Some Considerations on the Back Door Theorem and Conditional Randomization

1 Introduction

The main objective of this short report is to discuss the relationship between the back door theorem (Pearl 2000, 79) and the conditional randomization (or exchangeability) assumption. We will relate these two concepts through the d-separation rule, constructing both counterfactual variables and observed treatments in the same graph.

During the last years, several authors have been concerned with this problem. The twin directed acyclic graphs (DAGs), presented by Balke and Pearl (1994), allow simultaneous construction of observed and counterfactual variables.

Recently, Richardson and Robins (2013) presented a graphical theory based on single world intervention graphs (SWIGs), unifying causal directed graphs and potential outcomes. The present study, less ambitious, can be considered as a complementary work focusing in particular on the back door theorem, one of the most popular criteria to identify the distribution of counterfactual variables. We can present our results in a simple way, accessible to those who may be nonexperts in the mathematical technicalities, but still familiar with the field and with non-parametric structural equation models (NPSEM) (Pearl 2000, ch. 7). In this setting, Pearl has proposed a modified NPSEM where potential outcomes are defined by replacing the equations related to the treatment nodes by the constants corresponding to the desired intervention. Therefore, the observed treatment assignment and counterfactual variables do not occur together, neither in the model for the observed data nor in the modified model.

We propose here a new modified NPSEM model containing both treatments and counterfactual variables. Unlike the case of the twin graph, variables in this new model factorize according to the back door theorem graph, namely the DAG, where arrows emerging from nodes associated with the intervention are removed. We use this fact to prove that, for univariate treatment, the graphical assumptions of Pearl’s back door theorem (Pearl 2000, 79) imply conditional exchangeability. In this way, we establish a new proof of Pearl’s back door theorem and thus of identifiability of the mean of the counterfactual variables, a proof which can be understood by both proponents of the counterfactual and graphical approaches to causality.

It should be said that even if Richardson and Robins’ graphs differ from the graphs in this paper, our perspectives are rather similar. One difference is that, although they assume an underlying NPSEM, they do not assume independence of the disturbances (as we do). Instead, they prefer to work with the Finest Fully Randomized Causally Interpretable Structured Tree Graphs model (FFRCISTG), introduced in Robins (1986), but in the sense of Definition 2 (p. 23) presented in Richardson and Robins (2013).

In this work, we start assuming NPSEM with independent errors (NPSEM-IE). Although NPSEM-IE are less general than FFRCISTG models, we decided to present our proposal under this setting, considering that these models are commonly used. However, in Section 4 we show that all the results presented in this work remain valid if FFRCISTG models are assumed instead.

This work is organized as follows. In Section 2 we present a simple example with a three-node DAG, explaining the main idea to construct jointly the observed treatment assignment and counterfactual variables. We check in this example that the assumptions of the back door theorem imply conditional randomization. In Section 3 we generalize these results, first for the case of an intervention on one node, and then for many nodes. In Section 4 we discuss our results in the framework of FFRCISTG models.

To conclude this introduction, we would like to establish a subtle difference frequently omitted. Given a DAG G, we use V={V1,V2,…,Vn} to denote the nodes of a graph, whereas random variables associated with a given node Vi are denoted by Vi or by some perturbation of Vi, like Vi,t or Vit, as it will be explained later on.

2 Toy example – main idea

In the potential outcome framework (Rubin 1974), the identifiability of the average treatment effect is guaranteed under the assumption of conditional randomization (or ignorability). It states that there exists a vector L of observed variables such that Ya and A are independent given L, for a=t,c, where A is a binary treatment variable taking values in {t,c} while Ya denotes the potential outcomes under treatment level a. More precisely, Ya is the outcome variable that would have been observed in a hypothetical world in which all individuals received treatment level a. Denoting by Y the observed outcome and assuming that it satisfies Y=YtIA=t+YcIA=c, we get that the average treatment effect (ATE=E[Yt]−E[Yc]) is identified by the distribution of observed data (L,A,Y) by the formula ATE=EE[Y|A=t,L]−EE[Y|A=c,L].

The d-separation criterion (Pearl 2000, 18) is a graphical tool designed to check independence and conditional independence between coordinates (or sub-vectors) of a random vector whose distribution satisfies the Markov factorization with respect to a given DAG. Then, one is tempted to use such a tool to decide whether conditional ignorability can be assumed for the problem under consideration, studying the DAG associated with it.

For those who are familiar with DAGs, the back door theorem is a famous result used to identify the distribution of the counterfactual variables, and its assumptions give rise to the same formula presented under conditional exchangeability for identifying the average treatment effect. So, we asked ourselves whether the graphical conditions required by the back door theorem allow to prove conditional exchangeability using the d-separation criterion. To answer this question, we need to construct both counterfactual and treatment in the same DAG, in particular in the DAG involved in the back door theorem. To do so, we propose a simple modification to the approach presented by Pearl to define counterfactual variables. In the coming example, we outline the basic idea of our construction, which is generalized in the following section.

Assume that the causal diagram associated with the problem of interest is given by the DAG G (Figure 1).

Figure 1

The original DAG G

In terms of NPSEM (Pearl 2000, ch. 7 or 2009) this means that there exists a set of functions F={fL,fA,fY} and jointly independent disturbances U={UL,UA,UY}, which give rise to factual variables according to the following recursive system:

We use M=(F,U) to denote the model which defines the factual variables. To emulate the intervention do(a), Pearl (2000, ch. 7 or 2009) considers a model Ma where the function fA is replaced by the constant a, while the disturbances remain unchanged: Ma=(Fa,U), with Fa={fa,L,fa,A,fa,Y}, where

The variables obtained iterating the functions in model Ma using the same vector of disturbances U = {UL,UA, UY} are denoted with the subindex a: La, Aa and Ya. In this way, the counterfactual response of interest at level a is given by Ya.

Our proposal to represent counterfactual variables consists in the use of a new system of functions, in which the value a is inserted in lieu of the variable corresponding to the node A, every time this one is required by the recursion. To do so we change the functions related to each node having A as parent. In the present example, Ma=(Fa,U), with Fa={fLa,fAa,fYa}, where

Note that this new set of functions is compatible with the DAG GA_, where arrows emerging from A are removed (Figure 2).

Figure 2

GA_, constructed removing in G arrows emerging from A

The variables constructed iterating the functions in Fa and using the same vector of disturbances {UL,UA,UY} are denoted by the supraindex a: La, Aa and Ya. Then, we get that the distribution of (La,Aa,Ya) is compatible with GA_.

The following Lemma and Corollary summarize the main results of this section.

3 Intervention with constant regimes

4 FFRCISTG models

In the previous results, we used the rules of d-separation to detect independence or conditional independence between variables of a random vector. To do so, given a graph G, all we required from the joint distribution of our vector was compatibility with G. When variables are constructed following a NPSEM-IE, the Markov factorization induced by G holds automatically, and that is why our results are valid when the errors are independent.

However, the Markov factorization remains true under weaker conditions. For instance, let v=(v1,…,vn)∈∏j=1nVj and call vpaG(Vi) the subvector of v containing the coordinates related to the nodes in the set PAG(Vi), namely vpaG(Vi)=(vj:Vj∈PAG(Vi)). If

then, the distribution of the vector whose variables are constructed with M=(F,U) is compatible with the graph G. This condition mainly defines the FFRCISTG models (Richardson and Robins 2013).

It is worth noting that if M=(F,U) satisfies condition [9] relative to G, the intervened model Ma=(Fa,U), defined in Section 3.2, also satisfies condition [9] relative to GA_set, since

where vpaG(Vi)a denotes the vector that results from replacing vj with aj for {j:Aj∈Aset}. Then, the distribution of the variables constructed using the model Ma, requiring that the errors satisfy only condition [1], factors according to the graph GA_set, allowing the use of d-separation rules, and thus extending our results to this new model.