Causal model
In metaphysics and statistics, a causal model is a conceptual model that represents the causal mechanisms of a system. Causal models often employ formal causal notation, such as structural equation modeling or causal directed acyclic graphs, to describe relationships among variables and to guide inference.
By clarifying which variables should be included, excluded, or controlled for, causal models can improve the design of empirical studies and the interpretation of results. They can also enable researchers to answer some causal questions using observational data, reducing the need for interventional studies such as randomized controlled trials.
In cases where randomized experiments are impractical or unethical—for example, when studying the effects of environmental exposures or social determinants of health—causal models provide a framework for drawing valid conclusions from non-experimental data.
Causal models can help with the question of external validity. Causal models can allow data from multiple studies to be merged to answer questions that cannot be answered by any individual data set.
Causal models have found applications in signal processing, epidemiology, machine learning, cultural studies, and urbanism, and they can describe both linear and nonlinear processes.
Definition
defines a causal model as an ordered triple, 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
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".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 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. Instead scientists relied on correlations, partly at the behest of Wright's critic, Fisher. One exception was Burks, a student who in 1926 was the first to apply path diagrams to represent a mediating influence 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. 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, entails the sensing of regularities or patterns in the input data, expressed as correlations. The middle level, Intervention, predicts the effects of deliberate actions, expressed as causal relationships. The highest level, Counterfactuals, involves constructing a theory 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:or the probability of floss given 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.
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 because the price change may have been for some other reason that could itself affect the second event. Mathematically:where do is an operator that signals the experimental intervention. 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?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.
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 was said to cause another if it raises the probability of the other. Mathematically this is expressed as:
Such definitions are inadequate because other relationships 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:
where is the set of background variables and represents the values of those variables in a specific context. However, the required set of background variables is indeterminate, as long as probability is the only criterion.
Other attempts to define causality include Granger causality, a statistical hypothesis test that causality can be assessed by measuring the ability to predict the future values of one time series using prior values of another time series.