March 8, 2019

Testing Causal Theories

Plan for Today:

(1) Testing Causal Theories

  • hypotheses/empirical predictions
  • fundamental problem of causal inference

(2) "Gold Standard": Experiments

Testing Causal Theories


hypotheses/empirical predictions

statements about what we should observe if the causal claim is true, so stated in terms of independent and dependent variables

  • multiple hypotheses based on the overall causal claim and the causal logic
  • Hypotheses state relationships between \(X\) and potential outcomes of \(Y\)
    • When \(X\) is present(absent) for a case, \(Y\) is present(absent)
    • When \(X\) is present(absent) for a case, \(Y\) is more(less) likely
    • When \(X\) increases(decreases) for a case, \(Y\) increases(decreases)

Potential Outcomes and Hypotheses

Trump's causal claim (implicitly): "The wall caused El Paso to have fewer murders".

  • independent variable (\(X\)): presence of wall (\(1\)) or absence of wall (\(0\)) \(Wall_{El \ Paso}\)
  • dependent variable (\(Y\)): number of murders \(Murders_{El \ Paso}\)

So the hypothesis is…

potential outcomes of \(Murders_{El \ Paso}\) lower when \(Wall_{El \ Paso} = 1\)

\[Murders_{El \ Paso}^{Wall = Yes} < Murders_{El \ Paso}^{Wall = No}\]

Potential Outcomes and Hypotheses

causal claim: "Ownership of firearms by citizens reduces crime"

  • independent variable (\(X\)): fraction of citizens owning firearms \(Gun \ Rate_i\)
  • dependent variable (\(Y\)): crime victimization rate per 100 thousand \(Crime \ Rate_i\)

So the hypothesis is…

Across communities, \(i\) in \(1 \ldots n\), on average, potential outcomes of \(Crime \ Rate_i\) lower for higher values of \(Gun \ Rate_i\)

\[Crime \ Rate_i^{Gun \ Rate = High} < Crime \ Rate_{i}^{Gun \ Rate = Low}\]

A big problem

Hypotheses are empirical predictions: they are about what we should observe if \(X\) causes \(Y\).

Counterfactual causality means \(X\) causes \(Y\) in some specific case, only if potential outcomes of \(Y\) are different across levels of \(X\).

But for each case, only one potential outcome becomes "factual" and observable; the other potential outcome(s) remain counterfactual and unobservable

Fundamental Problem of Causal Inference

Fundamental Problem of Causal Inference

We can never observe all of the potential outcomes of \(Y\) for a case under different exposures to \(X\). We can only observe the potential outcome of a case that becomes factual where \(X\) takes on one, specific value. All other potential outcomes of \(Y\) remain counterfactual and unobservable.

For a single case, we can never empirically observe whether \(X\) causes \(Y\)

An Example:

causal claim: "Befriending an immigrant causes people to have positive attitudes of immigrants"

  • independent variable (\(X\)): person has immigrant friend \(Friend_i\): either \(1\) (yes) or \(0\) (no)
  • dependent variable (\(Y\)): self-reported "overall attitude" towards immigrants \(Attitude_i\) is \((-2,-1,0,1,2)\) for strongly negative, negative, neutral, positive, strongly positive

We collect data on 5 people.

Our hypothesis is:

Across persons, \(i\) in \(1 \ldots 5\), on average, potential outcomes of \(Attitude_i\) are higher for higher values of \(Friend_i\)

An Example:

We collect this data, and we observe this:

\(Person_i\) \(Friend_i\) \(Attitude_i\)
1 Yes Positive (1)
2 Yes Very Positive (2)
3 No Neutral (0)
4 No Neutral (0)
5 No Negative (-1)

Can we conclude \(Friend\) causes more positive \(Attitude\)s toward immigrants?

An Example:

No, we've forgotten potential outcomes

\(Person_i\) \(Friend_i\) \(Attitude_i^{Yes}\) \(Attitude_i^{No}\)
1 Yes Positive (1) ?
2 Yes Very Positive (2) ?
3 No ? Neutral (0)
4 No ? Neutral (0)
5 No ? Negative (-1)

An Example:

We want to know the difference in potential outcomes due to \(Friend\)ship: but FPCI means it is unobservable

\(Person_i\) \(Friend_i\) \(Attitude_i^{Yes}\) \(Attitude_i^{No}\) \(Attitude_i^{Yes} - Attitude_i^{No}\)
1 Yes Positive (1) ? ?
2 Yes Very Positive (2) ? ?
3 No ? Neutral (0) ?
4 No ? Neutral (0) ?
5 No ? Negative (-1) ?

An Example:

If we were omniscient deities, maybe we could know all potential outcomes: actually, no effect

\(Person_i\) \(Friend_i\) \(Attitude_i^{Yes}\) \(Attitude_i^{No}\) \(Attitude_i^{Yes} - Attitude_i^{No}\)
1 Yes Positive (1) Positive (1) 0
2 Yes Very Positive (2) Very Positive (2) 0
3 No Neutral (0) Neutral (0) 0
4 No Neutral (0) Neutral (0) 0
5 No Negative (-1) Negative (-1) 0

How do we solve this problem?

Making assumptions:

We make assumptions that let us treat \(Y\) for cases that we observe with on value of \(X\) as counterfactuals for cases with different values of \(X\).

How reasonable/believable are these assumptions?

One way: Experiments

Why do you think experiments allow us to infer causality?

One way: Experiments

We cannot know the causal effect of \(X\) on \(Y\) for a specific case.

Experiments (what you NEED to know)

Let us estimate (draw inferences about) the average causal effect of \(X\) on \(Y\) for a group of cases.

Three assumptions (you only need to know (1))

  1. Random Assignment to "Treatment" and "Control"
  2. Exclusion Restriction (only one thing is changing: \(X\))
  3. SUTVA (If I receive the treatment, it doesn't affect you)

One way: Experiments

Why do experiments work?

Intuition (you don't NEED to know this)

  • In example with immigrant friendship and attitudes about immigrants, people with immigrant friends already different in their potential outcomes (people who hate immigrants don't befriend them)
  • Experiments, through random assignment, ensure that potential outcomes of treated/untreated cases are the same (except for random sampling error)

One way: Experiments

Social Media Shaming and Voting

Last year's PR referendum

  • How many of you voted?

  • Would have voted if tagged in a post like this?

"Voting records are public! Records indicate that [name], [name], [name], [name], [name], [name], [name], [name], [name], and [name] have not yet voted in this year’s proportional representation referendum! Voting ends Friday. Click here for voting locations or how to vote from home. [Link]"

One way: Experiments

Proof (You REALLY DON'T NEED to know this)