Conceptual model in philosophy of science
In metaphysics, a causal model (or structural causal model) is a conceptual model that describes the causal mechanisms of a system. Several types of causal notation may be used in the development of a causal model. Causal models can improve study designs by providing clear rules for deciding which independent variables need to be included/controlled for.
They can allow some questions to be answered from existing observational data without the need for an interventional study such as a randomized controlled trial. Some interventional studies are inappropriate for ethical or practical reasons, meaning that without a causal model, some hypotheses cannot be tested.
Causal models can help with the question of "external validity" (whether results from one study apply to unstudied populations). Causal models can allow data from multiple studies to be merged (in certain circumstances) to answer questions that cannot be answered by any individual data set.
Causal models have found applications in signal processing, epidemiology and machine learning.
 ;templatestyles src="Template:TOC limit/styles.css" / ;
Definition.
 ;templatestyles src="Template:Blockquote/styles.css" / ;Causal models are mathematical models representing causal relationships within an individual system or population. They facilitate inferences about causal relationships from statistical data. They can teach us a good deal about the epistemology of causation, and about the relationship between causation and probability. They have also been applied to topics of interest to philosophers, such as the logic of counterfactuals, decision theory, and the analysis of actual causation.
—  Judea Pearl defines a causal model as an ordered triple formula_1, where U is a set of exogenous variables whose values are determined by factors outside the model; V is a set of endogenous variables whose values are determined by factors within the model; and E is a set of structural equations that express the value of each endogenous variable as a function of the values of the other variables in U and V.
History.
Aristotle defined a taxonomy of causality, including material, formal, efficient and final causes. Hume rejected Aristotle's taxonomy in favor of counterfactuals. At one point, he denied that objects have "powers" that make one a cause and another an effect. Later he adopted "if the first object had not been, the second had never existed" ("but-for" causation).
In the late 19th century, the discipline of statistics began to form. After a years-long effort to identify causal rules for domains such as biological inheritance, Galton introduced the concept of mean regression (epitomized by the sophomore slump in sports) which later led him to the non-causal concept of correlation.
As a positivist, Pearson expunged the notion of causality from much of science as an unprovable special case of association and introduced the correlation coefficient as the metric of association. He wrote, "Force as a cause of motion is exactly the same as a tree god as a cause of growth" and that causation was only a "fetish among the inscrutable arcana of modern science". Pearson founded "Biometrika" and the Biometrics Lab at University College London, which became the world leader in statistics.
In 1908 Hardy and Weinberg solved the problem of trait stability that had led Galton to abandon causality, by resurrecting Mendelian inheritance.
In 1921 Wright's path analysis became the theoretical ancestor of causal modeling and causal graphs. He developed this approach while attempting to untangle the relative impacts of heredity, development and environment on guinea pig coat patterns. He backed up his then-heretical claims by showing how such analyses could explain the relationship between guinea pig birth weight, "in utero" time and litter size. Opposition to these ideas by prominent statisticians led them to be ignored for the following 40 years (except among animal breeders). Instead scientists relied on correlations, partly at the behest of Wright's critic (and leading statistician), Fisher. One exception was Burks, a student who in 1926 was the first to apply path diagrams to represent a mediating influence ("mediator") and to assert that holding a mediator constant induces errors. She may have invented path diagrams independently.
In 1923, Neyman introduced the concept of a potential outcome, but his paper was not translated from Polish to English until 1990.
In 1958 Cox warned that controlling for a variable Z is valid only if it is highly unlikely to be affected by independent variables.
In the 1960s, Duncan, Blalock, Goldberger and others rediscovered path analysis. While reading Blalock's work on path diagrams, Duncan remembered a lecture by Ogburn twenty years earlier that mentioned a paper by Wright that in turn mentioned Burks.
Sociologists originally called causal models structural equation modeling, but once it became a rote method, it lost its utility, leading some practitioners to reject any relationship to causality. Economists adopted the algebraic part of path analysis, calling it simultaneous equation modeling. However, economists still avoided attributing causal meaning to their equations.
Sixty years after his first paper, Wright published a piece that recapitulated it, following Karlin et al.'s critique, which objected that it handled only linear relationships and that robust, model-free presentations of data were more revealing.
In 1973 Lewis advocated replacing correlation with but-for causality (counterfactuals). He referred to humans' ability to envision alternative worlds in which a cause did or not occur, and in which an effect appeared only following its cause. In 1974 Rubin introduced the notion of "potential outcomes" as a language for asking causal questions.
In 1983 Cartwright proposed that any factor that is "causally relevant" to an effect be conditioned on, moving beyond simple probability as the only guide.
In 1986 Baron and Kenny introduced principles for detecting and evaluating mediation in a system of linear equations. As of 2014 their paper was the 33rd most-cited of all time. That year Greenland and Robins introduced the "exchangeability" approach to handling confounding by considering a counterfactual. They proposed assessing what would have happened to the treatment group if they had not received the treatment and comparing that outcome to that of the control group. If they matched, confounding was said to be absent.
Ladder of causation.
Pearl's causal metamodel involves a three-level abstraction he calls the ladder of causation. The lowest level, Association (seeing/observing), entails the sensing of regularities or patterns in the input data, expressed as correlations. The middle level, Intervention (doing), predicts the effects of deliberate actions, expressed as causal relationships. The highest level, Counterfactuals (imagining), involves constructing a theory of (part of) the world that explains why specific actions have specific effects and what happens in the absence of such actions.
Association.
One object is associated with another if observing one changes the probability of observing the other. Example: shoppers who buy toothpaste are more likely to also buy dental floss. Mathematically:
formula_2
or the probability of (purchasing) floss given (the purchase of) toothpaste. Associations can also be measured via computing the correlation of the two events. Associations have no causal implications. One event could cause the other, the reverse could be true, or both events could be caused by some third event (unhappy hygienist shames shopper into treating their mouth better ).
Intervention.
This level asserts specific causal relationships between events. Causality is assessed by experimentally performing some action that affects one of the events. Example: after doubling the price of toothpaste, what would be the new probability of purchasing? Causality cannot be established by examining history (of price changes) because the price change may have been for some other reason that could itself affect the second event (a tariff that increases the price of both goods). Mathematically:
formula_3
where "do" is an operator that signals the experimental intervention (doubling the price). The operator indicates performing the minimal change in the world necessary to create the intended effect, a "mini-surgery" on the model with as little change from reality as possible.
Counterfactuals.
The highest level, counterfactual, involves consideration of an alternate version of a past event, or what would happen under different circumstances for the same experimental unit. For example, what is the probability that, if a store had doubled the price of floss, the toothpaste-purchasing shopper would still have bought it?
formula_4
Counterfactuals can indicate the existence of a causal relationship. Models that can answer counterfactuals allow precise interventions whose consequences can be predicted. At the extreme, such models are accepted as physical laws (as in the laws of physics, e.g., inertia, which says that if force is not applied to a stationary object, it will not move).
Causality.
Causality vs correlation.
Statistics revolves around the analysis of relationships among multiple variables. Traditionally, these relationships are described as correlations, associations without any implied causal relationships. Causal models attempt to extend this framework by adding the notion of causal relationships, in which changes in one variable cause changes in others.
Twentieth century definitions of causality relied purely on probabilities/associations. One event (formula_5) was said to cause another if it raises the probability of the other (formula_6). Mathematically this is expressed as:
formula_7.
Such definitions are inadequate because other relationships (e.g., a common cause for formula_5 and formula_6) can satisfy the condition. Causality is relevant to the second ladder step. Associations are on the first step and provide only evidence to the latter.
A later definition attempted to address this ambiguity by conditioning on background factors. Mathematically:
formula_10,
where formula_11 is the set of background variables and formula_12 represents the values of those variables in a specific context. However, the required set of background variables is indeterminate (multiple sets may increase the probability), as long as probability is the only criterion.
Other attempts to define causality include Granger causality, a statistical hypothesis test that causality (in economics) can be assessed by measuring the ability to predict the future values of one time series using prior values of another time series.
Types.
A cause can be necessary, sufficient, contributory or some combination.
Necessary.
For "x" to be a necessary cause of "y", the presence of "y" must imply the prior occurrence of "x". The presence of "x", however, does not imply that "y" will occur. Necessary causes are also known as "but-for" causes, as in "y" would not have occurred but for the occurrence of "x".
Sufficient causes.
For "x" to be a sufficient cause of "y", the presence of "x" must imply the subsequent occurrence of "y". However, another cause "z" may independently cause "y". Thus the presence of "y" does not require the prior occurrence of "x".
Contributory causes.
For "x" to be a contributory cause of "y", the presence of "x" must increase the likelihood of "y". If the likelihood is 100%, then "x" is instead called sufficient. A contributory cause may also be necessary.
Model.
Causal diagram.
A causal diagram is a directed graph that displays causal relationships between variables in a causal model. A causal diagram includes a set of variables (or nodes). Each node is connected by an arrow to one or more other nodes upon which it has a causal influence. An arrowhead delineates the direction of causality, e.g., an arrow connecting variables formula_13 and formula_14 with the arrowhead at formula_14 indicates that a change in formula_13 causes a change in formula_14 (with an associated probability). A "path" is a traversal of the graph between two nodes following causal arrows.
Causal diagrams include causal loop diagrams, directed acyclic graphs, and Ishikawa diagrams.
Causal diagrams are independent of the quantitative probabilities that inform them. Changes to those probabilities (e.g., due to technological improvements) do not require changes to the model.
Model elements.
Causal models have formal structures with elements with specific properties.
Junction patterns.
The three types of connections of three nodes are linear chains, branching forks and merging colliders.
Chain.
Chains are straight line connections with arrows pointing from cause to effect. In this model, formula_14 is a mediator in that it mediates the change that formula_13 would otherwise have on formula_20.
formula_21
Fork.
In forks, one cause has multiple effects. The two effects have a common cause. There exists a (non-causal) spurious correlation between formula_13 and formula_20 that can be eliminated by conditioning on formula_14 (for a specific value of formula_14).
formula_26
"Conditioning on formula_14" means "given formula_14" (i.e., given a value of formula_14).
An elaboration of a fork is the confounder:
formula_30
In such models, formula_14 is a common cause of formula_13 and formula_20 (which also causes formula_13), making formula_14 the confounder.
Collider.
In colliders, multiple causes affect one outcome. Conditioning on formula_14 (for a specific value of formula_14) often reveals a non-causal negative correlation between formula_13 and formula_20. This negative correlation has been called collider bias and the "explain-away" effect as formula_14 explains away the correlation between formula_13 and formula_20. The correlation can be positive in the case where contributions from both formula_13 and formula_20 are necessary to affect formula_14.
formula_46
Node types.
Mediator.
A mediator node