Causality I

POL51

Juan Tellez

jftellez@ucdavis.edu

UC Davis

October 31, 2024

Get ready

• Boot up Posit Cloud
• Download the script for today’s examples
  • Schedule ➡️ Example ➡️ Today
• Upload the script to Posit Cloud

Plan for today

Why causality? And what is it?

Fake data

The problem with causality

Prediction vs. causality

Why predict

We don’t know what’s going to happen in the future

Or in places/cases where we don’t have data

Even if in cases where we have data – what’s our best guess?

We can use models to make decisions informed by patterns in the data

Prediction is for making decisions

World Bank: what would happen to Jamaica if their GDP went up by 10k?

Moving forward

• There are lots of prediction resources online
• Kaggle hosts prediction competitions with interesting data
• Give it a shot!

A turning point

First half of course

• How to program, visualize data, modeling, relationships, etc.

• Look at all the functions you “learned”:

• group_by, tally, summarise, filter, mutate, %>%, distinct, lm, augment, tidy, ggplot, facet_wrap…

• There are thousands more!

First half of course

Use models to estimate the relationship between X and Y:

mod = lm(lifeExp ~ gdpPercap, data = gapminder)
tidy(mod)

# A tibble: 2 × 5
  term         estimate std.error statistic   p.value
  <chr>           <dbl>     <dbl>     <dbl>     <dbl>
1 (Intercept) 54.0      0.315         171.  0        
2 gdpPercap    0.000765 0.0000258      29.7 3.57e-156

But is this relationship causal? Would an increase in GDP cause an increase in life expectancy, on average?

Second half of the course

• “How do we know if X causes y?”

• Are our estimates causal? Academics fight about this all day!

• This question is at center of causal inference

• We will learn why it is so difficult to establish causality with data

• We will also learn potential solutions

Pictured: audience at a research presentation

Many interesting questions are causal

International relations: do peace-keepers reduce conflict in countries emerging from civil war, or are they ineffective?

Comparative politics: do elections reduce or increase corruption?

American politics: do voter ID laws hurt general turnout?

But not all

Many of the interesting questions people want to answer with data are causal

Some are not:

Instagram might want to know: “Is there a person in this photo?”

But not care about what factors cause the picture to be a photo of a person

Depends on the question; most why questions are causal

The value of causality

One of our comparative advantages

Not just academic; companies, governments, NGOs also need to answer “why” questions

Does this policy work (or not)? Did it do what was intended? How effective or counterproductive was it?

What do we mean by causality?

In this class, we say X causes Y if…

An intervention that changes the value of X produces a probabilistic change in Y

• Intervention = X is being changed or altered

• Probabilistic = Y should change, on average, but need not in every instance

Interventions and probability

How do the two parts of our definition fit here?

Aspirin causes a reduction in fever symptoms

Intervention = someone takes aspirin, we administer aspirin, we sneak it into someone’s food, etc.

Outcome = Taking aspirin doesn’t work 100% of the time; but in general, on average, more often than not, etc., taking aspirin \(\rightarrow\) less fever

What about this example? Democratic institutions reduce the incidence of interstate war

Terminology

We’ve seen this before:

Variable	Meaning	Examples
Y	The thing that is affected by a treatment	Employment, turnout, violence
X	The thing changing an outcome	Min. wage laws, voter ID laws, peacekeepers

The problem with causality

Obviously causal relationships

A heavier car has to work harder to get from A to B

Obviously causal relationships

A bigger house is more desirable

Not obvious: education and earnings

• Positive correlation between education and earnings, yet
• People who are already wealthy (or likely to become wealthy) get more education than people who are not wealthy
• In other words: people self-select into education levels

Not obvious: voter ID and turnout

States that pass these laws are different from states that don’t pass these laws

Spurious correlation

To make matters worse, correlation is common-place in nature:

The trouble with causality

We can’t directly observe a change in X causing a change in Y

This is true even in experiments, where we directly manipulate stuff

All we can see are correlations between X and Y

Some of those correlations are causal; some are not; how can we tell?

First step: yes, we can tell (sometimes)

People are obsessed with “correlation does not mean causation”

But sometimes it does! that’s the tricky part

Make up (simulate) data to convince ourselves

• A good way to see that we correlation can equal causation is to make up (simulate) data where we know that X causes Y
• We can make up data in R with the rnorm() function
• rnorm() draws random numbers from a normal distribution
• The numbers it makes up are pretty close to some mean, plus or minus a standard deviation

Drawing from a normal distribution

Generate 10 random numbers, most of which are +/- 2 away from 10:

rnorm(n = 10, mean = 10, sd = 2)

 [1] 11.906104 10.517202  8.150035 10.851565 12.973255 10.051876  8.300816
 [8] 10.742381  8.096446  7.913750

Generate 5 random numbers, most of which are +/- 10 away from 50:

rnorm(n = 5, mean = 50, sd = 10)

[1] 41.37935 49.31035 44.46696 48.74182 65.87727

Generate 4 random numbers, most of which are +/- 100 away from -10:

rnorm(n = 4, mean = -10, sd = 100)

[1] -71.71928  14.31545  23.79650  10.16522

Note

n = number of draws, mean = average, sd = standard deviation

Do winners attract funding?

Say we wanted to simulate data to show that doing well in elections (treatment) causes increases in campaign fundraising (outcome)

That is, that funders reward winners

First step is generating fake elections results (treatment)

Let’s say we have 500 fake elections that are pretty close to 50 (a tie), plus or minus 5 points

Faking (simulating) the treatment

We can use tibble() and rnorm() to simulate variables

elections (treatment) = 500 elections, vote share is 50% of the vote on average, +/- 5%

fake_election = tibble(party_share = rnorm(n = 500, mean = 50, sd = 5))
fake_election

# A tibble: 500 × 1
   party_share
         <dbl>
 1        47.4
 2        51.7
 3        48.2
 4        54.7
 5        53.3
 6        58.1
 7        48.4
 8        47.8
 9        41.5
10        46.7
# ℹ 490 more rows

And we can plot it

Making the outcome

Next we want to make up campaign fundraising (outcome)

Let’s say parties raise about 20k on average, plus or minus 4k

fake_election = tibble(party_share = rnorm(n = 500, mean = 50, sd = 5), funding = rnorm(n = 500,
    mean = 20000, sd = 4000))
fake_election

# A tibble: 500 × 2
   party_share funding
         <dbl>   <dbl>
 1        46.3  10644.
 2        44.9  24916.
 3        51.8  20045.
 4        50.8  19338.
 5        45.3  24607.
 6        53.9  15414.
 7        42.8  24798.
 8        68.3  15992.
 9        45.1  18680.
10        39.8  16282.
# ℹ 490 more rows

And we can plot it

Making X cause Y

We want to make it so that getting more votes increases campaign donations (winners attract funding)

Say for every percent of the vote a party gets, they get 2,000 (USD) more in donations

this is the causal effect of vote share (treatment) on campaign donations (outcome) \(\rightarrow\)

2,000 (USD) per percent of the vote a party gets in an election

Making X cause Y

We can do this in R like so:

fake_election = tibble(party_share = rnorm(n = 500, mean = 50, sd = 5), funding = rnorm(n = 500,
    mean = 20000, sd = 4000) + 2000 * party_share)
fake_election

# A tibble: 500 × 2
   party_share funding
         <dbl>   <dbl>
 1        47.6 115978.
 2        50.0 119467.
 3        48.7 121534.
 4        45.1 111531.
 5        54.8 131535.
 6        52.2 127072.
 7        45.4 106122.
 8        54.8 131403.
 9        55.9 134592.
10        44.4 114907.
# ℹ 490 more rows

As an equation, this looks like: \(Funding_{i} = Party\_share_{i} \times 2000\)

Eyeballing it

Is our fake data convincing? We can plot it to see:

Can we get the causal estimate back?

We know the effect of vote share on campaign donations: it’s 2k per percent of the vote

Can we use a model to get that estimate back?

Yes!

mod = lm(funding ~ party_share, data = fake_election)
tidy(mod)

# A tibble: 2 × 5
  term        estimate std.error statistic   p.value
  <chr>          <dbl>     <dbl>     <dbl>     <dbl>
1 (Intercept)   21235.    1806.       11.8 2.51e- 28
2 party_share    1978.      35.9      55.1 4.06e-214

Hope?

• There are at least some scenarios where X causes Y, and
  • This effect will be reflected in the data
  • And we can estimate the effect with a model

🕵️ Your turn: Michael Lacour 🕵️

• Lacour published a big study in Science
• Study showed talking to LGBT canvasser \(\rightarrow\) huge improvements in attitudes towards LGBT policy (gay marriage)
• Data was completely fabricated, using some of the same functions I showed you

🕵️ Your turn: Michael Lacour 🕵️

Using the steps I just went through above, make up some data that pre-confirms some pattern about the world you wish were true:

1. Change the treatment and outcome variables in the code to ones of your choosing

2. Alter the parameters in rnorm() so the values make sense for your variables (e.g., what is a reasonable distribution for age?)

3. Make a scatterplot with a trend line – use labs() to help us make sense of the plot axes!

4. Post your idea in Slack, winner (creativity + accuracy) will get small extra credit

15:00

So we (sometimes) can estimate causal effects

Where does it all go wrong then?

The Democratic Peace?

Does the spread of democracy reduce international conflict?

Theory: war is costly and the costs are borne by citizens; countries where citizens have more input \(\rightarrow\) less conflict

X here is whether the country is a democracy (versus autocracy)

Y is the number of wars the country is involved in

Ideal, imaginary approach: take a country, look at Y when democracy = 1, and then when democracy = 0

Do this for all countries, take the average

Potential outcomes

This magical world where we can compare the number of wars when a country is a democracy versus when it is not is called the potential outcomes

country	democracy	war
Canada	0	3
Canada	1	2
China	0	3
China	1	2
USA	0	2
USA	1	2

The fundamental problem of causality

In reality, we can only observe democracy at one value for each country

The US is a democracy, we can observe wars when democracy = 1, but not when democracy = 0

China is not a democracy, we can observe wars when democracy = 0, but not when democracy = 1

country	democracy	war
Canada	0	3
Canada	1	2
China	0	3
China	1	2
USA	0	2
USA	1	2

Potential vs. observed

We only observe the world on the right, but not the left

Potential outcomes world

country	democracy	war
Canada	0	3
Canada	1	2
China	0	3
China	1	2
USA	0	2
USA	1	2

Observed outcomes world

country	democracy	war
Canada	0	NA
Canada	1	2
China	0	3
China	1	NA
USA	0	NA
USA	1	2

The fundamental problem of causality

We have missing data on “what would have happened” had the US been an autocracy

“what would have happened” \(\rightarrow\) the counterfactual

Our goal in causal inference is to make as good a guess as possible as to what Y would have been had democracy = 0 instead of 1 (and vice versa)

Comparing apples and oranges

Why not just compare the number of wars for countries where democracy = 0 (autocracies) versus the countries where democracy = 1 (democracies)?

If democracies fight less, then democracy reduces conflict

observed %>%
    group_by(democracy) %>%
    summarise(wars = mean(war, na.rm = TRUE))

# A tibble: 2 × 2
  democracy  wars
      <dbl> <dbl>
1         0     3
2         1     2

Comparing apples and oranges

Implicitly, this means we are saying the countries that are autocracies are good counterfactuals for the countries that are democracies (and vice versa)

For instance, that China is a good counterfactual for the US

But China and the US are different in so many ways! They are not good counterfactuals of one another

We will see exactly why this is a problem in the following weeks

Why experiments work

In an experiment, we randomly expose participants to some treatment, while others are exposed to nothing (or a placebo)

Person	Shown an ad?	Democrats thermometer
1	Yes	105.64
2	No	4.92
3	Yes	33.22
4	No	33.88
5	No	87.75

We then compare the outcome of those who did and didn’t get the treatment

Why experiments work

Experiments have the same “problem” as the democracy and war example: we can’t observe the same person seeing and not seeing the ad

Person	Shown an ad?	Democrats thermometer
1	Yes	105.64
2	No	4.92

But since the experimental ad was randomly assigned, the people that did and didn’t see the ad are good counterfactuals of one another

it is, by definition, just as likely a person who received treatment could have instead received the control/placebo (note how this differs from democracy \(\rightarrow\) war)

The gold standard

This is why experiments are known as the gold standard of research

We have control over treatment, and we randomize it

With observational data that already exists in the world, we don’t have control over treatment, and we can’t randomize it

Experiments are great, but not feasible or ethical in most cases

So we’ll have to take other measures to try to overcome these problems with observational data

🧩 Your turn: counterfactuals 🧩

With neighbor, think through the counterfactual scenarios in these examples. What is the implicit counterfactual? What would a good counterfactual look like?

1. A study on whether being a victim of a crime makes someone more supportive of authoritarian leaders

2. A study on whether post-Dobbs abortion restrictions reduced abortions

10:00