Causal inference is the process of drawing a conclusion about a causal connection based on the conditions of the occurrence of an effect. The main difference between causal inference and inference of association is that the former analyzes the response of the effect variable when the cause is changed. The science of why things occur is called etiology. Causal inference is an example of causal reasoning.
Inferring the cause of something has been described as:
- “…reason[ing] to the conclusion that something is, or is likely to be, the cause of something else”.
- “Identification of the cause or causes of a phenomenon, by establishing covariation of cause and effect, a time-order relationship with the cause preceding the effect, and the elimination of plausible alternative causes.”
Epidemiological studies employ different epidemiological methods of collecting and measuring evidence of risk factors and effect and different ways of measuring association between the two. A hypothesis is formulated, and then tested with statistical methods (see Statistical hypothesis testing). It is statistical inference that helps decide if data are due to chance, also called random variation, or indeed correlated and if so how strongly. However, correlation does not imply causation, so further methods must be used to infer causation.
Epidemiology studies patterns of health and disease in defined populations of living beings in order to infer causes and effects. An association between an exposure to a putative risk factor and a disease may be suggestive of, but is not equivalent to causality because correlation does not imply causation. Historically, Koch’s postulateshave been used since the 19th century to decide if a microorganism was the cause of a disease. In the 20th century the Bradford Hill criteria, described in 1965 have been used to assess causality of variables outside microbiology, although even these criteria are not exclusive ways to determine causality.
A recent trend[when?] is to identify evidence for influence of the exposure on molecular pathology within diseased tissue or cells, in the emerging interdisciplinary field of molecular pathological epidemiology (MPE).[third-party source needed] Linking the exposure to molecular pathologic signatures of the disease can help to assess causality.[third-party source needed] Considering the inherent nature of heterogeneity of a given disease, the unique disease principle, disease phenotyping and subtyping are trends in biomedical and public health sciences, exemplified as personalized medicine and precision medicine.[third-party source needed]
Determination of cause and effect from joint observational data for two time-independent variables, say X and Y, has been tackled using asymmetry between evidence for some model in the directions, X → Y and Y → X. The primary approaches are based on Algorithmic information theory models and noise models.
Compare two programs, both of which output both X and Y.
- Store Y and a compressed form of X in terms of uncompressed Y.
- Store X and a compressed form of Y in terms of uncompressed X.
Incorporate an independent noise term in the model to compare the evidences of the two directions.
Here are some of the noise models for the hypothesis Y → X with the noise E:
- Additive noise:
- Linear noise:
- Heteroskedastic noise:
- Functional noise:
The common assumption in these models are:
- There are no other causes of Y.
- X and E have no common causes.
- Distribution of cause is independent from causal mechanisms.
On an intuitive level, the idea is that the factorization of the joint distribution P(Cause,Effect) into P(Cause)*P(Effect | Cause) typically yields models of lower total complexity than the factorization into P(Effect)*P(Cause | Effect). Although the notion of “complexity” is intuitively appealing, it is not obvious how it should be precisely defined. A different family of methods attempt to discover causal “footprints” from large amounts of labeled data, and allow the prediction of more flexible causal relations.
In statistics and economics, causality is often tested for using regression. Several methods can be used to distinguish actual causality from spurious indications of causality. First, the explanatory variable could be one that conceptually could not be caused by the dependent variable, thereby avoiding the possibility of being misled by reverse causation: for example, if the independent variable is rainfall and the dependent variable is the futures price of some agricultural commodity. Second, the instrumental variables technique may be employed to remove any reverse causation by introducing a role for other variables (instruments) that are known to be unaffected by the dependent variable. Third, the principle that effects cannot precede causes can be invoked, by including on the right side of the regression only variables that precede in time the dependent variable. Fourth, other regressors are included to ensure that confounding variables are not causing a regressor to spuriously appear to be significant. Correlation by coincidence, as opposed to correlation reflecting actual causation, can be ruled out by using large samples and by performing cross validation to check that correlations are maintained on data that were not used in the regression.
Graduate courses on causal inference have been introduced to the curriculum of many schools.
- Arizona State University, Department of Statistics
- Cornell University, Department of Government 
- Duke University, Department of Political Science
- Saint Louis University, College of Public Health & Social Justice
- Carnegie Mellon University, Department of Philosophy
- Harvard University, School of Public Health
- Johns Hopkins University, Department of Computer Science, Bloomberg School of Public Health
- London School of Hygiene & Tropical Medicine
- Karolinska Institutet, Department of Medical Epidemiology and Biostatistics
- McGill University, Department of Epidemiology, Biostatistics and Occupational Health
- New York University, Department of Applied Statistics, Social Sciences, and Humanities
- Northwestern University, Department of Sociology and Kellogg School of Management
- University of Pittsburgh, Department of Psychology in Education
- University of Groningen, Department of Statistics & Measurement Theory
- University of California, Los Angeles, Department of Epidemiology and Department of Computer Science
- University of California, Berkeley, School of Public Health
- University of California, Santa Barbara, Department of Computer Science
- University of Copenhagen, Department of Public Health
- University of Pennsylvania, Department of Biostatistics and Epidemiology
- University of Texas, Department of Educational Psychology 
- The University of British Columbia, School of Population and Public Health
- Vanderbilt University, Department of Leadership, Policy, and Organizations, Department of Biostatistics
- Stevens Institute of Technology, Department of Computer Science 
- University of North Carolina at Chapel Hill, Department of Biostatistics 
- University of California, Irvine, Department of Statistics 
- Granger causality
- Multivariate statistics
- Partial least squares regression
- Regression analysis
- Transfer entropy
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- Hill, Austin Bradford (1965). “The Environment and Disease: Association or Causation?”. Proceedings of the Royal Society of Medicine. 58 (5): 295–300. PMC 1898525. PMID 14283879.
- Kailash Budhathoki and Jilles Vreeken “Causal Inference by Compression” 2016 IEEE 16th International Conference on Data Mining (ICDM)
- Alexander Marx and Jilles Vreeken “Telling cause from effect by local and global regression” J. Knowl Inf Syst (2018)
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- Shimizu, Shohei, et al. “DirectLiNGAM: A direct method for learning a linear non-Gaussian structural equation model.” The Journal of Machine Learning Research 12 (2011): 1225-1248.
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- Lopez-Paz, David, et al. “Towards a learning theory of cause-effect inference” ICML. 2015
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