Learn about the fundamentals of causal inference.

Potential Outcomes Framework

Often, randomization of treatment is unethical. For example, it would be unethical to force hospital patients who don’t like animals to receive therapy animal services. Randomized experiments can also be expensive and demand a lot of resources. It is often more practical to use data that is already available. In these scenarios, the treatment group is not random but is based on other factors, such as personal preference.

When randomization is not possible or plausible, several sources of bias can impact the accuracy of estimated causal effects. One source of bias is _selection bias_. Selection bias is bias that happens because of how individuals were put into the treatment or control groups. 

In terms of our therapy animal example, selection bias might arise if:
* individuals choose whether they want the therapy (dog-lovers might be over-represented in this case)
* individuals in the control group come from a different hospital
* individuals are only able to receive the therapy based on another variable, such as insurance coverage

If any of these variables that are associated with treatment assignment are also related to the outcome variable (cortisol level), they are considered _confounders_, or _confounding variables_. These variables may lead us to incorrect conclusions about the impact of the treatment on the outcome.

The following is a graphical representation of how the confounder X is associated with both the treatment Z and outcome Y:

![Two diagrams with boxes labeled with letters and arrows connecting the boxes. Diagram 1 shows no confounding because the Z box has an arrow pointing to the Y box. Diagram 2 shows confounding because there is an additional X box with arrows pointing to both the Z and Y boxes.](https://static-assets.codecademy.com/Courses/causal-inference/conceptual-foundations/po-ex7-narr.svg)

Confounders and Selection Bias

In order to generalize our understanding of the potential outcomes framework, we will now introduce some notation that will be used throughout the rest of this course. 

 - Z represents the treatment or exposure condition. When Z is binary, there are only two possible treatment conditions: treatment (Z = 1) or control (Z = 0).
 
 (Z could also be a continuous variable, such as medication dosage or number of days in the hospital, but our example assumes that there are only two treatment conditions for simplicity.)
 - Y represents the observed value of the outcome variable.

In the potential outcomes framework, we consider how the Y values we observe would change if the treatment were different:

 - When Z is binary, the two potential outcomes are represented with Y1 and Y0:
 - Y1 is the potential outcome under the treatment condition (Z = 1).
 - Y0 is the potential outcome under the control condition (Z = 0).

In real life, an individual can only be in one treatment group at any specific point in time. Thus, we can never know the value of both potential outcomes for an individual. We will discuss how to deal with this problem in future exercises. For now, assume that we could know the values of both Y0 and Y1.

Potential Outcomes Notation

A diagram showing two boxes connected by an arrow that points from the left box to the right box. The left box is labeled "Identification" and states, "Determine the causal estimand. Consider whether causal assumptions have been met: 1 Conditional exchangeability, 2 SUTVA, 3 Overlap." The right box is labeled "Estimation" and states, "Carry out statistical modeling. Obtain an estimate of the treatment effect."

Now that you have some experience with the assumptions needed for causal inference as well as familiarity with a few causal estimands, we need to set up a structured way to apply what you’ve learned when approaching causal inference problems.

We can think of causal inference as a two-step process.
1. _Identification:_ During this stage we determine which causal estimand we will estimate based both on what we want to know and on what we are able to compute. We must also determine whether we will be able to meet the three assumptions in order to infer that the relationship is causal in nature.
2. _Estimation:_ Now that we've reasoned what can compute and that this measure will reflect a causal relationship between variables, we must carry out the statistical model to compute the treatment effect. We used some simple computations throughout this lesson, but we will soon use more complicated methods like multiple linear regression to obtain our estimand.

Causal Inference Process

Because we can never know both potential outcomes for an individual, we need to use a different method to estimate causal effects. The most accurate way to do this is to use _randomization_.

Randomization is a method of treatment group assignment that is essentially a coin flip to determine whether an individual receives the treatment or the control. This ensures that, for a large enough sample size, the treatment groups will be similar on average with respect to all factors EXCEPT for the treatment condition. While we still won’t know the counterfactual outcome for any individual, we can be reasonably confident that similar individuals received each treatment. This allows us to estimate the potential outcomes we weren’t able to observe.

When randomization is possible, we can estimate the ATE by taking the difference of the average observed outcome values in the treatment and control groups. In the example in the learning environment, the estimated ATE equals -5.4, which is very close to the true ATE of -5.8 that was calculated in the previous exercise.


Estimating the ATE

Chances are, you or someone you know is superstitious to some extent. Whether it’s wearing a lucky t-shirt to a sporting event or using a favorite pencil and eraser on exams, we believe in superstitions because we think our actions will lead to some desired&mdash;but usually unrelated&mdash;result. Superstitions are, in fact, extreme examples of assuming an associational relationship is actually causal in nature. 

One of the most important concepts in causal inference is the distinction between association and causation. Let’s formally define these two terms:

1. _Association_ is a general term to describe a relationship between variables. Association can describe the strength or pattern of a relationship, but it does not explain the mechanism behind the relationship. 
 
 One frequently used statistical measure of association is _correlation_. Correlation is typically used to describe the association between two variables with a linear pattern. The animation below shows what variables with different degrees of correlation look like.
 
<iframe src="https://static-assets.codecademy.com/Courses/causal-inference/conceptual-foundations/po-corr/index.html" width="350" height="350" frameBorder="0"></iframe>
 
2. _Causation_ describes not only the strength or pattern of a relationship but also the MECHANISM of a relationship. In a causal relationship, variable X CAUSES a change in variable Y; we know that X must happen before Y.

A line plot of swimming pool sales and forest fires over time, where both lines follow similar trends. The lines both rise in the summer months and fall in the colder months, meaning there are more swimming pool sales and more forest fires during warmer months.

Correlation is Not Causation

So how do we deal with confounders and estimate the ATE when the treatment assignment is not randomized? Let’s return to the therapy animal example once more. 

Suppose that instead of randomizing patients to receive therapy animal services, we allow the twelve hospital patients to CHOOSE whether or not they want therapy. Imagine that we also have data for a new confounding variable X that represents whether an individual does (X = 1) or does not (X = 0) have a diagnosis of anxiety disorder. Here, X is a confounder and impacts both the treatment and the outcome: patients who have anxiety might be more likely to choose to receive therapy animal services AND have higher cortisol levels generally. 

This confounding is problematic because it means there may be more people with anxiety in the treatment group than in the control group. More people with anxiety means the treatment group may have a higher average cortisol level compared to that of the control group before therapy animal services even occur!

When the treatment groups are unbalanced with respect to confounders, the treatment groups are not _exchangeable_: we would observe different outcomes if the treatment groups swapped treatment conditions.

To avoid making poor comparisons between potentially imbalanced treatment groups, we have to be able to assume _conditional exchangeablility_:
 - _Conditional exchangeability_ means that the treatment groups are exchangeable if we take into account confounding variables.
 - This is also called _ignorability_ or _unconfoundedness_.

By taking anxiety diagnosis (variable X) into account, we avoid getting a biased estimate of the cortisol levels produced by those receiving therapy animal services in comparison to those who do not.

Estimating the ATE with Confounders

The last exercise brought up one assumption used throughout causal inference: conditional exchangeability. In this lesson, we will learn a few more assumptions.

A second assumption made in causal inference is  _Stable Unit Treatment Value Assumption_ (SUTVA). The name is a mouthful, but it’s a pretty simple assumption that can be broken down into two components:
1. An individual’s treatment assignment doesn’t impact the outcome of other individuals. Using the example of the hospital patients, this would mean that one individual getting therapy animal services doesn’t impact the stress level of other individuals in the hospital.
2. The treatment (or control) is applied exactly the same way to all patients. For example, the patients receiving therapy animal services should receive therapy for the same amount of time and ideally from the same exact animal to ensure consistent treatment.

The next assumption we need to familiarize ourselves with is _overlap_. The assumption of overlap means that all subgroups of patients divided by their characteristics have a positive, non-zero probability of getting either treatment assignment. Overlap is also referred to as the _common support_ or _positivity_ assumption. 


A plot showing probability distributions for age in two situations. The left plot is labeled "Overlap" and shows treatment and control distributions that are nearly the same shape and location covering one another. The right plot is labeled "No Overlap" and shows the treatment distribution covers ages 0 to 38, while the control distribution covers ages 38 to 80.

Assumptions

A comic depicting two characters. Character 1: "Another huge study found no evidence that cell phones cause cancer. What was the W.H.O. thinking?" Character 2: "I think they just got it backward." Character 1: "Huh?" Character 2: "Well, take a look." A line chart of cancer incidence and cell phone users in the United States over time. the line for cancer rate increases to a high level first. Then the line for cell phone users begins to increase after. Character 1: "You're not... There are so many problems with that." Character 2: "Just to be safe, until I see more data I'm going to assume cancer causes cell phones."

As humans, we are hardwired to look for patterns and identify relationships between things we observe in the world around us. Our brains naturally tend to fill in details and come up with explanations for these relationships. Unfortunately, jumping to conclusions often leads us to see nonrandom in the random and to blur the lines between association and causation.

Before we get some hands-on experience using causal inference methods, we need to build up some intuition about what exactly causal inference is, when causal inference is appropriate to use, and what causal inference can (and cannot) do. In this lesson, we will gradually introduce you to the key ideas, statistical frameworks, and required assumptions that are fundamental to causal inference.

Causal Inference: An Introduction

A comic depicting a conversation between two characters. Character 1 says, "I used to think correlation implies causation. Then I took a statistics class. Now I don't." Character 2 says, "Sounds like the class helped." Character 1 responds, "Well, maybe."

Congratulations! You've completed the Potential Outcomes Framework lesson and now have a solid conceptual foundation for causal inference!

In this lesson, you learned:
 - Causation is different from association in that it implies a relationship where a change in one variable leads to a change in another variable.
 - If we could view both potential outcomes, we could accurately compute the true effect of a treatment.
 - The fundamental problem of causal inference is that we only get to view the observed outcome and not its counterfactual.
 - When randomization is not available to produce estimates of treatment effects, we must use other strategies to predict counterfactuals and find our estimand of interest.
 - The three main assumptions for causal inference are:
   - Conditional exchangeability
   - SUTVA
   - Overlap
 - The causal inference process can be thought of as two steps: identification and estimation.


Review

A depiction of the two possible realities. When the patient receives therapy animal services, Universe T is the factual outcome and Universe C is the counterfactual outcome. When the patient does not receive therapy animal services, Universe C is the factual outcome and Universe T is the counterfactual outcome.

The second important concept we must learn is _counterfactual thinking_. Counterfactual thinking is the process of asking, “What WOULD have happened if circumstances were different?” Let’s illustrate counterfactual thinking using the following example:

Dogs are often called “human’s best friend,” but did you know there may be a biological explanation behind this saying? Research has shown that just a few minutes of interacting with dogs or cats can reduce levels of cortisol, a hormone linked to stress that may lead to weight gain or a weakened immune system. Let’s say we’re interested in learning whether interacting with a trained therapy animal leads to decreased levels of cortisol in hospital patients. 

Approaching this using counterfactual thinking, we must consider what each patient’s cortisol level would be in two different “universes”: 
 - In one universe, a patient interacts with a therapy animal&mdash;we will call this “Universe T.” 
 - In the other universe, referred to as “Universe C,” the same patient does NOT interact with a therapy animal. 

The cortisol levels in these two parallel universes are called _potential outcomes_ because either could potentially be observed. But in reality, we can only observe one cortisol level for a particular patient at one specific moment in time.

Assume that a patient exists in Universe T and ACTUALLY interacted with a therapy animal. The cortisol level observed in this scenario is the _observed_ or _factual outcome_ because it is the outcome that was observed. The cortisol level that would have been observed if the patient existed in Universe C and did not interact with the therapy animal would be the _counterfactual outcome_. We can never actually observe the counterfactual outcome.

Using counterfactual thinking allows us to compare the exact same person at the exact same time under two different circumstances. Because the only difference between the two universes is that the patient received the treatment in Universe T and the control in Universe C, we could compare the cortisol levels from each universe to get an estimate of the effect of interacting with therapy animals.



Thinking Counterfactually

So far, we’ve only focused on the ATE, but there are many other useful estimands we can use to summarize the average causal effect of some intervention or exposure. In many situations, it is not realistic to meet the assumptions of conditional exchangeability, SUTVA, AND overlap. Other causal estimands allow us to relax some of these assumptions.

The ATE summarizes the causal effect of treatment across ALL treatment conditions. But, we may only be interested in the causal effect of treatment in a particular group. In these cases, we can use a different estimand:
 - _Average Treatment Effect of the Treated_ (ATT) is the average of Y1 - Y0 for all individuals assigned to the treatment condition (Z = 1).
 - _Average Treatment Effect of the Control_ (ATC) is the average of Y1 - Y0 for all individuals assigned to the control condition (Z = 0).

There are other estimands we may encounter in causal methods, but the ATE, ATT, and ATC are three that we see in a variety of situations.

Other Causal Estimands

If we knew both potential observations for every individual, we could use them to estimate several different statistics that summarize the effect of the treatment:
 - The _individual treatment effect_ (ITE) is computed as Y1 - Y0. This statistic directly compares the two potential outcomes for each individual.
 - The _Average Treatment Effect_ (ATE) is the average of all individual treatment effects, which can be calculated as the difference between the average of Y1 and the average of Y0.

Hold up! You may be wondering, “How are we supposed to calculate the individual treatment effect or true ATE if we can never observe the counterfactual outcome?” This question gets at the fundamental problem of causal inference. Causal inference is essentially a missing data problem: since we can only observe the outcome that actually happened, we are always missing the counterfactual outcome.