Causation and Correlation

The basics

Carolina Torreblanca

University of Pennsylvania

Global Development: Intermediate Topics in Politics, Policy, and Data

PSCI 3200 - Spring 2024

Logistics

Assignments

  • Did everyone find the readings for today?
  • Does everyone have R and RStudio installed and running?
  • Is everyone on the slack channel?

Agenda

  • Measuring statistics vs relationships
  • Summarizing relationships to predict
  • Causality to explain
    • Potential outcomes framework
    • FPCI

Why analyze data as a social scientist?

  • The type of social science we will study here heavily relies on data to . . .
  1. Measure
  2. Predict
  3. Explain

Measure

  • Life expectancy
  • Net exports
  • Share of votes for candidate A
  • The relationship between age and turnout

Measure

  • Measures of central tendency
    • Mean
    • Median
  • Measures of spread
    • Variance
    • Standard deviation

Central Tendency: Mean

\[ \mu_X = \frac{1}{n} \sum_{i}^{n} X_i \]

Spread: Variance

\[ \sigma^2_X = \frac{1}{N} \sum_{i}^{N} (X_i - \mu_X)^2 \]

  • What does the square in \(\sigma^2\) accomplish?
  • What are the implications for interpretation?

We can combine them to measure relationships.

  • Covariance \(\text{Cov}(X, Y) = \frac{1}{n} \sum_{i}^{N} (X_i - \bar{X})(Y_i - \bar{Y})\)
    • Range: unbounded
  • Correlation coefficient \(\text{Cor}(X, Y) = \frac{\text{Cov}(X, Y)}{\sigma_X \sigma_Y}\)
    • Range: -1 to 1
  • Slope \(\beta_X = \frac{\text{Cov}(X, Y)}{\sigma^2_X}\)
    • Expected change in \(Y\) with 1-unit change in \(X\)

Predict

  • Population growth
  • Future resource availability
  • Winner of an election!
  • Expected turnout rate by age group

Predict

To be able to use X to predict Y we need…

  • both variables to have a relationship.

  • We then can use statistical models to summarize the relationship.

  • And use the models to predict Y, given any X

  • Give me some examples of statistical models often used to summarize relationships.

Measure vs Predict

# Set up some fake data
set.seed(390)
# Predictor
x <- rnorm(100, mean = -3, sd = 1)
# Noise
error <- rnorm(100, mean = 0)
# Variable we want to explain
y<- x + error 
# Measure the relationship
cov(x, y) # covariance
[1] 1.105927
cor(x, y) # correlation
[1] 0.7318479
# or cov(x, y) / (sd(x)*sd(y))
mod <- lm(y ~ x)
coefficients(mod)[2]
       x 
1.098033

Measure vs Predict

# To predict we need the full model
coefficients(mod)
(Intercept)           x 
  0.3085034   1.0980332 
# or cov(x, y) / var(x)
hypothetical_x = 10
# predict using the ENTIRE model
coefficients(mod)[[1]] + coefficients(mod)[[2]]*hypothetical_x
[1] 11.28884

Measuring, predicting, and … explaining?

  • We can measure relationships

  • We can summarize relationships in models and use them to predict

  • For a variable X to be a good predictor of Y, X and Y need to have an association.

  • But predictively powerful models tell us nothing about why X and Y are associated.

Causality as a way to explain

  • When we are interested in explaining, we might want to know if X is just associated with Y or if X causes Y

  • The “causal effect of X on Y” is the change in the outcome Y produced by a change in the treatment X

    • Does exposure to misinformation cause polarization?

      • What is X? What is Y?

Causality as a way to explain

  • Broadly, two type of causal questions:

    1. Causes of consequences, e.g: Does democracy cause economic development?
    2. Consequences of causes: What is the causal effect of colonialism on the economic development of east African countries?

  • Which do you think is more common in policy work?

Causality: the FPOCI

To answer “does democracy cause economic development?” we need to:

  1. compare the economic development of democratic countries with
  2. the economic development of those same democratic countries had they not been democracies.

Causality: the FPOCI

To answer “does democracy cause economic development?” we need to:

  1. compare the economic development of democratic countries with
  2. the economic development of those same democratic countries had they not been democracies.

  • We can NEVER do this

  • We call this problem the fundamental problem of causal inference

Causality

One way to think about causality from this angle more formally is using the Potential Outcomes framework

  • Imagine n participants (indexed by i) in an experiment to test a new blood pressure medicine (X).

  • Participants can either take (X=1) or not take (X=0) the medicine.

Causality

Each participant, \(i\) has two potential health outcomes:

  • \(Y_i(X_i = 0)\) = Person i’s health outcome if they do not take the medicine and
  • \(Y_i(X_i = 1)\) = Person i’s health outcome if they take the medicine
Participant Medicine? (X) Observed BP Yi(0) Yi(1) \(Y_i(1) - Y_i(0)\)
1 1 121 125 121 -4
2 1 140 145 140 -5
3 0 120 120 119 -1

Causality

  • We never can observe \(Y_i\) when i took the medicine and when they did not!
    • We only observe the factual outcome, never the counterfactual outcome.
Participant Medicine? (X) Observed BP Yi(0) Yi(1)
1 1 121 ??? 121
2 1 140 ??? 140
3 0 120 120 ???

Causality: What to do?

  • What if we think in terms of average causal effects instead?
\[\begin{align*} \overline{\Delta Y} =& \frac{1}{n}\Sigma_{i = 1}^{n} \bigg(Y_i(1) - Y_i(0)\bigg)\\ \overline{\Delta Y} =& \frac{1}{n}\Sigma_{i = 1}^{n} Y_i(1) - \frac{1}{n}\Sigma_{i = 1}^{n} Y_i(0)\\ \overline{\Delta Y} =& \overline{Y(1)} - \overline{Y(0)} \end{align*}\]

Causality: on average

  • What if we could plug-in the mean Y value of observations assigned to the treatment as \(\overline{Y(1)}\) and the control observations as \(\overline{Y(0)}\)?

  • Well our problems would be solved! At least on average.

  • What would we need to assume about observations in treatment and control?

Causality: on average

  • What if we could plug-in the mean Y value of observations assigned to the treatment as \(\overline{Y(1)}\) and the control observations as \(\overline{Y(0)}\)?

  • Well our problems would be solved! At least on average.

  • What would we need to assume about observations in treatment and control?

  • That these two populations are, on average, the same in every relevant dimension.

Causality: on average

If X (receiving treatment) is independent of Y (outcome), then

\[\begin{align*} \overline{\Delta Y} =& \overline{Y(1)} - \overline{Y(0)} \\ \widehat{\overline{\Delta Y}} =& \overline{Y_{Treatment}} - \overline{Y_{Control}} \end{align*}\]

\(\widehat{\overline{\Delta Y}}\) is called the difference-in-means estimator.

  • Think about independence like this: knowing \(i\)’s value of Y provides no information about their value of X.

Causality: on average

  • Independence is a big assumption!

  • One way to build treatment and control samples that are comparable is to …

Causality: on average

  • Independence is a big assumption!

  • One way to build treatment and control samples that are plausibly comparable is to …

  • randomize treatment assignment.

  • Not the only way to estimate causal effects…

  • But ALL causal research designs require assumptions…

Next Meeting

  • More causality - RCTs
  • Causality with observational data
  • Some light R!
  • Assigment!! Please be sure to slack me your quarto file

jrspringman.github.io/psci3200-globaldev/