March 5, 2018

Recap

Causal Theory

• Concepts $$\xrightarrow{}$$ Variables
• Independent (Cause) and Dependent (Outcome) Variables
• Direction of effect
• Causal logic

Testing

Based on causal theory:

hypotheses: a statement about what we should expect to observe if a causal claim is true. (Also could call this empirical prediction)

Testing

Causal theory may generate several hypotheses:

• based on main causal claim
• based on steps in causal logic

What are hypotheses for causal claims?

Presence of cause and presence of effect. Absence of cause and absence of effect.

Higher/lower levels of X (cause) appear Higher/lower levels of Y (outcome).

Claim:

Claim:

Claim:

Claim:

Counterfactual Causality

X causes Y.

Implies:

If X does not happen then Y would not happen

Our examples as counterfactuals:

1. If economy had grown less (more), countries would have been less (more) democratic
2. If there were more (fewer) guns, there would be less (more) crime.
3. If there were more (less) economic growth, civil would be less (more) likely
4. If Nick Cage were in fewer (more) films, fewer (more) people would die by drowning

Thinking Counterfactually

X causes Y

Implies

If claim is true, then in one case with some level of X and some level of Y:

If that case had a different level of X, then the level of Y would have been different.

How do we observe this?

1. We can't make the same country have less economic growth and see if it doesn't democratize
2. We can't make the US gun ownership rate not change in the past and see if crime stays high
3. We can't make the same country have more economic growth to see if it prevents civil war
4. We can't unmake (or unsee) Ghost Rider 2 to see if fewer people would have died in 2012

Fundamental Problem of Causal Inference

We never can observe a case under the counterfactual condition: we can only observe a case under one (the factual) condition.

Let's see this in action:

We want to test whether $$X$$ causes higher $$Y$$.

For sake of simplicity, imagine a situation with $$5$$ cases.

Each observation is indexed by $$i$$.

$$X_i:$$ value of cause $$X$$ for case $$i$$. $$1$$ for cause is present, $$0$$ for cause is absent

$$Y_i:$$ The outcome (dependent variable) for $$i$$

What we can see

i $$Y_i$$ $$X_i$$
1 6 1
2 2 0
3 8 1
4 4 0
5 6 1

Evidence for the claim?

Take average of $$Y$$ for $$X = 1$$ and for $$X = 0$$ and take the difference:

1. $\frac{6 + 8 + 6}{3} - \frac{2 + 4}{2}$

2. $\frac{20}{3} - \frac{6}{2}$

3. $\frac{20}{3} - \frac{9}{3} = \frac{11}{3}$

Let's not get too excited

Fundamental Problem of Causal Inference:

• We don't know what cases would have looked like if they had the other value of $$X$$.
• That is: we don't know what they would look like under the counterfactual.

We can imagine though

Counterfactual causality means that every case has a value for the outcome that it would take under the factual and counterfactual conditions. We call these potential outcomes

We can imagine though

We want to test whether $$X$$ causes higher $$Y$$.

For sake of simplicity, imagine a situation with $$5$$ cases.

Each observation is indexed by $$i$$.

$$X_i:$$ value of cause $$X$$ for case $$i$$. $$1$$ for cause is present, $$0$$ for cause is absent

$$Y_i^0:$$ The outcome (dependent variable) for $$i$$ when $$X_i = 0$$

$$Y_i^1:$$ The outcome (dependent variable) for $$i$$ when $$X_i = 1$$

If we saw the potential outcomes:

i $$Y_i^0$$ $$Y_i^1$$ $$Y_i^1 - Y_i^0$$
1 6 6 0
2 2 2 0
3 8 8 0
4 4 4 0
5 6 6 0

Did our test work?

Our test: compare $$Y$$ for $$X = 1$$ and $$X = 0$$

We found that $$Y$$ was $$\frac{11}{3}$$ higher when $$X = 1$$

Truth: $$Y_i^1 - Y_i^0$$

The truth is the average difference for each case between condtion with $$X$$ and condition without $$X$$

Truth is that there is no difference

But can only see factual outcome

fundamental problem of causal inference:

i $$Y_i^0$$ $$Y_i^1$$ $$Y_i^1 - Y_i^0$$
1 ? 6 ?
2 2 ? ?
3 ? 8 ?
4 4 ? ?
5 ? 6 ?

What good is this?

• We can't trust comparing higher/lower levels of X and Y
• We can't observe cases in counterfactual condition

So, we still have fundamental problem of causal inference.

Can we learn anything about causality?

Causal claims and Hypotheses

We started with this:

1. Higher/lower levels of X (cause) appear with Higher/lower levels of Y (outcome).

But we need something else:

1. Comparison between cases that are very similar (next best thing to seeing case under both cause, absence of cause)

Why?

if causes deterministic: cases that have exact same set of causes acting on them behave exactly the same.

Mill's Method of Difference

Claim: $$X$$ is cause of $$Y$$

$$A,B,C$$ are other possible causes of $$Y$$.

Case 1 Case 2
X 1 0
A 1 1
B 0 0
C 1 1
Y 1 0

Case 1 Case 2
X 1 0
A 1 1
B 0 0
C 1 1
Y 1 0

Case 1 Case 2
X 1 0
A 1 1
B 0 0
C 1 1
Y 1 1

Case 1 Case 2
X 1 0
A 1 1
B 1 0
C 1 1
Y 1 0

Comparative Method:

If causal claim that $$X \rightarrow Y$$, then:

We generate empirical prediction:

If we observe two cases to be the same in all relevant respects except for value of $$X$$, then we should observe that the two cases differ in the value of $$Y$$

This empirical prediction based on a causal claim is called comparative method or method of difference (via John Stuart Mill)

Difficulty:

What are the relevant similarities? How many things need to be the same? How do we identify them?

Difficulty:

1. Society is complex, humans and our institutions have lots of possible attributes, exposure to lots of different causes
2. We might not know all possible causes
3. Very difficult to actually measure all of these causes
4. Even if we did, might not find case that is exactly the same

Worst Case Scenario:

Claim:

$$X \rightarrow Y$$

but also the case that $$\lbrace W_1, W_2, \ldots, W_\infty \rbrace \rightarrow Y$$

if true

1. We can't observe all $$W_1, W_2, \ldots, W_\infty$$
2. Unlikely to find cases that are the same on all $$\mathbf{W}$$

What do we do?

We've been trying to find a factual case $$j$$ that can be a counterfactual for the factual case $$i$$: identical on all attributes except the cause $$X$$.

But, we've had a devil of a time doing it.

Why? No reason to assume any case is exactly like another.

What do we do?

Maybe we've been doing this wrong.

Instead of finding counterfactual for each case $$i$$…

Solving the FPCI

The fundamental problem of causal inference states we can't observe both $$Y_i^1$$ (outcome when exposed to cause) and and $$Y_i^0$$ (outcome when not exposed to cause) for a given $$i$$.

But we can get around this if we look at many cases $$i \in \lbrace 1 \ldots n \rbrace$$ and take the average.

How?

We cannot know each individual causal effect: $$\tau_i = Y_i^1 - Y_i^0$$

But the average causal effect is the average of all individual causal effects:

$$ACE = \frac{1}{n}\sum\limits_i^n{\tau_i}$$

So it is also equal to this:

$$ACE = \frac{1}{n}\sum\limits_i^n (Y_i^1 - Y_i^0)$$

How?

Which means it can be the difference between the averages of both potential outcomes:

$$ACE = \Big( \frac{1}{n}\sum\limits_i^n Y_i^1 \Big) - \Big( \frac{1}{n}\sum\limits_i^n Y_i^0 \Big)$$

How?

But does this help us?

$$ACE = \Big( \frac{1}{n}\sum\limits_i^n Y_i^1 \Big) - \Big( \frac{1}{n}\sum\limits_i^n Y_i^0 \Big)$$

• $$\frac{1}{n}\sum\limits_i^n Y_i^1$$ is the average over all cases of the potential outcome when the cause is present.
• $$\frac{1}{n}\sum\limits_i^n Y_i^0$$ is the average over all cases of the potential outcome when the cause is absent.
• The FPIC means that for any case, we only observe $$Y_i^1$$ or $$Y_i^0$$, not both.

How?

At best we can observe:

1. $$\frac{1}{n}\sum\limits_i^n (Y_i^1 | X_i = 1)$$: The average of $$Y$$ when $$X$$ is present for the cases where $$X$$ is actually present
2. $$\frac{1}{n}\sum\limits_i^n (Y_i^0 | X_i = 0)$$: The average of $$Y$$ when $$X$$ is absent for the cases where $$X$$ is actually absent

And we know from above that this can go wrong:

i $$Y_i^0$$ $$Y_i^1$$ $$Y_i^1 - Y_i^0$$
1 ? 6 ?
2 2 ? ?
3 ? 8 ?
4 4 ? ?
5 ? 6 ?

Take average for $$X = 1$$ and for $$X = 0$$ and take the difference:

1. $\frac{6 + 8 + 6}{3} - \frac{2 + 4}{2}$

2. $\frac{20}{3} - \frac{6}{2}$

3. $\frac{20}{3} - \frac{9}{3} = \frac{11}{3}$

True effect is $$0$$ !

i $$Y_i^0$$ $$Y_i^1$$ $$Y_i^1 - Y_i^0$$
1 6 6 0
2 2 2 0
3 8 8 0
4 4 4 0
5 6 6 0

What do we need?

1. If the average outcome for cases with cause $$X$$ were similar to what the average outcome would be in the presence of $$X$$ for cases without $$X$$.
2. If the average outcome for cases without $$X$$ were similar to what the average outcome would be in the absence of $$X$$ for cases with $$X$$

Then:

• $$\frac{1}{n}\sum\limits_i^n (Y_i^1|X_i = 1) = \frac{1}{n}\sum\limits_i^n (Y_i^1)$$
• $$\frac{1}{n}\sum\limits_i^n (Y_i^0|X_i = 0) = \frac{1}{n}\sum\limits_i^n (Y_i^0)$$

What do we need?

That'd be great!

But we don't know the counterfactual outcomes, so how can we ensure they are similar?

How?

Wait…

Didn't we just see a problem like this?

• We had a population we want to describe, but it was too large to observe every case
• We wanted a sample that looked like the population

Random Sampling

A procedure that let us get a sample such that:

• Mean of sample is unbiased estimate of mean of population

What do we need?

If we randomly sampled

Some cases to get $$X$$ and some cases to not get $$X$$

Then…

1. The average $$Y^1$$ for cases with $$X$$ would be a sample of the average counterfactual $$Y^1$$ for cases without $$X$$
2. The average $$Y^0$$ for cases without $$X$$ would be a sample of the average counterfactual $$Y^0$$ for cases with $$X$$.

AND

$$ACE = \frac{1}{n}\sum\limits_i^n (Y_i^1 | X_i = 1) - \frac{1}{n}\sum\limits_i^n (Y_i^0 | X_i = 0)$$

All of which is observable because it is factual

What is this miraculous procedure?

If you haven't guessed it:

This is the precise logic of a randomized experiment.

It works because:

All cases have equal chance of being exposed to cause.

Recap:

Testing causal theories

We need to generate emprirical predictions or hypotheses about what we will observe if the causal theory is correct

Recap:

Testing causal theories

Causality implies a counterfactual.

Empirical Predictions for causal theories are that:

1. When we change the level of the independent variable (cause) for a specific case
2. The change in the level of the independent variable results in change in dependent variable (outcome)

Recap:

Testing causal theories

But we can't change the level of the independent variable for a case:

Fundamental Problem of Causal Inference: key part empirical prediction about causal theory is never observable

Recap:

Testing causal theories

Fundamental Problem of Causal Inference is solvable when

1. we look at groups of cases in aggregate
2. we randomly sample cases to get exposed to the cause/not

How do experiments work?

Randomly assigning exposure to cause

"Treatment" and "Control" group have same potential outcomes

because they are two random samples from the same population

• Treatment group factual outcomes are same as counterfactual outcomes for Control
• Control group factual outcomes are same as counterfactual outcomes for Treatment

How do experiments work?

Best way to understand is to do it!

Let's work through it

We have this table of potential outcomes:

i $$Y_i^0$$ $$Y_i^1$$ $$Y_i^1 - Y_i^0$$
1 5 9 4
2 4 8 4
3 3 7 4
4 2 6 4

Let's work through it

Could be: effect of campaign ad on campaign contributions:

i ($) without ad ($) with ad ($) with - ($) without
1 5 9 4
2 4 8 4
3 3 7 4
4 2 6 4

Let's work through it

What the true average causal effect?

i $$Y_i^0$$ $$Y_i^1$$ $$Y_i^1 - Y_i^0$$
1 5 9 4
2 4 8 4
3 3 7 4
4 2 6 4

Let's work through it

$\frac{4 + 4 + 4 + 4}{4} = 4$

Let's work through it

Let's run a randomized experiment:

• We will put 2 people in treatment (see the campaign ad)
• We will put 2 people in control (don't see the ad)

When we do this at random:

• we are choosing one possible partition of the sample
• there are several possible partitions into Treatment/Control
• each possible partition is equally likely

Let's work through it

Write down all possible treatment groups of size $$2$$ (e.g. (1,2))

i $$Y_i^0$$ $$Y_i^1$$ $$Y_i^1 - Y_i^0$$
1 5 9 4
2 4 8 4
3 3 7 4
4 2 6 4

Let's work through it

All possible treatment groups

V1 V2
1 2
1 3
1 4
2 3
2 4
3 4

Calculate the average $$Y^1$$ ($with ad) for every possible treatment (ad) group i $$Y_i^0$$ $$Y_i^1$$ $$Y_i^1 - Y_i^0$$ 1 5 9 4 2 4 8 4 3 3 7 4 4 2 6 4 Let's work through it Calculate the average $$Y^0$$ ($ with ad) for every corresponding control (no ad) group

i $$Y_i^0$$ $$Y_i^1$$ $$Y_i^1 - Y_i^0$$
1 5 9 4
2 4 8 4
3 3 7 4
4 2 6 4

T_Group Y_1 Y_0
1,2 8.5 2.5
1,3 8.0 3.0
1,4 7.5 3.5
2,3 7.5 3.5
2,4 7.0 4.0
3,4 6.5 4.5

Let's work through it

Calculate the average causal effect for every possible experiment (($) ad - ($) no ad) or $$Y^1 - Y^0$$

T_Group Y_1 Y_0
1,2 8.5 2.5
1,3 8.0 3.0
1,4 7.5 3.5
2,3 7.5 3.5
2,4 7.0 4.0
3,4 6.5 4.5

Let's work through it

T_Group Y_1 Y_0 ACE
1,2 8.5 2.5 6
1,3 8.0 3.0 5
1,4 7.5 3.5 4
2,3 7.5 3.5 4
2,4 7.0 4.0 3
3,4 6.5 4.5 2

Let's work through it

What is the average effect across all possible experiments?

T_Group Y_1 Y_0 ACE
1,2 8.5 2.5 6
1,3 8.0 3.0 5
1,4 7.5 3.5 4
2,3 7.5 3.5 4
2,4 7.0 4.0 3
3,4 6.5 4.5 2

Let's work through it

$\frac{6 + 5 + 4 + 4 + 3 + 2}{6} = 4$

Let's work through it

Why does this work?

If you wanted:

• you can compare $$Y^1$$ for the treatment groups and $$Y^1$$ for the control to see if they are the same
• you can compare $$Y^0$$ for the control groups and $$Y^0$$ for the treatment to see if they are the same

Let's work through it

Average of $$Y^1$$ for treated groups (factual outcome): 7.5

Average of $$Y^1$$ for control groups (counterfactual outcome): 7.5

Average of $$Y^0$$ for control groups (factual outcome): 3.5

Average of $$Y^0$$ for treated groups (counterfactual outcome): 3.5

Randomized Experiments:

Using random assignment to treatment and control:

• Lets us have two samples that are, on average, counterfactuals for each other
• Lets us address FPCI at aggregate, not individual level

Randomized Experiments:

What happens when we fail to randomize?

1 and 2 always get treated, 3 and 4 never do.

i $$Y_i^0$$ $$Y_i^1$$ $$Y_i^1 - Y_i^0$$ $$X_i$$
1 5 9 4 1
2 4 8 4 1
3 3 7 4 0
4 2 6 4 0

Randomized Experiments:

What happens when we fail to randomize?

$\frac{9+8}{2} - \frac{3+2}{2} = \frac{12}{2} \neq 4$

Randomized Experiments:

We can imagine cases with different potential outcomes have different attributes:

i $$Y_i^0$$ $$Y_i^1$$ $$Y_i^1 - Y_i^0$$ $$X_i$$ $$W_i$$
1 5 9 4 1 1
2 4 8 4 1 1
3 3 7 4 0 0
4 2 6 4 0 0

Randomized Experiments:

Presence of $$W$$ is fine if: $$W$$ unrelated to cause $$X$$

i $$Y_i^0$$ $$Y_i^1$$ $$Y_i^1 - Y_i^0$$ $$X_i$$ $$W_i$$
1 3 7 4 1 1
2 2 6 4 1 0
3 3 7 4 0 1
4 2 6 4 0 0

Randomized Experiments:

Presence of $$W$$ is fine if: $$W$$ unrelated to outcome $$Y$$

i $$Y_i^0$$ $$Y_i^1$$ $$Y_i^1 - Y_i^0$$ $$X_i$$ $$W_i$$
1 3 7 4 1 1
2 2 6 4 1 1
3 3 7 4 0 0
4 2 6 4 0 0

Randomized Experiments:

Randomization ensures that, on average, no relationship between $$X$$ and $$W$$.

Problems with Experiments:

Internal Validity

A study has internal validity when the causal effect of $$X$$ on $$Y$$ it finds is not biased (systematically incorrect).

Threats to internal validity

1. Selection Bias
2. Non-excludability
3. Systematic Measurement Error

Threats to Internal Validity

Selection Bias:

selection bias: When cases that receive a "treatment" or "cause" have different potential outcomes from those that do not.

Threats to Internal Validity

Selection Bias:

Consider this experiment:

• We want to evaluate the effect of emergency housing assistance on homelessness prevention.
• We give assistance to the first 100 people to show up and compare them to the second 100 people.

Problem?

• People with greater need OR better resources/capabilities show up first
• Different needs or different resources probably related to propensity to become homeless
• Exposure to treatment related to other factors that might affect outcome!

Threats to Internal Validity

Non-excludability

non-excludability occurs when the treatment in an experiment bundles multiple different treatments

Threats to Internal Validity

Non-excludability

Does an apple a day keep the doctor away?

• Some people assigned to eat apples, some are not.
• To ensure people eat apples, we remind them every day to eat apples and give them a financial incentive to eat it.

Problem?

• Are the effects we see due to apples? To being reminded every day? To being paid?
• Lots of differences between treatment and control.

Threats to Internal Validity

Systematic Measurement Error

systematic measurement error: error produced when our measurement procedure obtains scores that are, on average, too high or too low.

Threats to Internal Validity

Systematic Measurement Error

We randomly assign some people to be shamed into voting (we remind them of their past voting record)

We measure whether people exposed to this treatment have higher self-reported voting rates

Problem?

• If we see a difference between treatment and control, is it due to actual differences in voting?
• Or is it due to people who are shamed self-reporting differently?
• Treatment affects results of measurement procedures $$\rightarrow$$ measurement bias
• Only a problem when measurement bias is related to treatment

Experiments:

Randomization of cause

• Allows us to actually observe a counter-factual
• Directly tests causal theory
• Allows us to see whether the change in $$Y$$ due to $$X$$ is in the right direction
• Allows us to see whether $$Y$$ changes due to $$X$$ relative to the counterfactual.

Problems are solvable

• Selection bias, non-excludability, measurement bias all solvable
• Better design of experiment

Experiments:

One conclusion: only do experiments.

Problems with Experiments:

External Validity

external validity is the degree to which the causal relationship we find in a study matches the causal relationship and the context identified in a causal theory

• Study has external validity if the relationship found can generalize to all the cases to which our causal theory applies
• If our study suffers from sampling bias, then our study may lack external validity
• Study has external validity if the cause in the study is the same as the cause in our causal theory
• If our independent variable/cause in the study lacks validity then our study may lack external validity

External Validity:

Broockman and Kalla (2015): Does perspective taking change minds about minority groups?

Sample

• Invite tens of thousands of people to join a survey

Treatment

• One group gets canvasser for transgender rights
• Other gets canvasser for recycling

Outcome

• Survey responses on attitudes about transgender people, their rights

Problems with Experiments:

External Validity:

Broockman and Kalla (2015)

Sample

• Invite tens of thousands of people to join a survey

Problem:

• Will the effect of talking with canvasser hold for vast majority who won't join a survey?
• Sample in the experiment not the same as the population implied by the causal theory.

Problems with Experiments:

External Validity:

Does exposure to partisan media change people's political attitudes and make them more extreme? (E.g. Fox News, Breitbart)

Sample: Randomly chosen group of US adults

Treatment:

• Randomly exposed to 2 hours of partisan news or non-partisan movie

Outcome:

Problems with Experiments:

External Validity:

Treatment:

• Randomly exposed to 2 hours of partisan news or non-partisan movie

Problem:

• Is this short treatment the same as years of exposure on cable TV or internet news consumption?
• If we find no effect of 2 hours of watching, does this mean that emergence of partisan news didn't cause change in Americans' political attitudes?
• Cause we can manipulate in experiment not the same as cause in causal theory

Problems with Experiments:

More internal validity (unbiased calculation of causal effect) comes at the cost in external validity (relevance of study sample or cause to the causal theory)

Why?

Many relevant contexts for causal theories and interesting causes cannot or should not be manipulated at random:

• What causes democratization?
• What causes war or ethnic violence?
• What causes economic growth?
• Why were civil rights extended to oppressed minority groups?

Problems with Experiments:

What approaches can we use instead?

1. Comparative Method
2. Correlation

Aren't they flawed?

Yes, but now that we know how and why experiments work, we can make these better.

Comparative Method:

If causal claim that $$X \rightarrow Y$$, then:

We generate empirical prediction:

If we observe two cases to be the same in all relevant respects except for value of $$X$$, then we should observe that the two cases differ in the value of $$Y$$

This empirical prediction based on a causal claim is called comparative method or method of difference (via John Stuart Mill)

What do we know now?

1. We don't need cases to be identical on all attributes
2. They need to be the identical on attributes related to both cause ($$X$$) and outcome ($$Y$$)
• In short: identical on factors that affect $$Y$$ and are related to or affect $$X$$

Example:

Question:

What causes states to create laws designed to exclude and dominate people based on race?

Causal claim:

Attempts to resolve conflicts within the dominant (more powerful) racial group in a society lead to the creation of a legal regime that excludes other groups.

Example

What are other possible causes that could be related?

1. Some dominant groups more prone to in-fighting and exclusion?
• Maybe colonizing groups. diverse backgrounds $$\rightarrow$$ in-fighting
• Colonizers $$\rightarrow$$ more likely to use legal exclusion
2. Some relationships between dominant and oppressed groups more prone to in-fighting and exclusion?
• Slavery $$\rightarrow$$ conflicting interests within dominant group (owners, non-owners)
• Slavery $$\rightarrow$$ provides a framework to continue legal discrimination in the future

Example

What are other possible causes that could be related?

1. Legal exclusion more popular at some times rather than others:
• Racial exclusion easily justified in some eras
• Exclusion not seen as acceptable today, so other methods used

Example

In-fighting (cause, X) Exclusion (effect, Y)
Dominant group Settler Colonials from
diverse so more
divided
Settler Colonials
more likely to legally
exclude the colonized
Form of Domination Slavery creates conflict
b/t owners and non-owners
Slavery is legal category
basis for legal exclusion
Timing Newly created states
may have infighting
bc rules are new
Racial exclusion
more "acceptable"" in
19th and 20th c.

Example

Comparison

United States Brazil
White In-fighting YES NO
Dominant group White Europeans White Europeans
Settler Colonialism Yes Yes
Form of Domination Slavery Slavery
Era 1860s-1960s 1880s-1960s
Legal Discrimination YES NO

Example

Question:

What causes the spread of cholera?

Causal claim:

Contaminated water causes cholera outbreaks

Obvious?

To us, but not in mid-19th century England

Example

Background

19th Century London saw repeated outbreaks of cholera, with mass death

• Dominant view was that "miasmas" or bad air caused diseases like cholera

A novel idea…

No theory of germs, but

• John Snow, MD didn't buy into "miasmas"
• Published an essay On the Mode of Communication of Cholera in 1849
• Suggested that polluted water was means of communication