Data-Driven Causal Effect Estimation Based on Graphical Causal Modelling: A Survey

Causal relationships among variables are normally represented by a directed acyclic graph (DAG), denoted as $\mathcal{G}=(\mathbf{X},\mathbf{E})$, which contains a set of nodes (variables), directed edges between nodes, i.e. $\mathbf{E}$, and has no directed cycles. In a DAG, when there is a directed edge $X_{i}\rightarrow X_{j}$, $X_{i}$ is known as the parent of $X_{j}$ and $X_{j}$ is a child of $X_{i}$. A path between two nodes $(X_{i},X_{j})$ is a set of consecutive edges linking $(X_{i},X_{j})$ regardless of the edge directions, and a directed path from $X_{i}$ to $X_{j}$ contains a set of consecutive edges pointing towards $X_{j}$, i.e. $X_{i}\rightarrow\dots\rightarrow X_{j}$. $X_{i}$ is an ancestor of $X_{j}$ and $X_{j}$ is a descendant of $X_{i}$ if there is a directed path from $X_{i}$ to $X_{j}$. We use $Pa(X)$, $Ch(X)$, $An(X)$ and $De(X)$ to denote the sets of all parents, children, ancestors and descendants of $X$, respectively.

In a DAG, if an edge $X_{i}\rightarrow X_{j}$ denotes that $X_{i}$ is a direct cause of $X_{j}$, the DAG is termed a causal DAG. In a causal DAG $\mathcal{G}$, a directed path between two nodes is called a causal path and a non-directed path between two nodes is a non-causal path. Let $\ast$ be an arbitrary edge mark. $X_{i}$ is a collider on a path $\pi$ if $X_{k}\put(0.4,-2.5){*}\rightarrow X_{i}\leftarrow\put(-4.0,-2.5){*}X_{j}$ is a sub-path of $\pi$. A collider path is a path with every non-endpoint node being a collider. A path of length one is a trivial collider path.

Some assumptions are needed to use a causal DAG $\mathcal{G}$ to represent the data generation mechanism, including the Markov condition, the faithfulness assumption and causal sufficiency assumption.

The Markov condition indicates that the distribution of $X_{i}$ is determined by the distributions of its parents solely. With the Markov condition, $P(\mathbf{X})$ can be factorised as $P(\mathbf{X})=\prod_{i}P(X_{i}|Pa(X_{i}))$ according to $\mathcal{G}$.

When a distribution $P(\mathbf{X})$ and a DAG $\mathcal{G}=(\mathbf{X},\mathbf{E})$ are faithful to each other, if two variables $X_{i}$ and $X_{j}$ in $\mathcal{G}$ are $d$-separated by another set of variables $\mathbf{Z}$ in $\mathcal{G}$ as defined below, $X_{i}$ and $X_{j}$ are conditional independent given $\mathbf{Z}$. That is, conditional independence and dependence relationships between variables in $P(\mathbf{X})$ can be read off from the DAG based on $d$-separation or $d$-connection.

An exemplary DAG representing a causal mechanism (causal relationships between all variables in a problem domain) is shown in Fig. 2 (a). In the DAG, $X_{1}\to W\to Y$ is a causal path, and this path is $d$-separated by $W$. When we consider a $d$-separate set $\mathbf{Z}$ for the pair $(X_{1},Y)$, all five paths between the two variables need to be $d$-separated by $\mathbf{Z}$. $W$ should be in $\mathbf{Z}$ because it is the only variable $d$-separating the path $X_{1}\to W\to Y$. Now, $\mathbf{Z}$ $d$-connects $(X_{4},X_{2})$, $(X_{4},X_{3})$, $(X_{4},X_{6})$, $(X_{2},X_{6})$, and $(X_{3},X_{6})$ on the four other paths respectively since $W$ is a collider in every path. Let’s look into these four paths separately. Path $X_{1}\to W\leftarrow X_{6}\to X_{7}\leftarrow X_{8}\to Y$ is $d$-separated by $\emptyset$ since $X_{7}$ is a collider even when the collider $W$ is included in $\mathbf{Z}$. Path $X_{1}\to W\leftarrow X_{4}\to X_{5}\to Y$ is $d$-separated by set $\{X_{4}\}$ or $\{X_{5}\}$. Paths $X_{1}\to W\leftarrow X_{2}\to X_{3}\to Y$ and $X_{1}\to W\leftarrow X_{3}\to Y$ are $d$-separated by $\{X_{3}\}$. Consequently, the pair $(X_{1},Y)$ is $d$-separated by set $\mathbf{Z}=\{X_{3},X_{4},W\}$ or $\mathbf{Z}=\{X_{3},X_{5},W\}$.

In numerous real-world applications, latent common causes (a.k.a. latent confounders) are commonplace such that the causal sufficiency assumption is violated (Spirtes et al., 2000; Pearl, 2009a). In this situation, the ancestral graph is designed for representing causal ancestral relationships (Richardson and Spirtes, 2002, 2003). An ancestral graph is a mixed graph without directed cycles or almost directed cycles. An ancestral graph contains both directed edges $\rightarrow$ and bidirected edges $\leftrightarrow$ between variables. An almost directed cycles occurs if $X_{i}\leftrightarrow X_{j}$ in $\mathcal{G}$ and $X_{i}\in An(X_{j})$. Similar to $d$-separation in a DAG, $m$-separation serves as a general criterion to read off the independencies and dependencies between measured variables entailed by an ancestral graph (Richardson and Spirtes, 2002).

We take the ancestral graph in Fig. 2 (b) as an example for understanding $m$-separation. $X_{1}\to W\to Y$ is a causal path and the path is $m$-separated by $W$. When we consider an $m$-separated set $\mathbf{Z}$ for the pair $(X_{1},Y)$, all five paths between the two nodes need to be m-separated by $\mathbf{Z}$. $W$ should be in $\mathbf{Z}$ because it is the only variable $m$-separating the path $X_{1}\to W\to Y$. Path $X_{1}\to W\leftrightarrow X_{7}\leftrightarrow Y$ is $m$-separated by $\emptyset$ since $X_{7}$ is a collider. Path $X_{1}\to W\leftarrow X_{2}\to X_{3}\to Y$ is $m$-separated by set $\{X_{3}\}$. Paths $X_{1}\to W\leftarrow X_{3}\to Y$ and $X_{1}\to W\leftrightarrow X_{5}\to Y$ are $m$-separated by $\{X_{3},X_{5}\}$. Hence, the pair $(X_{1},Y)$ is $m$-separated by set $\mathbf{Z}=\{W,X_{3},X_{5}\}$.

When there are latent confounders in a system, a Maximal Ancestral Graph (MAG) is commonly used to represent causal relationships among measured variables (Richardson and Spirtes, 2002, 2003; Zhang, 2008a).

A bidirected edge $X_{i}\leftrightarrow X_{j}$ in a MAG indicates the presence of a latent confounder between $X_{i}$ and $X_{j}$, and the absence of a direct causal relationship between $X_{i}$ and $X_{j}$. A causal path in a MAG is a directed path without a bidirected edge. In a MAG, $X_{i}$ and $X_{j}$ are spouses if there exists a bidirected edge $X_{i}\leftrightarrow X_{j}$. The assumptions of Markov condition and faithfulness are still held in a MAG. Note that the causal relationships between measured variables in a MAG are still the same as in the underlying DAG over the measured and unmeasured variables (Richardson and Spirtes, 2002; Zhang, 2008b, a). Thus, in a MAG, we still use $Pa(X)$, $Ch(X)$, $An(X)$ and $De(X)$ to denote the sets of all parents, children, ancestors and descendants of $X$, respectively.

Another type of causal graphs to represent causal relationships between variables with latent confounders is the acyclic directed mixed graph (ADMG) (Tian and Pearl, 2002; Peña, 2018; Runge, 2021). ADMGs are generalisations of DAGs (Silva et al., 2011; Evans and Richardson, 2014), meaning that the causal relationships between observed variables in ADMGs can be encoded by marginalising causal DAGs (Richardson and Spirtes, 2002; Richardson, 2003). In an ADMAG, it is possible for two nodes to have more than one edge between them. For example, both a directed edge ($\rightarrow$) and a bidirected edge ($\leftrightarrow$) connect two nodes in an ADMG. Some works for using ADMG for causal discovery and inference can be found in (Silva et al., 2011; Evans and Richardson, 2014; Bhattacharya et al., 2020; Runge, 2021). However, it is challenging to learn an ADMG from data since it allows $W\rightarrow Y$ and $W\leftrightarrow Y$ simultaneously. To the best of our knowledge, there are no established data-driven approaches for learning ADMGs from data, which leads to the limited use of ADMGs in data-driven causal effect estimation. Consequently, we have refrained from reviewing ADMG-based methods in the subsequent sections.

Learning a DAG (or a MAG) directly from observational data is challenging. Instead, most existing structured learning algorithms (Neapolitan et al., 2004; Spirtes, 2010; Aliferis et al., 2010) are capable of learning a Markov equivalence class of DAGs (or MAGs) from the data (with latent confounders). A Markov equivalence class refers to a set of DAGs (or MAGs) encode the same set of conditional independence relations between measured variables.

A CPDAG (complete partial DAG) is used to represent a Markov equivalence class of DAGs (Pearl, 2009a; Spirtes, 2010; Aliferis et al., 2010), i.e. all the DAGs that encode the same d-separations as the underlying causal DAG. In a CPDAG, an undirected edge indicates that the orientation of this edge is uncertain. For example, the edge $W\;\put(0.9,2.1){\circle{2.5}}-\put(-0.9,2.1){\circle{2.5}}\;X_{1}$ in the learned CPDAG in Fig. 3 indicates a directed edge $W\rightarrow X_{1}$ or $W\leftarrow X_{1}$.

A PAG (Partial Ancestral Graph) is often used to represent a Markov equivalence class of MAGs (Spirtes et al., 2000; Richardson and Spirtes, 2003, 2002; Ali et al., 2009), i.e. all the MAGs that encode the same $m$-separations/$m$-connections as the underlying causal MAG. In a PAG, an undirected edge is allowed in addition to the edge types in a MAG. Like an undirected edge in a CPDAG, an undirected edge in a PAG also indicates uncertainty. For example, the edge $W\leftarrow\put(-2.0,2.5){\circle{2.5}}X_{2}$ in the PAG in Fig. 4 indicates a directed edge $W\leftarrow X_{2}$ or $W\leftrightarrow X_{2}$.

Because a PAG contains some edges that are not oriented, we need to introduce the term “definite status” for nodes and paths to add some certainty for $m$-separations and $m$-connections among nodes. In a PAG, a node $X_{j}$ is a definite non-collider on $\pi$ if there exists at least an edge out of $X_{j}$ on $\pi$, or both edges have a circle mark “$\circ$” at $X_{j}$ and the sub-path $\langle X_{i},X_{j},X_{k}\rangle$ is an unshielded triple (Zhang, 2008a). A node $X_{i}$ is known to have definite status if it is a collider or a definite non-collider on the path. A collider on a path $\pi$ is referred to as a definite collider since it is always of definite status. A path $\pi$ is of definite status if every node (excluding non-endpoint node) on $\pi$ is of definite status.

Let $W$ be a binary variable representing treatment status ($W$=1 for treated and $W$=0 for untreated) and $Y$ the outcome variable. The average causal effect (ACE) of $W$ on $Y$ is defined as $\operatorname{ACE}(W,Y)=\mathbf{E}(Y\mid do(w=1))-\mathbf{E}(Y\mid do(w=0))$, where the $do$-operator, $do(\cdot)$ (Pearl et al., 2009) represents an intervention by setting a variable to a specific value in an experiment.

Confounding adjustment is the most common way for obtaining unbiased causal effect estimation from observational data. Given an adjustment set $\mathbf{Z}$, $\operatorname{ACE}(W,Y)$ can be estimated unbiasedly as follows:

Therefore, the task of causal effect estimation is transformed into identifying an appropriate adjustment set. Note that, there may exist multiple adjustment sets, all leading to the unbiased estimation of $\operatorname{ACE}(W,Y)$.

Based on causal graphs, two main criteria have been developed and widely used, the back-door criterion (Pearl et al., 2009) and the generalised back-door criterion (Maathuis et al., 2015) for adjustment set identification in a causal DAG and MAG respectively.

We use the DAG in Fig. 2 (a) to illustrate the back-door criterion. The set $\{X_{3},X_{5}\}$ satisfies the back-door criterion and is a proper adjustment set w.r.t. $(W,Y)$, as it $d$-separates all three back-door paths: $W\leftarrow X_{4}\to X_{5}\to Y$, $W\leftarrow X_{2}\to X_{3}\to Y$, and $W\leftarrow X_{3}\to Y$. The other sets satisfying the back-door criterion are $\{X_{3},X_{4}\}$, $\{X_{2},X_{3},X_{4}\}$, $\{X_{2},X_{3},X_{5}\}$, $\{X_{3},X_{4},X_{5}\}$ and $\{X_{2},X_{3},X_{4},X_{5}\}$.

It is worth mentioning that in a given causal graph, the three do-calculus rules proposed by Pearl (Pearl, 1995a, 2009a) are sound and complete for determining whether a causal effect is identifiable (Shpitser and Pearl, 2006, 2008).

The generalised back-door criterion in a MAG is introduced as follows.

We use the MAG in Fig. 2 (b) to illustrate the generalised back-door criterion. The set $\{X_{2},X_{5}\}$ satisfies the generalised back-door criterion w.r.t. $(W,Y)$ as it $m$-separates all non-causal paths between $W$ and $Y$: $W\leftarrow X_{2}\rightarrow X_{3}\to Y$, $W\leftarrow X_{3}\to Y$, $W\leftrightarrow X_{5}\rightarrow Y$, and $W\leftrightarrow X_{7}\leftrightarrow Y$. The other sets satisfying the generalised back-door criterion are $\{X_{3},X_{5}\}$ and $\{X_{2},X_{3},X_{5}\}$.

Two other graphical criteria, the adjustment criterion (Shpitser et al., 2010) and the generalised adjustment criterion (Perković et al., 2018) extend the back-door criterion and the generalised back-door criterion, respectively, for the identification of adjustment sets. The adjustment criterion can be used to determine an adjustment set in a DAG, while the generalised adjustment criterion can be used to determine an adjustment set in four types of graphs, including DAG, CPDAG, MAG and PAG. The generalised adjustment criterion used for data-driven causal effect estimation will be introduced in Subsection 4.3.4.

Once an adjustment set is obtained, a confounding adjustment method can be used to remove confounding bias. Many confounding adjustment methods have been developed, such as Nearest Neighbor Matching (NNM) (Rosenbaum and Rubin, 1985; Sekhon, 2011; Cheng et al., 2022b), Propensity Score Matching (PSM) (Morgan and Harding, 2006; Rubin, 1973; Sekhon, 2011), Covariate Balance Propensity Score (CBPS) (Imai and Ratkovic, 2014), Inverse Probability of Treatment effect Weighting (IPTW) (Hirano et al., 2003; Robins et al., 2000a; Cole and Hernán, 2008; Hernán and Robins, 2020), Doubly Robust Learning (DRL) (Bang and Robins, 2005; Funk et al., 2011; Benkeser et al., 2017). For more details on confounding adjustment, please refer to the surveys (Stuart, 2010; Sauer et al., 2013; Imai and Ratkovic, 2014; Witte and Didelez, 2019; Yao et al., 2021; Guo et al., 2020), or the books (Pearl, 2009a; Koller and Friedman, 2009; Imbens and Rubin, 2015; Morgan and Winship, 2015; Hernán and Robins, 2020).

A proper adjustment set must include all variables that are sufficient to remove the confounding bias between $W$ and $Y$. However, when there is a latent common cause of $W$ and $Y$, confounding adjustment does not work. In such cases, the instrumental variable (IV) approach is a powerful way to estimate average causal effect (Hernán and Robins, 2006; Martens et al., 2006; Abadie, 2003; Angrist and Imbens, 1995).

As an example, in Fig. 5 (a), $S$ is a standard IV w.r.t. the pair of variables $(W,Y)$ since $S$ is a direct cause of $W$, all of its effect on $Y$ is through $W$, and $S$ and $Y$ do not share common causes.

When a standard IV is given, under the assumption that the data is generated by a linear structural equation (Angrist et al., 1996; Robins, 1997; VanderWeele and Shpitser, 2011), the $\operatorname{ACE}(W,Y)$ can be calculated as $\sigma_{sy}/\sigma_{sw}$, where $\sigma_{sy}$ and $\sigma_{sw}$ are the regression coefficients of $S$ when regressing $Y$ on $S$ and when regressing $Y$ on $W$, respectively. The two stage least square (TSLS) regression is one of the most commonly used IV estimators (Angrist and Imbens, 1995; Angrist et al., 1996; Imbens, 2014). The non-linear or non-parametric implementations of the IV estimators for calculating $\sigma_{sy}$ and $\sigma_{sw}$ can be found in (Hartford et al., 2017; Singh et al., 2019; Sjolander and Martinussen, 2019; Bennett et al., 2019). Data-driven causal effect estimation methods do not assume knowing IV. Thus the survey does not review the standard IV methods requiring a known standard IV. The reviews on the standard IV methods can be found in (Bowden and Turkington, 1990; Hernán and Robins, 2006; Martens et al., 2006; Guo et al., 2020).

A standard IV is difficult to find because of its stringent conditions. Ancestral IV (AIV) (Van der Zander et al., 2015) defined below relaxes some of the conditions of a standard IV. AIV is a practical definition of Conditional IV (CIV) and is used in data-driven causal effect estimation. More explanation is provided late.

An example of AIV is provided in Fig. 5 (b). We see that $S\rightarrow X_{3}\rightarrow Y$ violates requirement (ii) of a standard IV, and $S\leftarrow X_{1}\rightarrow X_{2}\rightarrow Y$ violates requirement (iii) of a standard IV. The set of variables $\mathbf{Z}=\{X_{1},X_{3}\}$ instrumentalises $S$ to be an AIV since $S\not\!\perp\!\!\!\perp_{d}W\mid\mathbf{Z}$ in the DAG, $S\perp\!\!\!\perp_{d}Y\mid\mathbf{Z}$ when $W\to Y$ is removed from the DAG, and all elements in $\mathbf{Z}$ are ancestors of $Y$.

Given an ancestral IV $S$ instrumentalised by $\mathbf{Z}$, the causal effect of $W$ on $Y$, $\operatorname{ACE}(W,Y)$, can be estimated by $\sigma_{sy\mathbf{z}}/\sigma_{sw\mathbf{z}}$, where $\sigma_{sy\mathbf{z}}$ and $\sigma_{sw\mathbf{z}}$ are the regression coefficient of $S$ when $Y$ is regressed on $S$ and $\mathbf{Z}$, and the regression coefficient of $S$ when $Y$ is regressed on $W$ and $\mathbf{Z}$, respectively.

An AIV is a CIV (Conditional IV) (Brito and Pearl, 2002), except that an AIV has the extra requirement that $\mathbf{Z}\subseteq(An(Y)\cup An(S))$. When a causal graph is not given and is to be discovered from data, a CIV may produce misleading results since it may be a variable that is uncorrelated with $W$ (Van der Zander et al., 2015). For example, in the DAG with $X_{1}\leftarrow U_{1}\rightarrow X_{2}\leftarrow U_{2}\rightarrow W\rightarrow Y$ and $W\leftarrow U\rightarrow Y$ where $\{U,U_{1},U_{2}\}$ are latent variables, $X_{1}$ is a CIV by definition, but is actually independent of $W$. This could result in a misleading conclusion. An AIV circumvents such problematic cases.

The IV approach has been extensively studied and developed, particularly in the field of economics (Arellano and Bover, 1995; Angrist et al., 1996; Abadie, 2003). There are two types of IV estimators: parametric and nonparametric. For a parametric IV estimator, the linearity and homogeneous causal effects assumptions are required to identify causal effects from data with latent confounders. The TSLS estimator is one of the most well-known parametric estimators (Angrist and Imbens, 1995). Without these assumptions, even if a valid IV exists, causal effects are non-identifiable in real-world scenarios. A nonparametric IV estimator relaxes these assumptions and can identify causal effects without relying on distribution assumptions and functional form restrictions (Hartford et al., 2017; Wu et al., 2022). Recently, some nonparametric IV estimators have been developed based on machine learning models, such as kernel-based IV regression (Singh et al., 2019), deep learning-based IV estimator (Hartford et al., 2017), and moment conditions-based IV estimator (Bennett et al., 2019).

The front-door criterion, introduced on page 83 of (Pearl, 2009a), is an alternative graphical criterion for causal effect estimation when there exists a latent confounder affecting both $W$ and $Y$. The front-door criterion can be used to identify a front-door adjustment set, which lies on the causal paths between $W$ and $Y$. The front-door criterion is to address the problem where the causal effect of $W$ on $Y$ is not identifiable by the back-door criterion or its extensions. For example, Fig. 1 (b) shows a case of front-door adjustment, in which $\mathbf{Z}$ satisfies the front-door criterion.

The front-door criterion and the IV approach are used to estimate the effects of $W$ and $Y$ in the presence of latent confounders (Pearl, 2009a). Broadly speaking, the IV approach selects a pretreatment variable as a valid IV, whereas the front-door criterion identifies a post-treatment variable as an adjustment set, their suitability depends on the specific research question and study design. For data-driven causal effect estimation, pretreatment variables are often assumed since otherwise, the discovery of causal structure around $W$ and $Y$ will be very challenging. So, there is not a data-driven method using the front-door criterion yet. Recently, Bhattacharya and Nabi (Bhattacharya and Nabi, 2022) used an auxiliary anchor variable and Verma constraint to test the satisfaction of the front-door criterion and showed an intersection model in which the IV and front-door conditions are satisfied can be tested in data. The results open a door for designing a data-driven solution using the front-door criterion.

Alternative to IV-based methods for dealing with latent confounders, sensitivity analysis can be employed to obtain bound estimates when latent confounders exist. Sensitivity analysis aims to explore the effect of residual latent confounders in causal effect estimation from observational data (VanderWeele and Ding, 2017; Robins et al., 2000b). An early work on sensitivity analysis by Cornfield et al. (Cornfield et al., 1959) focused on the relative prevalence of a binary latent factor (such as a gene switching on or off) among smokers and non-smokers and its ability to explain the observed association between smoking and lung cancer. Robin et al. (Robins et al., 2000b) provided a set of sensitivity parameters to capture the difference between the conditional distributions of the outcome of treated and control units. Mattei et al. (Mattei et al., 2014) proposed an approach to identifying causal effects of a binary treatment when the outcome is missing on a subset of units and nonresponse dependence on the outcome cannot be ruled out. Duarte et al. (Duarte et al., 2021) presented a novel approach to deriving the bound of causal effects in discrete settings and automating the process of partial identification. Scharfstein et al. (Scharfstein et al., 2021) used semiparametric theory to obtain a nonparametric efficient influence function of the ACE, for fixed sensitivity parameters. The sensitivity analysis methods are not strictly graphical causal modelling based, so in this survey, we do not cover the details of these methods. For those who are interested in sensitivity analysis methods, please refer to the literature in (Christopher Frey and Patil, 2002; Tortorelli and Michaleris, 1994; Ding and VanderWeele, 2016).

Latent variables exist in practical scenarios, because in a system some variables may be unmeasurable, or may not measured during data collection. In this survey, we consider that a latent variable is the unmeasured common cause of a pair of measured variables as stated in (Pearl, 2009a; Spirtes et al., 2000; Richardson and Spirtes, 2003, 2002). We also assume that there are no unmeasured selection variables due to selection bias cannot be addressed solely by confounding adjustment (van der Zander et al., 2014; Bareinboim et al., 2014; Pearl, 2009a; Perković et al., 2018). When there are latent variables in a system, the causal sufficiency assumption is violated since the assumption requires the common causes of every pair of measured variables in this system are also measured. When the causal sufficiency does not hold, the causal effect estimation from such system becomes challenging and difficult (Spirtes, 2010; Pearl et al., 2009). Latent variables mainly cause two difficulties in causal effect estimation discussed below.

The first difficulty is that latent variables introduce bias in the estimation of causal effects. For example, assume that the system modelled by Fig. 2 (b) (i.e. the underlying data generation mechanism is a causal MAG) contains three latent variables $X_{4}$, $X_{6}$ and $X_{8}$. If we do not consider the latent variables, and still try to learn a DAG from data, we will obtain a DAG as shown in Fig. 6 (b). When using the learned DAG in Fig. 6 (b), the estimated $\operatorname{ACE}(W,Y)$ will be biased since based on the learned DAG $X_{7}$ is on the back-door path between $W$ and $Y$ and thus is included the adjustment set. However, the inclusion of $X_{7}$ in an adjustment set introduces a spurious association between $W$ and $Y$ because $X_{7}$ is a collider in Fig. 6 (a). This is the well-known “$M$-bias” (Greene, 2003; Pearl, 2009b).

The second difficulty is that latent variables make causal effect impossible to estimate using the confounding adjustment approach when they are common (direct or indirect) causes of $W$ and $Y$. In this case, the unconfoundedness assumption used by confounding adjustment is violated (Imbens and Rubin, 2015; Rubin, 1974, 2007; Guo et al., 2020). Causal effect estimation in this case can be done using the instrumental variable (IV) approach with a standard IV or an ancestral IV (AIV). However, normally an IV needs to be identified by domain knowledge (Martens et al., 2006; Van der Zander et al., 2015; Silva and Shimizu, 2017; Sjolander and Martinussen, 2019; Cheng et al., 2022a), and it is difficult to identify IVs from data. There are only a few data-driven methods available for finding IVs from data (Chu et al., 2001; Kuroki and Cai, 2005; Cheng et al., 2023a), but they all require strong assumptions, such as, two variables or half of the variables in a system are standard IVs or AIVs.

From the above discussion, we also know that whether there is a latent confounder between $W$ and $Y$ determines the choice between the two totally different approaches (confounding adjustment and IV approach) for causal effect estimation. In the following, we introduce a graphical criterion for determining the visibility of $W\rightarrow Y$ or whether there is not a latent variable between $W$ and $Y$ in a MAG or a PAG.

There are two cases in a MAG (or a PAG) as illustrated in Fig. 7 in which $W\rightarrow Y$ is visible. When $W\rightarrow Y$ is invisible, there is a latent confounder affecting $W$ and $Y$, and the causal effect of $W$ on $Y$ is non-identifiable (Pearl et al., 2009; Zhang, 2008a), i.e. confounding adjustment is not able to recover an unbiased causal effect. When this is a valid IV in the underlying causal DAG, the IV approach can be used to obtain an unbiased causal effect. The details of data-driven methods are introduced in Section 4.

In real-world applications, the underlying causal graphs are rarely available and have to be recovered from data (Spirtes et al., 2000; Witte and Didelez, 2019). Thus, causal structure learning algorithms play a critical role in causal effect estimation from observational data when the causal DAGs/MAGs are unknown (Maathuis et al., 2009, 2010; Malinsky and Spirtes, 2017). A causal structure learning algorithm, such as PC algorithm (Spirtes et al., 2000) or RFCI algorithm (Colombo et al., 2012), learns from data a CPDAG or PAG, i.e. the output of such an algorithm is a CPDAG or PAG. More details on causal structure learning algorithms can be found in the surveys (Yu et al., 2021; Vowels et al., 2022; Glymour et al., 2019; Nogueira et al., 2022).

The uncertainty of the causal structures learned from data leads to uncertainty in causal effect estimation from observational data. We demonstrate this with an example in Fig. 3, where the Markov equivalence class of the DAGs is represented by the CPDAG in the second panel. The causal effect of $W$ on $Y$ based on DAG₃ is different from the causal effect based on DAG₁ since DAG₃ has two causal paths from $W$ to $Y$, while there is only one causal path $W\rightarrow Y$ in DAG₁. A real-world problem is more complex than this toy example and has more variables. The number of causal graphs in a Markov equivalence class could increase exponentially with the number of variables (Cheng et al., 2020, 2023b). Thus there may be a large number of estimated values for a causal effect from one dataset, leading to uncertainty in causal effect estimation.

Two approaches have been developed so far to deal with the non-uniqueness of causal structure learning (van der Zander et al., 2014; Perković et al., 2018). The first approach provides a bound estimation of a causal effect, i.e. a multiset of causal effects estimated based on the DAGs or MAGs enumerated from the CPDAG or PAG learned from data. The second approach produces a unique estimation of a causal effect. It has been shown that when a CPDAG or a PAG is adjustment amendable w.r.t. the pair of variables $(W,Y)$ as defined below, the causal effect of $W$ on $Y$ can be estimated uniquely if proper adjustment sets w.r.t. the pair of variables $(W,Y)$ are identified from the DAGs or MAGs encoded in the CPDAG or PAG.

For example, in Fig. 4, the PAG is adjustment amenable w.r.t. the pair of variables $(W,Y)$ and the two sets ${X_{2},X_{5}}$ and $\{X_{3},X_{5}\}$ are the proper adjustment sets for all MAGs in the learned PAG. The causal effects from all MAGs in the Markov equivalence class are consistent, and there is one unique estimate.

Some knowledge is required to determine whether a CPDAG or PAG is adjustment amendable or not. If we know that all the other variables in a system are pretreatment variables (De Luna et al., 2011; Entner et al., 2013; Häggström, 2018) w.r.t. the pair of variables $(W,Y)$ and a variable is correlated with $W$ but not $Y$, then the CPDAG or PAG learned from the data generated by the system is adjustment amendable relative to the pair of variables $(W,Y)$ (Cheng et al., 2023b).

In general, the amenability assumption can be satisfied in real-world applications. For example, it is possible that domain experts know there is a causal effect between $W$ and $Y$, and the directed edge between $W$ and $Y$ is the only causal path. In this case, if the edge $W\rightarrow Y$ is visible, and the amenability assumption is met.

The main computational cost for causal effect estimation is to recover causal graphs from data (or search for an adjustment set globally using properties in the underlying causal graph). An optimal search for a causal structure from data is fundamentally NP-complete (Chickering, 1996), and so is an exhaustive search for an adjustment set directly from data. If using a heuristic strategy for global causal structure learning, the optimal results may not be returned.

Alternatively, local causal structure learning (Aliferis et al., 2010; Yu et al., 2021) is faster and can handle hundreds of variables. Local causal structure learning algorithms aim to recover the local causal structure around a given variable instead of a complete causal DAG. In general, the local causal structure of a variable (referred to as $X$) contains the sets of parent nodes of the variable $Pa(X)$ and the child nodes of the variable $Ch(X)$. Some methods have been developed to search for an adjustment set in the local causal structure around $W$ and $Y$ (Perkovic et al., 2017; Fang and He, 2020; Cheng et al., 2023b, 2020).

Table 1 presents our classification of existing data-driven methods. We classify the existing data-driven causal effect estimation methods into three categories based on their approaches to handling uncertainty in structure learning, their time complexities, and whether they consider the presence of a latent variable between $W$ and $Y$.

A data-driven causal effect method may produce multiple estimated values for a causal effect due to the uncertainty in structure learning. Although these estimated values may contain the unbiased estimation of the causal effect, it is not clear which of them is the unbiased estimation. In this survey, we classify the existing causal effect estimation methods into two categories based on how they address this uncertainty: Bound estimation methods, which produce multiple estimated values of a causal effect, and Unique estimation methods, which produce a unique (single) estimated value of causal effect.

Since there is no known causal graph, data-driven methods need to recover the underlying causal graph from observational data using a causal discovery method, which can be a global structure learning method or a local structure learning method (Aliferis et al., 2010; Glymour et al., 2019). Therefore, we classify these methods as either Global or Local based on the causal discovery method used. Note that the former uses the global search, resulting in higher time complexity, while the latter uses the local search, which has lower time complexity.

Regarding the challenge of latent variables, we classify existing methods into two categories: those that do not consider latent confounders and those that do. For the methods that consider latent confounders (i.e. causal sufficiency does not hold), we further categorise them as either Not between $(W,Y)$, which consider latent confounders that affect either $W$ or $Y$ but not both (i.e. $W\rightarrow Y$ is visible in the corresponding MAG or PAG), or Between $(W,Y)$, which consider latent confounders that affect both $W$ and $Y$, such as $W\leftarrow U\rightarrow Y$ in the causal DAG (i.e. $W\rightarrow Y$ is invisible in the corresponding MAG or PAG).

In the bottom section of Table 1, we provide information on accessing the software packages for the methods that have been made publicly available online. Furthermore, Fig. 8 summarises the key milestones of data-driven causal effect estimation based on graphical causal modelling and graphical causal modelling theories supporting the algorithms. The theories were developed in the 1990s and early 2000s, while the methods largely emerged in the past ten years. The number of methods remains small relevant to the importance of causal effect estimation. There is a need for more methods to advance this important field.

In the following subsections, we will review the details of the data-driven methods following the divisions in Table 1, while using latent variables as the main storyline to structure the review.

Many data-driven methods based on graphical causal models have been developed to estimate causal effects from data without considering latent variables. These methods can be categorised into two types: IDA and its extensions, and data-driven adjustment set discovery methods. In this section, we will review the details of these two types of methods.

When there are latent variables in data, an ancestral graph is usually used to represent the causal relationships among measured variables. From the data, only an equivalence class of MAGs, i.e. a PAG, can be recovered. The generalised back-door criterion (or generalised adjustment criterion) can be used in a MAG (PAG) to determine a proper adjustment set when there exists such a proper adjustment set (Maathuis et al., 2015; Perković et al., 2018). In this section, we will review the data-driven methods considering latent variables and finding proper adjustment sets based on the graphical criteria for the causal effect estimation.

When there are latent confounders, i.e. latent variables affecting both $W$ and $Y$, the causal effect of $W$ on $Y$ is not identifiable via confounding adjustment. Instrumental variable (IV) approach is a powerful method for inferring the causal effect of $W$ on $Y$ from data with latent confounders even when there exist latent confounders affecting both $W$ and $Y$. If there exists a valid IV, the causal effect of $W$ on $Y$ can be recovered from data with latent confounders. It is challenging to determine an IV directly from data (Pearl, 1995b; Hernán and Robins, 2006). Traditionally, a valid IV is nominated based on domain knowledge. However, recently, research has emerged to work towards data-driven methods related to the IV approach. In this section, we review the emerging research on the IV-based approach.

Table 2 provides a summary of the properties and characteristics of the data-driven causal effect estimation methods listed in Table 1. The methods listed in Table 2 are used to estimate ACEs from observational data, and therefore, they can be widely applied in fields such as economics (Imbens and Rubin, 2015; Heckman, 2008), epidemiology (Sauer et al., 2013; Textor et al., 2016), healthcare (Robins, 1986), and bioinformatics (Maathuis et al., 2010; Zisoulis et al., 2012). Note that some of these methods are parametric and some are nonparametric (Pearl et al., 2009; Correa and Bareinboim, 2017; Jaber et al., 2019b, a). A parametric method assumes a functional form of the underlying data generation process, such as linearity, whereas a nonparametric method does not make such an assumption. As shown in Table 2, most methods are parametric, and only a few methods are nonparametric, including CET-SAT, DICE, EHS, GAC, DAVS, CEELS and modelIV. It is worth mentioning that in some cases, the causal effect may not be nonparametrically identifiable, and the best option is to derive a bound estimation by conducting sensitive analysis (Robins et al., 2000b; Scharfstein et al., 2021; Christopher Frey and Patil, 2002; Tortorelli and Michaleris, 1994). Researchers need to carefully consider these properties/characteristics in light of their specific research question, data, and resources, to select an appropriate method for their application.