The basics
Carolina Torreblanca
University of Pennsylvania
\[ \mu_X = \frac{1}{n} \sum_{i}^{n} X_i \]
\[ \sigma^2_X = \frac{1}{N} \sum_{i}^{N} (X_i - \mu_X)^2 \]
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.
(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] 11.28884We 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.
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?
Broadly, two type of causal questions:
Which do you think is more common in policy work?
To answer “does democracy cause economic development?” we need to:
To answer “does democracy cause economic development?” we need to:
We can NEVER do this
We call this problem the fundamental problem of causal inference
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.
Each participant, \(i\) has two potential health outcomes:
| 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 |
| Participant | Medicine? (X) | Observed BP | Yi(0) | Yi(1) |
|---|---|---|---|---|
| 1 | 1 | 121 | ??? | 121 |
| 2 | 1 | 140 | ??? | 140 |
| 3 | 0 | 120 | 120 | ??? |
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?
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.
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.
Independence is a big assumption!
One way to build treatment and control samples that are comparable is to …
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…