Counterfactuals and potential outcomes are key concepts in causal inference. They allow researchers to estimate the effects of treatments or interventions by comparing observed outcomes to hypothetical ones that would have occurred under different conditions.

The potential outcomes framework formalizes this approach, providing a way to define and estimate causal effects. It relies on assumptions like SUTVA and uses tools like the Rubin causal model to analyze data from experiments and observational studies.

Definition of counterfactuals

• Counterfactuals are hypothetical outcomes that would have occurred under different conditions or interventions
• They are the cornerstone of causal inference, allowing researchers to reason about cause-and-effect relationships
• Understanding counterfactuals is crucial for estimating the causal impact of treatments, policies, or interventions

Counterfactuals vs observed outcomes

• Observed outcomes are the actual, realized outcomes that are measured or recorded in a study or experiment
• Counterfactuals, on the other hand, are the potential outcomes that would have occurred under different treatment conditions
• The difference between the counterfactual and observed outcome for an individual is the basis for estimating causal effects

Counterfactuals in causal inference

• In causal inference, counterfactuals are used to define and estimate causal effects
• By comparing the counterfactual outcomes under different treatment conditions, researchers can quantify the causal impact of an intervention
• Counterfactuals allow for the conceptualization of cause-and-effect relationships, even when it is not possible to directly observe all potential outcomes

Potential outcomes framework

• The potential outcomes framework is a formal approach to defining and estimating causal effects using counterfactuals
• It provides a language and set of assumptions for reasoning about cause-and-effect relationships
• The framework is widely used in various fields, including statistics, economics, and social sciences

Defining potential outcomes

• Potential outcomes are the hypothetical outcomes that would be observed under different treatment conditions
• For each individual, there is a potential outcome for each possible treatment level or intervention
• Potential outcomes are denoted as $Y_i(t)$, where $i$ represents the individual and $t$ represents the treatment level

Potential outcomes vs observed outcomes

• Observed outcomes are the actual outcomes that are realized and measured in a study or experiment
• Potential outcomes, in contrast, include both the observed outcome and the counterfactual outcomes that would have occurred under different treatment conditions
• The observed outcome for an individual is only one realization of the potential outcomes, corresponding to the treatment level actually received

Assumptions of potential outcomes framework

• The potential outcomes framework relies on several key assumptions:
  1. Stable Unit Treatment Value Assumption (SUTVA): The potential outcomes for an individual are independent of the treatment assignment of other individuals
  2. Consistency assumption: The observed outcome for an individual under a given treatment level is equal to the potential outcome for that individual under the same treatment level
  3. No interference assumption: The treatment assignment of one individual does not affect the potential outcomes of another individual

Individual-level causal effects

• Individual-level causal effects are the differences between an individual's potential outcomes under different treatment conditions
• They represent the causal impact of a treatment or intervention on a specific individual
• Individual-level causal effects are the building blocks for estimating population-level causal effects

Definition of individual-level causal effects

• The individual-level causal effect is defined as the difference between an individual's potential outcomes under two different treatment levels
• Formally, the individual-level causal effect for individual $i$ is given by $\tau_i = Y_i(1) - Y_i(0)$, where $Y_i(1)$ and $Y_i(0)$ are the potential outcomes under treatment and control conditions, respectively
• Individual-level causal effects capture the unique effect of a treatment on each individual

Fundamental problem of causal inference

• The fundamental problem of causal inference arises because it is impossible to observe both potential outcomes for an individual simultaneously
• For each individual, only one potential outcome is realized and observed, depending on the actual treatment received
• This missing data problem makes it challenging to directly estimate individual-level causal effects

Estimating individual-level causal effects

• Due to the fundamental problem of causal inference, individual-level causal effects cannot be directly estimated
• However, under certain assumptions (e.g., randomization), it is possible to estimate average causal effects across a population
• Methods such as randomized experiments and observational studies with appropriate adjustments can be used to estimate average causal effects, which serve as approximations of individual-level effects

Average treatment effect (ATE)

• The average treatment effect (ATE) is a population-level causal effect that represents the average difference in potential outcomes between treatment and control conditions
• It is a key quantity of interest in many causal inference studies, as it summarizes the overall impact of a treatment or intervention
• The ATE is defined as $ATE = E[Y_i(1) - Y_i(0)]$, where $E[\cdot]$ denotes the expectation operator

Definition of ATE

• The ATE is the average difference in potential outcomes between the treatment and control conditions across all individuals in a population
• Mathematically, $ATE = \frac{1}{N} \sum_{i=1}^N (Y_i(1) - Y_i(0))$, where $N$ is the total number of individuals in the population
• The ATE represents the expected causal effect of a treatment if it were applied to the entire population

ATE vs individual-level causal effects

• The ATE is a population-level summary of individual-level causal effects
• While individual-level causal effects capture the unique impact of a treatment on each individual, the ATE represents the average of these effects across the population
• The ATE is more feasible to estimate than individual-level effects due to the fundamental problem of causal inference

Estimating ATE from potential outcomes

• To estimate the ATE, researchers typically rely on methods that leverage the potential outcomes framework
• Randomized experiments, where individuals are randomly assigned to treatment and control conditions, provide an unbiased estimate of the ATE
• In observational studies, statistical adjustments (e.g., propensity score methods, matching) can be used to estimate the ATE by controlling for confounding variables

Stable unit treatment value assumption (SUTVA)

• The stable unit treatment value assumption (SUTVA) is a key assumption in the potential outcomes framework
• SUTVA consists of two components: no interference and consistency
• Violations of SUTVA can lead to biased estimates of causal effects and complicate the interpretation of results

Definition of SUTVA

• No interference assumption: The potential outcomes of one individual are not affected by the treatment assignment of other individuals
• Consistency assumption: The observed outcome for an individual under a given treatment level is equal to the potential outcome for that individual under the same treatment level
• SUTVA ensures that the potential outcomes for each individual are well-defined and independent of the treatment assignment of others

Importance of SUTVA in causal inference

• SUTVA is crucial for the validity of causal effect estimates in the potential outcomes framework
• When SUTVA holds, the observed outcomes can be used to estimate the potential outcomes and causal effects
• Violations of SUTVA can introduce bias and make it difficult to attribute causal effects solely to the treatment of interest

Violations of SUTVA

• Interference: When the treatment assignment of one individual affects the potential outcomes of another individual (spillover effects)
  • Example: In an educational intervention, the treatment of one student may influence the outcomes of their classmates
• Inconsistency: When the observed outcome under a given treatment level differs from the potential outcome under the same treatment level
  • Example: Different versions of a treatment (e.g., different doses of a medication) may lead to different observed outcomes, violating consistency

Rubin causal model

• The Rubin causal model, named after Donald Rubin, is a framework for causal inference based on the potential outcomes approach
• It provides a formal language and set of assumptions for estimating causal effects from observational or experimental data
• The Rubin causal model has been widely influential in the development of modern causal inference methods

Overview of Rubin causal model

• The Rubin causal model defines causal effects in terms of potential outcomes
• It emphasizes the role of treatment assignment mechanisms and the importance of considering all potential outcomes
• The model relies on key assumptions, such as SUTVA and ignorability, to identify and estimate causal effects

Potential outcomes in Rubin causal model

• In the Rubin causal model, each individual has a set of potential outcomes corresponding to different treatment levels
• The potential outcomes are considered fixed, and the observed outcome is determined by the actual treatment received
• The causal effect for an individual is defined as the difference between their potential outcomes under different treatment conditions

Assumptions of Rubin causal model

• The Rubin causal model relies on several assumptions to identify and estimate causal effects:
  1. SUTVA: No interference between units and consistency of treatment
  2. Ignorability (or unconfoundedness): Treatment assignment is independent of the potential outcomes, given the observed covariates
  3. Positivity (or overlap): Each individual has a non-zero probability of receiving each treatment level
• These assumptions allow for the identification and estimation of causal effects from observational or experimental data

Neyman-Rubin causal model

• The Neyman-Rubin causal model is an extension of the Rubin causal model that incorporates ideas from Jerzy Neyman's work on randomization-based inference
• It combines the potential outcomes framework with the concept of randomization as a basis for causal inference
• The Neyman-Rubin causal model is particularly relevant for analyzing data from randomized experiments

Comparison to Rubin causal model

• The Neyman-Rubin causal model shares many similarities with the Rubin causal model, including the use of potential outcomes and the focus on treatment assignment mechanisms
• However, the Neyman-Rubin model places a stronger emphasis on randomization as a key tool for identifying and estimating causal effects
• It also considers the role of finite population inference and the estimation of average causal effects

Randomization in Neyman-Rubin causal model

• Randomization is a central concept in the Neyman-Rubin causal model
• By randomly assigning individuals to treatment and control conditions, randomization ensures that the treatment assignment is independent of the potential outcomes
• Randomization provides a basis for unbiased estimation of average causal effects and allows for finite population inference

Estimating causal effects in Neyman-Rubin model

• In the Neyman-Rubin causal model, causal effects are typically estimated using the difference in average outcomes between the treatment and control groups
• The average treatment effect (ATE) can be estimated as the difference in sample means: $\hat{ATE} = \bar{Y}_1 - \bar{Y}_0$, where $\bar{Y}_1$ and $\bar{Y}_0$ are the sample means of the treatment and control groups, respectively
• Randomization-based inference, such as permutation tests and randomization-based confidence intervals, can be used to assess the uncertainty of the estimated causal effects

Applications of counterfactuals

• Counterfactuals and the potential outcomes framework have numerous applications across various fields
• They provide a powerful tool for understanding and quantifying the causal impact of interventions, policies, and treatments
• The use of counterfactuals allows researchers to answer "what if" questions and make informed decisions based on causal evidence

Counterfactuals in policy evaluation

• Counterfactuals are widely used in policy evaluation to assess the impact of policies or interventions on outcomes of interest
• By comparing the observed outcomes under a implemented policy to the counterfactual outcomes that would have occurred in the absence of the policy, researchers can estimate the causal effect of the policy
• Examples: Evaluating the impact of minimum wage laws on employment, assessing the effectiveness of educational interventions on student achievement

Counterfactuals in medical research

• In medical research, counterfactuals are used to estimate the causal effects of treatments or interventions on patient outcomes
• Randomized controlled trials (RCTs) are the gold standard for estimating causal effects in medicine, as they ensure that the treatment assignment is independent of potential outcomes
• Counterfactuals allow researchers to compare the potential outcomes of patients under different treatment conditions and make informed treatment decisions

Counterfactuals in social sciences

• Counterfactuals are increasingly used in social sciences to study the causal impact of social interventions, policies, and programs
• By comparing the observed outcomes of individuals or groups exposed to an intervention to the counterfactual outcomes that would have occurred without the intervention, researchers can estimate the causal effect of the intervention
• Examples: Evaluating the impact of job training programs on employment outcomes, assessing the effectiveness of crime prevention strategies on crime rates