Prospective and retrospective causal inferences based on the potential outcome framework

2.1 Confounders

The problem about confounders has been explored for a long time, especially in epidemiology. However, the criteria for assessing confounders and confounding in the epidemiological literature have been inconsistent [15–21]. Using Neyman’s potential outcome framework, confounding bias B is defined as the difference between the counterfactual probability of potential outcome without exposure in the exposed population and the probability of observed outcome in the unexposed population [18,22], i.e.,

By adjusting the distribution pr ( C = k ∣ X = 0 ) of covariate C in the unexposed population to pr ( C = k ∣ X = 1 ) , a standardized probability pr Δ ( Y 0 = 1 ∣ X = 0 ) is defined as

A confounder is defined as a risk factor whose control can reduce the confounding bias [15,23–26]. Replacing the counterfactual probability pr ( Y 0 = 1 ∣ X = 0 ) in (1) by the adjusted pr Δ ( Y 0 = 1 ∣ X = 0 ) , Geng et al. [25] defined a confounder as a covariate C for which

This definition states that the standardized probability pr Δ ( Y 0 = 1 ∣ X = 0 ) adjusted for a confounder C is closer to the counterfactual probability pr ( Y 0 = 1 ∣ X = 1 ) than the observed probability pr ( Y = 1 ∣ X = 0 ) . For a case of C with multiple covariates, C may be recategorized by a single categorical variate C ′ with the same number of categories as C . VanderWeele and Shpitser [26] considered a similar definition of confounders for the overall effect of the exposure on the whole population rather than the effect of the exposure on the exposed population. Note that these definitions of confounders do not need any assumption such as subpopulation-comparability or a known causal diagram. With this definition, we can determine that a covariate is not a confounder when pr Δ ( Y 0 = 1 ∣ X = 0 ) = pr ( Y 0 = 1 ∣ X = 0 ) , but we cannot confirm that it is a confounder since pr ( Y 0 = 1 ∣ X = 1 ) is not identifiable without further assumptions.

2.2 Surrogate endpoints

In many scientific studies, true endpoint variables cannot be measured or observed due to being expensive, inconvenient, or impractical within a short time span. For example, in clinical trials, CD4 count is used as a surrogate endpoint for survival time in acequired immune deficiency syndrome (AIDS) studies, and bone mass is used as a surrogate endpoint for fracture in osteoporosis studies. However, Fleming and Demets [27] pointed out that in many real clinical trials, surrogates failed to evaluate the treatment effects on true endpoints.

Chen et al. [28] introduced and formulated the surrogate paradox, where a treatment has a positive effect on a surrogate endpoint, which in turn has a positive effect on a true endpoint, but the treatment has a negative effect on the true endpoint. Even by conducting two randomized experiments, we can separately prove probabilistically both that a variate X has a positive causal effect on a variable Y and that the variate Y has a positive causal effect on a variable Z , but we cannot judge that X has a positive causal effect on Z , even if X does not have any direct causal effect on Z . Even if the intermediate variable Y breaks all causal paths from X to Z , the probabilistic causal relationships may not be transitive, although the individual causal relationships may be. The surrogate paradox implies that the sign of treatment effect on the endpoint cannot be predicted by the sign of treatment effect on the surrogate and the sign of causal effect of surrogate on the endpoint. Therefore, the logical reasoning may not be applied to the probabilistic results of causal inference. Jiang et al. [29] also discussed the transitivity of different associations under the conditional independence of X and Z given Y and showed that the finer an association measure is, the stronger its transitivity is.

Prentice [30] proposed a criterion for a statistical surrogate Y , which requires both a strong association between treatment X and the surrogate Y and the conditional independence of the true endpoint Z and treatment X given Y , denoted as X ⊥ ⊥ Z ∣ Y in the notation by Dawid [31]. The conditional independence means that the surrogate Y can break the association between the treatment X and the endpoint Z , and thus, X ⊥ ⊥ Y implies X ⊥ ⊥ Z . Frangakis and Rubin [32] presented the criterion for a principal surrogate Y , which should possess the causal necessity: a treatment X has a causal effect on an endpoint Z only if the treatment X has a causal effect on the surrogate Y . Lauritzen [33] proposed the criterion for a strong surrogate Y , which breaks all causal paths from X to Z in a causal diagram (Figure 1). Chen et al. [28] showed that the surrogate paradox cannot be avoided, even by such strong criteria of the statistical surrogates, the principal surrogates, and the strong surrogates. A surrogate Y is an intermediate variable in a causal path from X to Z , and variable X is an instrumental variable when Y is a strong surrogate. A more proper name for the paradox may be “the intermediate variable paradox” to reflect its wider generality. The same paradox applies to other situations. For example, the paradox can be called “the instrumental paradox” in the use of the instrumental variable. The surrogate paradox also points out an issue of the transitivity of causal effects on a causal path. Jiang et al. [34] proposed approaches to identifying the principal stratification causal effects by multiple trials and provided the criteria for surrogates that avoid the surrogate paradox.

Figure 1

Criterion for a strong surrogate.

Moore [35] provided a real-world example of the surrogate paradox. Doctors knew that irregular heartbeat was a risk factor for sudden death and presumed that correcting irregular heartbeat would prevent sudden death. Therefore, they used “correction of heartbeat” as a surrogate, and several drugs (Enkaid, Tambocor and Ethmozine) were approved by FDA (Food and Drug Administration). However, the Cardiac Arrhythmia Suppression Trial [36] showed that these drugs did not improve survival times but increased mortality.

Chen et al. [28], Ju and Geng [37], Wu et al. [38], and VanderWeele and Shpitser [26] proposed consistent surrogates and criteria to avoid the surrogate paradox. These criteria apply to single surrogates only. However, in many applications, a treatment may affect the endpoint through multiple pathways, and thus, a single surrogate cannot break all of these pathways. For example, a drug may reduce the risk of death from AIDS through two pathways: by decreasing human immunodeficiency virus type 1 ribonucleic acid (HIV-1 RNA) concentrations and by increasing CD4 count. In this case, a single surrogate may not satisfy any criterion of the statistical, principal, strong, or consistent surrogates. Both HIV-1 RNA concentrations and CD4 count should be used as multiple surrogates for the risk of death from AIDS. Joffe [39] suggested that it is meaningful to generalize the criteria for a single surrogate to multiple surrogates. Luo et al. [40] proposed a criterion for multiple surrogates Y = ( Y 1 , … , Y p ) based on stochastic orders of random vectors. All of these criteria for surrogates require some knowledge of causality or associations among the observed variables X and Y and the unobserved variable Z . Therefore, these criteria are not falsifiable without untestable assumptions or observed data for Z .

3.1 Probabilities of causation

Pearl [11] provided counterfactual definitions of causation to capture how necessary and/or sufficient a cause is capable of producing a given effect or outcome. Dawid et al. [5] highlighted the distinction between the effects of causes and the causes of effects, and proposed the probability of causation to make inference about the causes of effects. Inferring the causes of effects requires more subtle logic and stronger assumptions than inferring the effects of causes. In the following, we focus on binary variables. First, we introduce probabilities of causation for the case of a single effect variable Y and a single cause X . To measure how possible X is a cause of an effect Y , Dawid et al. [5] defined the probability of causation as

To measure how necessary X is a cause of an occurred effect Y = 1 , Pearl [41] defined the probability of necessary causation as

Lu et al. [13] proposed the posterior causal effects given observed evidence to measure the probabilities of causes and treated evaluating effects of causes and discovering causes of effects from the same perspective. Let C denote a pretreatment variable prior to treatment X (i.e., a covariate). Let O = o denote the observed evidence for the target individual, where o is an observed value of O . For an individual case, we can sometimes observe only a subset O of variables { X , Y , C } . Li et al. [14] defined the posterior total causal effect given O = o as

For the evidence O = ( C = c ) , the posterior total causal effect E ( Y X = 1 − Y X = 0 ∣ C = c ) is an average causal effect conditional on C = c ; for the evidence O = ( X = 1 ) , the posterior total causal effect E ( Y X = 1 − Y X = 0 ∣ X = 1 ) is an average causal effect in a treated subpopulation; for the evidence O = ( X = 1 , Y = 1 ) , the posterior total causal effect is equal to PN :

Thus, the posterior total causal effect can be used not only to evaluate effects of causes for a prospective causal inference, but also to assess causes of effects for a retrospective causal inference. For example, let X denote smoking and Y lung cancer. By the posterior total causal effect E ( Y X = 1 − Y X = 0 ∣ X = 1 , Y = 1 ) , we evaluate the causal effect of smoking on lung cancer in the subpopulation of smokers with lung cancer, which measures the probability that individuals in the subpopulation would not have developed lung cancer if they had not smoked.

The posterior intervention causal effect of X on Y proposed by Zhao et al. [14] given the observed evidence O = o is defined as

Different from PostTCE, PostICE measures the change of Y ’s expectation if X is removed. When the observed evidence O = o contains X = 1 , we have PostTCE ( X ⇒ Y ∣ X = 1 , … ) = PostICE ( X ⇒ Y ∣ X = 1 , … ) . When the observed evidence O = o only contains Y = 1 , we have PostICE ( X ⇒ Y ∣ Y = 1 ) = PN(X ⇒ Y ) pr ( X = 1 ∣ Y = 1 ) . Therefore, PostICE is different from PN. In disease diagnosis, PostICE considers not only the probability that the disease X is the cause of symptoms Y but also the posterior probability of the disease given the occurrence of symptoms.

Next, we extend the aforementioned case to the case of multiple effect variables Y = ( Y 1 , … , Y q ) and multiple cause variables X = ( X 1 , … , X p ) , and we define the posterior causal effect of simultaneously intervening on a subset of X on Y . In real applications, available evidence may include multiple observed effect variables, and thus, they can be used simultaneously to more accurately deduce the causes. For example, in medical diagnosis, the more symptoms of a patient are available, the more accurately a doctor can diagnose the patient’s disease. Without loss of generality, we assume that the causes are arranged in a topological order such that X l is not a cause of X k for k < l , and that Y 1 , … , Y q subsequent to X are arranged in a topological order such that Y j is not a cause of Y m for m < j . Let g ( y ) be a known function weighting the importance of multiple effects in Y . For example, an additive weighting function is g ( Y ) = ∑ i = 1 q a i × Y i , where a i is a weight for Y i . Let X S denote a subvector of X , where S is the subset of indexes { 1 , … , p } , and let x S 1 ≽ x S 0 denote x i 1 ≥ x i 0 for each i ∈ S . For a treated group of X S = x S ≠ 0 versus a control group of X S = 0 , where 0 denotes ( 0 , … , 0 ) , we define the posterior total causal effect of multiple causes X S = x S on multiple effects Y = ( Y 1 , … , Y q ) as

Differing from the conditional counterfactual causal effect, defined by Zhao et al. [42], which restricts x S = ( 1 , … , 1 ) , the aforementioned definition does not require this restriction. Comparing PostTCE of X S on Y with different values x S and x ′ S , we can obtain various posterior controlled direct causal effects and interaction effects. For example, given the observed evidence o = ( X 1 = 1 , X 2 = 1 , X 3 = 1 , Y = 1 ) , a controlled direct causal effect of a set ( X 1 , X 2 ) on Y by controlling for X 3 = x 3 can be measured by

Comparing PostTCE across different subsets X S and X S ′ , we contrast whether an event should be attributed more to X S or to X S ′ . For example, given the observed evidence o = ( X 1 = 1 , X 2 = 1 , X 3 = 1 , Y = 1 ) , comparing PostTCE [ X 1 ( 1 ) ⇒ Y ∣ O = o ] , PostTCE [ X 2 ( 1 ) ⇒ Y ∣ O = o ] , and PostTCE [ X 3 ( 1 ) ⇒ Y ∣ O = o ] , we can argue that the event Y = 1 should be attributed most to X 1 , X 2 , or X 3 .

It can be shown that for an additive weighting function g ( Y ) = ∑ j = 1 q a j × Y j ,

In terms of PostTCE, we can do the attributions of multiple effects to multiple causes with interaction effects.

The posterior intervention causal effect of X S on Y is defined as

When the evidence O includes some ( Y j = 1 ) ’s, comparing PostICE ( X S ⇒ Y ∣ O = o ) with PostICE ( X S ′ ⇒ Y ∣ O = o ) , we can retrospectively judge which of X S and X S ′ might make the happened effects more likely. For g ( y 1 , … , y q ) = ∑ j = 1 q y j , the posterior intervention causal effect PostICE ( X S ⇒ Y ∣ O = o ) measures the expected number of outcomes eliminated by removing risk factors in X S .

3.2 Identification assumptions of posterior causal effects

Let ( X , Y ) = ( V 1 , … , V p + q ) be arranged in a causal order and V r : s = ( V r , V r + 1 , … , V s ) be a subvector of V for r ≤ s . Let ( V s ) v 1 : s − 1 denote the potential outcome of V s if V 1 : s − 1 were intervened to v 1 : s − 1 To identify these posterior causal effects, we need to follow the monotonicity and no-confounding assumptions [13].

This assumption is often expressed as “no prevention” in epidemiology and states that no individual can be helped by exposure to a risk factor. For example, let V 1 , V 2 , and V 3 denote poor diet, high blood pressure and stroke, respectively. The monotonicity assumption means that poor diet and high blood pressure are two potential risk factors for stroke. Exposures to them are not preventive for stroke, and a poor diet is also not preventive for high blood pressure. The validity of monotonicity cannot be directly tested, but this assumption imposes testable restrictions on the probability distribution of observed data in certain cases. Similar assumptions are often made in studies of imperfect compliance of treatment.

Assumption 2 (i) means that the potential outcomes of each variable are independent of its precedent variables arranged in the causal order. If V s has a causal structural model V s = f s ( V 1 : s − 1 , ε s ) and an error variable ε s ⊥ ⊥ ε 1 : s − 1 , then Assumption 2 is equivalent to the absence of latent confounders. Assumption 2 excludes the presence of unobserved confounders between variables X and Y . However, each variable X k may still confound the relationships between Y and X l or between X l and X s for where k < l , s . When there exists a set C containing observed background variables that are not influenced by X , the independence in Assumption 2 can be relaxed to those conditional on C .

When the evidence does not contain any effect variable Y i , the identification of posterior causal effects only requires Assumption 2 of no confounding. But when the evidence contains some effect variables, the identification of posterior causal effects requires both Assumptions 1 and 2. First, consider the case with a single X and a single Y . For the case of a single X and a single Y , Assumption 1 of monotonicity means Y X = 0 ≤ Y X = 1 , and PN has the following equation:

The numerator is the treatment effect on treated. Under Assumption 2 of no confounding, we have pr ( Y 0 = 1 ∣ X = 1 ) = pr ( Y = 1 ∣ X = 0 ) , and thus, PN is identifiable. Similarly, under Assumptions 1 and 2 of monotonicity and no-confounding, it can be shown that the aforementioned posterior causal effects defined by intervening on a single cause X k are identifiable [13,14]. When simultaneously intervening on a set X S = x S of multiple causes, for the identification of PostTCE [ X S ( x S ) ⇒ Y ∣ O = o ] , we further need the restriction on the relationship between x S and o . Let X S * = X S ∩ O , and x S * and x S * ′ are the value of X S * in x S and o , respectively. Let X S ′ = X S \ X S * , and x S ′ is the value of X S ′ . When q = 1 , i.e., Y = Y 1 , and the evidence contains the effect variable Y = y , we have the following theorem.

For the case of q > 1 and that the evidence O includes Y k , let X O = O ∩ X and Y O = O ∩ { Y 1 , … , Y k − 1 } . The following equality holds from Zhao et al. [42]:

Therefore, for an additive function g ( Y ) , we have

When each item of the aforementioned equation is identifiable, PostTCE [ X S ( x S ) ⇒ Y ∣ O = o ] is identifiable.

3.3 Relationship between posterior causal effect and population attributable risk

Greenland [10] pointed out that there are many incorrect equations regarding the probabilities of causation and the population attributable risks. The population attributable risks are used to measure the proportional amounts by which a disease risk would be reduced if risk factors were eliminated from a population [43]. For example, how much of the disease burden due to leukemia in a population could be eliminated if the exposures of benzene and ionizing radiation were eliminated from the population. In the following, we explain the relation of the posterior causal effects to the population attributable risks. For a case of a single Y and multiple causes X = ( X 1 , … , X p ) , the population attributable risk is defined by Bruzzi et al. [44] as follows:

It measures the proportional amount by which a disease risk would be reduced if all risk factors were eliminated from a population. Under Assumption 2 of no confounding and a weak monotonicity assumption that Y X = 0 ≤ Y X = x ≤ Y X = 1 for any x , the population attributable risk is equal to the posterior causal effect of multiple causes X on Y given the evidence of Y = 1 , i.e.,

AR does not measure how much the disease Y is attributed to a specified risk factor X k . The adjusted attributable risk for X k is defined by adjusting for the remaining risk factors X − k = X \ { X k } [44]:

Note that the set X − k should not contain any intermediate factor between X k and Y since eliminating X k can affect the intermediate factors in the condition X − k . Let A k = ( X 1 , … , X k − 1 ) denote the variable set, which is prior to X k in a topological causal order, and thus, AR ( X k ∣ A k ) is a proper adjusted attributable risk. Lu et al. [13] showed that under Assumption 2 of no confounding and a weak monotonicity assumption Y X k = 0 ≤ Y X k = 1 , the attributable risk of X k on Y adjusted for the set A k is equal to the posterior total effect of X k on Y given the evidence of Y = 1 :

Equations (2) and (3) show the relationships between the posterior causal effects and the population and adjusted attributable risks, and they explain the causal meaning of the attributable risks in terms of the potential outcome framework. The equations also give other identification equations of posterior causal effects PostTCE ( X ⇒ Y ∣ Y = 1 ) and PostTCE ( X k ⇒ Y ∣ Y = 1 ) under the weaker monotonicity assumptions than Assumption 1 of monotonicity.

3.4 Diagnostic approaches based on Bayesian posterior probabilities and posterior causal effects

In the following, we discuss the problem about whether the medical diagnosis should be based on Bayesian posterior probabilities or the posterior causal effects. Bayesian posterior probabilities measure the uncertainty of past events given the later observed evidence, but they do not capture the causal relationships between these events. Posterior causal effects, on the other hand, measure the uncertainty of past events that have causal effects on the later happened evidence.

In the field of medical diagnosis, a probabilistic expert system based on Bayesian posterior probabilities was developed by Lauritzen and Spiegelhalter [45] and Spiegelhalter et al. [46]. This system computes the posterior probabilities of diseases given the observed symptoms, which depend on the prior probabilities of the diseases. The diagnosis based on the maximum posterior probability minimizes the misdiagnosis error [47]. However, this approach does not account for the causal relationships between diseases and symptoms.

As Encyclopaedia Britannica [48] defines, the diagnostic process is the method by which health professionals select one disease over another, identifying one as the most likely cause of a person’s symptoms. Richens et al. [49] pointed out that most existing diagnostic algorithms, including Bayesian model-based and deep learning methods, rely on associative inference, and they identified diseases based on how correlated they are with a patient’s symptoms and medical history. This contrasts with how doctors perform medical diagnosis, selecting the diseases that offer the best causal explanations for the patient’s symptoms. They argued that disease diagnostic reasoning should satisfy three principles concerning not only the posterior probability, but also causality and simplicity. They proposed an approach based on a noisy-operation model, but their model is restricted to the case where neither diseases nor symptoms can affect each other. Li et al. [14] proposed a medical diagnostic approach based on posterior intervention causal effects PostICE, which satisfies the aforementioned principles for medical diagnosis. For disease diagnosis, the evidence O contains some symptoms and backgrounds of a patient, but does not contain the status of diseases X k . Since PostICE ( X k ⇒ Y ∣ O = o , X k = 0 ) = 0 , we can obtain

This equation means that the diagnostic approach based on PostICE considers not only Bayesian posterior probability pr ( X k = 1 ∣ o ) , but also the posterior causal effect of the disease X k on the symptoms Y in the subpopulation with the disease X k = 1 , which is ignored by the approach based on Bayesian posterior probability. For a patient given the evidence O = o , we diagnose the patient with a disease X k , which has the largest value in { PostICE ( X j ⇒ Y ∣ o ) , ∀ j } . It means that the number of symptoms could be eliminated at most if the patient had not gotten the disease X k .

When a patient may have multiple diseases simultaneously, Bayesian approach diagnoses the patient with multiple diseases based on the maximum posterior probabilities pr ( X = x ∣ O = o ) . The approach based on posterior causal effects uses PostICE ( X S ⇒ Y ∣ O = o ) for diagnosis. For a patient given the evidence O = o , we diagnose the patient with multiple diseases X S , which has the largest value in { PostICE ( X S ′ ⇒ Y ∣ o ) , ∀ S ′ ⊆ { 1 , … , p } } . Bayesian posterior probabilities and posterior intervention causal effects have the following equation:

If the diagnostic result is used further for eliminating the symptoms in the future, the Bayesian diagnostic approach may not be optimal, and the diagnostic approach based on posterior causal effects requires an assumption of invariance maybe require the following assumption of invariant relationships between causes and effects in the past and the future.

This assumption can be weakened to that the average causal effects of the diseases on symptoms are invariant across time in the subpopulations of O = o , i.e., E ( Y X = 1 − Y X = 0 ∣ O = o ) = E ( Z W = 1 − Z W = 0 ∣ O = o ) . This invariance assumption may hold for many real scenarios, e.g., in a specified room, X denotes a switch on or off, and Y a light on or not. But the assumption may not hold in some real scenarios, e.g., X denotes that Jack drank poison and Y = 1 denotes that he died.

In the following, we use a numerical example of medical diagnosis to compare the approach based on Bayesian posterior probabilities with that based on the posterior causal effects.

Figure 2

Causal diagram of two diseases X 1 and X 2 and a symptom Y .

Figure 3

Misclassification probabilities of two approaches.

Figure 4

PostICEs of two approaches.

To diagnose possible diseases, regardless of whether or not they are the causes of the occurred symptoms, Bayesian diagnosis based on posterior probabilities always has the minimum misclassification probability [47]. One drawback of the Bayesian diagnostic approach is that it may not identify the causes of occurred symptoms. To identify the causes, we argue that the approach based on posterior causal effects is a better choice. In the aforementioned numerical example, we assume that the probabilities of the causal mechanism are known. A limitation of the approach based on posterior causal effects is that the identifiability of the posterior causal effects requires Assumptions 1 and 2 of monotonicity and no confounding. Under Assumption 1 of monotonicity, patients with symptom Y = 1 are always diagnosed with certain diseases and are not diagnosed with no disease since PostICE ( X S = ∅ ⇒ Y ∣ Y = 1 ) = 0 has the least value.