Intro to Causal Inference

class: center, middle, inverse, title-slide

.title[
# Intro to Causal Inference
]
.author[
### Yue Jiang
]
.date[
### STA 440 / Spring 25
]

---

.question[
Who is depicted here?
]

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### Potential outcomes

For now, we will consider a treatment `$A$` taking on values of 0 or 1, and an
outcome `$Y$` taking on values 0 or 1.

Define a .vocab[potential outcome] `$Y^{a = 1}$` as the outcome that *would be observed* if treatment `$a = 1$` was given (this might not have happened in reality). `$Y^{a = 0}$` is the potenetial outcome that would be observed if treatment `$a = 0$` was given.

These potential outcomes are known as .vocab[counterfactuals]. Before treatment `$A$` happens, there are two potential outcomes. After `$A$` happens, then a counterfactual exists.

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### Counterfactuals

You've likely dealt with counterfactuals your whole life!

- What if I stayed in bed this morning?
- What if we shot better during the Elite 8 game?
- What if I didn't take that Tylenol?
- If this patient got a liver transplant 3 years ago, would they sitll have been alive now?

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### Potential outcomes

(A famous (and very reproduced) table from Hernan and Robins)

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### Individual vs. average causal effects

.question[
What does it mean to have an individual "causal effect"? What about an average causal effect?
]

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### Individual causal effects

In this table, we see that some benefit, some are harmed, some are "immune," and some are "doomed."

Consider the individual potential outcomes `$Y_i^{a = 1}$` and `$Y_i^{a = 0}$` for the `$i$`th observation. For individual `$i$`, we say that `$A$` has an effect on `$Y$` if `$Y_i^{a = 1} \neq Y_i^{a = 0}$`.

.question[
The .vocab[sharp null hypothesis] states that there is no effect for *all* individuals. What does it mean for `$A$` to have no effect on `$Y$` in the language of potential outcomes?
]

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### Individual causal effects

The sharp null hypothesis states that `$Y_i^{a = 1} = Y_i^{a = 0}$` for all `$i$`.

We will also introduce .vocab[causal consistency], which states that the observed outcome `$Y$` is related to the observed treatment and counterfactuals by `$Y = Y^{a = 1}A + Y^{a = 0}(1 - A)$`.

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### Individual causal effects

We usually care about some difference in potential outcomes, for instance `$Y_i^{a = 1} - Y_i^{a = 0}$`.

.question[
Is it possible to identify individual causal effects from the observed data `$Y$` and `$A$` for observation `$i$`?
]

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### Quick aside: variability

So far, we've assumed that potential outcomes are deterministic based on the treatment, but note that we could (somewhat) straightforwardly define potential outcomes probabilistically as well.

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### Potential outcomes vs. observed data

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### Some assumptions

.vocab[Infererence] occurs when one individual's treatment affects the outcome of another individual. Having no interference is an important part of SUTVA (next slide).

.question[
What are some examples of interference?
]

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### Some assumptions

The .vocab[Stable Unit Treatment Value Assumption] (SUTVA) essentially states that there are only two potential outcomes: there is no interference, and that there is only one version of treatment and one version of non-treatment (or that multiple versions of each of these are irrelevant).

.vocab[
What is an example of having multiple "versions" of treatment? How about an example where this is irrelevant? Why would SUTVA be important philosophically?
]

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### Average causal effects

Let's now move on to average causal effects, defined in terms of expectations. An .vocab[average causal effect] exists in the population if `$E(Y^{a = 1}) \neq E(Y^{a = 0})$`. We say there is no average causal effect if `$E(Y^{a = 1}) = E(Y^{a = 0})$`.

.question[
Does having no average causal effect imply the sharp null? Does the sharp null imply no average causal effect? Explain why or provide counterexamples.
]

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### Causal vs. associational effects

We might care about various measures of causal effects, such as causal risk differences, odds ratios, etc. These are defined in terms of potential outcomes. *Associational* effects are based on observed data. E.g.:

- Causal risk difference: `$E(Y^{a = 1}) \neq E(Y^{a = 0})$`
- *Associational* risk difference: `$E(Y | A = 1) - E(Y|A = 0)$`

.question[
When we might we be able to claim an association is actually causation?
]

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### Causal vs. associational effects

(that's the whole point of causal inference!)

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### A quick calculation

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### A quick calculation

.vocab[
What is `$P(Y^{a = 0} = 1 | A = 1)$`? What is `$P(Y^{a = 0} = 1 | A = 0)$`? What is the implication of this?
]

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### A quick calculation

Those that received treatment were at higher risk of death than the untreated in the counterfactual world in which no one receives treatment.

.question[
But we don't get to observe the counterfactuals. What then? What can we do?
]

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### Exchangeability

.vocab[Exchangeability] is the property that the counterfactual outcome and actual treatment received are independent for all `$a$`: `$Y^a \perp A$` (note that like "causal consistency," "exchangeability" does not have the usual probabilistic definition of exchangeability).

Be careful, this does *not* mean `$Y \perp A$` (in fact, we probably wouldn't want this at all for many studies!).
 
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### Exchangeability

Here, exchangeability implies that we can "exchange" group assignment without altering the outcome. For instance, if in the treatment group you have sicker patients than another, we will not have exchangeability - if they instead got placebo instead, our results would be different.

> *When the groups of exposed and unexposed are exchangeable, the risk of the outcome in the treated group would have been the same as the risk of the outcome in the untreated group had the treatment been switched between the groups*

.right[--Hernan]

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### Exchangeability

Similarly, .vocab[mean exchangeability] is a weaker, but analogous property on the expectations: `$E(Y^a | A = 1) = E(Y^a | A = 0)$` (exchangeability implies mean exchangeability).

.question[
Why is (mean) exchangeability important? Did we have exchangeability in the example earlier? Why or why not?
]

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### Exchangeability

Mean exchangeability means that `$E(Y^{a = 1}) = E(Y^{a = 1} | A = 1)$`, which by causal consistency is equal to `$E(Y | A = 1)$` (which *is* identifiable).

Similarly, it means that `$E(Y^{a = 0}) = E(Y | A = 0)$`.

.vocab[
What is the implication of this fact?
]

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### Exchangeability

Under exchangeability the causal risk difference (and risk ratio and odds ratio) equal the associational risk difference (or risk ratio or odds ratio).

.question[
How might we introduce exchangeability to a study?
]

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### Conditionally randomized experiments

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### Conditionally randomized experiments

We saw last time that we did *not* have (mean) exchangeability in these data. The counterfactual outcomes would have been different if the treatment assignments were swapped.

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### Conditionally randomized experiments

Suppose there is some baseline covariate `$L$`, for instance `$L = 1$` if the individual is in critical condition at baseline. We might consider a .vocab[conditionally randomized] experiment: perhaps we randomly assign 3/4 of the patients for which `$L = 1$` to treatment, and only 1/2 of patients for which `$L = 0$`.

.question[
Why will we **not** have exchangeability for such an experiment?
]

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### Conditionally randomized experiments

We see that the "sicker" patients at baseline were more likely to receive treatment; if they were more likely to die if they did *not* receive treatment, then we see that (mean) exchangeability is violated (their potential outcomes under the opposite treatment would not be the same on average).

However, we can view this as two separate marginal experiments: one among the sicker patients `$L = 1$`, and one among the less sick patients, `$L = 0$`. We have .vocab[conditional exchangeability] (for `$a, l = 0, 1$`):

`$$Y^{a} \perp A | L.$$`

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### Conditionally randomized experiments

We might be able to view this as potential outcomes being MAR instead - if we were able to know L, then we would be able to control for it somehow and make value inferences: conditional on `$L$`, expected potential outcomes are the same regardless of what treatment the patient received (we can swap them out with no difference in expected potential outcomes).

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### Conditionally randomized experiments

`\begin{align*}
E(Y^a) &= \sum_{l \in \mathscr{L}} E(Y^a | L = l)P(L = l)\\
&= \sum_{l \in \mathscr{L}} E(Y^a | A = a, L = l)P(L = l)\\
&= \sum_{l \in \mathscr{L}} E(Y | A = a, L = l)P(L = l)
\end{align*}`

by conditional exchangeability and causal consistency. Again, the right side of the equation is fully identifiable from observed data - this is just a weighted average.

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### Conditionally randomized experiments

Thus, we know that this weighted average, the .vocab[standardized mean], is the counterfactual expectation if in fact all patients received treatment `$a$`. We can define causal risk estimands, such as the risk difference

`$$E(Y^{a = 1}) - E(Y^{a = 0})$$`

and estimate them with observed data:

`$$\sum_l E(Y | L = l, A = 1)P(L = l) - E(Y | L = l, A = 0)P(L = l)$$`

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