Lecture 3 - Correlation to Causation

Introduction to Correlation vs. Causation

Definitions

  • Correlation: A statistical measure that indicates the extent to which two variables fluctuate together.

  • Causation: The act of causing; it implies a relationship where one event (the cause) leads to the outcome of another event (the effect).

  • Importance of distinguishing between correlation (when two events occur together) and causation (where one event directly causes another).

Volcano Eruptions and Causality

Example:

  • Eruption of Mount Tambora (1815): Recorded as the biggest volcanic eruption in human history, resulting in substantial ejecta affecting global conditions.

    • Volume of Ejecta: 19 cubic miles.

  • Subsequent impact: the Year Without a Summer (1816), leading to significant temperature changes across the globe.

    • Consequences: The eruption invoked cultural shifts, including the emergence of literary works like Mary Shelley's "Frankenstein."

      • Creates the question; Did the eruption of Tambora cause Frankenstein?

Defining Causation Coherently

Goal for Causation Definition:

  • Causal Effect: A tangible change in one feature of the world as a result of a change in another.

    • Examples of a Causal Question:

      • Does smoking cause cancer?

Different attempted Definitions of Causation

Correlation Approach:

  • If knowing one variable allows prediction of another, does this indicate causation?

    • Example: Stop global warming become a pirate.

    • Issue: Global warming is increasing due to industrialization, and due to that same reason, there are less pirates as it is harder for them to survive

    • Significance: even though there is a correlation, is there a causal relationship?

      • Answer: no

Problems

  • Correlation without causation:

    • Common cause or confounding cause.

      • Meaning: there is a third variable that is responsible or is the cause for the treatment occurring (Example: industrialization).

  • Direction

    • Correlation of X with Y is the same as correlation of Y with X

      • Meaning: reverse causation (Example: windmills cause wind because when they spin there is wind, instead of wind causing windmills to spin).

Regularity Approach:

  • If X happens, Y follows, but not the other way around.

    • This solves the issue of direction.

Problem

  • Deterministic:

    • Meaning: implies that a something happening is absolutely determined by something else always happening (Example: if shot you die, but that doest always happen).

  • Trivial relationships:

    •  Meaning: establish relationships between variables that don’t determine the reality of them existing (Example: every time I ring a bell, i’m human).

Temporal Order:

  • A must occur before B to assert that A causes B.

  • Examples of problematic relationships exist (e.g., Christmas cards causing Christmas).

Problem

  • Points the arrow in the wrong direction:

    • In general: prediction ≠ causation

Physical Connection Approach

  • We can experience it with our senses

  • The effect occurs due to some physical effect.

Problem

  • Hard to verify

  • Requires more convoluted stories

Counterfactual Dependence Approach (Correct definition)

  • X causes Y if and only if

    • Y occurs when X occurs (X and Y demonstrate some level of correlation)

    • Y would not have occurred in the counterfactual world where X did not occur

      • Under this definition, the only possible cause of the change in Y is the change in X.

      • Example: Under this definition, the following would have to be argued, in a world in which mount Tamaro does not exist, Frankenstein would not have existed.

        • The problem os that this is very hard to argue/prove

  • The only requirement for this definition is tat we need to be able to imagine a counterfactual world.

  • Regularity is not required

  • Causal stories should have a clear connection to counterfactual dependence.

Model of Potential Outcomes

  • Corresponding model to the Model of Counterfactual Dependence

  • For each possible value of the causal variables (person or observation), there is an associated measure of the outcome. These values are the potential outcomes.

Counterfactual world

  • The counterfactual world is one identical to the observable one except for a single change in X (everything is identical up until our treatment occurs, the only thing thats differs is that X is different).

    • Example: easiest to image if the casual variable can only take one of two values (Example: taking the pill or not taking the pill)

    • Other examples include more than two categories, like the dosage of a drug.

Example

  • X(binary event) = College

    • X = 1 ‘treated’ —> Y1 = Wage 10 years after graduating

    • X = 0 ‘untreated’ —> Y0 = Wage 10 years after graduating High School

      • What is a reasonable way to asses the effect of X on Y?

      • Answer = Y1 - Y0 (The difference in the potential outcomes is equal to the casual impact)