Graphical models, like Directed Acyclic Graphs (DAGs), are powerful tools for visualizing causal relationships. They help us understand how variables interact and influence each other in complex systems. DAGs use nodes to represent variables and directed edges to show causal effects.

D-separation and the backdoor criterion are key concepts in causal inference using DAGs. They allow us to determine conditional independence relationships and identify sets of variables that, when adjusted for, enable unbiased estimation of causal effects from observational data.

Graphical models of causality

• Graphical models provide a visual representation of the causal relationships between variables in a system
• Enable reasoning about independence relationships and identifiability of causal effects from observational data
• Fundamental tool in causal inference for representing expert knowledge and assumptions about a domain

Directed acyclic graphs (DAGs)

• DAGs are a specific type of graphical model used to represent causal relationships
• Consist of nodes representing variables and directed edges representing direct causal effects
• Must not contain any directed cycles, meaning no variable can be a cause of itself (directly or indirectly)
• Encode independence assumptions and enable application of graphical criteria for causal inference

Nodes, edges, and paths

• Nodes in a DAG represent variables, which can be observed or unobserved (latent)
• Directed edges between nodes represent direct causal effects, with the arrow pointing from cause to effect
• Paths are sequences of edges connecting nodes, which can be either directed (causal) or undirected (associational)
  • Directed paths follow the direction of the arrows and represent causal relationships
  • Undirected paths do not necessarily follow the direction of the arrows and represent associational relationships

Causal Markov assumption

• The causal Markov assumption is a fundamental assumption in causal inference using DAGs
• States that a variable is independent of its non-descendants given its direct causes (parents) in the graph
• Implies that the joint distribution of the variables factorizes according to the DAG structure
• Allows for the derivation of testable implications and identification of causal effects from observational data

d-separation

• d-separation is a graphical criterion for determining conditional independence relationships in a DAG
• Provides a way to read off independence statements directly from the graph structure
• Fundamental concept for assessing identifiability of causal effects and selecting adjustment sets

Conditional independence in DAGs

• Two variables are conditionally independent given a set of variables Z if they are independent in every subpopulation defined by Z
• In a DAG, conditional independence relationships are implied by the graph structure and can be determined using d-separation
• If two variables are d-separated by a set Z, then they are conditionally independent given Z in any probability distribution that factorizes according to the DAG

Blocked vs unblocked paths

• A path between two variables in a DAG is blocked by a set of variables Z if it contains a collider that is not in Z or has a descendant in Z, or a non-collider that is in Z
  • Colliders are variables with two incoming arrows on the path (common effects)
  • Non-colliders are variables with at least one outgoing arrow on the path (common causes or mediators)
• If all paths between two variables are blocked by Z, then the variables are d-separated and conditionally independent given Z
• If there exists an unblocked path between two variables given Z, then they are d-connected and potentially dependent given Z

Three basic configurations

• The three basic configurations in a DAG that determine d-separation relationships are chains, forks, and colliders
  • Chains: X → M → Y, where M is a non-collider. Conditioning on M blocks the path and makes X and Y independent
  • Forks: X ← M → Y, where M is a non-collider. Conditioning on M blocks the path and makes X and Y independent
  • Colliders: X → M ← Y, where M is a collider. Conditioning on M (or a descendant of M) unblocks the path and makes X and Y dependent

Formal definition and rules

• Two variables X and Y are d-separated by a set Z in a DAG G if and only if every path between X and Y is blocked by Z
• The rules for d-separation can be summarized as follows:
  1. A path is blocked if it contains a non-collider that is in Z
  2. A path is blocked if it contains a collider that is not in Z and has no descendant in Z
  3. A path is unblocked if it contains a collider that is in Z or has a descendant in Z
• These rules can be applied systematically to determine all d-separation relationships in a DAG

Backdoor criterion

• The backdoor criterion is a graphical criterion for determining whether a set of variables is sufficient for adjusting for confounding in a causal effect estimation
• Provides a way to identify admissible adjustment sets from the DAG structure alone, without requiring knowledge of the underlying probability distribution
• Fundamental tool for causal effect identification and estimation in observational studies

Confounding and backdoor paths

• Confounding occurs when there is a common cause of the treatment and outcome variables, leading to spurious associations
• In a DAG, confounding is represented by the presence of backdoor paths between the treatment and outcome
  • Backdoor paths are paths that start with an arrow into the treatment variable and end with an arrow into the outcome variable
• Adjusting for a sufficient set of variables that blocks all backdoor paths can eliminate confounding and enable unbiased estimation of the causal effect

Adjusting for confounders

• To adjust for confounding, we need to condition on a set of variables that blocks all backdoor paths between the treatment and outcome
• Conditioning on a variable can be done through stratification, matching, or regression adjustment
• The goal is to create subpopulations within which the treatment assignment is unconfounded, and then aggregate the estimates across subpopulations
• Proper adjustment for confounders requires careful selection of variables based on the causal DAG and the backdoor criterion

Formal definition of criterion

• A set of variables Z satisfies the backdoor criterion relative to a treatment variable X and an outcome variable Y in a DAG G if:
  1. No variable in Z is a descendant of X
  2. Z blocks all backdoor paths from X to Y
• If a set Z satisfies the backdoor criterion, then the causal effect of X on Y can be identified by adjusting for Z, i.e., P(Y|do(X)) = ∑_z P(Y|X,Z)P(Z)

Relationship to d-separation

• The backdoor criterion is closely related to the concept of d-separation in DAGs
• If a set Z satisfies the backdoor criterion relative to X and Y, then X and Y are d-separated by Z in the graph G_X, which is the graph obtained by removing all arrows emanating from X
• This connection allows for the use of d-separation algorithms to find admissible adjustment sets and assess the identifiability of causal effects

Applications of d-separation and backdoor

• d-separation and the backdoor criterion are fundamental tools in causal inference with wide-ranging applications
• They provide a principled way to reason about conditional independence relationships and identifiability of causal effects from observational data
• Enable the selection of appropriate adjustment sets for confounding control and the assessment of the feasibility of causal effect estimation

Determining identifiability of causal effects

• The backdoor criterion can be used to determine whether a causal effect is identifiable from observational data given a causal DAG
• If there exists a set of variables that satisfies the backdoor criterion, then the causal effect can be identified by adjusting for that set
• If no such set exists, then the causal effect may not be identifiable without additional assumptions or data (e.g., instrumental variables)

Selecting variables for adjustment

• d-separation and the backdoor criterion provide guidance on which variables should be included in an adjustment set for confounding control
• They help avoid the inclusion of colliders or mediators, which can introduce bias or block the effect of interest
• By systematically applying these criteria, researchers can select adjustment sets that are sufficient for eliminating confounding while avoiding unnecessary or harmful adjustments

Examples in practice

• Epidemiological studies often use DAGs and the backdoor criterion to identify potential confounders and select adjustment variables (e.g., in studies of the effect of smoking on lung cancer, adjusting for age and sex)
• In social sciences, researchers use these tools to assess the feasibility of estimating causal effects from observational data and guide the design of studies (e.g., in studies of the effect of education on earnings, adjusting for family background and ability)

Limitations and assumptions

• The application of d-separation and the backdoor criterion relies on the correctness of the assumed causal DAG
• If important variables are omitted or the causal relationships are misspecified, the resulting conclusions may be invalid
• These tools also assume that there is no unmeasured confounding, which may not always hold in practice
• Sensitivity analyses and triangulation of evidence from multiple sources can help assess the robustness of findings to potential violations of assumptions

Advanced topics and extensions

• The concepts of d-separation and the backdoor criterion have been extended and generalized to handle more complex causal structures and identification scenarios
• These advances have expanded the range of causal questions that can be addressed using graphical models and have deepened our understanding of the foundations of causal inference

Frontdoor criterion and adjustment

• The frontdoor criterion is another graphical criterion for identifying causal effects in the presence of unmeasured confounding
• It applies when there is no set of variables that satisfies the backdoor criterion, but there exists a mediator variable that is not affected by the unmeasured confounders
• Adjusting for the frontdoor variable can enable the identification of the causal effect by decomposing it into the product of two identifiable components

m-separation and MAGs

• Maximal ancestral graphs (MAGs) are a generalization of DAGs that allow for the representation of unmeasured confounding and selection bias
• m-separation is a generalization of d-separation that applies to MAGs and provides a graphical criterion for conditional independence in the presence of unmeasured variables
• These extensions enable the identification of causal effects in a wider range of settings and provide a foundation for causal discovery algorithms

Causal discovery algorithms

• Causal discovery algorithms aim to learn the causal structure of a system from observational data
• They use the principles of d-separation and m-separation to infer the presence or absence of causal relationships between variables
• Examples include the PC algorithm, which uses conditional independence tests to construct a DAG, and the FCI algorithm, which learns a MAG in the presence of unmeasured confounding
• These algorithms provide a data-driven approach to causal structure learning and can complement expert knowledge in the specification of causal models

Challenges with unmeasured confounding

• Unmeasured confounding is a major challenge in causal inference from observational data
• When important confounders are not measured or included in the analysis, the resulting estimates of causal effects may be biased
• Sensitivity analysis techniques, such as the E-value and bias formulas, can help assess the robustness of findings to potential unmeasured confounding
• Instrumental variable methods and other approaches that leverage natural experiments or quasi-random variation can also help address unmeasured confounding in some settings