INUS

Causality (also referred to as causation, or cause and effect) is influence by which one event, process, state, or object (''a'' ''cause'') contributes to the production of another event, process, state, or object (an ''effect'') where the cause is partly responsible for the effect, and the effect is partly dependent on the cause. In general, a process has many causes, which are also said to be ''causal factors'' for it, and all lie in its past. An effect can in turn be a cause of, or causal factor for, many other effects, which all lie in its future. Some writers have held that causality is metaphysically prior to notions of time and space.

Causality is an abstraction that indicates how the world progresses. As such a basic concept, it is more apt as an explanation of other concepts of progression than as something to be explained by others more basic. The concept is like those of agency and efficacy. For this reason, a leap of intuition may be needed to grasp it. Accordingly, causality is implicit in the logic and structure of ordinary language.

In English studies of Aristotelian philosophy, the word "cause" is used as a specialized technical term, the translation of Aristotle's term αἰτία, by which Aristotle meant "explanation" or "answer to a 'why' question". Aristotle categorized the four types of answers as material, formal, efficient, and final "causes". In this case, the "cause" is the explanans for the explanandum, and failure to recognize that different kinds of "cause" are being considered can lead to futile debate. Of Aristotle's four explanatory modes, the one nearest to the concerns of the present article is the "efficient" one.

David Hume, as part of his opposition to rationalism, argued that pure reason alone cannot prove the reality of efficient causality; instead, he appealed to custom and mental habit, observing that all human knowledge derives solely from experience.

The topic of causality remains a staple in contemporary philosophy.

Concept

Metaphysics

The nature of cause and effect is a concern of the subject known as metaphysics. Kant thought that time and space were notions prior to human understanding of the progress or evolution of the world, and he also recognized the priority of causality. But he did not have the understanding that came with knowledge of Minkowski geometry and the special theory of relativity, that the notion of causality can be used as a prior foundation from which to construct notions of time and space.

Ontology

A general metaphysical question about cause and effect is what kind of entity can be a cause, and what kind of entity can be an effect.

One viewpoint on this question is that cause and effect are of one and the same kind of entity, with causality an asymmetric relation between them. That is to say, it would make good sense grammatically to say either "''A'' is the cause and ''B'' the effect" or "''B'' is the cause and ''A'' the effect", though only one of those two can be actually true. In this view, one opinion, proposed as a metaphysical principle in process philosophy, is that every cause and every effect is respectively some process, event, becoming, or happening. An example is 'his tripping over the step was the cause, and his breaking his ankle the effect'. Another view is that causes and effects are 'states of affairs', with the exact natures of those entities being less restrictively defined than in process philosophy.

Another viewpoint on the question is the more classical one, that a cause and its effect can be of different kinds of entity. For example, in Aristotle's efficient causal explanation, an action can be a cause while an enduring object is its effect. For example, the generative actions of his parents can be regarded as the efficient cause, with Socrates being the effect, Socrates being regarded as an enduring object, in philosophical tradition called a 'substance', as distinct from an action.

Epistemology

Since causality is a subtle metaphysical notion, considerable intellectual effort, along with exhibition of evidence, is needed to establish knowledge of it in particular empirical circumstances. According to David Hume, the human mind is unable to perceive causal relations directly. On this ground, the scholar distinguished between the regularity view on causality and the counterfactual notion. According to the counterfactual view, ''X'' causes ''Y'' if and only if, without ''X, Y'' would not exist. Hume interpreted the latter as an ontological view, i.e., as a description of the nature of causality but, given the limitations of the human mind, advised using the former (stating, roughly, that ''X'' causes ''Y'' if and only if the two events are spatiotemporally conjoined, and ''X'' precedes ''Y'') as an epistemic definition of causality. Having an epistemic concept of causality is needed to distinguish between causal and noncausal relations. The contemporary philosophical literature on causality can be divided into five big approaches to causality. These include the (mentioned above) regularity, probabilistic, counterfactual, mechanistic, and manipulationist views. The five approaches can be shown to be reductive, i.e., define causality in terms of relations of other types. According to this reading, they define causality in terms of, respectively, empirical regularities (constant conjunctions of events), changes in conditional probabilities, counterfactual conditions, mechanisms underlying causal relations, and invariance under intervention.

Geometrical significance

Causality has the properties of antecedence and contiguity. These are topological, and are ingredients for space-time geometry. As developed by Alfred Robb, these properties allow the derivation of the notions of time and space. Max Jammer writes "the Einstein postulate ... opens the way to a straightforward construction of the causal topology ... of Minkowski space." Causal efficacy propagates no faster than light.

Thus, the notion of causality is metaphysically prior to the notions of time and space. In practical terms, this is because use of the relation of causality is necessary for the interpretation of empirical experiments. Interpretation of experiments is needed to establish the physical and geometrical notions of time and space.

Volition

The deterministic world-view holds that the history of the universe can be exhaustively represented as a progression of events following one after as cause and effect.Sklar, L. (1995). Determinism, pp. 117–119 in ''A Companion to Metaphysics'', edited by Kim, J. Sosa, E., Blackwell, Oxford UK, pp. 177–181. The incompatibilist version of this holds that there is no such thing as " free will". Compatibilism, on the other hand, holds that determinism is compatible with, or even necessary for, free will.

Necessary and sufficient causes

Causes may sometimes be distinguished into two types: necessary and sufficient. A third type of causation, which requires neither necessity nor sufficiency in and of itself, but which contributes to the effect, is called a "contributory cause".

;Necessary causes: If ''x'' is a necessary cause of ''y'', then the presence of ''y'' necessarily implies the prior occurrence of ''x''. The presence of ''x'', however, does not imply that ''y'' will occur.
;Sufficient causes: If ''x'' is a sufficient cause of ''y'', then the presence of ''x'' necessarily implies the subsequent occurrence of ''y''. However, another cause ''z'' may alternatively cause ''y''. Thus the presence of ''y'' does not imply the prior occurrence of ''x''.
;Contributory causes: For some specific effect, in a singular case, a factor that is a contributory cause is one among several co-occurrent causes. It is implicit that all of them are contributory. For the specific effect, in general, there is no implication that a contributory cause is necessary, though it may be so. In general, a factor that is a contributory cause is not sufficient, because it is by definition accompanied by other causes, which would not count as causes if it were sufficient. For the specific effect, a factor that is on some occasions a contributory cause might on some other occasions be sufficient, but on those other occasions it would not be merely contributory.

J. L. Mackie argues that usual talk of "cause" in fact refers to INUS conditions (insufficient but non-redundant parts of a condition which is itself unnecessary but sufficient for the occurrence of the effect). An example is a short circuit as a cause for a house burning down. Consider the collection of events: the short circuit, the proximity of flammable material, and the absence of firefighters. Together these are unnecessary but sufficient to the house's burning down (since many other collections of events certainly could have led to the house burning down, for example shooting the house with a flamethrower in the presence of oxygen and so forth). Within this collection, the short circuit is an insufficient (since the short circuit by itself would not have caused the fire) but non-redundant (because the fire would not have happened without it, everything else being equal) part of a condition which is itself unnecessary but sufficient for the occurrence of the effect. So, the short circuit is an INUS condition for the occurrence of the house burning down.

Contrasted with conditionals

Conditional statements are ''not'' statements of causality. An important distinction is that statements of causality require the antecedent to precede or coincide with the consequent in time, whereas conditional statements do not require this temporal order. Confusion commonly arises since many different statements in English may be presented using "If ..., then ..." form (and, arguably, because this form is far more commonly used to make a statement of causality). The two types of statements are distinct, however.

For example, all of the following statements are true when interpreting "If ..., then ..." as the material conditional:

# ''If Barack Obama is president of the United States in 2011, then Germany is in Europe.''
# ''If George Washington is president of the United States in 2011, then .''

The first is true since both the antecedent and the consequent are true. The second is true in sentential logic and indeterminate in natural language, regardless of the consequent statement that follows, because the antecedent is false.

The ordinary indicative conditional has somewhat more structure than the material conditional. For instance, although the first is the closest, neither of the preceding two statements seems true as an ordinary indicative reading. But the sentence:
* ''If Shakespeare of Stratford-on-Avon did not write Macbeth, then someone else did.''
intuitively seems to be true, even though there is no straightforward causal relation in this hypothetical situation between Shakespeare's not writing Macbeth and someone else's actually writing it.

Another sort of conditional, the counterfactual conditional, has a stronger connection with causality, yet even counterfactual statements are not all examples of causality. Consider the following two statements:

# ''If A were a triangle, then A would have three sides.''
# ''If switch S were thrown, then bulb B would light.''

In the first case, it would not be correct to say that A's being a triangle ''caused'' it to have three sides, since the relationship between triangularity and three-sidedness is that of definition. The property of having three sides actually determines A's state as a triangle. Nonetheless, even when interpreted counterfactually, the first statement is true. An early version of Aristotle's "four cause" theory is described as recognizing "essential cause". In this version of the theory, that the closed polygon has three sides is said to be the "essential cause" of its being a triangle. This use of the word 'cause' is of course now far obsolete. Nevertheless, it is within the scope of ordinary language to say that it is essential to a triangle that it has three sides.

A full grasp of the concept of conditionals is important to understanding the literature on causality. In everyday language, loose conditional statements are often enough made, and need to be interpreted carefully.

Questionable cause

Fallacies of questionable cause, also known as causal fallacies, ''non-causa pro causa'' (Latin for "non-cause for cause"), or false cause, are informal fallacies where a cause is incorrectly identified.

Theories

Counterfactual theories

Counterfactual theories define causation in terms of a counterfactual relation. These theories can often be seeing as "floating" their account of causality on top of an account of the logic of counterfactual conditionals. This approach can be traced back to David Hume's definition of the causal relation as that "where, if the first object had not been, the second never had existed." More full-fledged analysis of causation in terms of counterfactual conditionals only came in the 20th century after development of the possible world semantics for the evaluation of counterfactual conditionals. In his 1973 paper "Causation," David Lewis proposed the following definition of the notion of ''causal dependence'':

:An event E ''causally depends'' on C if, and only if, (i) if C had occurred, then E would have occurred, and (ii) if C had not occurred, then E would not have occurred.

Causation is then defined as a chain of causal dependence. That is, C causes E if and only if there exists a sequence of events C, D₁, D₂, ... D_k, E such that each event in the sequence depends on the previous. This chain may be called a ''mechanism''.

Note that the analysis does not purport to explain how we make causal judgements or how we reason about causation, but rather to give a metaphysical account of what it is for there to be a causal relation between some pair of events. If correct, the analysis has the power to explain certain features of causation. Knowing that causation is a matter of counterfactual dependence, we may reflect on the nature of counterfactual dependence to account for the nature of causation. For example, in his paper "Counterfactual Dependence and Time's Arrow," Lewis sought to account for the time-directedness of counterfactual dependence in terms of the semantics of the counterfactual conditional. If correct, this theory can serve to explain a fundamental part of our experience, which is that we can only causally affect the future but not the past.

Probabilistic causation

Interpreting causation as a deterministic relation means that if ''A'' causes ''B'', then ''A'' must ''always'' be followed by ''B''. In this sense, war does not cause deaths, nor does smoking cause cancer or emphysema. As a result, many turn to a notion of probabilistic causation. Informally, ''A'' ("The person is a smoker") probabilistically causes ''B'' ("The person has now or will have cancer at some time in the future"), if the information that ''A'' occurred increases the likelihood of ''B''s occurrence. Formally, P≥ P where P is the conditional probability that ''B'' will occur given the information that ''A'' occurred, and P is the probability that ''B'' will occur having no knowledge whether ''A'' did or did not occur. This intuitive condition is not adequate as a definition for probabilistic causation because of its being too general and thus not meeting our intuitive notion of cause and effect. For example, if ''A'' denotes the event "The person is a smoker," ''B'' denotes the event "The person now has or will have cancer at some time in the future" and ''C'' denotes the event "The person now has or will have emphysema some time in the future," then the following three relationships hold: P ≥ P, P ≥ P and P ≥ P. The last relationship states that knowing that the person has emphysema increases the likelihood that he will have cancer. The reason for this is that having the information that the person has emphysema increases the likelihood that the person is a smoker, thus indirectly increasing the likelihood that the person will have cancer. However, we would not want to conclude that having emphysema causes cancer. Thus, we need additional conditions such as temporal relationship of ''A'' to ''B'' and a rational explanation as to the mechanism of action. It is hard to quantify this last requirement and thus different authors prefer somewhat different definitions.

Causal calculus

When experimental interventions are infeasible or illegal, the derivation of a cause-and-effect relationship from observational studies must rest on some qualitative theoretical assumptions, for example, that symptoms do not cause diseases, usually expressed in the form of missing arrows in causal graphs such as Bayesian networks or path diagrams. The theory underlying these derivations relies on the distinction between ''conditional probabilities'', as in $P(cancer, smoking)$, and ''interventional probabilities'', as in $P(cancer, do(smoking))$. The former reads: "the probability of finding cancer in a person known to smoke, having started, unforced by the experimenter, to do so at an unspecified time in the past", while the latter reads: "the probability of finding cancer in a person forced by the experimenter to smoke at a specified time in the past". The former is a statistical notion that can be estimated by observation with negligible intervention by the experimenter, while the latter is a causal notion which is estimated in an experiment with an important controlled randomized intervention. It is specifically characteristic of quantal phenomena that observations defined by incompatible variables always involve important intervention by the experimenter, as described quantitatively by the observer effect. In classical thermodynamics, processes are initiated by interventions called thermodynamic operations. In other branches of science, for example astronomy, the experimenter can often observe with negligible intervention.

The theory of "causal calculus"Pearl, Judea (2000).
Causality: Models, Reasoning, and Inference
'', Cambridge University Press. (also known as do-calculus, Judea Pearl's Causal Calculus, Calculus of
Actions) permits one to infer interventional probabilities from conditional probabilities in causal Bayesian networks with unmeasured variables. One very practical result of this theory is the characterization of confounding variables, namely, a sufficient set of variables that, if adjusted for, would yield the correct causal effect between variables of interest. It can be shown that a sufficient set for estimating the causal effect of $X$ on $Y$ is any set of non-descendants of $X$ that $d$-separate $X$ from $Y$ after removing all arrows emanating from $X$. This criterion, called "backdoor", provides a mathematical definition of "confounding" and helps researchers identify accessible sets of variables worthy of measurement.

Structure learning

While derivations in causal calculus rely on the structure of the causal graph, parts of the causal structure can, under certain assumptions, be learned from statistical data. The basic idea goes back to Sewall Wright's 1921 work on path analysis. A "recovery" algorithm was developed by Rebane and Pearl (1987) which rests on Wright's distinction between the three possible types of causal substructures allowed in a directed acyclic graph (DAG):

# $X \rightarrow Y \rightarrow Z$
# $X \leftarrow Y \rightarrow Z$
# $X \rightarrow Y \leftarrow Z$

Type 1 and type 2 represent the same statistical dependencies (i.e., $X$ and $Z$ are independent given $Y$) and are, therefore, indistinguishable within purely cross-sectional data. Type 3, however, can be uniquely identified, since $X$ and $Z$ are marginally independent and all other pairs are dependent. Thus, while the ''skeletons'' (the graphs stripped of arrows) of these three triplets are identical, the directionality of the arrows is partially identifiable. The same distinction applies when $X$ and $Z$ have common ancestors, except that one must first condition on those ancestors. Algorithms have been developed to systematically determine the skeleton of the underlying graph and, then, orient all arrows whose directionality is dictated by the conditional independencies observed.

Alternative methods of structure learning search through the ''many'' possible causal structures among the variables, and remove ones which are strongly incompatible with the observed correlations. In general this leaves a set of possible causal relations, which should then be tested by analyzing time series data or, preferably, designing appropriately controlled experiments. In contrast with Bayesian Networks, path analysis (and its generalization, structural equation modeling), serve better to estimate a known causal effect or to test a causal model than to generate causal hypotheses.

For nonexperimental data, causal direction can often be inferred if information about time is available. This is because (according to many, though not all, theories) causes must precede their effects temporally. This can be determined by statistical time series models, for instance, or with a statistical test based on the idea of Granger causality, or by direct experimental manipulation. The use of temporal data can permit statistical tests of a pre-existing theory of causal direction. For instance, our degree of confidence in the direction and nature of causality is much greater when supported by cross-correlations, ARIMA models, or cross-spectral analysis using vector time series data than by cross-sectional data.

Derivation theories

Nobel laureate Herbert A. Simon and philosopher Nicholas Rescher claim that the asymmetry of the causal relation is unrelated to the asymmetry of any mode of implication that contraposes. Rather, a causal relation is not a relation between values of variables, but a function of one variable (the cause) on to another (the effect). So, given a system of equations, and a set of variables appearing in these equations, we can introduce an asymmetric relation among individual equations and variables that corresponds perfectly to our commonsense notion of a causal ordering. The system of equations must have certain properties, most importantly, if some values are chosen arbitrarily, the remaining values will be determined uniquely through a path of serial discovery that is perfectly causal. They postulate the inherent serialization of such a system of equations may correctly capture causation in all empirical fields, including physics and economics.

Manipulation theories

Some theorists have equated causality with manipulability.Collingwood, R. (1940) ''An Essay on Metaphysics.'' Clarendon Press.von Wright, G. (1971) ''Explanation and Understanding.'' Cornell University Press. Under these theories, ''x'' causes ''y'' only in the case that one can change ''x'' in order to change ''y''. This coincides with commonsense notions of causations, since often we ask causal questions in order to change some feature of the world. For instance, we are interested in knowing the causes of crime so that we might find ways of reducing it.

These theories have been criticized on two primary grounds. First, theorists complain that these accounts are circular. Attempting to reduce causal claims to manipulation requires that manipulation is more basic than causal interaction. But describing manipulations in non-causal terms has provided a substantial difficulty.

The second criticism centers around concerns of anthropocentrism. It seems to many people that causality is some existing relationship in the world that we can harness for our desires. If causality is identified with our manipulation, then this intuition is lost. In this sense, it makes humans overly central to interactions in the world.

Some attempts to defend manipulability theories are recent accounts that do not claim to reduce causality to manipulation. These accounts use manipulation as a sign or feature in causation without claiming that manipulation is more fundamental than causation.Woodward, James (2003) ''Making Things Happen: A Theory of Causal Explanation''. Oxford University Press,

Process theories

Some theorists are interested in distinguishing between causal processes and non-causal processes (Russell 1948; Salmon 1984).Salmon, W. (1984)
Scientific Explanation and the Causal Structure of the World
''. Princeton University Press.Russell, B. (1948) ''Human Knowledge''. Simon and Schuster. These theorists often want to distinguish between a process and a pseudo-process. As an example, a ball moving through the air (a process) is contrasted with the motion of a shadow (a pseudo-process). The former is causal in nature while the latter is not.

Salmon (1984) claims that causal processes can be identified by their ability to transmit an alteration over space and time. An alteration of the ball (a mark by a pen, perhaps) is carried with it as the ball goes through the air. On the other hand, an alteration of the shadow (insofar as it is possible) will not be transmitted by the shadow as it moves along.

These theorists claim that the important concept for understanding causality is not causal relationships or causal interactions, but rather identifying causal processes. The former notions can then be defined in terms of causal processes.

A subgroup of the process theories is the mechanistic view on causality. It states that that causal relations supervene on mechanisms. While the notion of mechanism is understood differently, the definition put forward by the group of philosophers referred to as the 'New Mechanists' dominate the literature.

Fields

Science

For the scientific investigation of efficient causality, the cause and effect are each best conceived of as temporally transient processes.

Within the conceptual frame of the scientific method, an investigator sets up several distinct and contrasting temporally transient material processes that have the structure of experiments, and records candidate material responses, normally intending to determine causality in the physical world. For instance, one may want to know whether a high intake of carrots causes humans to develop the bubonic plague. The quantity of carrot intake is a process that is varied from occasion to occasion. The occurrence or non-occurrence of subsequent bubonic plague is recorded. To establish causality, the experiment must fulfill certain criteria, only one example of which is mentioned here. For example, instances of the hypothesized cause must be set up to occur at a time when the hypothesized effect is relatively unlikely in the absence of the hypothesized cause; such unlikelihood is to be established by empirical evidence. A mere observation of a correlation is not nearly adequate to establish causality. In nearly all cases, establishment of causality relies on repetition of experiments and probabilistic reasoning. Hardly ever is causality established more firmly than as more or less probable. It is most convenient for establishment of causality if the contrasting material states of affairs are precisely matched, except for only one variable factor, perhaps measured by a real number.

Physics

One has to be careful in the use of the word cause in physics. Properly speaking, the hypothesized cause and the hypothesized effect are each temporally transient processes. For example, force is a useful concept for the explanation of acceleration, but force is not by itself a cause. More is needed. For example, a temporally transient process might be characterized by a definite change of force at a definite time. Such a process can be regarded as a cause. Causality is not inherently implied in equations of motion, but postulated as an additional constraint that needs to be satisfied (i.e. a cause always precedes its effect). This constraint has mathematical implications
such as the Kramers-Kronig relations.

Causality is one of the most fundamental and essential notions of physics. Causal efficacy cannot 'propagate' faster than light. Otherwise, reference coordinate systems could be constructed (using the Lorentz transform of special relativity) in which an observer would see an effect precede its cause (i.e. the postulate of causality would be violated).

Causal notions appear in the context of the flow of mass-energy. Any actual process has causal efficacy that can propagate no faster than light. In contrast, an abstraction has no causal efficacy. Its mathematical expression does not propagate in the ordinary sense of the word, though it may refer to virtual or nominal 'velocities' with magnitudes greater than that of light. For example, wave packets are mathematical objects that have group velocity and phase velocity. The energy of a wave packet travels at the group velocity (under normal circumstances); since energy has causal efficacy, the group velocity cannot be faster than the speed of light. The phase of a wave packet travels at the phase velocity; since phase is not causal, the phase velocity of a wave packet can be faster than light.

Causal notions are important in general relativity to the extent that the existence of an arrow of time demands that the universe's semi-Riemannian manifold be orientable, so that "future" and "past" are globally definable quantities.

Engineering

A causal system is a system with output and internal states that depends only on the current and previous input values. A system that has ''some'' dependence on input values from the future (in addition to possible past or current input values) is termed an acausal system, and a system that depends ''solely'' on future input values is an anticausal system. Acausal filters, for example, can only exist as postprocessing filters, because these filters can extract future values from a memory buffer or a file.

Biology, medicine and epidemiology

Austin Bradford Hill built upon the work of Hume and Popper and suggested in his paper "The Environment and Disease: Association or Causation?" that aspects of an association such as strength, consistency, specificity, and temporality be considered in attempting to distinguish causal from noncausal associations in the epidemiological situation. (See Bradford-Hill criteria.) He did not note however, that temporality is the only necessary criterion among those aspects. Directed acyclic graphs (DAGs) are increasingly used in epidemiology to help enlighten causal thinking.

Psychology

Psychologists take an empirical approach to causality, investigating how people and non-human animals detect or infer causation from sensory information, prior experience and innate knowledge.

;Attribution
Attribution theory is the theory concerning how people explain individual occurrences of causation. Attribution can be external (assigning causality to an outside agent or force—claiming that some outside thing motivated the event) or internal (assigning causality to factors within the person—taking personal responsibility or accountability for one's actions and claiming that the person was directly responsible for the event). Taking causation one step further, the type of attribution a person provides influences their future behavior.

The intention behind the cause or the effect can be covered by the subject of action. See also accident; blame; intent; and responsibility.

;Causal powers
Whereas David Hume argued that causes are inferred from non-causal observations, Immanuel Kant claimed that people have innate assumptions about causes. Within psychology, Patricia Cheng attempted to reconcile the Humean and Kantian views. According to her power PC theory, people filter observations of events through an intuition that causes have the power to generate (or prevent) their effects, thereby inferring specific cause-effect relations.

;Causation and salience
Our view of causation depends on what we consider to be the relevant events. Another way to view the statement, "Lightning causes thunder" is to see both lightning and thunder as two perceptions of the same event, viz., an electric discharge that we perceive first visually and then aurally.

;Naming and causality
David Sobel and Alison Gopnik from the Psychology Department of UC Berkeley designed a device known as ''the blicket detector'' which would turn on when an object was placed on it. Their research suggests that "even young children will easily and swiftly learn about a new causal power of an object and spontaneously use that information in classifying and naming the object."

;Perception of launching events
Some researchers such as Anjan Chatterjee at the University of Pennsylvania and Jonathan Fugelsang at the University of Waterloo are using neuroscience techniques to investigate the neural and psychological underpinnings of causal launching events in which one object causes another object to move. Both temporal and spatial factors can be manipulated.

See Causal Reasoning (Psychology) for more information.

Statistics and economics

Statistics and economics usually employ pre-existing data or experimental data to infer causality by regression methods. The body of statistical techniques involves substantial use of regression analysis. Typically a linear relationship such as

:$y_i = a_0 + a_1x_ + a_2x_ + \dots + a_kx_ + e_i$

is postulated, in which $y_i$ is the ''i''th observation of the dependent variable (hypothesized to be the caused variable), $x_$ for ''j''=1,...,''k'' is the ''i''th observation on the ''j''th independent variable (hypothesized to be a causative variable), and $e_i$ is the error term for the ''i''th observation (containing the combined effects of all other causative variables, which must be uncorrelated with the included independent variables). If there is reason to believe that none of the $x_j$s is caused by ''y'', then estimates of the coefficients $a_j$ are obtained. If the null hypothesis that $a_j=0$ is rejected, then the alternative hypothesis that $a_ \ne 0$ and equivalently that $x_j$ causes ''y'' cannot be rejected. On the other hand, if the null hypothesis that $a_j=0$ cannot be rejected, then equivalently the hypothesis of no causal effect of $x_j$ on ''y'' cannot be rejected. Here the notion of causality is one of contributory causality as discussed above: If the true value $a_j \ne 0$, then a change in $x_j$ will result in a change in ''y'' ''unless'' some other causative variable(s), either included in the regression or implicit in the error term, change in such a way as to exactly offset its effect; thus a change in $x_j$ is ''not sufficient'' to change ''y''. Likewise, a change in $x_j$ is ''not necessary'' to change ''y'', because a change in ''y'' could be caused by something implicit in the error term (or by some other causative explanatory variable included in the model).

The above way of testing for causality requires belief that there is no reverse causation, in which ''y'' would cause $x_j$. This belief can be established in one of several ways. First, the variable $x_j$ may be a non-economic variable: for example, if rainfall amount $x_j$ is hypothesized to affect the futures price ''y'' of some agricultural commodity, it is impossible that in fact the futures price affects rainfall amount (provided that cloud seeding is never attempted). 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; this principle is invoked, for example, in testing for Granger causality and in its multivariate analog, vector autoregression, both of which control for lagged values of the dependent variable while testing for causal effects of lagged independent variables.

Regression analysis controls for other relevant variables by including them as regressors (explanatory variables). This helps to avoid false inferences of causality due to the presence of a third, underlying, variable that influences both the potentially causative variable and the potentially caused variable: its effect on the potentially caused variable is captured by directly including it in the regression, so that effect will not be picked up as an indirect effect through the potentially causative variable of interest. Given the above procedures, coincidental (as opposed to causal) correlation can be probabilistically rejected if data samples are large and if regression results pass cross-validation tests showing that the correlations hold even for data that were not used in the regression. Asserting with certitude that a common-cause is absent and the regression represents the true causal structure is ''in principle'' impossible.

Apart from constructing statistical models of observational and experimental data, economists use axiomatic (mathematical) models to infer and represent causal mechanisms. Highly abstract theoretical models that isolate and idealize one mechanism dominate microeconomics. In macroeconomics, economists use broad mathematical models that are calibrated on historical data. A subgroup of calibrated models, dynamic stochastic general equilibrium (DSGE) models are employed to represent (in a simplified way) the whole economy and simulate changes in fiscal and monetary policy.

Management

For quality control in manufacturing in the 1960s, Kaoru Ishikawa developed a cause and effect diagram, known as an Ishikawa diagram or fishbone diagram. The diagram categorizes causes, such as into the six main categories shown here. These categories are then sub-divided. Ishikawa's method identifies "causes" in brainstorming sessions conducted among various groups involved in the manufacturing process. These groups can then be labeled as categories in the diagrams. The use of these diagrams has now spread beyond quality control, and they are used in other areas of management and in design and engineering. Ishikawa diagrams have been criticized for failing to make the distinction between necessary conditions and sufficient conditions. It seems that Ishikawa was not even aware of this distinction.

Humanities

History

In the discussion of history, events are sometimes considered as if in some way being agents that can then bring about other historical events. Thus, the combination of poor harvests, the hardships of the peasants, high taxes, lack of representation of the people, and kingly ineptitude are among the ''causes'' of the French Revolution. This is a somewhat Platonic and Hegelian view that reifies causes as ontological entities. In Aristotelian terminology, this use approximates to the case of the ''efficient'' cause.

Some philosophers of history such as Arthur Danto have claimed that "explanations in history and elsewhere" describe "not simply an event—something that happens—but a change". Like many practicing historians, they treat causes as intersecting actions and sets of actions which bring about "larger changes", in Danto's words: to decide "what are the elements which persist through a change" is "rather simple" when treating an individual's "shift in attitude", but "it is considerably more complex and metaphysically challenging when we are interested in such a change as, say, the break-up of feudalism or the emergence of nationalism".

Much of the historical debate about causes has focused on the relationship between communicative and other actions, between singular and repeated ones, and between actions, structures of action or group and institutional contexts and wider sets of conditions. John Gaddis has distinguished between exceptional and general causes (following Marc Bloch) and between "routine" and "distinctive links" in causal relationships: "in accounting for what happened at Hiroshima on August 6, 1945, we attach greater importance to the fact that President Truman ordered the dropping of an atomic bomb than to the decision of the Army Air Force to carry out his orders." He has also pointed to the difference between immediate, intermediate and distant causes. For his part, Christopher Lloyd puts forward four "general concepts of causation" used in history: the "metaphysical idealist concept, which asserts that the phenomena of the universe are products of or emanations from an omnipotent being or such final cause"; "the empiricist (or