Some Statistical Aspects of Causality

European Sociological Review,Vol. 17 No. 1, 65^74 65

D.R. Coxand NannyWermuth

A general review of approaches to causality is given from a statistical perspective. Three broad notions are distinguished. In the ¢nal part of the paper the challenges of reaching potentially causal representations are outlined for a study of some German political and social attitudes.

Generalities There is a long history in the philosophical literature Symmetricand Directed Relations. Association is a sym- of discussions of causality.It typically regards a cause metric relation between two or possibly more as necessary and su¤cient for an e¡ect: all children features, but causality is asymmetric.That is, if C is of divorced parents have behavioural problems and associated with R then R is associated with C,butif all children with behavioural problems have C is a cause of R then R is not a cause of C. Thus, divorced parents. In virtually all sociological con- given any two features C and R, we need to distin- texts, however, the concern is with multiple causes, guish the possibilities where even if one is predominant.Thus explicit or implicit 1 C and R are to be treated on an equal footing and statistical considerations are inescapable to evaluate dealt with symmetrically in any interpretation; empirical evidence. Even within the statistical view 2 One of the variables, say C, is to be regarded as of causality there are a number of di¡erent formula- explanatory to the other variable, R, regarded as tions. Here we review three. In the ¢nal section of a response. Then, if there is a relation, it is the paper a speci¢c illustration is sketched. Refer- regarded asymmetrically. ences to work quoted by authors only are in The distinction here is not about statistical signi¢- Holland %1986). cance but rather is concerned with substantive interpretation.

Notions of Causality GraphicalRepresentation. Auseful graphical represen- tation shows two variables X1 and X2, regarded on Causality as Stable Association an equal footing, if associated, as connected by an undirected edge, whereas two variables such that C Suppose that there is clear evidence that two features is explanatory of R, if connected, are done so by a of the individuals %people, communities, house- directed edge: see Figures 1a and 1b. holds, etc.) under investigation are associated, a There are two possible bases for the distinction possible cause C and a possible response R.For between explanatory and response variables and instance, individuals with high values of C may thus for using a directed edge. One is that features tend to have high values of R and vice versa. Thus referring to an earlier time-point are explanatory to C and R might be test scores for individuals at a features referring to a later time-point.Thus aspects given age in arithmetic and language or they may of previous education may be possible explanatory be the unemployment rate and level of crime in a variables for subsequent career performance. In community.What might it mean to conclude that C such situations the relevant time is not the time is a cause of a response R? when the observation is made but the time to

& Oxford University Press 2001 66 D.R.COX AND NANNY WERMUTH

Figure 1. "a)Undirected edge between two variables X1, X2 on an equal footing "b)Directed edge between explanatory variable Cand response variable R "c)General dependence ofresponse Ron B, C "d) Specialsituation with R ?? C j B "e)Specialsituation with B ?? C corresponding inparticular to randomization of C

which the features refer although, of course, obser- are typicallyassessed empiricallyby some form of re- vations recorded retrospectively are especially gression analysis. In such a situation one would not subject to recall biases. The second is a subject- regard C as a cause of R even though in an analysis matter working hypothesis based for example on without the background variable B there is a statisti- theory or on empirical data from other kinds of cal dependence between C and R. investigation. Suppose that cross-sectional data are This discussion leads to one de¢nition used in the collected on current salary and on attitudes to var- literature of C being a cause of R, namely that there ious social and political issues. It might then be is a dependence between C and R and that the sign reasonable provisionally to regard income as expla- of that dependence is unaltered whatever variables natory of attitudes, of course only as part of a more B1,B2, ... themselves explanatory to C are consid- complex network of relationships. ered simultaneously with C as possible sources of In summary, the ¢rst step towards causality is to dependence. This de¢nition has a long history but require good reasons for regarding C as explanatory is best articulated by I. J. Good and P. Suppes. A of R as a response and that any notion of causal con- corresponding notion for time series is due to N. nection between C and R is that C is a cause of R and Wiener and C.W.Granger.This de¢nition underlies not the other way round. much empirical statistical analysis in so far as it aims to achieve causal explanation. CommonExplanatoryVariables. Next consider the pos- The de¢nition entertains all possible alternative sibility of one or more common explanatory explanatory variables. In any particular study one variables. For this, suppose that a background vari- can atbest checkthatthe measured background variable B is potentially explanatory of C and hence also ablesB do notaccountfor the dependencebetweenC of R.There are a number of possibilities of which the and R.The possibility that the dependence could be most general is shown in Figure 1c with directed explained by variables explanatory to C that have not edges from B to C, from C to R and also directly been measured, i.e. by so-called unobserved con- from B to R. On the other hand, if the relation were founders, is less likely the larger the apparent e¡ect that represented schematically in Figure 1d,theonly and can be discounted only by general plausibility dependencebetweenC andR isthatinducedbytheir arguments about the ¢eld in question. Sensitivity both depending on B.ThenC and R are said to be analysis may be helpful as it involves calculating conditionally independent given B,sometimescon- what the properties of an unobserved confounder veniently written R ?? C j B.There is no direct path would have to be to explain away the dependence in from C to R that does not pass via B. Such relations question.When the empirical dependence found is SOME STATISTICAL ASPECTS OF CAUSALITY 67

Figure 2. "a)Intermediate variable Iaccounting foroverall e¡ectof Cafter ignoring I; R ?? C j I "b)Associated variables C,C* on an equalfooting and both explanatory to R very strong it may be possible to argue that it is C would not be enough in assessing whether such a implausible that there is an unmeasured variable path exists. If R is independent of C given an inter- itself with a strong enough dependence to explain mediate variable I, but dependent on I,thenC may away completely the observed e¡ect. For further still have caused I and I may be a cause of R. details see Rosenbaum %1995). Forinstance suppose thatC representsan aspectof primary school education and I some feature of sec- RoleofRandomization. Althoughphysical randomiza- ondary education, which in turn a¡ects some aspect tion by the investigator is rarely possible in ofemployment; see Figure 2a.Doestheaspectof pri- sociological investigations, it is conceptually impor- mary education cause a change in R?IfR is tant to consider brie£y its consequences for conditionally independent of C given I it would be interpretation. In this case in the scheme sketched reasonable tosay that primaryeducation does cause a in Figure 1e there can be no edge between the Bsand change in R and that this change appears to be C, since such dependence would be contrary to ran- explained via what happens in secondary education. domization,i.e.to each individualunder studybeing equallylikely toreceive each treatment possibility.In Explanatory Variables on an Equal Footing. An even this situation an apparent dependence between C more delicate situation arises with variables CÃ on and R cannot be explained by a background variable an equal footing with the variable C whose causal as in Figure 1d. It is in this sense that causality can be status is under consideration; see Figure 2b.Ifthe inferred from randomized experiments and not role of C is essentially the same whether or not CÃ from observational studies as sometimes stated, is conditioned, i.e. whether or not CÃ is included in especially in the statistical literature. While, other the regression equation, there is no problem, at least things being equal, randomized experiments are at a qualitative level. On the other hand it is relatively greatly tobe preferred to observationalstudies, di¤- common to ¢nd clear dependence on C; CÃ as a culties of interpretation, sometimes serious ones, pair, but that either variable on its own is su¤cient remain.The most important are possible interactive to explain the dependence. There are then broadly e¡ects, especially with unobserved explanatory vari- three routes to interpretation: ables, and unanticipated future interventions in the 1 To regard C; CÃ collectively as the possibly system under study that remain unnoticed when the causal variables; ¢nal response is recorded. 2 To present at least two possibilities for interpretation, one based on C and one on CÃ; IntermediateVariables. Inthe above discussionthevari- 3 To obtain further information clarifying the ables B have been supposed to be explanatory of C relation between C and CÃ, establishing for and hence of R. For judging a possible causal e¡ect instance that CÃ is explanatory of C and that of C it would be wrong to consider in the same way the appropriate interpretation is to ¢x CÃ when variables intermediate between C and R, i.e. vari- analysing variations in R. ables I that are responses to C and explanatory of R. For example, suppose that C and CÃ are respec- Although theyare valuable in clarifying the nature of tively measures of educational performance in any indirect path between C and R, the use of I as an arithmetic and language of a child both measured explanatory variable in a regression analysis of R on at the same age and that the response is some adult 68 D.R.COX AND NANNY WERMUTH

feature.Then the third possibility is inapplicable; the value of R is a so-called counterfactual whose intro- ¢rst possibility is to regard the variables as a two- duction is, however, useful to capture the notion dimensional measure of educational performance hinted at above of a deeper meaning of causality. and to abandon, at least temporarily, any notion of separating the role of arithmetic and language. By Formalizing Di¡erences in Counterfactuals. The simplest taking the second route we would recognize two and leastdemanding relation between the two values alternative simple explanations consistent with the of R is that over some population of individuals data, one based on arithmetic and one on language. under study the average of R exceeds that of This is typically studied via the careful examination pres Rabs.This is a notion of an average e¡ect and is testa- of empirical regression equations, for example for ble empirically in favourable circumstances. A much binary R via logistic regression. stronger requirement is that the required inequality holds for every individual in the population of con- Causality as the E¡ect of Intervention cern. Stronger still is the requirement that the di¡erence between the two values of R is the same Counterfactuals. The concept of causality discussed for all individuals, i.e. that for all individuals above is important and is connected with the approach adopted in many empirical statistical Rpres À Rabs D: studies. It does not, however, directly capture a stronger interpretation of the word causal. This is In the language of the theoryof the design of experi- connected with the idea of hypothetical interven- ments this is called the assumption of unit-treatment tion or modi¢cation. Suppose for simplicity of additivity. exposition that C takes just two possible forms, to Now these last two assumptions are clearly not be called presence and absence. Thus presence directly testable and can be objected to on that might be the implementation of some programme account.The assumptions are indirectly testable, to of intervention, and absence a suitable control a limited extent at least. If the individuals are divided state. For monotone relations one may say that the into groups, for example on the basis of one or more presence of C causes an increase in the response R of the background variables B, the assumptions if an individual with C present tends tohave a higher imply that for each individual observed that the dif- R than that same individual would have had if C had ference between the two levels of R hasthe same sign been absent, other things being equal. in the ¢rst case and the second case. However, over- Slightly more explicitly let B denote all variables looking the fact that a possiblycausal variable C has a possibly explanatory to C and suppose that there are very di¡erent e¡ect on di¡erent individuals can have no variables CÃ to be considered on an equal footing severe consequences. to C. Consider for each individual two possible values of R, Rpres and Rabs, that would arise as C IntrinsicVariables or Attributes. There is an important takes on its two possible values present and absent restriction implicit in this discussion. It has to be and B is ¢xed. Then the presence of C causes, say, meaningful in the context in question to suppose an increase in R if Rpres is in some sense systemati- that a potential cause C for an individual might cally greater than Rabs. The notion of other things have been di¡erent from how it in fact is.This is re- being equal is captured by holding B ¢xed. levant only to variables that appear solely as This notion is in part a translation of J. Neyman's explanatory variables. For example they may be vari- and R. A. Fisher'sworkon the design of experiments ables measured at some base-line, i.e. at entry into a into a more general setting, including an observa- study. Purely explanatory variables can be divided tional one. It is connected with the work of H. A. into intrinsic variables or attributes, sometimes also Simon and has been systematically studied and very called structural variables, which are essentially de- fruitfully applied by D. B. Rubin. ¢ning characteristics of the individual, and potential For a given individual only one of Rpres and Rabs explanatory variables which might play the role of C can be observed, corresponding to the value of C in the present discussion. Intrinsic variables should actually holding for that individual. The other not be regarded as potentially causal in the present SOME STATISTICAL ASPECTS OF CAUSALITY 69 sense. For example the gender of an individual is in that such generating processes are not merely most contexts an intrinsic characteristic. The ques- hypothesized. Causality is not to be established by tion what would R have been for this woman had merely calling a statistical model causal. she been a man other things being held ¢xed is in Explanations of phenomena in terms of under- many, although not quite all, contexts meaningless. lying processes are inevitably provisional. Nevertheless they are the cornerstone of the natural Variables to be Held Fixed. Finally, care is essential in sciences. We suggest that statistical analysis should de¢ning what is to be held ¢xed under hypothetical aim towards establishing processes that are poten- changes in C. In terms of statistical analysis this is the tially causal. issue of what other variables should be included as explanatory variables in the regression equation for R in addition to C itself. Certainly responses I to C Special Issues are not ¢xed.Variables, B, explanatory of C are held Interaction Involving a Potentially Causal ¢xed.There is an essential ambiguity for variables CÃ Variable on an equal footing with C.Todistinguish changing C to a given level with certain other features held We now turn to the issue of interactions with a ¢xed from the probabilistic notion of conditioning potentiallycausal variable.The graphical representa- Pearl %1995) has introduced the terminology ofsetting tions used above to show the structure of various C to its required level; see also Pearl %2000) and kinds of dependency and independency holding Lauritzen %2000) for further discussion. between a set of variables have the limitation, in the form used here, that they do not represent interaction, in particular that an e¡ect of C may be Causality as Explanation of a Process systematically di¡erent for di¡erent groups of indi- There is a third notion of causality that is in some viduals. For example if B is an intrinsic feature such ways most in line with normal scienti¢c usage. In as gender, we consider whether the e¡ect of C is dif- this context causality implies that there is some ferent for men and for women. In particular, if the understanding, albeit provisional, of the process e¡ects of C are in opposite directions for di¡erent that leads from C to R.This understanding typically levels of B we say there is a qualitative interaction, a comes from theory, or often from knowledge at a possibility of special importance for interpretation. hierarchical level lower than the data under immedi- Note especially that evenwhen C represents a ran- ate analysis. Sometimes, it may be possible to domized treatment which is automatically represent such a process by a graph without directed decoupled from preceding possibly unobserved cycles and tovisualize the causal e¡ect by the tracing variables B the possibility of serious interactions of paths from C to R via variables I intermediate with B cannot in general be ignored. between C and R. Thus the e¡ect of interventions Viewed slightly di¡erently, absence of interaction at a community level might be related to ideas of is important not only in simplifying interpretation individual psychology. but also in enhancing generalizability and speci¢- This last notion of causality as concerned with city.That is, an e¡ect that has been shown to have generating processes is to be contrasted with the sec- no serious interaction with a range of potential vari- ond viewof causalityas concerned with the e¡ects of ables is more likely to be reproduced in some new intervention and with the ¢rst view of causality as situation and more likely to have a stable subject- stable statistical dependence. These views are com- matter interpretation. plementary not con£icting. Goldthorpe %this volume) has argued for this third view of causality Unwanted Unobserved Intermediate Variable as the appropriate one for sociology with explanation via rational choice theory as an important Consider further the role of variables I referring to route for interpretation. time points after the implementation of C.Asub- To be satisfactory there needs to be evidence, ject-matter distinction can be drawn between, on typically arising from studies of di¡erent kinds, the one hand, intermediate variables that are 70 D.R.COX AND NANNY WERMUTH

responses to C and that are explanatory to R and are essential, especially but not only in observational part of some natural process, and, on the other hand, studies.The most widely quoted conditions tending interventions into the system that may depend on C to make a causal interpretation more likely are those and which may be explanatory of R but which in of Hill %1965) put forward in connection with the some sense are unwanted or inappropriate for inter- interpretation of epidemiological studies. Hill pretation. Thus in studying the e¡ect of emphasized their tentative character. For a critical modi¢cations in inner-city housing policies on discussion of these conditions, see Rothman and satisfaction it may be necessary to take account of Greenland %1998). Because they are usually men- interventions other than those which are the tioned in an epidemiological context we reproduce immediate object of study. Another example is in them in outline in a slightly revised version %Cox evaluations of study programmes whenever stu- and Wermuth, 1996; sect. 8.7). dents in only one of the programmes receive According to these conditions a dependency is unplanned intensive encouragement during the eva- more likely to be causal luation period. ö if an a priori subject-matter explanation of it is available; ö if a convincing subject-matter explanation is Aggregation found retrospectively although such is typically So far little has been said about the choice of obser- less convincing than an a priori explanation; vational units for study. At a fundamental research ö if the e¡ect is a large one, it then being less level it may be wise to choose individuals showing likely that there is an alternative explanation the e¡ects of interest in their simplest and most strik- via an unmeasured confounder; ing form. More generally, however, the choice has to ö if the dependencyhas a natural monotonic rela- be considered at two levels. There is the level at tion with levels of the explanatory variable in which ultimate interpretation and action is required question; and the level at which careful observation of largely ö if the e¡ect is found repeatedly in independent decoupled individuals is available. For example, a studies especially if these are of somewhat dif- criminologist comparing di¡erent sentencing or ferent form; policing policies is interested in individual o¡enders ö if there is no major interaction with intrinsic but may be able to observe only di¡erent commu- features; nities or policing areas. A nutritional epidemio- ö if the dependence is the consequence of a mas- logist comparing di¡erent diets is interested in sive intervention in the system. individual people but may have to rely, in part at The bacteriologist Koch gave conditions for least, on consumption and mortality data from inferring causality when the potential cause can be whole countries.The assumption that a dependence applied, withdrawn, and reapplied in a relatively established on an aggregate scale, for example at a controlled way and the pattern of response country level, has a similar interpretation at a observed. Similar ideas were used in psychology small-scale level, for example for individual persons, following early experiments by Pavlov with cond- involves the assumption that there are no confoun- itioning stimuli and the extinction of responses after ders B at the person level that would account for the withdrawing the stimulus. apparent dependency. This will typically be very hard or even impossible to check awith any accuracy from country-level data. A Sociological Case Study

Causal formulations are common in social science Bradford Hill's Conditions contexts whenever empirical research is planned to study explanations for the development of opinions, The above discussion implicitly emphasizes that, attitudes, and judgements or behaviour. In these while causal understanding is the aim of perhaps cases those statistical models and analyses are espe- nearly all research work, a cautious approach is cially helpful which permit the exclusion of prior SOME STATISTICAL ASPECTS OF CAUSALITY 71 expectations on a developmental process since they context variables on the far right to the response provide interpretations that are compatible with variable of primary interest on the far left: attitude causal explanations.We give an example of deriving towards state interventions, Y, a sum score formed a graphical Markov model that concerns the devel- from two questions with high values denoting opment of attitudes towards interventions by the agreement with interventions. Relations among state. In the case of increasing unemployment context variables are known from many studies; should the state intervene? For the general well- they are taken to be ¢xed in the present analysis, i.e. being of society should the state provide a social they are not analysed here, they are conditioned on. safety net? Two context variables are taken as quantitative mea- The data used are from the German General surements: age, W, and education of parents, V, Social Surveys %Central Archive for Empirical Social %0=none , 1=one, or 2=both have at least 12 years Research, 1985; 1992). They are cross-sectional sur- of formal schooling); the three other context vari- veys which started in 1980 and are typically carried ables are binary: own education, B, %1=at least 12 out every second year. A special additional survey years of formal schooling), marital status, C, was conducted in 1991 in both East and West Ger- %1=married), and gender, D, %1=female), all taken many. Since questions on attitudes towards state to have values zero and one. interventions and on their possible determinants Risk of exclusion from the workforce, A, %1=yes) were asked only in 1991 and in 1984, i.e. before uni- is a rough indicator constructed from information ¢cation, only West Germans are included in the available in the data on unemployment, on the qua- analyses. And, since we study the risk of exclusion li¢cations necessary to proceed to university,and on from the workforce as one of the possible determi- completed vocational training. It is listed alone in nants the data are restricted to individuals who are one of the middle boxes, indicating that it is an inter- between 18 and 65 years old. mediate variable, regarded both as a possible Figure 3 shows an initial ordering of the variables response to each of the context variables %to its with four boxes indicating four possible stages of right) and as a possible explanatory variable for the development, ranging from given background or remaining variables %to its left). Two attitudes are

Figure 3. An initial ordering ofthe variables ofpossibledeterminantsofattitudestowards interventions by thestate 72 D.R.COX AND NANNY WERMUTH

expected to be of a more stable kind: judgement of and 1727 cases in 1984) appeared essentially own social position, X1, %1=lower class to 5= upper unchanged.The main changes from 1984 to 1991 in class) and judgement on which factors are important univariate distributions are an increase in the per- forgetting ahead in society, X2.There are three sepe- centage of persons with at least 12 years of formal cial aspects of the latter: chance, X21, others, X22, schooling, from 21 to 27 per cent, and a decrease in %wealthy family, general connections %in 1991), pro- the percentage of persons at risk of exclusion from tection %in 1984), political connections, money, and the workforce, from 25 to 20 per cent. opportunism), or oneself, X23, %own education, hard Figure 4 summarizes the analyses conducted sepa- work, initiative, and intelligence). For each of these rately for the two years by showing an arrow for each measurements a high value indicates strong agree- variable which turned out to be an important expla- ment that the aspect is important. natory factor in either year. For instance the Since we do not consider any two of the four jud- contributions of each variable with a missing arrow gements to relate as response and explanatory to the response of primary interest would have been variables they are treated as being on an equal foot- very minor, i.e. corresponding to a t-value smaller in ing, i.e. they are listed in the same box, so that some absolute value than1.5 if included next as an explana- joint response model will be used for their analysis. tory variable in the regresssion. A dashed line Jointly they are intermediate variables, i.e. possibly indicates that a substantial correlation among two explanatory for a respondent's attitude towards state joint responses remained after accounting for all the interventions and possible responses to the risk and e¡ects of variables being important for either one. context variables. None of the considered context variables was expla- The ordering of the boxes provides a plan for the natory for the getting-ahead scales. This can be type of analyses to be carried out: for instance a lin- interpretedpositivelyasthelackofitemandscalebias. ear regression for the attitudes towards state By usingknown relations among the context vari- intervention on all other variables, a logistic regres- ables in addition to the regression results, parts of sion for the risk on all the context variables, and a which are shown inTables1to 3, one of the pathways multivariate regression for the joint responses of development in Figure 4 %Y; X1; A; B; V)canbe given the risk and the context variables. For the pur- interpreted as follows: if a respondent's parents have pose of checking for independences we replace the less formal education it is more likely that his or her multivariate regression of the X-variables by uni- own formal educationwill be shorter.The less formal variate analyses for each response taken in turn education one has the higher is the risk of exclusion with all %or a subset of) the variables to the right as fromtheworkforce. A high riskofexclusion fromthe explanatory variables. workforce makes it more likely that the respondent's Standard checks for non-linear relations and own social position is judged to be low. Persons who interactions did not reveal very strong e¡ects of judge their own social position to be low tend to either type. Univariate distributions for the raw agree strongly with the type of state interventions data and for reduced sets of data with complete considered here. Additional factors for predicting observations on all 11 variables %693 cases in 1991 high agreement with state interventions are the

Figure 4. Achain graph developed as a resultofanalyses SOME STATISTICAL ASPECTS OF CAUSALITY 73

Tabl e 1. Regression coe¤cientsfor responseY, attitudetowards state interventions

Explanatory variable 1984 1991 ......

X1, social position À0.35 À0.46 X2, getting ahead X21, chance 0.11 0.15 X22, others 0.07 0.04 constant 5.99 6.35

Tabl e 2. Regression coe¤cientsfor response X1, judgementofown socialposition

Explanatory variable 1984 1991 ...... A, risk for low social position À0.17 À0.18 B, own education 0.39 0.40 C,married 0.04 0.15 V, education of parents 0.22 0.17 constant 2.68 2.67

Table 3: Regression coe¤cientsfor response A, risk ofexclusionfrom workforce

Explanatory variable 1984 1991 ...... B, own education À1.16 À0.97 C,married À0.31 À0.73 D, gender 0.74 0.23 constant À1.11 À0.86 beliefs that chance or others determine what is tal status has increased over the years.Thus, the type important to get ahead in society. of dependencies are replicated except when political Probably the most remarkable ¢nding of this changes can explain changes in dependencies. study is the strong agreement in qualitative conclu- The interpretation in terms of tracing paths is sions in the years 1984 and 1991. The regression close to the original suggestions of Sewall Wright coe¤cients shown for directly important factors for path analyses in linear systems. Graphical Mar- determining attitudes towards state interventions kov models are one extension of path analysis in %in Table 1) are essentially identical. For the judge- which responses, intermediate and explanatory vari- ment of one's own social position %in Table 2) the ables may be quantitative or categorical variables and general level and the e¡ects of the predictors of in which joint instead of only single responses may risk for exclusion from the workforce and own edu- be modelled. Linear structural equation models are cation are unchanged over the years, while the a di¡erent extension of path analysis.The main dif- importance of parents'education has decreased and ferences from a chain graph model such as that in the e¡ect of marital status is strong only in 1991. Figure 4 are twofold: the way in which categorical Although the overall risk of exclusion from the response variables are modelled, and the inter- workforce has decreased in Germany from 1984 to pretation of parameters. In chain graphs every edge 1991 the direction of dependencies has remained in the graph, missing or present, can be interpreted unchanged for own education, marital status, and as a particular conditional relationship, so that the gender.The risk is higher for less formal schooling, vanishing of an edge means a conditional indepen- for females, and for the unmarried. Only the relative dence statement.The interpretation of parameters in importance of gender has diminished; that of mari- a linear structural equation model may have to be 74 D.R.COX AND NANNY WERMUTH

derived from scratch if it does not coincide with a %1992) ALLBUS Basisumfrage 1991. Codebuch ZA- chain graph model. Nr.1990. Zentralarchiv, Ko« ln. In summary,the analysis sketched here aims to ¢nd Cochran, W. G. %1965) The planning of observational pathways of dependence from the baseline variables studies of human populations %with discussion). through the intermediate variables to the ¢nal Journal of the Royal Statistical Society A 128, 234^265. Cox, D. R., Wermuth, N. %1996) MultivariateDependencies: response variable.While, of course, conclusions from Models, Analysis and Interpretation. Chapman and Hall, asingle observationalstudyare inevitablyprovisional, London. the approach brings us a step closer to the notions of Dawid, A. P. %2000) Causality without counterfactuals causality discussed in the ¢rst part of the paper.The %with discussion). Journal of the American Statistical analysis gives more insight than, for example, only Association 95, 407^448. direct regression analysis of the ¢nal response on all Hill, A. B. %1965) The environment and disease: associa- available potentially explanatory variables. tion or causation. Proceedings of the Royal Society of Medicine, 58, 295^300. Holland, P. %1986) Statistics and causal inference %with discussion). Journal of the American Statistical Association, Some References on Causality 81, 945^970. The extensive and growing literature on statistical Lauritzen, S. L. %2000) Causal inference from graphical aspects of causality is best approached via the discus- models. In Klu« ppelberg, C., Barndor¡-Nielsen, O. E. and Cox, D. R. %eds.) Complex Stochastic sion paper of by Holland %1986); see also Cox and Systems. Chapman and Hall/CRC, London. Wermuth %1996; sect. 8.7). For general issues about Pearl, J. %1995) Causal diagrams for empirical research observationalstudies,seeCochran%1965)andRosen- %with discussion). Biometrika, 82, 669^710. baum %1995). For a philosophical perspective, see Pearl, J. %2000) Causality: Models, Reasoning and Inference. Simon %1972) and Cartwright %1989). For an interven- Cambridge University Press, Cambridge. tionist view, see Rubin %1974) and for a more formal Rosenbaum, P. R. %1995) Observational Studies. Springer, approach still from a social science viewpoint Sobel New York. %1995). For a systematic account based on directed Rothman, K. J., Greenland, S. %1998) %eds.) Modern Epi - acyclic graphs, see Pearl %1995, 2000) and for the gen- demiology, 2nd edn. Raven-Lippincott, Philadelphia. eral connections with graph theory see Lauritzen Rubin, D. B. %1974) Estimating causal e¡ects of treatment %2000). For an approach based on a complete speci¢- in randomized and nonrandomized studies. Journal of Educational Studies, 66, 688^701. cation of all independencies between a set of Simon, H. A. %1972) Causation. In Kruskal, W. H., Tanur variables followed by a computer-generated listing and J. M. %eds.) Encyclopedia of Statistics, i. Free Press, of all directed acyclic graphs consistent with those New York, pp. 35^41. independencies, see Spirtes et al.%1993).Theuseof Sobel, M. E. %1995) Causal inference in the social and beha- counterfactuals is criticized by Dawid %2000). Be vioral sciences. In Arminger, G., Clogg, C. C. and aware that many rather di¡erent interpretations of Sobel, M. E. %eds.) Handbook of Statistical Modeling for causality are involved in these discussions. the Social and Behavioral Sciences. Plenum, New York. Spirtes, P., Glymour, C., and Scheines, R. %1993) Causa- tion, Prediction and Search. Springer, New York. Wermuth, N. %1998) Graphical Markov models. In Kotz, References S., Read, C. and Banks, D. %eds) Encyclopedia of Cartwright, N. %1989) Nature's Capacities and their Statistical Sciences. Wiley, New York: 2nd Update Measurement. Clarendon Press, Oxford. vol., pp. 284^300. Central Archive for Empirical Social Research %Zentra- larchiv fu« r Empirische Sozialforschung) and ZUMA Author's Address %1985) ALLBUS Basisumfrage 1984. Codebuch ZA- Nr.1340. Zentralarchiv, Ko« ln. Professor D. R. Cox, Nu¤eld College, Oxford OX1 1NF, Central Archive for Empirical Social Research %Zentra- U.K. Tel.: +44 %0) 1865 278690; fax: +44 %0) 1865 larchiv fu« r Empirische Sozialforschung) & ZUMA 278621; e-mail: [email protected].