258.Fisher, . A. (1926), The Arrangementf ield xperiments," ournal

ofMinistry fAgriculture,3,503-513.-- (1957), Letter o the ditor," ritish edical ournal, , July ,43.

Florens, . P., andMouchart, . (1985), A Linear heory orNoncau-sality," conometrica,3,157-175.

Goldberger, . S., andDuncan, . D. (1973), tructuralquation odelsin the ocial ciences, ewYork: eminar ress.

Granger, . W. J. 1969), Investigating ausal Relations y Econo-metric odels ndCross-Spectral ethods," conometrica, 7,424-438.

(1980), Testing or ausality: Personal iewpoint," ournalofEconomic ynamics nd Control, , 329-352.

Hamilton, . A. (1979), Choosing Parameter or x 2Table r2 x2 x 2TableAnalysis," merican ournal fEpidemiology,09, 62-375.

Hill,A. B. (1965), TheEnvironmentndDisease:Associationr Cau-sation," roceedingsf heRoyal ociety f Medicine,8,295-300.

Holland, .W., ndRubin, . B.(1980, Causal nferencenProspectiveand Retrospective tudies," ddress iven t the Jerome ornfieldMemorial ession of the American tatistical ssociation nnualMeeting, ugust.

(1983), On Lord'sParadox," n Principals f Modern sycho-logicalMeasurement,ds. H. Wainer nd S. Messick, illsdale, J:Lawrence rlbaum.

Hume, . (1740),A Treatise n HumanNature.(1748),An nquiry oncerning umanUnderstanding.

Kempthorne, . (1952),TheDesign ndAnalysis f Experiments, ewYork:JohnWiley.

(1978), Logical, pistemologicalndStatistical spects f Na-ture-Nurture ata Interpretation," iometrics,4,1-24.

Locke,J. 1690),AnEssayConcerning umanUnderstanding,ook I,Chapter XVI.

McCurdy, R. (1957), "Letter to the Editor," BritishMedicalJournal, ,July 0.

Mill,J. S. (1843),A System f Logic.Neyman, J. (with Iwaszkiewicz, K., and Kolodziejczyk, S.) (1935),

"Statistical Problems in Agricultural xperimentation" with dis-cussion), upplement f Journal f the RoyalStatistical ociety, ,107-180.

Powers, D. E., and Swinton, S. S. (1984), "Effects f Self-Study orCoachableTest tem Types," Journal f EducationalMeasurement, 6,266-278.

Rosenbaum, P. R. (1984a), "From Association to Causation n Obser-vational Studies: The Role of Tests of Strongly gnorable TreatmentAssignment," ournal f heAmerican tatistical ssociation,9, 41-48.

(1984b), "The Consequences of Adjustment or ConcomitantVariable That Has Been Affected y the Treatment," Journal f theRoyal Statistical ociety, er. A, 147,656-666.

(1984c),"Conditional ermutation ests nd the Propensity corein Observationaltudies," ournal f heAmerican tatistical ssoci-ation, 79, 565-574.

Rosenbaum, P. R., and Rubin, D. B. (1983a), "The Central Role of thePropensity core in Observational tudies for Causal Effects," Bio-metrika, 0, 41-55.

(1983b), "Assessing Sensitivity o an Unobserved Binary Co-variate n an Observational tudy With Binary Outcome," Journal f

theRoyal tatistical ociety, er. B, 45, 212-218.(1984a), Discussion of "On the Nature and Discoveryof Struc-ture," byJ. W. Pratt nd R. Schlaifer, ournal ftheAmerican tatisticalAssociation, 9, 26-28.

(1984b), "Reducing Bias in Observational Studies Using Sub-classification n the Propensity core," Journal f the American ta-tistical ssociation,9,516-524.

(1985a),"Constructing Control roupUsingMultivariate atchedSamplingMethodsThat ncorporate he Propensity core," TheAmer-ican tatistician,9,33-38.

(1985b), "The Bias Due to Incomplete Matching," Biometrics,41, 103-116.

Rubin, D. B. (1974), "Estimating ausal Effects f Treatments n Ran-domized and Nonrandomized tudies," Journal f Educational Psy-chology, 6, 688-701.

(1977), "Assignment f Treatment Group on the Basis of a Co-variate," ournal fEducationaltatistics, , 1-26.

(1978),"Bayesian nference or Causal Effects: he Role of Ran-domization," TheAnnals of Statistics, , 34-58.(1980),Discussion of "Randomization Analysis f Experimental

Data: The Fisher Randomization Test," by D. Basu, Journal f theAmerican tatistical ssociation,5,591-593.

Saris, W., and Stronkhorst, . (1984), Causal Modelling in Non-experimental Research, Amsterdam: Sociometric Research Foun-dation.

Smith,R. Jeffrey 1980), "Government aysCancer Rate Is Increasing,"Science,227, 998-1002.

Suppes, P. C. (1970),A Probabilistic heory f Causality, Amsterdam:North-Holland.

Yerushalmy, J., and Palmer, C. E. (1959), "On the Methodology fInvestigations f Etiologic Factors n Chronic Diseases," Journal fChronic iseases, 0,27-40.

This content downloaded from 14 7 8 31 43 on Mon 19 Aug 201 3 03:19:41 AM
http://www.jstor.org/page/info/about/policies/terms.jsp

w4 holland 1986

Documents