Potential Outcomes Framework: Causality

The Potential Outcomes Framework

• Examines potential outcomes and counterfactuals to define unit-specific causal effects.
• Addresses the fundamental problem of causal inference across multiple units.
• Explores solutions to the fundamental problem through simple and conditional randomization.
• Covers ignorability of treatment assignment assumption, observational data, and strong ignorability.
• Discusses the Stable Unit Treatment Value Assumption (SUTVA).
• Focuses estimating treatment effects: (ATE), average treatment effect on the treated (ATET), and ATE conditional (ATEC).
• Presents the importance of conceptual frameworks in causality studies.

Big Picture of Framework for Understanding Causality

• Aims to review the basic framework for understanding causality, with the goal of helping create frameworks that mathematical notation can be applied to.
• It aids in understanding when a measure of association can lead to a causal effect.
• Allows the separation of research design from estimation.
• It will cover ways to estimating causal effects that follow from the definition.

Basic Concepts in Framework

• Causality is linked to a manipulation, such as a treatment, intervention, action, or strategy applied to a unit.
• A unit can be a person, firm, hospital, nursing home, country, county, classroom, etc.
• It refers to the entity that receives the action or is manipulated.
• For simplicity, scenarios often involve only two possibilities: receiving or not receiving treatment (treatment vs. control unit).
• The ideas extend to multilevel treatments, but the notation becomes more complex.
• A unit’s treatment status is linked to a potential outcome.

Potential Outcomes for Units

• The concept refers to the idea that only one outcome is realized after an intervention.
• Dictionary definition: Potential refers to having or showing the capacity to become or develop into something in the future.
• Before the intervention, there are two potential outcomes.
• For example, a person may or may not receive Medicaid coverage, with the outcome being income or health status one year after the program, which can be binary.
• Each person will either get medicaid or not.

Notation Used in Framework

• The observed or realized outcome for a unit i is Y;
• The treatment received by a unit i is D,, which can be 0 or 1, so D, ∈ {0,1}
• There are two potential outcomes: Y¹ if the unit received treatment (D; = 1) and Y if not (D; = 0)
• The index i helps keep track of the data structure.
• Other common notations include W for treatment, used also in the Stata manual, and tother alternative notations for potential outcomes such as Yoi or Y,(0) which Stata uses yo etc.

Definition of Causal Effects

• The effect of receiving treatment for a unit i involves a comparison of potential outcomes.
• This can be done by comparing Y¹ - Yº or measuring the effect as a relative measure, such as Y¹/Yº.
• The definition of causal effects does not depend on action taken.
• Causal effect of receiving Medicare coverage for person A can be defined as Y¹ - Y°.

Further Discussion of Causal Effects

• Comparison can take many forms depending on how the outcome is measured.
• The outcome can be defined as the probability of being depression-free, with the causal effect being Pr(Y¹) – Pr(Yº), which is a marginal effect.
• The odds-ratio method can be used, where you dividethe probability of an event occurring by the probability of it not occurring: Pr(Y) / 1-Pr(Y)
• The way we make comparisons is related to how some statistical models measure those comparisons.

Real-World Examples of Causal Effects

• Thinking about causal effects matches everyday thinking.
• The causal effect of studying hard for class is the comparison of two potential outcomes: grade if studying hard versus grade if not.
• Many single time-travel movies utilize the concept of one action causing an event to happen, and using this to "fix" the cause and future outcomes.

The Fundamental Problem of Causal Inference

• It is that we do not observe both potential outcomes.
• This is one of the biggest issues in program evaluation.
• The non-observable or not-realized outcome is called the counterfactual outcome.
• The understanding of causal effects is now called the Rubin causal model or the Neyman-Rubin causal model of causality.

Deeper Look in Causal Inference

• The main issue is there is no way to know the counterfactual.
• The same unit receiving a treatment at a different time is effectively a different unit.
• A conceptual framework and story is also needed to find causal effects in situations like allergy testing.

Counterfactuals

• Going back to our notation, we can write Y₁ = Y° + (Y – Y)D;
• Is D, = 1 then the observed outcome is Y₁ = Y¹.
• If D, = 0 then the observed outcome is Y₁ = Y
• In this case, it is understood that a unit either receives the treatment or not, never both.
• We can now re-frame the causal inference problem as a prediction problem.

Using Multiple Units for Causal Inference

• Causal inference problems can be solved with predicting outcomes with multiple units.
• Doing this in everyday situations involves using other units.
• It can also involve the same individual changing the outcome at different times to figure out methods that work.
• To estimate the counterfactual, we must ensure something: comparing similar units and situations.

Comparisons and Assignment Bias

• With multiple units, some receive a binary treatment (D₁ = 1 if treated) and a continuous outcome is observed.
• The average observed outcomes can then be compared through calculating conditional expectation of: δₙₐᵢᵥₑ = E[Y¡|D₁ = 1] – E[Y¡|D₁ = 0]
• Bias needs to be identified in order to find causal effects.
• Understanding assignment mechanism is key figuring out casual effects.

More on Comparison and Assignment Bias

• The naive difference in expected values can be rewritten as: E[Y₁|D, = 1] – E[Y°\D, = 1] + E[Y°\D; = 1] – E[Y°\D; = 0]
• The mean of the observed outcome for the treated is the equal to the mean potential outcome for the treated when treated.
• In the same way, E[Y;|D; = 0] = E[Y°|D; = 0]
• Rewriting observed expectations in terms of potential outcome expectations is key.

Association versus Causation

• To interpret comparisons (δₙₐᵢᵥₑ = E[Y¡|D₁ = 1] – E[Y¡|D₁ = 0]), the bias must be zero in order to have a causal effect.
• If the outcome for the treated when they are not treated [counterfactual] is the same as the observed outcome for the control units when they are not treated, then a simple comparison of average outcomes can provide an estimate of average treatment effects.
• To identify causal effects, the method of treatment assignment must be independent of potential outcomes.

Conditioning & Flip Side of Assignment Bias

• One form of assignment bias is: δₙₐᵢᵥₑ = δₐₜ𝒸 + E[Y;|D₁ = 1] – E[Y;|D;}\D₁ = 0]
• With both forms of assignment bias it follows that: δₙₐᵢᵥₑ = δₐₜ𝒸 = δₐₜₜ = δₐₜ𝒸
• In other words, each group provides a counterfactual.

Independence

• This is a refresher on statistical independence.
• Two random variables are independent if and only if: fx.y(x,y) = fx(x)fy(y).
• For discrete random variables: P(X = x, Y = y) = P(X = x)P(Y = y)
• In terms of events: P(A∩B) = P(A)P(B) (their joint probability is the same as the multiplication of the individual probabilities).
• Conditional probability is: P(A\B) = P(A∩B) / P(B) = P(A)P(B) / P(B) = P(A)

Conditional Independence

• Conditional independence is important regression models and conditional independence assumption.
• Events A and B are conditionally independent if P(A∩B|Z) = P(A|Z)P(B|Z)
• In other words, knowing B does not tell us about P(A) given Z.

For Causality, Key Understanding is Conditional Independence

• If you randomly assign patients to treatment based on severity of illness.
• and people with severe illness are more likely to receive treatment, and age A is related to the illness.
• Therefore, D and A are not independent but D and A are conditionally independent given Z where P(D\Z, A) = P(D|Z)
• once we know the severity of the illness, Z, age A doesn't tell anything about D.

Two Experimental Models

• In simple randomization units are randomly divided into groups.
• Then in conditional randomization treatment assignment is conditional.
• Aim is to the study being as conditional randomization as possible, and understanding what influences treatment.
• This involves designing observational studies that approximate conditional randomization as closely as possible.
• Emphasizes that the key is understanding why some units ended up receiving treatment.

Further Elaborating Ways to Solve Causal Inference

• a) Simple randomization: Control groups are best when they have no assignment bias.
• After the manipulation in one group, the other provides the counterfactual.
• Or precisely the potential outcomes independence from particular group assignments.
• Groups are "exchangeable" so which group receives treatment doesn't matter.

Randomization and Independence

• The observed can come form the counterfactual through: E[Y;|D₁ = 1] – E[Y;|D₁ = 0] = E[Y₁i|D₁ = 1] – E[Yoi|D₁ = 0]
• If causal inference is a prediction problem, randomization solves it.

Even More Info on Solving Causal Inference

• With conditional randomization "condition" on the variables that influence treatment assignment.
• So these need to be true: E[Y;|D; = 0, Z = 1] = E[Y°|D; = 1, Z = 1] and E[Y; D; = 0, Z = 0] = E[Y°|D; = 1, Z = 0]

Average Treatment Versus Individual Effects

• Only average treatment effects are obtained from the causal with multiple units.
• Must assume that the effects are average to estimate individual effects, and that they aren't heterogeneous.

Treatment Effects Described: ATE, ATET, LATE

• The equation is: E[Y;|D; = 1] – E[Y;|D; = 0] = E[Y1;|D; = 1] – E[Yo¡|D; = 1] + E[Yo;|D; = 1] – E[Yo¡|D; = 0]
• Observed/naive average treatment effect = Average treatment effect on the treated + selection bias 1
• Observed/naive average treatment effect = Average treatment effect on the control + selection bias 2
• Under randomization, ATE (average treatment effect) = ATET (average treatment effect on the treated) = ATEC (average treatment effect, conditional)

More Ways Of Solving Causal Inference

• b) Statistical adjustment: The provides intuition with (some form of) regression or other forms of adjustment.
• We can experiment with randomization through severty classified patients 80% rate of treatment for severe, 20% rate of treatment if severity is low.
• So a severity score can used with regression.

Conditional Randomization with Regression

• The selection bias is not zero when the treated have different expected outcomes with E[Yo¡|D; = 1] ≠ E[Y;|D; = 0]
• Model = Y; = βο + β₁ D; + €¡ so severity Z is part of e.
• A simple fix is to run model like Y; = Yo + Y1 D¡ + Y3Z; + n; and run the other versions assuming the model is correct.

Fundamental Assumption for Statistical Adjustment

• In order to justify statistical adjustment.
• Observational variables must be present for the assignment of units to experimental groups to be independent.
• This also can be looked at with ignorability: with the inability to control confounders meaning correlation doesn't mean causality, with the assumption that no unmeasured confounders are present.

Confounders

• A variable in a statistical model that correlates with dependent variable and independent variable.
• Another way: a confounder predicts both a covariate and outcome
• Though antidepressants can be, caution must also be taken as all confounders aren't created equal.

Definitions for Moderator, Mediator, Confounder

• Moderator: A variable/factor that could change the treatment effect.
• Mediator: A variable factor that explains effects and the mechanism relating to the variable's connection between the two variables. Also called a "pacifier"
• Without a framework of mediator, it is hard to truly discern relationships.

Assumptions

• To determine biases, comparing what variables are affected and knowing their relationship to treatment and outcome is needed.
• More nuance is also needed especially in instances when covariates are used.

Heterogeneous Treatment Effect

• The model for treatment effect: Y; = βο + β₁ D; + β2Z; + €¡
• Stratified models are needed to compare severty as 1 for different treatment, zero in other cases.
• Another model for this is Y¡ = Yo + Y1 D; + Y2Z; + ¥3D¡ × Z¡ + Ni
• We need this extended definition to better understand what variables cause the treatment.

Regression Adjustment Requires Another Assumption: Overlap

• The group has been actually need another assumption that, has oversight, which also helps with causal effects by reading with observational data
• If the data only includes people of either extreme, it will have an affect.
• For the assumption, the overlap should have no restrictions. All are required, also the covariate needs to a defined definition from its proper city area. This includes a certain score.

Stable Unit Treatment Value Assumption (SUTVA)

• This assumes that the outcome of the treatment will have no interference.
• This is needed when it is shown to be related more.
• It is important to consider that all assumptions can have exclusions.

Other Important Concepts

• Internal validy helps determine how well its effects can be.
• This could include some sort of instrumental design.
• So the pop up, might be hard to hard with difficulty but always with the valid assessment.