Causality

Data Analytics and Visualization with R
Session 4

Viktoriia Semenova

University of Mannheim
Spring 2023

Warm Up

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Quiz: Which of these statements are correct?

04:00
  1. Regression line represents a conditional mean of the explanatory variable X given the value of the outcome variable Y.
  2. Extreme values of correlation coefficient (i.e. close to -1 or 1) imply that there is a large substantive effect of X on Y.
  3. Correlation between X and Y implies there is a causal relationship between them.
  4. Causal relationship between X and Y implies there is a correlation between them.
  5. Causal relationship between X and Y implies there is an association between them.

Association vs. Correlation

Correlation is a type of association and measures increasing or decreasing trends quantified using correlation coefficients.

Causality

Data Generating Process

  • An unknown process in the real world that “generates” the data we are interested in
  • In social sciences, DGP is often not very precise
  • Our understanding of DGP comes from the theory and subject knowledge

Causality

  • A variable \(X\) is a cause of a variable \(Y\) if \(Y\) in any way relies on \(X\) for its value…. \(X\) is a cause of \(Y\) if \(Y\) listens to \(X\) and decides its value in response to what it hears (Pearl, Glymour, and Jewell 2016, 5–6)

  • This incorporates:

    • association between \(X\) and \(Y\)
    • time ordering: cause precedes outcome
    • nonspuriousness: there is plausible relationship
  • Causal effect is the change in variable Y that would result from a change in variable X

Example: Boston Commuters Experiment

  • Question: How does intergroup contact impact the immigration attitudes?
  • Unit of analysis (indexed by \(i\)): individuals
  • Treatment variable \(T\): exposure to Spanish-speakers on a train platform (yes or no)
  • Treatment group (treated units): individuals exposed to Spanish-speakers
  • Control group (untreated units): individuals not exposed to Spanish-speakers
  • Outcome variable \(Y\): immigration attitudes
    • Let’s simplify for now and say \(Y\) is binary: pro- or anti-immigration

Causal Effects & Counterfactuals

  • Two potential outcomes:
    • \(Y_{i}(1)\): would commuter \(i\) report pro-immigration attitudes if exposed to Spanish-speakers (\(T = 1\))?
    • \(Y_{i}(0)\): would commuter \(i\) report pro-immigration attitudes if not exposed to Spanish-speakers (\(T = 0\))?
  • Causal effect: \(Y_{i}(1) -Y_{i}(0)\) (aka treatment effect)
    • \(Y_{i}(1) -Y_{i}(0) = 0\): exposure to Spanish-speakers has no impact on attitudes
    • \(Y_{i}(1) -Y_{i}(0) = +1\): exposure to Spanish-speakers leads to pro-immigration attitudes
    • \(Y_{i}(1) - Y_{i}(0) = -1\): exposure to Spanish-speakers leads to anti-immigration attitudes

Potential Outcomes

Attitude if Treated Attitude if Control
Jack Pro-immigration Anti-immigration


More formally:

\(Y_{i}(1)\) \(Y_{i}(0)\)
Jack 1 0

Fundamental Problem of Causal Inference

Causal Effect
\(Y_{i}(1)\) \(Y_{i}(0)\) \(Y_{i}(1) - Y_{i}(0)\)
Jack 1 0 1
  • We cannot observe \(Y_{i}(1) - Y_{i}(0)\) in real life though:
    • We only observe one of the two potential outcomes \(Y_{i}(1)\) or \(Y_{i}(0)\)
    • To infer causal effect, we need to infer the missing counterfactuals

Multiple Units

\(Y_{i}(1)\) \(Y_{i}(0)\) \(Y_{i}(1) - Y_{i}(0)\)
Jack 1 0 1
Dan 0 0 0
Anne 1 0 1
Yao 0 0 0
Judy 0 1 -1
  • Individual treatment effects: value of \(Y_{i}(1) - Y_{i}(0)\) for each \(i\)
  • Average treatment effect: mean of all the individual causal effects \(ATE = \frac{1 + 0+ 1+0+(-1)}{5} = 0.2\)

Back to Real World…

\(Y_{i}(1)\) \(Y_{i}(0)\) \(Y_{i}(1) - Y_{i}(0)\)
Jack ? 0 ?
Dan 0 ? ?
Anne 1 ? ?
Yao 0 ? ?
Judy ? 1 ?

Randomized Experiment as a Solution

  • Each unit’s treatment assignment is determined by chance
  • Randomization ensures balance between treatment and control group:
    • they are identical on average
    • we shouldn’t see large differences between treatment and control group on pretreatment variable

ATE vs. Difference-in-Means


We want to estimate the average causal effects over all units:

\[\text{Average Treatment Effect} = \frac{\sum^n_{i=1} (Y_{i}(1) - Y_{i}(0))}{n}\] But we can only estimate instead:

\[ \text{Difference in means} = \overline Y_{i}(1) - \overline Y_{i}(0) \]

This is a pretty good estimate of ATE if randomization worked!

Casual Diagrams

Directed Acyclic Graphs (DAGs)

Nodes: variables in the DGP
Arrows: causal relationships in the DGP (associations)
Direction: from the cause variable to the caused variable

Directed: Each node has an arrow that points to another node

Acyclic: You can’t cycle back to a node (and arrows only have one direction)

Graph: Well…it is a graph.

D a X b Y a->b c Z c->a c->b

Major Types of Association

Confounding
(Fork)

D a X b Y a->b c Z c->a c->b

Common cause

Causation
(Chain)

D a X b Y a->b c Z a->c c->b

Mediation

Collision
(Inverted Fork)

D a X b Y a->b c Z a->c b->c

Selection / endogeneity

Confounding

Effect of money on elections

  1. Find the part of campaign money that is explained by quality, remove it.

  2. Find the part of win margin that is explained by quality, remove it.

  3. Find the relationship between the residual part of money and residual part of win margin. This is the causal effect.

Campaign Example

Collider

Height is unrelated to basketball skill… among NBA players

- Colliders can create fake causal effects

  • Colliders can hide real causal effects

Causal Identification

  • DAGs help us with the process of identification

  • Causal effect is identified if the association between treatment and outcome is properly stripped and isolated

  • Identification implies that:

    • All alternative stories are ruled out
    • We have enough information to answer a specific causal inference question
  • Sometimes we cannot identify the effect with our data alone

D a Education b Health a->b b->a c Money c->a c->b

Studying Example

D a Hours Spent Studying b Exam Performance a->b d Y a->d e Z a->e b->e c X c->a c->b d->b