Imbalance (2)
Imbalance Regression Table
Overlap
- Overlap describes how similar the range of the data is across groups.
- Overlap means that for every combination of observed covariate Z, there is a non-zero probability of observing both treatment and control.
- A lack of overlap means that there are no equivalents in the two groups , and we may have to extrapolate beyond the support of the data.
Blocked Experiment(3)
A Glorified Flowchart?
To give you an intuition before we start with the theory, a DAG looks like this:
- They are similar in concept to flowcharts, but they are not the same.
- You might occasionally have seen them in informal use, e.g. in Econometrics I.
- In this example, the arrow represents a causal effect from \(\text{cause}\) on \(\text{effect}\).
A Short Intro to Graph Theory
What you see on the right is what we call a graph.
This graph has three nodes. They are labeled \(i\), \(j\), and \(k\). Sometimes, we call the nodes “vertices,” “agents,” “points,” etc.
Some of the nodes in a graph are usually connected to each other, while others are not. We call those connections edges. Alternatively, they can be called “links,” “connections,” “lines,” etc.
Edges are pairs of two nodes. In the second graph, there is one edge from \(i\) to \(j\). We call this edge \(\{i,j\}\).
A Short Intro to Graph Theory
This edge does not have a direction.
However, we can easily give edges a direction. We call an edge like this a directed edge. When an edge is directed, the corresponding pair of nodes is no longer an unordered pair, but an ordered pair: \(\{j,i\}\neq\{i,j\}\).
A walk is a sequence of edges that joins a sequence of nodes. A cycle is a special case of a walk where all edges are distinct and the initial and final node are equal. In this graph, \(\left\{\{a,b\},\{b,c\},\{c,a\}\right\}\) is a cycle.
Directed Graphs, Acyclic Graphs
A graph that does not contain any cycles is called an acyclic graph.
If a graph contains only directed edges, we call it a directed graph.
The following graph is both directed and acyclic. We therefore call it a
Directed Acyclic Graph (DAG).
Think
Why is \(\{\{A,B\},\{B,C\},\)\(\{C,E\},\{E,A\}\}\) not a cycle?
DAGs for Causal Modeling
Why do we talk about DAGs in an Econometrics class? Because they are really useful for causal modeling.
In the following DAG, nodes represent (random) variables, and edges represent (hypothesized) causal effects.
Missing edges also convey information: the assumption of no causal effect.
The Basics of Causal Inference with DAGs
- Is \(Y\) related to \(U\)?
- Is \(X\) related to \(U\)? Can we randomize treatment?
- Are there other important variables?
It turns out that there are two paths from \(X\) to \(Y\),
- one direct path \(X \rightarrow Y\)
- and one backdoor path \(X \leftarrow U \rightarrow Y\).
We call it a backdoor path because it enters \(X\) trough the “back door,” via an arrow pointed at \(X\).
Confounders
In this DAG, when we want to isolate the effect \(X\rightarrow Y\), there is one open backdoor path.
This path confounds the causal effect of interest. We therefore call the variable \(U\) a confounder.
Confounders and Backdoors
If we just look at the connection between \(X\) and \(Y\), two effects are mixed together:
- The effect of \(X\) on \(Y\), our effect of interest.
- The effect of \(U\) on \(Y\) via \(X\).
We can close the backdoor by controlling for the confounder. We only run into problems when we cannot control for the confounder.
We would run a regression along the lines of:
\[ \boldsymbol{y} \sim \boldsymbol{x} + \boldsymbol{u}. \]
Colliders
Now imagine a different situation: There is a third variable, \(V\), that is jointly influenced by \(X\) and \(Y\).
Effects of both variables collide at \(V\). We therefore call \(V\) a collider. There is again one direct path and one backdoor path, but since the backdoor collides at \(V\), it is already closed.
Closing Backdoors
Open backdoors between two variables introduce systematic, non-causal correlation between them. If we want to estimate a causal effect, we need to close them. There are three cases we have to consider:
Confounders
We close backdoor paths by controlling for confounders.
Colliders
We can (and need to) leave colliders alone. The backdoor path is already closed.
Mediators
A mediator mediates part of the effect. If we control for the mediator, we remove the mediated effect and leave only the direct effect.
Enumerating Paths
How does this framework look like if we apply it to an example? Let us look at the following graph on the effect of gender (\(F\)) based discrimination (\(X\)) on earnings (\(Y\)).
We account for occupation (\(O\)) and aptitude (\(A\)).
Note that aptitude is not observed.
How many paths from \(X\) to \(Y\) can we enumerate?
Enumerating Paths
How many paths between \(X\) and \(Y\) can we enumerate?
- \(X\rightarrow Y\),
- \(X \rightarrow O \rightarrow Y\),
- \(X \rightarrow O \leftarrow A \rightarrow Y\),
- \(X \leftarrow F \rightarrow O \rightarrow Y\),
- \(X \leftarrow F \rightarrow O \leftarrow A \rightarrow Y\).
Which models can we use to isolate the effect of interest?
- \(Y \sim F\): We get a compound effect of \(X\) and \(O\) (1, 2, 4).
- \(Y \sim X\): We get the effects of \(X\), but they are confounded by \(F\) (4).
- \(Y \sim X, O\): We get rid of the confounder \(F\) and separate the effects of \(X\) (1, 2), but they are now confounded by \(A\) (3, 5).
Without \(A\), we cannot isolate the causal effect of \(X\) on \(Y\) in this model. DAGs can highlight what cannot be done.
Does Smoking During Pregnancy Protect Your Child?
- The debate about whether smoking causes cancer was settled by the 1960s. Its conclusion had been delayed by multiple years because scientists disagreed on the meaning of “to cause,” and no ways of discerning causal effects from observational data were available.
- But even afterwards, one paradox remained. Some researchers argued that smoking during pregnancy was actually good – if the unborn child was underweight.
- The paradox was not resolved until 2006. Pearl & Mackenzie (2018) argue that it took so long because precise language of causality was not yet available.
Smoking Mothers
- Underweight infants were found to have a death rate twenty times higher than normal-weight newborns.
- Babies of smokers during pregnancy were on average 200 grams lighter than those of non-smokers.
- However, underweight babies of smoking mothers had a higher survival rate than underweight babies of non-smoking mothers.
How come?
Smoking Mothers
- Scientists at the time cautiously concluded that smoking may not affect the development of the fetus.
- However, another explanation makes much more sense:
- There are (thinking in a blatantly simplified way) two possible causes for a low birth weight: Smoking and having a birth defect.
- If a mother does not smoke, a low birth weight points much more strongly to a birth defect.
- Birth weight acts as a collider.
- The original paradox becomes a (literal) textbook case of collider bias.
The Berkeley Admissions Paradox
- In 1978, an associate dean at the University of California noticed that 44 percent of applying men were admitted, but only 35 percent of women.
- Admission decisions were made by individual departments.
- The university surveyed all departments, and found that in every department, admission decisions were more favorable to women than to men.
How is this possible?
The Berkeley Admissions Paradox
- It turns out that the \(\text{gender}\rightarrow\text{outcome}\) relation has an important mediator.
- Discrimination is a causal concept, and thus a causal graph can help understand the situation.
- Women were applying to different departments/majors than men.
- More women applied to humanities departments, which were harder to get into.
- The choice of department is a mediator. Whether we want to condition on the mediator depends on the specific question we want to answer.
- In this case, it depends on our understanding of discrimination as well as whether we ask about a societal phenomenon or whether the university is at fault.
Simpson’s Paradox
| No Drug | Took Drug | |||
|---|---|---|---|---|
| Heart Attack | No Heart Attack | Heart Attack | No Heart Attack | |
| Female | 1 | 19 | 3 | 37 |
| Male | 12 | 28 | 8 | 12 |
| Total | 13 | 47 | 11 | 49 |
- Assume a fictional doctor, Dr. Smith, reads about a drug that reduced the probability of a heart attack among the subjects that took the drug (no randomized trial).
- However, both in men and in women, the drug seemed to increase the propensity to suffer from a heart attack.
How does this “Bad-Bad-Good (BBG)” drug paradox arise?
Simpson’s Paradox
| No Drug | Took Drug | |||
|---|---|---|---|---|
| Heart Attack | No Heart Attack | Heart Attack | No Heart Attack | |
| Female | 1 | 19 | 3 | 37 |
| Male | 12 | 28 | 8 | 12 |
| Total | 13 | 47 | 11 | 49 |
- Actually, the drug is just plain bad.
- Since gender is a confounder and affects both the propensity to take a drug and the chance of a heart attack, we need to control for it when comparing totals.
- In the example, we can do this by looking at both groups separately and then averaging. We find out that there is a negative effect for both women and men, and so the aggregate effect is also negative.