Abstract
The literature on causality is wide-ranging, difficult at times, and controversial.1 Rather than delving into all of the nuances of this material, this chapter and the next have a much more modest goal. Through the analysis of examples, these chapters aim to sensitize the reader to various conceptions of causality discussed by social and statistical scientists. With this overview in mind, the reader of this book will be better able to critique assertions about causal effects in the subsequent chapters and in other research reports. By reading the cited material, the reader can gain a more detailed understanding of causality.
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Notes
- 1.
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Cox and Wermuth have referred to the categories of their three-category classification using different terms. Their 2004 article defines zero-, first-, and second-level causality. Their 2001 article defines causality as stable association, causality as the effect of an intervention, and causality as explanation of a process. Their 1996 book distinguishes these categories as follows (page 220):
There are essentially three interrelated senses in which causality arises in the contexts considered in this book:
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As a statistical dependence, which cannot be removed by alternative acceptable explanatory variables;
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As the inferred consequences of some intervention in the system;
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As the above, augmented by some understanding of a process or mechanism accounting for what is observed.
All these notions are valuable; we favor restricting the word to the final and most stringent form.
They emphasize (1996, 219) that causality rarely can be inferred from the results of one study, especially observational studies.
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- 3.
Morgan and Winship (2007, 286–287) refer to the first aspect of assessment of stable association as Mode 1: Associational Analysis. It is a prerequisite to causal analysis in that there is no correlation without association. They distinguish this mode from their Mode 2: Conditional Association Analysis, which researchers apply after establishing that an association exists in order to eliminate obvious sources of spuriousness. They note that conditioning variables may be determinants of the response variable y and not causes of the key explanatory variable x (such variables are on equal footing with x) or they may be causes of both x and y (such variables are prior to x and to y). The controls for variables on equal footing with x purify the nonspurious effect of x on y; the controls for variables prior to both x and y test directly for spurious association.
- 4.
Cox (1992, 292–293) refers to this operation as causality via association (i.e., zero-level causality) and formalizes testing for spuriousness as follows (1992, 292–293); this logic applies to the procedures for assessing causality of Suppes and Granger discussed later in this chapter:
Let C and E be binary events and B be a third variable or collection of variables. We may say that C is a candidate cause of E if C and E are positively associated, i.e. if
$$ {P(E|C) > P(E|not\,C).} $$We can regard the cause as spurious if B explains the association in that
$$ {P(E|C\,\,and\,B) = P(E|not\,\,C,\,\,and\,\,B),} $$i.e. if E and C are conditionally independent given B. The cause is confirmed if C is a candidate cause that is not spurious, i.e. which cannot be so explained via any B.
In contrast, Pearl (personal communication, 5/2/2010) defines spurious correlation in terms of his do(x) operator as follows:
C is spuriously related to E if you find correlation without causation:
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C and E are dependent, and
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P(e|do(c)) = P(e) for all c
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P(c|do(e)) = P(c) for all e
For dichotomous C and E: P(e|do(C = 1) = P(e|do(C = 0) = P(e) and P(c|do(E = 1) = P(c|do(E = 0) = P(c). The probabilities in each set of tables are equal and equal the marginal probability.
Then, by implication C causes E if you find causation (Pearl 2002, 208):
1. P(e|do(c)) ≠ P(e) for some values c and e
For dichotomous C and E: P(e|do(C = 1) ≠ P(e) or P(e|do(C = 0) ≠ P(e). A probability in each set of tables does not equal the marginal probability.
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- 5.
Lazarsfeld ([1946] 1972) used panel data to form 16-fold tables that enabled him to ascertain the mutual effects of x and y: whether the effect of x on y is larger than the effect of y on x. Coleman (1964, 162–188) developed causal models for such two-attribute systems for over time data; his computer program provided parameter estimates. Goodman ([1973] 1978, 173–229) developed causal models based on loglinear models for such data.
- 6.
Susser (1973, 111–135) clarifies classic epidemiological approaches for assessing spuriousness and elaborating analyses. Greenland, Pearl, and Robins (1999a) show how investigators can apply graphical methods to determine which variables in a system are sufficient when controlled so as to remove bias from the estimate of the causal effect of exposure on a disease. Greenland, Robins, and Pearl (1999b) provide an overview of confounding based on a counterfactual model of causation. Advancing these introductory expositions, Robins and his colleagues apply highly abstract, state-of-the-art Bayesian networks and graphical methods to contemporary epidemiological problems. Greenland (2004, 6–8) discusses Robins’s g-estimation approach; for a list of Robins’s publications on this topic, see his website: http://www.biostat.harvard.edu/~robins/research.html
- 7.
Let Y i be the realized empirical outcome for individual i, Y i (1) be the potential outcome if individual i would receive the experimental treatment, Y i (0) be the potential outcome if individual i would receive the control treatment, and W i be an indicator such that W i = 1 implies that the individual would receive the experimental treatment and W i = 0 implies that the individual would receive the control treatment. Then (Imbens and Wooldridge 2009, 9): Y i = Y i (W i ) = Y i (0)(1 − W i ) + Y i (1)(W i ). When W i = 1, then Y i = Y i (1). When W i = 0, then Y i = Y i (0). A specific manipulation W i would make one of the potential outcomes (Y i (0), Y i (1)) the realized outcome Y i .
- 8.
Cox highlights limitations to the applications of Rubin’s causal model, distinguishing treatments from intrinsic properties of individuals under study, and from nonspecific factors such as countries and organizations (1992, 296):
In this discussion, only those variables which in the context in question can conceptually be manipulated are eligible to represent causes, i.e. it must make sense, always in the context in question, that for any individual the causal variable might have been different from the value actually taken. Thus in most situations gender is not a causal variable but rather an intrinsic property of the individual. The study of sex-change operations and of possible discriminatory employment practices would be exceptions. Again, the passage of time as such is not a causal variable.
- 9.
Note that R 1 − R 0 = Δ implies that R 1 = R 0 + Δ; the model assumes unit-treatment additivity, as does Coleman’s model for attributes. That is, “the response that would be observed under C = 1 differs by a constant Δ from the response that would be observed on that same unit were it to receive C = 0. … Δ [is] the causal effect of changing C from 0 to 1” (Cox 1992, 295).
- 10.
Imbens and Wooldridge (2009, 10–11) discuss five advantages that the potential outcomes approach (POA) has over the analysis of realized outcomes. They essentially state that the POA: (1) allows the definition of causal effects before specifying the assignment mechanism, and without making functional form or distributional assumptions; (2) links the analysis of causal effects to explicit manipulations; (3) separates the modeling of the potential outcomes from that of the assignment mechanism; (4) allows formulation of probabilistic assumptions in terms of potentially observable variables, rather than in terms of unobserved components; and (5) clarifies where the uncertainty in the estimators comes from.
- 11.
Harding (2003) established that people who are exposed to different levels of neighborhood poverty between the ages of 11 and 20 have different rates of dropping out of school and, for females, teenage pregnancies. Being exposed to poverty as a teenager can be conceptualized as a contextual attribute of the person. With this conceptualization, Harding’s study exemplifies a causal analysis of the effects of an attribute.
- 12.
Explication of overlaps between Coleman’s and Rubin’s models of causality appeared in the author’s unpublished paper (Smith 1999a). Portions of this present explication appear in Smith (2003a, 346–353; 2006, xi-xxviii) and are used here, respectively, with the permission of Kluwer Academic Publishers and AldineTransaction, who graciously allow their authors to reuse their work in their own publications. These earlier expositions and Chapter 9 conceptualize the introduction of a new computer system as a treatment.
- 13.
Greenland (2004, 4) cites this passage from David Hume that is perhaps the original statement of the potential outcomes notion of causality; it bears on Rubin’s and Coleman’s models:
“We may define a cause to be an object, followed by another,…if the first object had not been, the second had never existed.”
Here, if the person has the attribute “working in a target office” then that person has a high probability of being against computerized fraud detection. However, if that person had the attribute “working in a control office,” then that person would have a lower probability of being against computerized fraud detection. All things being equal, the causal effect of the attribute “working in a target office” is the difference between these probabilities.
- 14.
Coleman’s model is thus clearly in the tradition of Kurt Lewin; the movement depends on relative sizes of these forces (Lewin [1951] 1997, 136). Coleman was very knowledgeable about Lewin’s notions of psychological forces: He had written an unpublished paper with Allen Barton that explicated the concept of cohesiveness as developed by Festinger et al. (1950).
- 15.
Heckman’s (2005, 12) simplest statement of the potential outcomes model is that: “the individual-level treatment effect for person w comparing outcomes from treatment s with outcomes from treatment s’ is Y(s, w) − Y(s′, w), s ≠ s′. Initially, Rubin restricted the treatments to be manipulated variables, whereas Coleman allows the treatments to be different categories of an attribute.
- 16.
Temporal stability asserts the constancy of the response over time (Holland 1986, 948).
- 17.
The assumption of unit homogeneity implies that Y t (u 1) = Y t (u 2) = Y t (u n) and Y c(u 1) = Y c(u 2) = Y c(u 3) (Holland 1986, 948). In Coleman’s model, this means that each person in one treatment cell is characterized by the same causal force. The assumption of constant effect implies that the effect of t on every unit is the same: T = Y t (u) − Y c(u) for all u in U (Holland 1986, 949). For the treatment cell, unit-treatment additivity implies that the treatment t adds a constant amount T to the level of the control response for each unit, as in Coleman’s model.
- 18.
Causal transience asserts that there is a “washout effect,” the person’s response to the treatment c at an earlier time does not affect the person’s response to the treatment t at a later time.
- 19.
SUTVA implies: “that the value of Y for unit u when exposed to treatment t will be the same no matter what mechanism is used to assign treatment t to unit u and no matter what treatment the other units receive.” (Rubin 1986, 961). If in fact there are two treatments being applied to some units in the treatment group, or if there is leakage of the treatment t to the units in the control group, then SUTVA is violated.
- 20.
Hellevik (2009) offers a spirited defense of the use of the OLS linear regression model rather than a logistic regression model when the response is a dichotomy. However, he does not consider the transformation of odds ratios into probabilities p using the inverse link function; briefly p = odds/(1 + odds).
- 21.
The propensity scores standardized by their overall mean can be included as a covariate that would appear in the intercept term along with the other covariates, see Chapter 12 for an example.
- 22.
To avoid these problems, prior to obtaining the final predicted proportions the data can be made linear by applying Goldberger’s method; see Achen (1986, 41) for instructions. Chapter 12 also provides an example; it calculates propensity scores using ordinary least squares after using Goldberger’s method and then compares these scores to those obtained from logistic regression. After conducting an OLS regression, a first step is to change a unit’s predicted value p greater than 1 to.99 and p less than 0 to.01. For each unit set, q = 1–p and s = (pq)½. Then, for each variable on a unit divide each variable’s score by that unit’s s. In the subsequent “no intercept” regression, the intercept is set to 1/s. Then, apply OLS to the transformed data to get the “Goldbergerized” regression coefficients and their standard errors. Logistic models that do not converge may do so when applied to linearized data if Firth’s (1993, 1995) penalized likelihood method is applied; also see Chapter 12 for an example. The OLS and logistic estimates correlated.999.
- 23.
Given an equation similar to (289), Coleman developed this strategy for obtaining the effect parameter from a logistic regression. He asks (1981, p. 31): “What amount is added to p when x 1 = 1 if, when x 1 = 0, p is 0.5?” He defines this transformation αk of β k such that 0.5 + αk = 1/(1 + e−(0 + βk). By solving for α k , Coleman’s transformation is obtained:
$$ {{\alpha_k} = \left( {{{\hbox{e}}^{\beta k}} - { 1}} \right)/{2}\left( {{{\hbox{e}}^{{ }\beta \,\rm{k}} + {1}}} \right)} $$When β k = 1.7771 is substituted in the expression above, αk = 4.9127/13.8254 = 0.36, which is about equal to the average treatment effect Δ = 0.38.
- 24.
Coleman (1964, 145–149) applies a variant of his causal model to experimental data in which one group receives the treatment and the other group a null treatment. The measure of the causal effect is similar to those in this chapter.
- 25.
Gelman and Hill (2007, 186–194) strongly advise against controlling for post-experimental-treatment variables that intervene between the randomized treatment and the response because this weakens the causal (i.e., potential outcomes) interpretation of the effect of the treatment. Regarding observational studies, they state:
Researchers often already know to control for many predictors. So it is possible that these predictors will mitigate some of the problems we have discussed. On the other hand, studying intermediate outcomes involves two ignorability problems to deal with rather than just one, making it all the more challenging to obtain trustworthy results.
Ignorability implies that conditional on the confounding covariates used in a regression, the distribution of units across treatment conditions is, in essence, random. That is, the distribution of the potential outcomes is the same across levels of the treatment variable, when the confounding covariates are controlled (Gelman and Hill 2007, 182–184).
- 26.
For a recent discussion of the general form of additive linear models, see Moreno and Martinez (2008, 597–604).
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Acknowledgements
The author derived Causal Models for Attributes, a section of this chapter, from portions of his earlier writings: Introduction to James S. Coleman's Mathematics of collective action (vii–li, © Aldine Transaction 2006) and Inferential causal models (Quality & Quantity, vol. 37, issue 4, 337–361; © Kluwer2003). The author thanks these publishers for their permission to reuse this material here.
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Smith, R.B. (2011). Stable Association and Potential Outcomes. In: Multilevel Modeling of Social Problems. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-9855-9_3
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