1. Conditioning:

• What is it
• How does it work?
• Assumptions

We want to use correlation to provide evidence of causation:

• but we know that confounding (a form of bias) can lead us astray.
• if we can reasonably make some assumptions, correlation can be strong evidence of causality without bias.

• What are the assumptions needed to infer correlation \(\to\) causation?
• How can we check/interrogate those assumptions?

Every way of using correlation as evidence for causality makes trade-off between:

• internal validity: how plausible are assumptions that set aside confounding?
• external validity: how much does this causal relationship speak to the causal claim/question of interest?

conditioning

when we observe \(X\) and \(Y\) for multiple cases, we examine the correlation of \(X\) and \(Y\) within groups of cases that are the same on confounding variables \(W, etc. \ldots\)

How does conditioning solve the problem?

• Cases compared have same values on confounding variable(s) \(W\)
• In these groups, \(W\) cannot affect \(X\) or \(Y\) (because \(W\) is not moving, it can’t move \(X\) or \(Y\))
• “Backdoor path” from \(X\) to \(Y\) is “blocked”

Feinberg, Branton, and Martinez-Ebers compare hate crimes in counties with and without Trump rallies, but condition on (hold constant):

• percent Jewish
• number of hate groups
• crime rate
• 2012 Republican vote share
• percent university educated
• region

Feinberg, Branton, and Martinez-Ebers find that, even after conditioning, Trump rallies increase the risk of hate crimes by 200%!

• Lots of news headlines like this

Economics PhD Candidates show that conditioning on the same variables…

• Clinton rallies increased hate crimes by nearly 250%!!

• What is going on here??

We would be wrong to use observed hate crimes in county \(b\) (without a rally) to substitute in for counterfactual hate crimes in county \(a\) (without a rally).

Population differences \(\to\) difference in hate crimes regardless of rally.

After also conditioning on population (a confounder): no correlation.

In order to use conditioning to infer \(X\) causes \(Y\) if \(X,Y\) correlated …

Must Assume

1. There are no other confounding variables (that you have not conditioned on)
   • i.e. you have conditioned on ALL confounding variables
   • sometimes called “ignorability assumption”: you assume you can “ignore” other variables without confounding

How can we tell whether this assumption is correct?

DISCUSS

1. Assume there are no other confounding variables (that you have not conditioned on)

• this can never be proven to be true: a “strong” assumption
• we can’t know whether all confounders are blocked because we don’t know true causal graph
• compare this against the assumption of randomization in experiments

In wake of mass shootings, we might ask:

Do mass shooting events cause people to become more supportive of stricter gun control policies?

Newman and Hartman (2017) examine whether exposure to mass shootings cause an increase in support for stricter gun control

• Use large political attitudes survey (CCES) to look at individual responses to: “In general, do you feel that laws covering the sale of firearms should be made more strict, less strict, or kept as they are?”
• Measure exposure to mass shooting as “distance to nearest mass shooting in recent years”

• What might be possible confounding variables?

Which variables do we NEED to condition on? Which variables do we NOT NEED to condition on?

Must Assume

1. There are no other confounding variables (that you have not conditioned on)

• i.e. you have conditioned on ALL confounding variables
• this can never be proven to be true: a “strong” assumption

But…

• don’t need to condition on ALL variables that might affect \(X\) or affect \(Y\): only confounders.

Which variables do we NEED to condition on? Which variables do we NOT NEED to condition on?

Must Assume

1. There are no other confounding variables (that you have not conditioned on)

• i.e. you have conditioned on ALL confounding variables
• this can never be proven to be true: a “strong” assumption

But…

• don’t need to condition on ALL variables that might affect \(X\) or affect \(Y\): only confounders.
• don’t need to condition on ALL variables on a backdoor path. Just one variable per backdoor path is needed.

Newman and Hartman (2017):

Compare proximity to mass shooting and gun control attitudes, conditioning on (holding constant:

• community level factors: income, education, partisanship, racial composition, murder, firearms stores, population density, population
• individual level factors: education, income, age, gender, race, property ownership, having children, in military, military family, partisanship, ideology, religiosity, region of birth

• Are there any potentially really important confounders that might not be included?

Proximity to mass shootings increases support for stricter gun laws… assuming no other confounders

Must Assume

1. There are no other confounding variables (that you have not conditioned on)

But…

• don’t need to condition on ALL variables that might affect \(X\) or affect \(Y\): only confounders.
• don’t need to condition on ALL variables on a backdoor path. Just one variable per backdoor path is needed.
• we can give reasons/arguments for why we think any remaining confounding is likely small OR biased against the correlation (small bias may not be enough to change our conclusions, or bias is against our conclusions)

In order to infer \(X\) causes \(Y\) if \(X,Y\) correlated after conditioning

Must Assume

2. Variables used to condition relationship between \(X\) and \(Y\) are measured without error.

• even random measurement error in confounding variables leads conditioning to not remove bias.
• Why? You are no longer comparing like-with-like.

Imagine we want to condition on gun ownership, when examining correlation of mass shootings and gun attitudes.

What if we measure gun ownership with random measurement error?

Let’s see what happens… BOARD

In order to infer \(X\) causes \(Y\) if \(X,Y\) correlated after conditioning

Must Assume

3. We can find cases that are the same on confounding variables \(W\) and different in \(X\).

Let’s say we want to examine the effect of gun laws on gun violence across countries:

What are some factors that might affect strictness of gun regulation and gun violence?

Is there, e.g., country that is similar to the US on all confounders?

In order to infer \(X\) causes \(Y\) if \(X,Y\) correlated after conditioning

Must Assume

3. There are cases that are the same on confounding variables \(W\) and different in \(X\).

• as we want to condition on more and more variables…
• less and less likely we find cases that exactly match
• so, cannot condition on all confounding variables

Conditioning:

• examine correlation of \(X\) and \(Y\) for cases that are the same on confounders \(W\)
• Assumptions:
  • no other confounding variables: can’t check
  • no measurement error: can check
  • similar cases with different \(X\): can check
• While imperfect, we should not dismiss this approach.

Did Trump rallies increase hate crimes?

1. If we know confounding variables, can we find cases with and without rallies that are the same on many confounding variables?

2. If we don’t know or can’t measure confounding variables, may still be differences between places with and without rallies that produce confounding.

• Is there a better comparison we can make that permits us hold (potentially unmeasurable) confounders constant and where we can easily find a “similar” case that didn’t have a rally?

We can make Before and After comparisons:

• we compare the same case before and after it experiences some cause

Taking the same data from Feinberg, Branton, and Martinez-Ebers…

• we focus only on counties that ever had a Trump (Clinton) rally
• compare the month after the rally to the month before the rally

DISCUSS

If we compare counties to themselves before and after rallies…

1. Which confounding variables are held constant?

2. What are confounding variables that might NOT be addressed in this comparison?

Conditioning removes confounding by:

• identifying possible confounding variables
• measuring confounding variables
• relationship b/t \(X\) and \(Y\) for cases with similar value of confounding variables \(W\).

Design-based solutions remove confounding by:

• selecting cases for comparison in order to eliminate many known/unknown as well as measurable/unmeasurable confounding variables.
• the nature of the comparison holds constant classes of confounding variables, not specific confounding variables.
• by a “class” we mean all confounding variables that share certain properties

Which of these possible confounders are held constant in a before-and-after comparison?

Confounding Solved?…

All confounding variables (affect whether a rally occurs; affect hate crimes) that are unchanging over time (before to) are held constant.

• because held constant, cannot produce confounding
• e.g., demographic features, political leaning, location/geography, long-term economic trends, 8chan white nationalists
• any variable that does not change in the time period of the comparison (in this case, two months) held constant
• does not matter if we can think of or even measure the confounders