📊Causal Inference Unit 2 – Potential outcomes framework

Key Concepts

• Potential outcomes framework provides a formal language for defining causal effects and identifying assumptions needed for causal inference
• Focuses on individual-level causal effects rather than average effects across a population
• Introduces the concept of counterfactuals, which are potential outcomes under different treatment conditions
• Defines causal effects as comparisons between potential outcomes under different treatment conditions for the same individual
• Emphasizes the importance of clearly specifying the treatment, outcome, and population of interest
• Highlights the role of assumptions in identifying causal effects from observed data
• Provides a framework for discussing the strengths and limitations of different research designs and estimation methods

Historical Context

• Developed by Donald Rubin and others in the 1970s and 1980s as a formal framework for causal inference
• Builds on earlier work by Jerzy Neyman on randomized experiments and potential outcomes
• Influenced by the Rubin Causal Model, which emphasizes the role of treatment assignment mechanisms in causal inference
• Has become a dominant framework for causal inference in statistics, economics, and other social sciences
• Complements other approaches to causal inference, such as structural equation modeling and directed acyclic graphs
• Has been extended to handle more complex settings, such as time-varying treatments and interference between units

Fundamental Assumptions

• Stable Unit Treatment Value Assumption (SUTVA) states that the potential outcomes for each individual are unaffected by the treatment assignment of other individuals and that there are no different versions of the treatment
  • Implies that there is no interference between units and no hidden variations in treatment
• Consistency assumption states that the observed outcome for an individual under a given treatment condition is equal to the potential outcome for that individual under that treatment condition
• Positivity assumption requires that each individual has a non-zero probability of receiving each treatment condition
• Ignorability assumption states that treatment assignment is independent of potential outcomes given observed covariates
  • Also known as the unconfoundedness assumption or the no unmeasured confounders assumption
• These assumptions are often untestable and must be justified based on substantive knowledge and the design of the study

Counterfactuals and Potential Outcomes

• Counterfactuals are hypothetical outcomes that would have been observed under different treatment conditions
• Potential outcomes are the values of the outcome variable that would be observed under each possible treatment condition
  • Denoted as $Y_i(0)$ and $Y_i(1)$ for binary treatments, where $i$ indexes individuals
• The fundamental problem of causal inference is that we can only observe one potential outcome for each individual, corresponding to the treatment they actually received
• Causal effects are defined as comparisons between potential outcomes under different treatment conditions for the same individual
  • E.g., the individual-level causal effect is $Y_i(1) - Y_i(0)$
• The key challenge in causal inference is estimating causal effects from observed data, which requires assumptions about the missing potential outcomes

Causal Effects and Estimands

• Causal effects can be defined at the individual level or averaged across a population
• The average treatment effect (ATE) is the average difference in potential outcomes across all individuals in the population
  • $ATE = E[Y_i(1) - Y_i(0)]$
• The average treatment effect on the treated (ATT) is the average difference in potential outcomes among individuals who actually received the treatment
  • $ATT = E[Y_i(1) - Y_i(0) | D_i = 1]$, where $D_i$ indicates treatment receipt
• Other estimands include the average treatment effect on the untreated (ATU) and quantile treatment effects
• The choice of estimand depends on the research question and the population of interest
• Estimands are typically defined in terms of potential outcomes, which are not directly observable

Treatment Assignment Mechanisms

• Treatment assignment mechanisms describe the process by which individuals are assigned to treatment conditions
• Randomized experiments assign treatments randomly, ensuring that treatment assignment is independent of potential outcomes
  • Enables unbiased estimation of causal effects by comparing outcomes between treatment groups
• Observational studies rely on non-random treatment assignment, which can lead to confounding bias if treatment assignment is related to potential outcomes
• Unconfoundedness assumption justifies treating observational data as if treatments were randomly assigned conditional on observed covariates
  • Requires that all confounders are measured and included in the analysis
• Different assignment mechanisms (e.g., stratified randomization, matching) can be used to improve balance and reduce bias in observational studies

Estimation Methods

• Various methods can be used to estimate causal effects under the potential outcomes framework
• Regression adjustment controls for confounders by including them as predictors in a regression model
  • Assumes that the relationship between confounders and outcomes is correctly specified
• Propensity score methods model the probability of treatment assignment given observed covariates
  • Can be used for matching, stratification, or weighting to balance treatment groups
• Matching methods pair treated and untreated individuals with similar covariate values to estimate causal effects
  • Assumes that matched individuals are comparable on both observed and unobserved factors
• Doubly robust methods combine regression adjustment and propensity score weighting to provide unbiased estimates if either model is correctly specified
• Instrumental variables methods use a variable that affects treatment assignment but not outcomes to estimate causal effects
  • Requires strong assumptions about the instrument's validity and relevance

Limitations and Challenges

• The potential outcomes framework relies on strong assumptions that may not hold in practice
  • SUTVA violations can occur due to interference between units or variations in treatment implementation
  • Unconfoundedness assumption is often untestable and may be violated if important confounders are unmeasured
• Estimating individual-level causal effects is impossible without strong assumptions, as only one potential outcome is observed for each individual
• Causal effects may vary across individuals, making it difficult to summarize the overall impact of a treatment
• Missing data and measurement error can bias estimates of causal effects and complicate inference
• Sensitivity analyses are important for assessing the robustness of causal estimates to violations of key assumptions
• Careful design of studies and data collection is crucial for supporting credible causal inference

Applications in Research

• The potential outcomes framework has been widely applied in social and biomedical sciences to estimate causal effects of interventions
• Randomized controlled trials are considered the gold standard for causal inference, as they ensure unconfounded treatment assignment
  • Examples include clinical trials of medical treatments and field experiments in economics and psychology
• Observational studies are more common in many fields, as randomization may be infeasible or unethical
  • Examples include studies of the effects of education on earnings, the impact of pollution on health, and the consequences of public policies
• The framework has been extended to handle complex settings, such as longitudinal studies with time-varying treatments and studies with interference between units
• Causal inference methods based on the potential outcomes framework have been implemented in various statistical software packages (e.g., R, Stata)
• Careful application of the framework can help researchers design more informative studies, assess the credibility of causal claims, and communicate the strengths and limitations of their findings