📊Causal Inference Unit 9 – Causal Graphs & Structural Models

Key Concepts & Definitions

• Causal inference aims to understand the causal relationships between variables and estimate the effects of interventions
• Causal graphs, also known as directed acyclic graphs (DAGs), visually represent the causal relationships between variables
• Nodes in a causal graph represent variables, while edges represent causal relationships between variables
• Structural models mathematically describe the causal relationships between variables and the functional form of these relationships
• Confounding occurs when a variable influences both the treatment and the outcome, leading to biased estimates of causal effects
• Colliders are variables that are influenced by two or more other variables in a causal graph
• Mediators are variables that lie on the causal path between the treatment and the outcome, transmitting part of the causal effect
• Counterfactuals are hypothetical scenarios that describe what would have happened under different treatment conditions

Causal Graph Fundamentals

• Causal graphs consist of nodes (variables) and directed edges (arrows) representing causal relationships
• Directed edges in a causal graph indicate the direction of causality, with the arrow pointing from the cause to the effect
• Causal graphs must be acyclic, meaning there cannot be any feedback loops or cycles in the graph
• The absence of an edge between two nodes in a causal graph implies that there is no direct causal relationship between the variables
• Causal graphs can be used to identify confounding, colliders, and mediators in a causal system
• The graphical structure of a causal graph encodes the conditional independence relationships between variables
  • Variables that are d-separated by a set of other variables are conditionally independent given those variables
  • Conditional independence relationships can be used to test the compatibility of a causal graph with observed data
• Causal graphs provide a framework for reasoning about the effects of interventions and the identifiability of causal effects

Types of Structural Models

• Linear structural models assume that the causal relationships between variables are linear and additive
  • Example: $Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \epsilon$, where $Y$ is the outcome, $X_1$ and $X_2$ are the causes, and $\epsilon$ is the error term
• Non-linear structural models allow for more complex, non-linear relationships between variables
  • Example: $Y = \exp(\beta_0 + \beta_1 X_1 + \beta_2 X_2^2) + \epsilon$, which includes a quadratic term for $X_2$
• Structural equation models (SEMs) are a general class of models that can incorporate both linear and non-linear relationships, as well as latent variables
• Generalized linear models (GLMs) extend linear models to accommodate non-normal outcomes (binary, count, etc.) using link functions and appropriate error distributions
• Time-varying structural models account for the temporal dynamics of causal relationships, allowing for feedback loops and time-dependent confounding
• Structural nested models (SNMs) are used to estimate the effects of time-varying treatments in the presence of time-dependent confounding
• Marginal structural models (MSMs) estimate the causal effects of time-varying treatments by weighting observations based on the probability of receiving the observed treatment history

Building Causal Graphs

• Causal graphs can be constructed based on domain knowledge, expert opinion, or data-driven methods
• When building causal graphs based on domain knowledge, it is important to consider all relevant variables and their potential causal relationships
• Causal discovery algorithms, such as the PC algorithm and the FCI algorithm, can be used to learn the structure of causal graphs from observational data
  • These algorithms rely on conditional independence tests to infer the presence or absence of edges in the graph
• Causal graphs should be assessed for plausibility and consistency with prior knowledge and scientific understanding
• Sensitivity analyses can be conducted to evaluate the robustness of causal graphs to alternative specifications or omitted variables
• Causal graphs should be iteratively refined as new evidence or knowledge becomes available
• It is important to consider the possibility of unmeasured confounding when building causal graphs, as omitted variables can bias the estimated causal effects
• Causal graphs can be extended to incorporate selection bias, measurement error, and other common challenges in causal inference

Identifying Causal Effects

• The identification of causal effects depends on the structure of the causal graph and the available data
• The causal effect of a treatment on an outcome can be identified if all confounding variables are measured and controlled for
  • This is known as the backdoor criterion, which requires that all backdoor paths between the treatment and outcome are blocked by conditioning on a sufficient set of variables
• Front-door adjustment can be used to identify causal effects when there are unmeasured confounders, but a mediator variable is available that satisfies certain conditions
• Instrumental variables (IVs) can be used to identify causal effects when there are unmeasured confounders, but a variable exists that affects the treatment but not the outcome directly
  • Example: using proximity to a hospital as an IV for the effect of hospital treatment on patient outcomes
• Causal effects can be estimated using various methods, such as regression adjustment, propensity score matching, inverse probability weighting, and doubly robust estimation
• The choice of estimation method depends on the causal graph, the available data, and the assumptions about the functional form of the relationships between variables
• Sensitivity analyses should be conducted to assess the robustness of causal effect estimates to potential violations of assumptions or alternative specifications

Common Challenges & Pitfalls

• Unmeasured confounding is a major challenge in causal inference, as it can bias the estimated causal effects
  • Sensitivity analyses can be used to assess the potential impact of unmeasured confounders on the results
• Selection bias occurs when the sample used for analysis is not representative of the target population, leading to biased estimates of causal effects
  • Example: studying the effect of a job training program on earnings, but only observing outcomes for those who complete the program
• Measurement error in the treatment, outcome, or confounding variables can bias the estimated causal effects and reduce statistical power
• Spillover effects occur when the treatment of one unit affects the outcomes of other units, violating the stable unit treatment value assumption (SUTVA)
  • Example: estimating the effect of a vaccination program, but vaccinated individuals reduce the risk of infection for unvaccinated individuals in the same community
• Causal graphs can be misspecified, leading to incorrect conclusions about the presence or absence of causal effects
• Extrapolating causal effects to new populations or settings requires careful consideration of the similarities and differences between the original and target contexts
• Causal inference methods rely on assumptions, such as exchangeability, positivity, and consistency, which may not always hold in practice
• Interpreting causal effect estimates requires careful consideration of the units of analysis, the time horizon, and the specific intervention being studied

Practical Applications

• Causal inference methods are widely used in epidemiology to study the effects of exposures or interventions on health outcomes
  • Example: estimating the causal effect of smoking on lung cancer risk, while controlling for potential confounders like age and occupation
• In social sciences, causal inference is used to evaluate the impact of policies, programs, or interventions on various outcomes
  • Example: assessing the effect of a job training program on employment and earnings, using a causal graph to identify and control for confounding factors
• Causal inference is important in business and marketing to understand the drivers of consumer behavior and the effectiveness of marketing strategies
  • Example: using a causal graph to analyze the impact of advertising campaigns on product sales, while accounting for factors like price and competitor actions
• In personalized medicine, causal inference can be used to estimate the heterogeneous treatment effects of medical interventions across different subgroups of patients
• Causal inference methods are applied in environmental studies to assess the impact of pollutants or climate change on various outcomes
  • Example: using instrumental variables to estimate the causal effect of air pollution on respiratory health, addressing potential confounding by socioeconomic factors
• In policy evaluation, causal inference is used to estimate the impact of interventions on economic, social, or health outcomes
  • Example: employing difference-in-differences methods to evaluate the effect of a minimum wage increase on employment and income levels
• Causal inference is crucial in the development and evaluation of AI and machine learning systems to ensure fairness, accountability, and transparency
  • Example: using causal graphs to identify and mitigate biases in algorithmic decision-making systems, such as those used in hiring or lending

Advanced Topics & Extensions

• Causal discovery algorithms, such as the PC algorithm and the FCI algorithm, can be used to learn the structure of causal graphs from observational data
  • These algorithms rely on conditional independence tests and can handle the presence of latent variables and selection bias
• Bayesian networks extend causal graphs by incorporating probability distributions over the variables, allowing for probabilistic reasoning and inference
• Structural causal models (SCMs) provide a formal framework for representing and reasoning about causal relationships, incorporating both graphical and functional components
• Counterfactual reasoning is a key concept in causal inference, allowing for the estimation of individual-level causal effects and the assessment of treatment effect heterogeneity
  • Example: estimating the effect of a drug on a specific patient's blood pressure, considering their individual characteristics and potential outcomes under different treatment scenarios
• Mediation analysis aims to decompose the total causal effect into direct and indirect effects, mediated through intermediate variables
  • Example: analyzing the extent to which the effect of education on earnings is mediated by job experience and skills
• Interference and spillover effects can be addressed using causal inference methods for dependent data, such as network or spatial models
• Causal inference with time-varying treatments and confounders requires specialized methods, such as marginal structural models and structural nested models
• Machine learning techniques, such as causal forests and causal regularization, can be used to estimate heterogeneous treatment effects and improve the performance of causal inference methods
• Sensitivity analysis methods, such as the E-value and the Rosenbaum bounds, can be used to assess the robustness of causal effect estimates to potential unmeasured confounding