Causal model

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:

${\displaystyle P(floss\vline toothpaste)}$

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 ).[5]

This level asserts specific causal relationships between events. Causality is assessed by experimentally performing some action that affects one of the events. Example: if we doubled 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:

${\displaystyle P(floss\vline do(toothpaste))}$

where do is an operator that signals the experimental intervention (doubling the price).[5] 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.[7]

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?

${\displaystyle P(floss\vline toothpaste,price*2)}$

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).[5]

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.[3]

Twentieth century definitions of causality relied purely on probabilities/associations. One event (${\displaystyle X}$) was said to cause another if it raises the probability of the other (${\displaystyle Y}$). Mathematically this is expressed as:

${\displaystyle P(Y\vline X)>P(Y)}$.

Such definitions are inadequate because other relationships (e.g., a common cause for ${\displaystyle X}$ and ${\displaystyle Y}$) 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.[5]

A later definition attempted to address this ambiguity by conditioning on background factors. Mathematically:

${\displaystyle P(Y\vline X,K=k)>P(Y|K=k)}$,

where ${\displaystyle K}$ is the set of background variables and ${\displaystyle k}$ 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.[5]

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.[5]

A cause can be necessary, sufficient, contributory or some combination.[8]

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.[9] Necessary causes are also known as "but-for" causes, as in y would not have occurred but for the occurrence of x.[5]^: 261

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.[9]

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.[10]

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 ${\displaystyle A}$ and ${\displaystyle B}$ with the arrowhead at ${\displaystyle B}$ indicates that a change in ${\displaystyle A}$ causes a change in ${\displaystyle B}$ (with an associated probability). A path is a traversal of the graph between two nodes following causal arrows.[5]

Causal diagrams include causal loop diagrams, directed acyclic graphs, and Ishikawa diagrams.[5]

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.[5]

Causal models have formal structures with elements with specific properties.[5]

The three types of connections of three nodes are linear chains, branching forks and merging colliders.[5]

Chains are straight line connections with arrows pointing from cause to effect. In this model, ${\displaystyle B}$ is a mediator in that it mediates the change that ${\displaystyle A}$ would otherwise have on ${\displaystyle C}$.[5]^: 113

${\displaystyle A\rightarrow B\rightarrow C}$

In forks, one cause has multiple effects. The two effects have a common cause. There exists a (non-causal) spurious correlation between ${\displaystyle A}$ and ${\displaystyle C}$ that can be eliminated by conditioning on ${\displaystyle B}$ (for a specific value of ${\displaystyle B}$).[5]^: 114

${\displaystyle A\leftarrow B\rightarrow C}$

"Conditioning on ${\displaystyle B}$" means "given ${\displaystyle B}$" (i.e., given a value of ${\displaystyle B}$).

An elaboration of a fork is the confounder:

${\displaystyle A\leftarrow B\rightarrow C\rightarrow A}$

In such models, ${\displaystyle B}$ is a common cause of ${\displaystyle A}$ and ${\displaystyle C}$ (which also causes ${\displaystyle A}$), making ${\displaystyle B}$ the confounder.[5]^: 114

In colliders, multiple causes affect one outcome. Conditioning on ${\displaystyle B}$ (for a specific value of ${\displaystyle B}$) often reveals a non-causal negative correlation between ${\displaystyle A}$ and ${\displaystyle C}$. This negative correlation has been called collider bias and the "explain-away" effect as ${\displaystyle B}$ explains away the correlation between ${\displaystyle A}$ and ${\displaystyle C}$.[5]^: 115 The correlation can be positive in the case where contributions from both ${\displaystyle A}$ and ${\displaystyle C}$ are necessary to affect ${\displaystyle B}$.[5]^: 197

${\displaystyle A\rightarrow B\leftarrow C}$

A mediator node modifies the effect of other causes on an outcome (as opposed to simply affecting the outcome).[5]^: 113 For example, in the chain example above, ${\displaystyle B}$ is a mediator, because it modifies the effect of ${\displaystyle A}$ (an indirect cause of ${\displaystyle C}$) on ${\displaystyle C}$ (the outcome).

A confounder node affects multiple outcomes, creating a positive correlation among them.[5]^: 114

An instrumental variable is one that:[5]^: 246

• has a path to the outcome;
• has no other path to causal variables;
• has no direct influence on the outcome.

Regression coefficients can serve as estimates of the causal effect of an instrumental variable on an outcome as long as that effect is not confounded. In this way, instrumental variables allow causal factors to be quantified without data on confounders.[5]^: 249

For example, given the model:

${\displaystyle Z\rightarrow X\rightarrow Y\leftarrow U\rightarrow X}$

${\displaystyle Z}$ is an instrumental variable, because it has a path to the outcome ${\displaystyle Y}$ and is unconfounded, e.g., by ${\displaystyle U}$.

In the above example, if ${\displaystyle Z}$ and ${\displaystyle X}$ take binary values, then the assumption that ${\displaystyle Z=0,X=1}$ does not occur is called monotonicity.[5]^: 253

Refinements to the technique include creating an instrument by conditioning on other variable to block the paths between the instrument and the confounder and combining multiple variables to form a single instrument.[5]^: 257

Definition: Mendelian randomization uses measured variation in genes of known function to examine the causal effect of a modifiable exposure on disease in observational studies.[11][12]

Because genes vary randomly across populations, presence of a gene typically qualifies as an instrumental variable, implying that in many cases, causality can be quantified using regression on an observational study.[5]^: 255

Independence conditions are rules for deciding whether two variables are independent of each other. Variables are independent if the values of one do not directly affect the values of the other. Multiple causal models can share independence conditions. For example, the models

${\displaystyle A\rightarrow B\rightarrow C}$

and

${\displaystyle A\leftarrow B\rightarrow C}$

have the same independence conditions, because conditioning on ${\displaystyle B}$ leaves ${\displaystyle A}$ and ${\displaystyle C}$ independent. However, the two models do not have the same meaning and can be falsified based on data (that is, if observational data show an association between ${\displaystyle A}$ and ${\displaystyle C}$ after conditioning on ${\displaystyle B}$, then both models are incorrect). Conversely, data cannot show which of these two models are correct, because they have the same independence conditions.

Conditioning on a variable is a mechanism for conducting hypothetical experiments. Conditioning on a variable involves analyzing the values of other variables for a given value of the conditioned variable. In the first example, conditioning on ${\displaystyle B}$ implies that observations for a given value of ${\displaystyle B}$ should show no dependence between ${\displaystyle A}$ and ${\displaystyle C}$. If such a dependence exists, then the model is incorrect. Non-causal models cannot make such distinctions, because they do not make causal assertions.[5]^{: 129–130}

An essential element of correlational study design is to identify potentially confounding influences on the variable under study, such as demographics. These variables are controlled for to eliminate those influences. However, the correct list of confounding variables cannot be determined a priori. It is thus possible that a study may control for irrelevant variables or even (indirectly) the variable under study.[5]^: 139

Causal models offer a robust technique for identifying appropriate confounding variables. Formally, Z is a confounder if "Y is associated with Z via paths not going through X". These can often be determined using data collected for other studies. Mathematically, if

${\displaystyle P(Y|X)\neq P(Y|do(X))}$

then we say that X and Y are confounded (by some confounder variable Z).[5]^: 151

Earlier, allegedly incorrect definitions of confounder include:[5]^: 152

• "Any variable that is correlated with both X and Y."
• Y is associated with Z among the unexposed.
• Noncollapsibility: A difference between the "crude relative risk and the relative risk resulting after adjustment for the potential confounder".
• Epidemiological: A variable associated with X in the population at large and associated with Y among people unexposed to X.

The latter is flawed in that given that in the model:

${\displaystyle X\rightarrow Z\rightarrow Y}$

Z matches the definition, but is a mediator, not a confounder, and is an example of controlling for the outcome.

In the model

${\displaystyle X\leftarrow A\rightarrow B\leftarrow C\rightarrow Y}$

Traditionally, B was considered to be a confounder, because it is associated with X and with Y but is not on a causal path nor is it a descendant of anything on a causal path. Controlling for B causes it to become a confounder. This is known as M-bias.[5]^: 161

For analysing the causal effect of X on Y in a causal model we need to adjust for all confounder variables (deconfounding). To identify the set of confounders we need to (1) block every noncausal path between X and Y by this set (2) without disrupting any causal paths and (3) without creating any spurious paths.[5]^: 158

Definition: a backdoor path from variable X to Y is any path from X to Y that starts with an arrow pointing to X.[5]^: 158

Definition: Given an ordered pair of variables (X,Y) in a model, a set of confounder variables Z satisfies the backdoor criterion if (1) no confounder variable Z is a descendent of X and (2) all backdoor paths between X and Y are blocked by the set of confounders.

If the backdoor criterion is satisfied for (X,Y), X and Y are deconfounded by the set of confounder variables. It is not necessary to control for any variables other than the confounders.[5]^: 158 The backdoor criterion is a sufficient but not necessary condition to find a set of variables Z to decounfound the analysis of the causal effect of X on y.

When the causal model is a plausible representation of reality and the backdoor criterion is satisfied, then partial regression coefficients can be used as (causal) path coefficients (for linear relationships).[5]^: 223 [13]

${\displaystyle P(Y|do(X))=\textstyle \sum _{z}\displaystyle P(Y|X,Z=z)P(Z=z)}$[5]^: 227

If the elements of a blocking path are all unobservable, the backdoor path is not calculable, but if all forward paths from ${\displaystyle X\to Y}$ have elements ${\displaystyle z}$ where there are no open paths ${\displaystyle z\to Y}$, then we can use ${\displaystyle Z}$, the set of all ${\displaystyle z}$s, to measure ${\displaystyle P(Y|do(X))}$. Effectively, there are conditions where ${\displaystyle Z}$ can act as a proxy for ${\displaystyle X}$.

Definition: a frontdoor path is a direct causal path for which data is available for all ${\displaystyle z\in Z}$,[5]^: 226 ${\displaystyle Z}$ intercepts all directed paths ${\displaystyle X}$ to ${\displaystyle Y}$, there are no unblocked paths from ${\displaystyle Z}$ to ${\displaystyle Y}$, and all backdoor paths from ${\displaystyle Z}$ to ${\displaystyle Y}$ are blocked by ${\displaystyle X}$. [14]

The following converts a do expression into a do-free expression by conditioning on the variables along the front-door path.[5]^: 226

${\displaystyle P(Y|do(X))=\textstyle \sum _{z}\left[\displaystyle P(Z=z|X)\textstyle \sum _{x}\displaystyle P(Y|X=x,Z=z)P(X=x)\right]}$

Presuming data for these observable probabilities is available, the ultimate probability can be computed without an experiment, regardless of the existence of other confounding paths and without backdoor adjustment.[5]^: 226

Queries are questions asked based on a specific model. They are generally answered via performing experiments (interventions). Interventions take the form of fixing the value of one variable in a model and observing the result. Mathematically, such queries take the form (from the example):[5]^: 8

${\displaystyle P({\text{floss}}\vline do({\text{toothpaste}}))}$

where the do operator indicates that the experiment explicitly modified the price of toothpaste. Graphically, this blocks any causal factors that would otherwise affect that variable. Diagramatically, this erases all causal arrows pointing at the experimental variable.[5]^: 40

More complex queries are possible, in which the do operator is applied (the value is fixed) to multiple variables.

The do calculus is the set of manipulations that are available to transform one expression into another, with the general goal of transforming expressions that contain the do operator into expressions that do not. Expressions that do not include the do operator can be estimated from observational data alone, without the need for an experimental intervention, which might be expensive, lengthy or even unethical (e.g., asking subjects to take up smoking).[5]^: 231 The set of rules is complete (it can be used to derive every true statement in this system).[5]^: 237 An algorithm can determine whether, for a given model, a solution is computable in polynomial time.[5]^: 238

The calculus includes three rules for the transformation of conditional probability expressions involving the do operator.

Rule 1 permits the addition or deletion of observations.:[5]^: 235

${\displaystyle P(Y|do(X),Z,W)=P(Y|do(X),Z)}$

in the case that the variable set Z blocks all paths from W to Y and all arrows leading into X have been deleted.[5]^: 234

Rule 2 permits the replacement of an intervention with an observation or vice versa.:[5]^: 235

${\displaystyle P(Y|do(X),Z)=P(Y|X,Z)}$

in the case that Z satisfies the back-door criterion.[5]^: 234

Rule 3 permits the deletion or addition of interventions.:[5]

${\displaystyle P(Y|do(X))=P(Y)}$

in the case where no causal paths connect X and Y.[5]^{: 234  : 235}

The rules do not imply that any query can have its do operators removed. In those cases, it may be possible to substitute a variable that is subject to manipulation (e.g., diet) in place of one that is not (e.g., blood cholesterol), which can then be transformed to remove the do. Example:

${\displaystyle P({\text{Heart disease}}|do({\text{blood cholesterol}}))=P({\text{Heart disease}}|do({\text{diet}}))}$

Definition: A potential outcome for a variable Y is "the value Y would have taken for individual u, had X been assigned the value x". Mathematically:[5]^: 270

${\displaystyle Y_{X=x}(u)}$ or ${\displaystyle Y_{x}(u)}$.

The potential outcome is defined at the level of the individual u.[5]^: 270

The conventional approach to potential outcomes is data-, not model-driven, limiting its ability to untangle causal relationships. It treats causal questions as problems of missing data and gives incorrect answers to even standard scenarios.[5]^: 275

In the context of causal models, potential outcomes are interpreted causally, rather than statistically.

The first law of causal inference states that the potential outcome

${\displaystyle Y_{X}(u)}$

can be computed by modifying causal model M (by deleting arrows into X) and computing the outcome for some x. Formally:[5]^: 280

${\displaystyle Y_{X}(u)=Y_{Mx}(u)}$

Examining a counterfactual using a causal model involves three steps.[15] The approach is valid regardless of the form of the model relationships, linear or otherwise. When the model relationships are fully specified, point values can be computed. In other cases (e.g., when only probabilities are available) a probability-interval statement, such as non-smoker x would have a 10-20% chance of cancer, can be computed.[5]^: 279

Given the model:

${\displaystyle Y\leftarrow X\rightarrow M\rightarrow Y\leftarrow U}$

the equations for calculating the values of A and C derived from regression analysis or another technique can be applied, substituting known values from an observation and fixing the value of other variables (the counterfactual).[5]^: 278

Apply abductive reasoning (logical inference that uses observation to find the simplest/most likely explanation) to estimate u, the proxy for the unobserved variables on the specific observation that supports the counterfactual.[5]^: 278 Compute the probability of u given the propositional evidence.

For a specific observation, use the do operator to establish the counterfactual (e.g., m=0), modifying the equations accordingly.[5]^: 278

Calculate the values of the output (y) using the modified equations.[5]^: 278

Direct and indirect (mediated) causes can only be distinguished via conducting counterfactuals.[5]^: 301 Understanding mediation requires holding the mediator constant while intervening on the direct cause. In the model

${\displaystyle Y\leftarrow M\leftarrow X\rightarrow Y}$

M mediates X's influence on Y, while X also has an unmediated effect on Y. Thus M is held constant, while do(X) is computed.

The Mediation Fallacy instead involves conditioning on the mediator if the mediator and the outcome are confounded, as they are in the above model.

For linear models, the indirect effect can be computed by taking the product of all the path coefficients along a mediated pathway. The total indirect effect is computed by the sum of the individual indirect effects. For linear models mediation is indicated when the coefficients of an equation fitted without including the mediator vary significantly from an equation that includes it.[5]^: 324

In experiments on such a model, the controlled direct effect (CDE) is computed by forcing the value of the mediator M (do(M = 0)) and randomly assigning some subjects to each of the values of X (do(X=0), do(X=1), ...) and observing the resulting values of Y.[5]^: 317

${\displaystyle CDE(0)=P(Y=1|do(X=1),do(M=0))-P(Y=1|do(X=0),do(M=0))}$

Each value of the mediator has a corresponding CDE.

However, a better experiment is to compute the natural direct effect. (NDE) This is the effect determined by leaving the relationship between X and M untouched while intervening on the relationship between X and Y.[5]^: 318

${\displaystyle NDE=P(Y_{M=M0}=1|do(X=1))-P(Y_{M=M0}=1|do(X=0))}$

For example, consider the direct effect of increasing dental hygienist visits (X) from every other year to every year, which encourages flossing (M). Gums (Y) get healthier, either because of the hygienist (direct) or the flossing (mediator/indirect). The experiment is to continue flossing while skipping the hygienist visit.

The indirect effect of X on Y is the "increase we would see in Y while holding X constant and increasing M to whatever value M would attain under a unit increase in X".[5]^: 328

Indirect effects cannot be "controlled" because the direct path cannot be disabled by holding another variable constant. The natural indirect effect (NIE) is the effect on gum health (Y) from flossing (M). The NIE is calculated as the sum of (floss and no-floss cases) of the difference between the probability of flossing given the hygienist and without the hygienist, or:[5]^: 321

${\displaystyle NIE=\sum _{m}[P(M=m|X=1)-P(M=m|X=0)]xxP(Y=1|X=0,M=m)}$

The above NDE calculation includes counterfactual subscripts (${\displaystyle Y_{M=M0}}$). For nonlinear models, the seemingly obvious equivalence[5]^: 322

${\displaystyle {\mathsf {Total\ effect=Direct\ effect+Indirect\ effect}}}$

does not apply because of anomalies such as threshold effects and binary values. However,

${\displaystyle {\mathsf {Total\ effect}}(X=0\rightarrow X=1)=NDE(X=0\rightarrow X=1)-\ NIE(X=1\rightarrow X=0)}$

works for all model relationships (linear and nonlinear). It allows NDE to then be calculated directly from observational data, without interventions or use of counterfactual subscripts.[5]^: 326