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Causal Inference

Notes on Causal Inference following the course by Brady Neal.

Author

Affiliation

Skip Moses

Published

May 31, 2022

DOI

In this post we preview the course on Causal Inference. This post will give a brief overview of the main concepts.

What is causal inference?

Inferring the effects of any treatment/policy/intervention/ect.
Examples:
- Effect of treatment on a disease
- Effect of climate change policy on emissions
- Effect of social media on mental health
- In general, effect of $X$ on $Y$ .

Motivating Example

Suppose we have some disease we are tying to treat with treatment A and treatment B. Our only goal is minimizing death. Suppose treatment B is much more scarce than treatment A.

Treatment T: A (0) and B (1)
Condition C: mild (0) or severe (1)
Outcome Y : alive (0) dead (1)

Data at Treatment Level

Treatment	Total
A	240/1500 16%
B	105/550 19%

Note this column is just $\mathbb{E}[Y \vert T]$
On average 16% (240 out of 1500) died after receiving treatment A.
On average 19% (105 out of 550) died after receiving treatment B.
It appears treatment A is the best option, because fewer patients that received treatment A died.

Data at the Condition Level

Treatment	Mild	Severe	Total
A	210/1400 15%	30/100 30%	240/1500 16%
B	5/50 10%	100/500 20%	105/550 19%

Note the two new columns are $\mathbb{E}[Y \vert T,C]$ .
Now that we have conditioned on condition we see our conclusion is flipped. Fewer patients died that received treatment B in both mild and severe patients.
This is Simpson’s Paradox: Our conclusions seem to depend on how we partition our data.
We can think of these numbers the following way:

$\frac{1400}{1500}(0.15) + \frac{100}{1500}(0.30) = 0.16$

$\frac{50}{550}(0.10) + \frac{500}{550}(0.20) = 0.19$

Now, the fractions are like weights on our percentages. We see the 19% of patients that received treatment B had a severe condition.
Likewise, the most patients that received treatment A only had a mild condition.

Which treatment should you choose?

The answer will depend on the causal structure of the problem.
- In this case, treatment B is the correct choice.
- Since treatment B was scarcer, doctors administer the more readily available treatment A.
- But if a more severe case is identified the doctor administers treatment B.
- In this case, treatment A is the correct choice.
- Because of the scarcity of treatment B, it is possible you will have to wait for treatment B.
- Alternatively, you don’t have to wait to receive treatment A.
- A patient with a mild condition, might progress to a severe condition because they have to wait.
- In order to account for effect of treatment through condition, we consider the total numbers.

Correlation does not imply causation

Observations can be correlated by chance, or if there is a common cause of both.
- Suppose sleeping with shoes on is highly correlated with waking up with a headache.
- You might conclude that sleeping with your shoes on causes a headache in the morning.
- However, a large majority of those who slept with their shoes on also drank the night before.
- In this example drinking the night before is counfounding the association between sleeping with shoes on and waking up with a headache.
Total association (e.g. correlation) is a mixture of causal and confounding association.

What does imply causation?

Potential Outcomes

Suppose you have a headache, and you know if you take a pill your headache will go away; if you don’t your headache will not go away.
- You might say the pill causes the headache to go away.
- If your headache goes away without taking the pill, you might conclude the pill didn’t do anything.

The causal effect of the treatment on the outcome is defined to be difference between potential outcomes. $Y_i(1) - Y_i(0)$

Fundamental problem of causal inference

Suppose you do not take the pill, then $Y_i(0)$ is the Factual.
The problem is we cannot compute the Counterfactual, $Y_i(1)$ .
Therefore, we cannot compute the causal effect.

Work around (Average Treatment Effect ATE)

Leverage linearity of expectation to
Denote the individual treatment effect (ITE) by $Y_i(1)-Y_i(0)$ .
The ATE is $\mathbb{E}[Y_i(1) - Y_i(0)] = \mathbb{E}[Y_i(1)] - \mathbb{E}[Y_i(0)]$ .
Note, potential outcomes are not actual outcomes, so

$\mathbb{E}[Y_i(1)] - \mathbb{E}[Y_i(0)] \neq \mathbb{E}[Y\vert T=1] - \mathbb{E}[Y\vert T = 0]$ The left hand side is causal, while the right hand side is causal and confounding.

This is where randomized trials come in. This allows us to remove any causal relationship between treatment and condition.
When there is no confounding: $\mathbb{E}[Y_i(1)] - \mathbb{E}[Y_i(0)] = \mathbb{E}[Y\vert T=1] - \mathbb{E}[Y\vert T = 0]$
Randomization is very powerful, because it also removes any causal effects of unobserved variables.

Observational Sudies

Can’t always be randomized.
- Randomization could be unethical/infeasible/impossible.
How do we measure causal effect in observational studies?
- We adjust/control for the right variables $W$ .
- If $W$ is a sufficient adjustment set, we have

$\mathbb{E}[Y(t) \vert W = w] := \mathbb{E}[Y\vert do(T=t), W =w] = \mathbb{E}[Y\vert t,w]$ - This still depends on $w$ , so we take the marginal

$\mathbb{E}[Y(t)] := \mathbb{E}[Y\vert do(T=t)] = \mathbb{E}_W\mathbb{E}[Y\vert t, W]$

Example

Treatment	Mild	Severe	Total
A	210/1400 15%	30/100 30%	240/1500 16%
B	5/50 10%	100/500 20%	105/550 19%

Here are sufficient adjustment is Condition.

$\mathbb{E}[Y\vert do(T=t)] = \mathbb{E}_W\mathbb{E}[Y\vert t, C] = \sum_{c\in C} \mathbb{E}[Y\vert t,c]P(c)$

$\frac{1450}{2050}(0.15) + \frac{600}{2050}(0.30) \approx 0.194$

$\frac{1450}{2050}(0.10) + \frac{600}{2050}(0.20) \approx 0.129$