# 2 The Potential Outcomes Framework

## 2.1 The Basic Idea

The potential outcomes framework, also known as the Rubin Causal Model, provides a formal mathematical approach to defining and estimating causal effects. This framework, developed by Donald Rubin building on work by Jerzy Neyman, is central to modern causal inference and has become increasingly important in business data science. At its core, the potential outcomes framework posits that each unit (e.g., person, company, product) has a set of potential outcomes corresponding to each possible treatment condition. For instance:

- A tech company testing a new app interface might consider:
- \(Y(1)\): user engagement if exposed to the new interface
- \(Y(0)\): user engagement if exposed to the old interface

- An e-commerce platform implementing a recommendation system might examine:
- \(Y(1)\): customer purchase amount with personalized recommendations
- \(Y(0)\): customer purchase amount without personalized recommendations

- A SaaS company offering a free trial could look at:
- \(Y(1)\): conversion rate if offered a 30-day free trial
- \(Y(0)\): conversion rate if not offered a free trial

The causal effect for an individual is then defined as the difference between these potential outcomes: \(Y(1) - Y(0)\). However, we face what Holland (1986) termed the “fundamental problem of causal inference” - we can only observe one of these potential outcomes for each unit. If a customer is exposed to the new interface, we observe \(Y(1)\) but \(Y(0)\) remains unobserved (and vice versa). This makes causal inference inherently a missing data problem, a concept we’ll explore further later in this chapter.

## 2.2 Key Concepts and Estimands

Several important concepts and estimands are central to the potential outcomes framework. The Average Treatment Effect (ATE) is the average causal effect across the entire population, defined as \(E[Y(1) - Y(0)]\). This gives us an overall measure of the treatment’s impact.

When we’re interested in how the treatment effect varies across different subgroups, we look at the Conditional Average Treatment Effect (CATE). This is defined as \(E[Y(1) - Y(0) | X]\), where X represents a specific set of covariates. CATE allows us to understand how the treatment effect might differ for various segments of our population.

Sometimes, we’re particularly interested in the effect on those who actually received the treatment. This is captured by the Average Treatment Effect on the Treated (ATT), defined as \(E[Y(1) - Y(0) | W = 1]\), where W is the treatment indicator. In certain scenarios, such as when using instrumental variables, we might focus on the Local Average Treatment Effect (LATE), which represents the average treatment effect for a specific subpopulation of compliers.

A crucial assumption in many causal analyses is **ignorability**. This assumes that treatment assignment is independent of the potential outcomes given observed covariates. Mathematically, this can be expressed as: \((Y(1), Y(0)) ⊥ W | X\) where \(W\) is the treatment assignment and \(X\) are the observed covariates. For instance, in our e-commerce recommendation system example, ignorability would mean that whether a customer sees personalized recommendations (W) is independent of how much they would potentially purchase with or without recommendations (\(Y(1), Y(0)\)), once we account for observed factors like browsing history, past purchases, etc. (\(X\)).

## 2.3 Experimental vs Observational Studies

The potential outcomes framework can be applied to both experimental and observational studies, each with its own strengths and challenges:

**Experimental Studies:** In randomized controlled trials, treatment assignment is controlled by the researcher. This control ensures that the ignorability assumption holds by design. For example, when A/B testing a new website design, the randomization of which users see which version ensures that potential outcomes are independent of the assignment. This makes causal inference more straightforward but may not always be feasible in business settings due to ethical, practical, or cost constraints.

**Observational Studies:** These are more common in business contexts but present more challenges. For instance, if we want to study the effect of a loyalty program on customer retention, customers typically choose whether to join the program rather than being randomly assigned. In these cases, we need to carefully consider and account for potential confounders to approximate the conditions of an experiment. This often involves sophisticated statistical techniques to adjust for differences between the treatment and control groups, such as propensity score matching or inverse probability weighting.

## 2.4 Heterogeneous Treatment Effects

In business applications, it’s crucial to consider that the effect of an intervention might vary across different subgroups of customers or products. This heterogeneity can be masked when looking only at average effects. For example:

- A new marketing strategy might have a positive effect on one customer segment but a negative effect on another.
- A product feature might significantly boost engagement for power users but have minimal impact on casual users.

Understanding these heterogeneous effects can lead to more targeted and effective business strategies, such as personalized marketing campaigns or tailored product features.

## 2.5 Selection Bias

Selection bias occurs when the individuals who select into the treatment group differ systematically from those who do not. In business contexts, this is a common issue. For example:

- Customers who choose to use a new feature might be systematically different from those who don’t, making it challenging to isolate the true effect of the feature on outcomes like engagement or sales.
- Early adopters of a product might have different characteristics and behaviors compared to later adopters, potentially skewing our understanding of the product’s impact.

Recognizing and addressing selection bias is crucial for making valid causal inferences in business settings.

## 2.6 Connections to Missing Data

The link between causal inference and missing data is profound. In the potential outcomes framework, we’re always missing at least one potential outcome for each unit. This is similar to the problem of missing data in surveys or experiments where some values are unobserved.

Methods developed for handling missing data have direct analogues in causal inference. For example, multiple imputation techniques can be adapted to impute missing potential outcomes. Inverse probability weighting, commonly used in missing data problems, is analogous to propensity score weighting in causal inference.

The assumptions underlying missing data methods also have parallels in causal inference. The assumption of “Missing At Random” (MAR) in missing data literature is similar to the ignorability assumption in causal inference. Both assume that the missingness (or treatment assignment) is independent of the unobserved data, given the observed data.

Understanding these connections can provide valuable insights and tools for addressing the inherent missing data problem in causal inference. By leveraging techniques from both fields, researchers can develop more robust methods for estimating causal effects in a variety of real-world scenarios.

## 2.7 Conclusion

The potential outcomes framework provides a powerful tool for business data scientists to approach causal questions rigorously. By understanding the fundamental concepts, key estimands, and challenges associated with this framework, data scientists can make more informed decisions about experimental design, analysis methods, and interpretation of results. As we delve deeper into specific techniques and applications in the following chapters, keep these foundational ideas in mind – they will serve as the bedrock for more advanced causal inference methods in business contexts.

Cunningham (2021) Causal Inference: The Mixtape. Potential Outcomes Causal Model