A graphical description of partial exchangeability

Abstract

Partial exchangeability is sufficient for identification of some causal effects of interest. Here, we review the use of common graphical tools and the sufficient component cause model in the context of partial exchangeability. We illustrate the utility of single world intervention graphs (SWIGs) in depicting partial exchangeability and provide an illustrative example of when partial exchangeability might hold in the absence of complete exchangeability.

Introduction

Exchangeability is often used as a condition for the identification of average treatment effects (1,2). Informally, for a binary time-fixed treatment, we say that exchangeability is complete when (i) the outcomes of the treated are representative of the outcomes of the untreated had they been treated, and (ii) the outcomes of the untreated are representative of the outcomes of the treated had they been untreated. In randomized experiments with perfect compliance, we expect that complete exchangeability will hold unconditionally; in observational studies, we might assume that complete exchangeability holds within levels of measured covariates (3).

Complete exchangeability, however, is not necessary for the identification of some causal effects. The weaker condition (i), which we refer to as partial exchangeability under treatment can be used to identify the average treatment effect in the treated, and condition (ii), which we refer to as partial exchangeability under no treatment, can be used to identify the average treatment effect in the untreated (1).

Here we use graphical tools and the sufficient component cause model to provide an illustrative example of a setting in which a partial exchangeability condition holds even when complete exchangeability does not.

Causal graphs and partial exchangeability

Let us denote the random variable A as a time-fixed binary treatment, Y^a=1 as an individual’s potential (counterfactual) outcome under treatment, and Y^a=0 as an individual’s potential outcome under no treatment. Partial exchangeability under treatment (i) holds if Y^a=1 ⫫ A, and partial exchangeability under no treatment (ii) if Y^a=0 ⫫ A. Complete exchangeability holds when partial exchangeability holds for both treatment levels.

Causal directed acyclic graphs (DAGs), which have nodes representing real-world (factual) variables and edges representing direct causal effects, can describe complete exchangeability (4,5). Throughout we assume that causal DAGS represent Finest Fully Randomized Causally Interpreted Structural Tree Graph (FFRCISTG) models, which only make counterfactual independence assumptions that are, at least in principle, experimentally testable (6).

Under faithfulness, and in the absence of selection and measurement bias, complete exchangeability is equivalent to the absence of any unblocked backdoor paths between treatment and outcome on the causal DAG (5). However, there is no equivalence between partial exchangeability and a graphical criterion on the causal DAG.

To see this, suppose that an unmeasured covariate U₁ affects both treatment A and outcome Y, as represented by the causal diagram of Figure 1a. Then, we can infer that complete exchangeability does not hold because the causal DAG shows an unblocked backdoor path. Now suppose that U₁ affects an individual’s outcome Y only when the individual receives treatment (A=1) but not when the individual remains untreated (A=0). Then partial exchangeability holds under no treatment but not under treatment (i.e. Y^a=1 EQ\o (⫫,\) A, Y^a=0 ⫫ A). However, the causal DAG would still show U₁ as a common cause of A and Y. That is, the DAG alone does not allow us to conclude that the average treatment effect in the treated is identified.

Figure 1:

(a) A causal directed acyclic graph (DAG) in which A represents a received binary treatment, Y a binary outcome, and U₁ the unmeasured confounder of the effect of A on Y. The backdoor criterion is not met due to the presence of a backdoor path A← U₁→Y which includes an unmeasured variable, U₁. (b) A pair of single world intervention graphs (SWIGs) consistent with the DAG in (a) and in which a split node indicates an intervention on the treatment variable, A, and the potential (counterfactual) outcomes Y^a=1 and Y^a=0 are indicated on the graphs. The backdoor criterion is not met due to the presence of a backdoor path A←U₁→Y^a on both graphs, which includes the unmeasured variable, U₁. (c) A pair of SWIGs consistent with the DAG in (a) describing partial exchangeability in the absence of treatment. The backdoor criterion is met for the graph representing a=0. (d) A pair of SWIGs consistent with the DAG in (a) describing partial exchangeability in the presence of treatment. The backdoor criterion is met for the graph representing a=1.

In contrast, single world intervention graphs (SWIGs) (7,8) can represent partial exchangeability. Briefly, SWIGs can be derived from DAGs by ‘splitting’ treatment nodes and replacing variables affected by treatment by their counterfactual versions. SWIGs are often expressed as templates, implying that the relevant causal structure is invariant between single worlds under different treatment levels. However, this need not be the case. Consider the pairs of SWIGs in Figures 1b, 1c, and 1d. Each pair of SWIGs is compatible with the DAG in Figure 1a and, importantly, the latter two pairs (1c and 1d) graphically describe independencies consistent with partial exchangeability but not complete exchangeability. Specifically, in these graphs, U₁ has a direct causal effect on the potential outcome under one level of treatment but not the other.

An example of partial exchangeability

Consider a hypothetical randomized trial in which adults with acute uncomplicated appendicitis are randomized to either appendectomy or a course of antibiotics, and are then followed for 30 days to assess the presence of complications. Let A be an indicator for the actual treatment received, Y an indicator for the presence of a complication at 30-days, and U₁ an unmeasured indicator for genetic susceptibility to malignant hyperthermia, which predisposes individuals to complications (Y=1) in the form of life-threatening reactions to volatile anesthetic agents (9). Some individuals with U₁ defy their assignment and switch treatment groups. Particularly, individuals genetically susceptible to malignant hyperthermia may be apprehensive about surgery, perhaps because they have family members who had adverse reactions to surgery, and preferentially choose antibiotic treatment even when randomly assigned to surgery. In this example, U₁ has a direct effect on the treatment received.

The causal DAG in Figure 1a represents this trial with imperfect compliance. Because susceptibility to malignant hyperthermia U₁ affects both the treatment received A and the presence of complications Y, the causal DAG shows that there is no complete exchangeability. However, the causal DAG cannot represent the lack of an effect of susceptibility to malignant hyperthermia on complications Y if all individuals received antibiotics (a=0). In contrast, the pair of SWIGs in Figure 1c naturally represent this situation. That is, partial exchangeability Y^a=0 ⫫ A holds. To further illustrate when this condition might hold in simplified settings, it is useful to link our discussion thus far with the sufficient component cause model.

Sufficient component causes and partial exchangeability

Suppose we assume a simple deterministic model for disease occurrence where the set of sufficient causes in Figure 2 represent the only possible covariate patterns sufficient for the outcome to occur. Therefore, individuals can be classified based on their covariate patterns into four causal types. Individuals with U₁=1, U₂=0, and U₃=0 are “causal”– they would only get the outcome under treatment. Individuals with U₂=1, or both U₁=1 and U₃=1 are “doomed” – they would get the outcome regardless of treatment. Individuals with U₁=0, U₂=0, and U₃=1 are “preventive” – they would only get the outcome in the absence of treatment. Individuals with U₁=0, U₂=0, and U₃=0 are “immune” – they would never get the outcome.

Figure 2:

Sufficient causes in which an outcome can occur under any of three covariate patterns: (U₁=1, A=1); (U₂=1); and (U₃=1, A=0), but will not otherwise occur.

Complete exchangeability holds when the proportion of “doomed” plus “preventive” type individuals (i.e., those who would get the outcome in the absence of treatment) and “doomed” plus “causative” type individuals (i.e., those who would get the outcome under treatment) is the same in the treated and untreated groups (10,11). But partial exchangeability requires only one of the two conditions. Specifically, if the proportion of “doomed” plus “preventive” type individuals is the same in the treated and untreated groups, then the average treatment effect only amongst the treated can be computed.

In Table 1, we present hypothetical observed data from the trial described above (see columns labeled “With non-compliance”), alongside hypothetical data from the same trial, had all individuals complied with their assigned treatments (see columns labeled “Under perfect compliance”). For illustrative purposes, individuals in the table are stratified based on covariate patterns: their values for unmeasured causes of the outcome (U₁, U₂, and U₃).

Table 1.

Partial exchangeability after non-ignorable treatment non-compliance in a hypothetical randomized controlled trial

		Treatment group
		Under perfect compliance				With non-compliancec
		Antibiotics (A=0)		Surgery (A=1)		Antibiotics (A=0)		Surgery (A=1)
Covariate Patterna	Causal typeb	n	p	n	p	n	p	n	p
U1=1, U2=0, U3=0	Causal	1800	0.20	2700	0.20	3600	0.32	900	0.08
U1=1, U2=0, U3=1	Doomed	150	0.15	225	0.15	300	0.16	75	0.14
U1=1, U2=1, U3=0		150		225		300		75
U1=1, U2=1, U3=1		150		225		300		75
U1=0, U2=1, U3=1		450		675		450		675
U1=0, U2=1, U3=0		450		675		450		675
U1=0, U2=0, U3=1	Preventive	450	0.05	675	0.05	450	0.04	675	0.06
U1=0, U2=0, U3=0	Immune	5400	0.60	8100	0.60	5400	0.48	8100	0.72
Factual and counterfactual probabilities, conditional on treatment group
P(Y=1 | A=a)		0.20		0.35		0.20		0.22
P(Ya=1 =1 | A=a)		0.35		0.35		0.48		0.22
P(Ya=0 =1 | A=a)		0.20		0.20		0.20		0.20

Y = complications (1: yes / 0: no); A = received treatment (1: surgery / 0: antibiotics); U₁ = malignant hyperthermia (1: yes / 0: no); U₂ = arbitrary risk factor (1: yes / 0: no); U₃ = arbitrary risk factor (1: yes / 0: no). U₁ is independent of U₂ and U₃. The data have been generated for illustrative purposes.

Under perfect compliance, P(Y=1 | A=0) = P(Causal Type = doomed or preventive | A=0) = P(Y^a=0=1 | A=0) = P(Y^a=0=1 | A=1) = 0.15 + 0.05 = 0.20, and P(Y=1 | A=1) = P(Causal Type = doomed or causal | A=1) = P(Y^a=1=1 | A=1) = P(Y^a=1=1 | A=0) = 0.15 + 0.20 = 0.35.

Under imperfect compliance, P(Y=1 | A=0) = P(Causal Type = doomed or preventive | A=0) = P(Y^a=0=1 | A=0) = 0.16 + 0.04 = P(Y^a=0=1 | A=1) = 0.14 + 0.06 = 0.20. However, P(Y=1 | A=1) = P(Causal Type = doomed or causal | A=1) = P(Y^a=1=1 | A=1) = 0.14 + 0.08 = 0.22 ≠ P(Y^a=1=1 | A=0) = 0.32 + 0.16 = 0.48.

It can be seen that under initial randomization and with perfect compliance, complete exchangeability Y^a ⫫ A holds since P(Y^a=1=1 | A=1) = P(Y^a=1=1 | A=0) = 0.35, and P(Y^a=0=1 | A=1) = P(Y^a=0=1 | A=0) = 0.20. As such, all counterfactual risks (P(Y^a=1=1 | A=1), P(Y^a=0=1 | A=1), P(Y^a=1=1 | A=0), P(Y^a=0=1 | A=0)) are identified from the observed data (P(Y=1 | A=1), P(Y=1 | A=0)).

In this numerical example, two-thirds of individuals with U₁=1 who are initially assigned to surgery switch to antibiotic treatment, and switching is independent of other covariates (i.e. U₂ and U₃). Because switching is independent of other prognostic factors, and because U₁ is only a prognostic factor for those treated surgically, even after switching P(Y^a=1=1 | A=1) and P(Y^a=0=1 | A=1) can be identified using the observed data, allowing calculation of the effect of surgical treatment amongst those treated surgically. Note that this is not the case for those treated with antibiotics. Specifically, partial exchangeability holds under treatment with antibiotics, Y^a=0 ⫫ A, since P(Y^a=0=1 | A=1) = P(Y^a=0=1 | A=0) = 0.20, but not under treatment with surgery, Y^a=1 EQ \o (⫫,\) A since P(Y^a=1=1 | A=1) ≠ P(Y^a=1=1 | A=0)

^aIn this simple deterministic model, complications (Y) are sufficiently determined by the following covariate patterns: (U₁=1, A=1); (U₂=1); and (U₃=1, A=0).

^bIndividuals are classified in one of four principal strata: causal (Y^a=1=1, Y^a=0=0); doomed (Y^a=1=1, Y^a=0=1); preventive (Y^a=1=0, Y^a=0=1); and immune (Y^a=1=0, Y^a=0=0). Principal strata are deterministic functions of U₁, U₂, and U₃ (see footnote 1)

^cTwo thirds (2/3) of individuals assigned to surgery (A=1) with U₁ = 1 did not adhere to their assigned treatment, switching to antibiotics (A=0); non-compliance was independent of U₂ and U₃. This setting is represented in Figure 1.

Under perfect compliance, as is evident in Table 1, the proportions of covariate patterns (and causal types), and therefore the average potential outcomes, are equal in the two treatment groups, and would be expected to be so in any randomized experiment with perfect compliance. That is, complete exchangeability holds.

Alternatively, under imperfect compliance, because of confounding by U₁, the proportions of covariate patterns (and causal types) are not all equal between exposure groups. Despite this, as a consequence of our simplified deterministic model and the specific nature of confounding in this case, the proportion of “doomed” plus “preventive” type is equal across received treatment groups, and thus so is the average potential outcome under a=0 . This is because the prognostic factor that affects compliance (U₁) is present only in sufficient causes with a single level of treatment (Figure 2) and is not associated with other prognostic factors (as is evident upon careful inspection of Table 1). Under these conditions, in general, partial exchangeability is expected to hold for received treatment. Because malignant hyperthermia, U₁, only causes complications in those treated with surgery (A=1) and its accompanying anesthesia, we say that A and U₁ interact to cause Y.

Therefore, under these conditions, we can use the outcomes of the group treated with antibiotics to learn what would have happened to the group treated with surgery had they been treated with antibiotics, thus identifying the effect of surgery versus antibiotics among those actually treated with surgery. However, the reverse is not true. We would expect the antibiotic group, had they been treated surgically, to have a higher risk of complications than the risk actually experienced by the surgically treated group (also evident in Table 1). As such, the outcomes of the group treated with surgery cannot be used to learn about the potential outcomes of the group treated with antibiotics. Note that the effect of treatment on the surgically treated group is relevant in this example because surgically treated individuals differ substantively from the population at large (specifically, in their probability of being predisposed to malignant hyperthermia).

Discussion and Conclusion

We described an example with partial, but not complete, exchangeability – a scenario that can be represented by SWIGs, but not by regular DAGs. Our scenario assumed that our simple deterministic model, with only a single predictor of noncompliance, was correct. But more complex scenarios could also be represented by SWIGS. For example, we could consider several measured prognostic variables that are common causes of noncompliance and the outcome in addition to the unmeasured variable. Then, after all measured confounders have been adjusted for, the average treatment effect in the treated can still be identified. If the unmeasured confounder causes the outcome only in the absence of treatment, then the average treatment effect in the untreated can be computed.