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...all units, u. Hence (13) E(Y, I S = t) = T + E(Yc I S = t), (14) so TPF = T + {E(Yc I s = t) - E(Yc I s = c)}. (15) The term in braces in (15) is not 0 in general, that is, if the independence assumption is not true. It is easy to show that the stronger assumption of unit homogeneity does imply equality between T and TPF· 4.5 Causal Inference In Nonrandomized Observational Studies It is beyond the scope of this article to apply the model for causal inference to nonrandomized studies. This has been done extensively, and the reader is referred to Rubin (1974, 1977, 1978), Rosenbaum (1984a,b,c), Rosenbaum and Rubin (1983a,b, 1984a,b, 198Sa,b ), and Holland and Rubin (1980, 1983). An important emphasis in these papers is on the ways that pre-exposure variables can be used to replace the independence assumption with less stringent conditional independence assumptions that are useful in observational studies. Rosenbaum and Rubin referred to one such assumption as "strong ignorability." 5. COMMENTS ON SELECTED PHILOSOPHERS So much has been written about causality by philosophers that it is impossible to give an adequate coverage of the ideas that they have expressed in a short article. This section views some of ...

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..., if specificity exists we may be able to draw conclusions without hesitation; if it is not apparent, we are not thereby necessarily left sitting on the fence. (p. 297) Of course, specificity does not guarantee that the independence assumption is valid, but it does not directly contradict this assumption in the way that a lack of specificity does. 8.2 Granger Causation In Economics The primary source of data that is available to economists is so-called "time series" data in which measurements of a variable or set of variables are made repeatedly on an economic entity over time. For such data, Granger (1969) developed a particular notion of causality that some economists have found useful in their analyses. In my opinion, however, Granger's essential ideas involving causation do not require the time-series setting he adopted. I will try to restate his theory in terms of the types of models used in Sections 2 and 3-that is, variables defined on a population of units. Granger formulated his theory around the idea of prediction-a "cause" ought to improve our ability to predict an effect in a probabilistic system. In Granger's theory a variable causes another variable; that is, the values of one vari...

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...cause', and that which is produced, 'effect'." Although it is evident that these definitions refer to the same kinds of things that concern the model in Section 3, they do little more than suggest that the model is not out of line with an ancient philosophical tradition. It should be noted, however, that Aristotle emphasized the causes of a thing rather than the effects of causes. Locke seems a little more even-handed. Bunge (1959) gave a very accessible discussion of the history of many ideas about the essential meaning of causation. 5.1 Hume When we tum to the analysis of causation given by Hume (1740, 1748) we find a critical basis for examining Rubin's model. Hume's analysis of causality is generally regarded to be an important contribution to the literature of this subject. Hume emphasized that causation is a relation between experiences rather than one between facts. He argued that it is not empirically verifiable that the cause produces the effect, but only that the experienced event called the cause is invariably followed by the experienced event called the effect. Hume's empirical stance can be regarded as sympathetic with the classical statistical view that the role of statistics i...

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...imentation is such a powerful scientific and statistical tool and one that often introduces clarity into discussions of specific cases of causation, I unabashedly draw on the language and framework of experiments for the model for causal inference. It is not that I believe an experiment is the only proper setting for discussing causality, but I do feel that an experiment is the simplest such setting. The purpose is to construct a model that is complex enough to allow us to formalize basic intuitions concerning cause and effect. The point of departure is the analysis of causal effects given in Rubin (1974, 1977, 1978, 1980). It will be sufficient for our purposes, however, to deal with a simplified, population-level version of Rubin's model. This simplified model was used in Holland and Rubin (1980) to analyze causal inference in retrospective, case-control studies used in medical research and in Holland and Rubin (1983) to analyze Lord's "analysis of covariance" paradox. I refer to this as "Rubin's model" even though Rubin would argue that the ideas behind the model have been around since Fisher. I think that Rubin (1974) was the place where these ideas were first applied to the study of caus...

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...e inoculant is in fairly exact terms. In a sense, Postulate II is a way of minimizing measurement error in the treatment (t) that is exposed to the units. Medicine is more difficult when the biological theory is less well developed. As an example I now consider several suggestions made by Sir Austin Bradford Hill to those who might wish to separate association from causation in the study of the environment and disease. He had spent a lifetime in public health and was among the first to a:rgue, quantitatively, for the causal link between smoking and lung cancer (Doll and Hill1950, 1952, 1956). Hill (1965) named nine factors that he felt were useful in such work for deciding that the most likely interpretation of an observed association is causation. I will consider these in an order that differs from Hill's. Temporality. "Which is the cart and which the horse?" (Hill1965, p. 297). Hill felt that while the time sequence of events, cause preceding effect, might not be difficult to establish in many cases, "it certainly needs to be remembered, particularly with selective factors at work in industry" (p. 298). Clearly, temporal succession is a given for Hill. Experiment. In this category Hill plac...

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... it into the notation of Section 3, the units are the plots, u;i, and X;i(k) = Yk(u;i), where Yk(u) is defined as in the previous discussion of Kempthorne. Neyman also had an explicit expression for the average value of X;i(k) over all of the units, uii· It is X..(k). In the notation of Section 3 this is X..(k) = E(Yk)· Hence it is clear that by the time Neyman was writing the idea of multiple versions of the response, one for each treatment, was established. It seems to have been used by writers concerned about the details of the effects of randomization in specific experimental plans (e.g., Cox 1958; Kempthorne 1952) but is generally not a part of the standard statistical notation of many other writers [an exception is Hamilton (1979)]. The Neyman (1935) reference is also relevant to the model in Section 3 because of the controversy between Fisher and Neyman that it engendered. The controversy revolves around the choice of null hypothesis in experiments such as randomized block designs. Fisher was quite clear that the null hypothesis that he proposed is that the causal effect (as we have defined it) is 0 for each unit. For example, in the famous discussion at the end of Neyman (1935) Fis...

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..., in my opinion, to get to the heart of the notion of causality in the same way that Suppes's theory of causality fails. Granger's "causes" are always only temporarily in that category. If an analyst simply gathers more information, that is, changes Z, an X that was once a Granger cause of Y might be shown to be only a spurious cause in exactly the same spirit as in Suppes's theory. 8.3 Causal Models In Social Science No discussion of causal inference would be complete without some reference to the expanding literature on causal modeling, that is, Blalock (1971), Goldberger and Duncan (1973), Duncan (1975), and Saris and Stronkhorst (1984). Little work has been done to relate Rubin's model to those used in the causal modeling literature-an exception is Rosenbaum (1984b), in which the average causal effect in a population is related to coefficients that arise in certajn linear path models. The relationship between these two types of models is a natural research topic, since both causal models and Rubin's model were developed to deal with the same problem-causal inference in nonexperimental research. In this section I will hint at some possible points of contact between the path diagrams that are...

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...voted to a brief examination of the writings of a few statisticians to see in what way the idea of multiple versions of the response, that is, Y1 and Yo has appeared before. I find that many people have difficulty with the idea of distinguishing Y 1 and Yc from Y or Ys and perhaps this look at earlier work may help clarify this assumption. Unfortunately, the exact idea is never stated explicitly, so there is a need for a certain amount of detective work to find it. I hope I will not be held guilty of wrongly reinterpreting the work of others. A fairly clear statement of this idea was given by Kempthorne (1952) in a discussion of the analysis of randomized block designs. (A randomized block design is a typical agricultural experimental plan in which larger tracts of land, called blocks, are each subdivided into p plots and then one of the experimental treatments is applied at random to each of the p plots within each block.) For example, Kempthorne (1952, p. 136) first defined yields as follows: "We shall denote the yield with treatment k ... on plot j ... of block i ... by Yijk·" He then wrote: In fact we do not observe the yield of treatment k on plot j but merely the yield of treatment k on a ran...

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...consistency, and specificity of the association in question. Strength. This is Hill's first factor-"First upon my list I would put the strength of the association" (p. 295). This may be viewed as simple acceptance of Mill's method of concomitant variation in practical terms or of the scientific utility of the prima facie causal effect. Although there is no guarantee for this, it is often more likely that a larger prima facie causal effect will hold up when a controlled study is performed than will a smaller prima facie causal effect. A relevant result in this regard is the inequality given in Cornfield et al. (1959) that bounds the influence of unmeasured factors on the relative risk (a form of prima facie causal effect). Consistency. Hill's second significant factor concerns the generality of the association across populations of units. This might be viewed as a weakened form of constant conjunction. At the very least, an association that is present in one population and absent in another suggests variable causal effects. I think that there is a clear bias against calling variable causal effects "causal" by scientists, even though those who must deal with heterogeneous units, such as humans, will genera...

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...ually received it would not still have given the higher yield? (p. 504) It is fairly clear in this quotation that he could consider the possibility that had a different treatment (i.e., no manure) been applied to the field the resulting yield might have been the same. This clearly concerns the null hypothesis of no treatment effect and, more generally, Fisher came closest to the idea of multiple versions of the response in his discussions of the relationship between the null hypothesis and randomization. The earliest explicit reference that I have found to multiple versions of the response is Neyman (1935). In his paper (read before the Industrial and Agricultural Re954 search Section of the Royal Statistical Society in March of 1935) Neyman gave an explicit statement of the idea of multiple versions of the response (which is for Neyman the yield from an experimental plot of land in an agricultural experiment). Unfortunately, Neyman's discussion also introduced the notion of a stochastic element that is added to Y to allow for "technical errors" that are due to inaccuracies of experimental technique. If we ignore this problem of measurement error and assume zero "technical errors," then Neyman'...

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... appear to be primarily technical rather than conceptual. He did not reject the idea of variable causal effects, however, and discussed ways in which causal effects might depend "on the value of some supplementary measurement that can be made on each unit" (p. 18). Curiously, R. A. Fisher, who founded the modern theory of experimental design, never dealt directly with the idea of multiple versions of the response. Instead, he gave examples that are so laced with specific details that it is not always clear what level of generality he meant to convey. For example, in the first article in which Fisher (1926) attempted to set out the principles of the design of field experiments in agriculture we find this question in a discussion of a hypothetical experiment to evaluate the apparent productive value of treating a given acre of ground with a manurial treatment: What reason is there to think that, even if no manure had been applied, the acre which actually received it would not still have given the higher yield? (p. 504) It is fairly clear in this quotation that he could consider the possibility that had a different treatment (i.e., no manure) been applied to the field the resulting yield might hav...

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...ect to Rubin's model, it fails, in my opinion, to get to the heart of the notion of causality in the same way that Suppes's theory of causality fails. Granger's "causes" are always only temporarily in that category. If an analyst simply gathers more information, that is, changes Z, an X that was once a Granger cause of Y might be shown to be only a spurious cause in exactly the same spirit as in Suppes's theory. 8.3 Causal Models In Social Science No discussion of causal inference would be complete without some reference to the expanding literature on causal modeling, that is, Blalock (1971), Goldberger and Duncan (1973), Duncan (1975), and Saris and Stronkhorst (1984). Little work has been done to relate Rubin's model to those used in the causal modeling literature-an exception is Rosenbaum (1984b), in which the average causal effect in a population is related to coefficients that arise in certajn linear path models. The relationship between these two types of models is a natural research topic, since both causal models and Rubin's model were developed to deal with the same problem-causal inference in nonexperimental research. In this section I will hint at some possible points of contact between the path di...

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...ssible, in this article I take the position that causes are only those things that could, in principle, be treatments in experiments. The qualification "in principle" is important because practical, ethical, and other considerations might make some experiments infeasible, that is, limit us to contemplating hypothetical experiments. For example, in the medical and social world we might be able to conceive of an experiment, but no one would ever try to carry it out. Instead, we might have to wait for a "natural experiment" to occur. "Observational study" is the term used by statisticians (e.g., Cochran 1983) to refer to studies for which "The objective is to study the causal effects of certain agents" but "For one reason or another the investigator can not . . . impose on . . . or withhold from the subject, a treatment whose effects he desires to discover" (p. 1). I believe that the notion of cause that operates in an experiment and in an observational study is the same. The difference is in the degree of control an experimenter has over the phenomena under investigation compared with that which an observer has. In Rubin's model this is expressed by the joint distribution of S with Y, and Yc. Tot...

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... Y,(u) = Yc(u) + T for all units, u. Hence (13) E(Y, I S = t) = T + E(Yc I S = t), (14) so TPF = T + {E(Yc I s = t) - E(Yc I s = c)}. (15) The term in braces in (15) is not 0 in general, that is, if the independence assumption is not true. It is easy to show that the stronger assumption of unit homogeneity does imply equality between T and TPF· 4.5 Causal Inference In Nonrandomized Observational Studies It is beyond the scope of this article to apply the model for causal inference to nonrandomized studies. This has been done extensively, and the reader is referred to Rubin (1974, 1977, 1978), Rosenbaum (1984a,b,c), Rosenbaum and Rubin (1983a,b, 1984a,b, 198Sa,b ), and Holland and Rubin (1980, 1983). An important emphasis in these papers is on the ways that pre-exposure variables can be used to replace the independence assumption with less stringent conditional independence assumptions that are useful in observational studies. Rosenbaum and Rubin referred to one such assumption as "strong ignorability." 5. COMMENTS ON SELECTED PHILOSOPHERS So much has been written about causality by philosophers that it is impossible to give an adequate coverage of the ideas that they have expressed in a short art...

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...use in an experiment, because the notion of potential exposability does not apply to it. The only way for an attribute to change its value is for the unit to change in some way and no longer be the same unit. Statements of "causation" that involve attributes as "causes" are always statements of association between the values of an attribute and a response variable across the units in a population. In (A) all that is meant is that the performance of women on the exam exceeds, in some sense, that of men. Examples of the confusion between attributes and causes fill the social science literature. Saris and Stronkhorst (1984) gave the following example of a causal hypothesis: "Scholastic achievement affects the choice of secondary school" (p. 13). These authors clearly intended for this hypothesis to state that an attribute of a student (i.e., scores on tests, performance in primary school) can cause (i.e., affect) the student's choice of a particular type of secondary school. It is difficult to conceive of how scholastic achievement could be a treatment in an experiment and, therefore, be a "cause" in the sense used in this article. A somewhat stronger statement of my point was given by Kempthorne (1978, p. 15): ...

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...nt of specific bacteria as the cause of specific infectious diseases. Yerushalmy and Palmer (1959) described the situation in the following terms: Almost from the very beginning, when bacteria were first found to cause disease, bacteriologists felt the need for a set of rules to act as guideposts in investigation of ba~teria as possible causal agents in disease. (p. 28) These two authors described three postulates formulated by the great bacteriologist, Robert Koch, who discovered, among other things, the tuberculosis bacillus in 1882. Koch's postulates [also called the Koch-Henle postulates, Evans (1978)] are simple, no-nonsense criteria for deciding when a microscopic organism is implicated in a disease. According to Yerushalmy and Palmer (1959), "while there is no single formulation of Koch's postulates-they can be stated as consisting essentially of the following: I. The organism must be found in all cases of the disease in question. 956 II. It must be isolated from patients and grown in pure culture. III. When the pure culture is inoculated into susceptible animals or man, it must reproduce the disease." (p. 30) Rubin's model applies rather clearly to Postulates I and III. Postulate I is ...

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...the material-these are the R causes, t' and c'. We might then be able to ascertain whether the students did or did not study the material-these are the S causes, t and c. The response variable is the score Y on the test given subsequent to these events. Diagram (43) indicates that encouragement can affect studying and possibly the test scores and that studying can affect the scores. For example, one might hypothesize that encouragement really does not affect test scores directly. This would be expressed in the model by (44) for all u in U and s = tor c. For more on "encouragement designs" see Powers and Swinton (1984). The essential point I wish to make about these diagrams is that they are easily interpreted in terms of Rubin's model when they are not causally meaningless. The causal model literature has not been careful in separating meaningful and meaningless causal statements and path diagrams, in Holland: Statistics and Causal Inference my opinion. For a similar view see Kempthorne (1978}. One expects that the application of Rubin's model will help clarify the meaning of complex causal models and their path diagrams. 9. SUMMARY This article has covered a variety of topics that involve causation, but t...

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...ose values are determined prior to, at, or after a given point in time. I will simply adopt these temporal distinctions in the definitions of the variables that arise. Granger (1969, p. 430) clearly accepted the idea of temporal succession in his analysis: "In the author's opinion there is little use in the practice of attempting to discuss causality without introducing time." It is the past values of a variable that cause, in Granger's sense, the future values of another variable. Although Granger originally formulated his theory in terms of one variable causing another, later writers (e.g., Florens and Mouchart 1985) restated it in terms of noncausality and I will follow that approach. In reformulating his theory I will also shift from his emphasis on a particular type of predictor, that is, "the optimum, unbiased, leastsquares predictor" (p. 428), to the more generally applicable notion of conditional statistical independence. This means that instead of limiting attention to the inability of a specific predictor to predict the values of a variable, I will use the stronger condition that no predictor can predict the desired values. Although this is a stronger type of noncausality than Granger defined I do...

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...of the entire debate was given by Cook (1980). 8. COMMENTS ON CAUSAL INFERENCES IN VARIOUS DISCIPLINES This section will briefly consider discussions of causation in three disciplines-medicine, economics, and "causal modeling." In each case an attempt will be made to relate the discussion to Rubin's model for causal inference, but no attempt is made to be exhaustive or even representative in the selection of topics considered. 8.1 Causation and Medicine We begin with a simple, yet basic, example from medicine-the establishment of specific bacteria as the cause of specific infectious diseases. Yerushalmy and Palmer (1959) described the situation in the following terms: Almost from the very beginning, when bacteria were first found to cause disease, bacteriologists felt the need for a set of rules to act as guideposts in investigation of ba~teria as possible causal agents in disease. (p. 28) These two authors described three postulates formulated by the great bacteriologist, Robert Koch, who discovered, among other things, the tuberculosis bacillus in 1882. Koch's postulates [also called the Koch-Henle postulates, Evans (1978)] are simple, no-nonsense criteria for deciding when a microscopic organism is implica...

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...e ideas in the context of Rubin's model for causal inference given in Sections 3 and 4. It makes no attempt to be exhaustive or even representative. Aristotle distinguished four "causes" of a thing in his Physics: The material cause (that out of which the thing is made), the formal cause (that into which the thing is made), the efficient cause (that which makes the thing), and the 950 final cause (that for which the thing is made). It is his notion of efficient cause that is relevant to our discussion and to most discussions of causation that grow out of inquiries into the methods of science. Locke (1690) proposed these definitions: "That which produces any simple or complex idea, we denote by the general name 'cause', and that which is produced, 'effect'." Although it is evident that these definitions refer to the same kinds of things that concern the model in Section 3, they do little more than suggest that the model is not out of line with an ancient philosophical tradition. It should be noted, however, that Aristotle emphasized the causes of a thing rather than the effects of causes. Locke seems a little more even-handed. Bunge (1959) gave a very accessible discussion of the history of man...

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..." The response to Fisher's criticism can also be viewed as attempting to show that smoking should be thought of in causal terms rather than as indicative of a genetic attribute of subjects. For example, among his responses to Fisher, McCurdy (1957) pointed out that lung cancer rates increase with the amou,nt of smoking and that subjects who stopped smoking had lower lung cancer rates than those who did not. Both of these arguments can be viewed as emphasizing the causal aspects of smoking-one can do more or less of it and one might stop doing it. A discussion of the entire debate was given by Cook (1980). 8. COMMENTS ON CAUSAL INFERENCES IN VARIOUS DISCIPLINES This section will briefly consider discussions of causation in three disciplines-medicine, economics, and "causal modeling." In each case an attempt will be made to relate the discussion to Rubin's model for causal inference, but no attempt is made to be exhaustive or even representative in the selection of topics considered. 8.1 Causation and Medicine We begin with a simple, yet basic, example from medicine-the establishment of specific bacteria as the cause of specific infectious diseases. Yerushalmy and Palmer (1959) described the si...

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...elected philosophers, medical researchers, statisticians, econometricians, and proponents of causal modeling. KEY WORDS: Causal model; Philosophy; Association; Experiments; Mill's methods; Causal effect; Koch's postulates; Hill's nine factors; Granger causality; Path diagrams; Probabilistic causality. 1. INTRODUCTION The reaction of many statisticians when confronted with the possibility that their profession might contribute to a discussion of causation is immediately to deny that there is any such possibility. "That correlation is not causation is perhaps the first thing that must be said" (Barnard 1982, p. 387). Possibly this evasive action is in response to all of those needling little headlines that pop up in the most unexpected places, for example, "If the statistics cannot relate cause and effect, they can certainly add to the rhetoric" (Smith 1980, p. 998). One need only recall that a well-designed randomized experiment can be a powerful aid in investigating causal relations to question the need for such a defensive posture by statisticians. Randomized experiments have transformed many branches of science, and the early proponents of such studies were the same statisticians who founded...

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...l other circumstances is modeled by considering the same unit. Thus Y1(u) - Yc(u) = 1, so the causal effect of the circumstance on the unit is 1 and corresponds to Mill's statement that the circumstance is "the cause or an indispensable part of the cause of the phenomenon." Mill also considered reversing the process to look for causes of given effects. This is a well-known scientific technique-for example, it occurs often in epidemiological studies of public health problems. It is beyond the scope of this article to apply the model to such a case, but some work along this line can be found in Hamilton (1979) and Holland and Rubin (1980). The Method of Residues. This method also applies fairly simply to the model. Its statement is Subduct from any phenomenon such part as is known by previous inductions to be the effect of certain antecedents, and the residue of the phenomenon is the effect of the remaining antecedents. (p. 460) To place this into the context of the model let the antecedents (i.e., causes) be denoted by a = those whose effect is known and b = the remaining antecedents. The causal effect of ab relative to a is simply Yab(u) - Ya( u ), which is the residue Mill tells us to compute. I...

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...nd framework of experiments for the model for causal inference. It is not that I believe an experiment is the only proper setting for discussing causality, but I do feel that an experiment is the simplest such setting. The purpose is to construct a model that is complex enough to allow us to formalize basic intuitions concerning cause and effect. The point of departure is the analysis of causal effects given in Rubin (1974, 1977, 1978, 1980). It will be sufficient for our purposes, however, to deal with a simplified, population-level version of Rubin's model. This simplified model was used in Holland and Rubin (1980) to analyze causal inference in retrospective, case-control studies used in medical research and in Holland and Rubin (1983) to analyze Lord's "analysis of covariance" paradox. I refer to this as "Rubin's model" even though Rubin would argue that the ideas behind the model have been around since Fisher. I think that Rubin (1974) was the place where these ideas were first applied to the study of causation. This model also begins with a population of units, U. Units in the model for causal inference are the objects of study on which causes or treatments may act. The terms cause and treatment wil...

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...lity; Path diagrams; Probabilistic causality. 1. INTRODUCTION The reaction of many statisticians when confronted with the possibility that their profession might contribute to a discussion of causation is immediately to deny that there is any such possibility. "That correlation is not causation is perhaps the first thing that must be said" (Barnard 1982, p. 387). Possibly this evasive action is in response to all of those needling little headlines that pop up in the most unexpected places, for example, "If the statistics cannot relate cause and effect, they can certainly add to the rhetoric" (Smith 1980, p. 998). One need only recall that a well-designed randomized experiment can be a powerful aid in investigating causal relations to question the need for such a defensive posture by statisticians. Randomized experiments have transformed many branches of science, and the early proponents of such studies were the same statisticians who founded the modern era of our field. This article takes the view that statistics has a great deal to say about certain problems of causal inference and ought to play a more significant role in philosophical analyses of causation than it has heretofore. In additi...