Causal Assumptions

# Causal Assumptions

**Session 4**

]

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# Dichotomous exposure & outcome

.box-inv-1.medium[X (1: exposed, 0: unexposed)]

.box-inv-1.medium[Y (1: died, 0: survived)]

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# Potential Outcome

What *will* happen to you in the future given you have a particular exposure

]

.box-inv-1.medium[

**fixed** but **unknown**

]
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# Potential Outcomes

.box-1.medium[

`$\large Y(x)$`

]

.box-inv-1.medium[

The outcome `$(Y)$` under a specific exposure `$(X=x)$`

]
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# Potential Outcomes

.box-1.medium[

`$\large Y(1)$`

]

.box-inv-1.medium[

The outcome `$(Y)$` with exposure `$(X=1)$`

]

---

# Potential Outcomes

.box-1.medium[

`$\large Y(0)$`

]

.box-inv-1.medium[

The outcome `$(Y)$` without exposure `$(X=0)$`

]

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# Potential Outcomes

.box-1.medium[

`$\large Y(1) = 0$`

]

.box-inv-1.medium[

The outcome `$(Y)$` with exposure `$(X=1)$` is **"survival"**

]

---

# Potential Outcomes

.box-1.medium[

`$\large Y(0) = 1$`

]

.box-inv-1.medium[

The outcome `$(Y)$` without exposure `$(X=0)$` is **"death"**

]

---

.box-inv-1.medium[
Will eating ice cream make me happy?
]

.box-1.medium[Four options:]

(1) I am happy only with ice cream

---

.box-inv-1.medium[
Will eating ice cream make me happy?
]

.box-1.medium[Four options:]

(2) No effect of ice cream, I am happy either way

---

.box-inv-1.medium[
Will eating ice cream make me happy?
]

.box-1.medium[Four options:]

(3) No effect of ice cream, I am angry either way

.box-inv-1.medium[
Will eating ice cream make me happy?
]

.box-1.medium[Four options:]

(4) I am happy only without ice cream

---

# Potential Outcomes

.pull-left[
.box-1.medium[No Causal Effect]
 
.box-inv-1[
`$Y$`(🍦) = 😠, `$Y$`(no 🍦) = 😠
]
 
.box-inv-1[
`$Y$`(🍦) = 😀, `$Y$`(no 🍦) = 😀
]
]

.pull-right[
.box-1.medium[Causal Effect]
 
.box-inv-1[
`$Y$`(🍦) = 😀, `$Y$`(no 🍦) = 😠
]
 
.box-inv-1[
`$Y$`(🍦) = 😠, `$Y$`(no 🍦) = 😀
]
]
---

# Potential Outcome

What *will* happen to you in the future given you have a particular exposure

]

.box-inv-1.medium[

**fixed** but **unknown**

]

---

# What is a causal effect

.box-inv-1.medium[The causal effect is the comparison of the potential outcomes for the same unit at the same moment in time post-exposure]

.box-inv-1.medium[Whether the effect is **causal** depends on the potential outcomes but it **does not** depend on which outcome is actually observed]

---
class: title title-1 center

# Outcome for an individual

.box-1.medium[
Individual *i* has a pair of potential outcomes
]

--

.box-inv-1.medium[
`$(Y_i(1), Y_i(0))$`
]

---

# Outcome for an individual

.box-1.medium[
Individual *i* has a pair of potential outcomes
]

.box-inv-1.medium[
`$Y_i$`(🍦), `$Y_i$`(no 🍦)
]

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# Causal Effect

.box-inv-1.medium[

`$\Huge Y(1) - Y(0)$`
]

---

# Causal Effect

.box-inv-1.medium[

`$\Huge Y_i(1) - Y_i(0)$`
]

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# Causal Estimand

.box-inv-1.medium[

`$$\large \frac{1}{N}\sum_{i=1}^N(Y_i(1) - Y_i(0))$$`
]

---

# Potential Outcome

What *will* happen to you in the future given you have a particular exposure

]

.box-inv-1.medium[

again...**fixed** but **unknown**

]

---

# What do we observe?

.box-inv-1.medium[Recall: The causal effect is the comparison of the potential outcomes for the same unit at the same moment in time post-exposure]

---

# What do we observe?

.box-inv-1.medium[Recall: The causal effect is the comparison of the potential outcomes for the same unit **at the same moment** in time post-exposure]

---

---

.pull-right[
.box-inv-1[
Individual `$i$` does **not** eat 🍦
]
 
.box-1.medium[
`$Y_i^{obs}$`(no 🍦)
]
]

---

---

.pull-right[
.box-inv-1[
Individual `$i$` does **not** eat 🍦
]
 
.box-1.medium[
`$Y_i^{obs}$`(no 🍦)
]
]

---

---

---

---

---

# Outcomes for an individual

.box-inv-1[
`$Y_i^{obs}= Y_i(X_i)=\begin{cases}Y_i(0)&\textrm{if }X_i=0\\Y_i(1)&\textrm{if }X_i = 1\end{cases}$`
]

--

.box-1[
`$Y_i^{mis}= Y_i(1-X_i)=\begin{cases}Y_i(1)&\textrm{if }X_i=0\\Y_i(0)&\textrm{if }X_i = 1\end{cases}$`
]

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# Causal inference is fundamentally a missing data problem

---
class: title title-1

# Does ice cream make us happy?

.box-1.medium[Let's add some data!]

---

# Does ice cream make us happy?

---
class: title title-1

# Does ice cream make us happy?

.pull-left[
.box-inv-1.medium[My possibilities]
 
.box-1.small[
`$Y_{me}$`(🍦) = 😀, `$Y_{me}$`(no 🍦) = 😠
]

.box-1.small[
`$Y_{me}$`(🍦) = 😀, `$Y_{me}$`(no 🍦) = 😀
]

.box-1.small[
`$Y_{me}$`(🍦) = 😠, `$Y_{me}$`(no 🍦) = 😠
]

.box-1.small[
`$Y_{me}$`(🍦) = 😠, `$Y_{me}$`(no 🍦) = 😀
]
]

---

# Does ice cream make us happy?

.pull-left[
.box-inv-1.medium[My possibilities]
 
.box-1.small[
`$Y_{me}$`(🍦) = 😀, `$Y_{me}$`(no 🍦) = 😠
]

.box-1.small[
`$Y_{me}$`(🍦) = 😀, `$Y_{me}$`(no 🍦) = 😀
]

.box-1.small[
`$Y_{me}$`(🍦) = 😠, `$Y_{me}$`(no 🍦) = 😠
]

.box-1.small[
`$Y_{me}$`(🍦) = 😠, `$Y_{me}$`(no 🍦) = 😀
]
]

.pull-right[
.box-inv-1.medium[Your possibilities]
 
.box-1.small[
`$Y_{you}$`(🍦) = 😀, `$Y_{you}$`(no 🍦) = 😠
]

.box-1.small[
`$Y_{you}$`(🍦) = 😀, `$Y_{you}$`(no 🍦) = 😀
]

.box-1.small[
`$Y_{you}$`(🍦) = 😠, `$Y_{you}$`(no 🍦) = 😠
]

.box-1.small[
`$Y_{you}$`(🍦) = 😠, `$Y_{you}$`(no 🍦) = 😀
]
]

---

# Does ice cream make us happy?

.box-inv-1.medium[Exposure possibilities]

.box-1.medium[We both eat ice cream]

.box-1.medium[We both do not eat ice cream]

.box-1.medium[I eat ice cream you do not]

.box-1.medium[I do not eat ice cream, you do]

---

# Potential Outcomes

--

---

## Stable Unit Treatment Value Assumption

.box-inv-1[
The potential outcomes for any unit do not vary with the exposures assigned to other units, and, for each unit, there are no different forms or versions of each exposure level, which lead to different potential outcomes.]

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## Stable Unit Treatment Value Assumption

.box-inv-1[
**The potential outcomes for any unit do not vary with the exposures assigned to other units**, and, for each unit, there are no different forms or versions of each exposure level, which lead to different potential outcomes.]

.box-1.medium[No Interference]

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# No interference

.box-inv-1.medium[This is a strong assumption!]

.box-inv-1.medium[These are examples of **interference**!]

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# Vaccines: a classic interference example

.box-inv-1.medium[If everyone else is immunized with a perfect vaccine, your immunization will have no effect]

.box-1.medium[In these cases, we can]

---

## Stable Unit Treatment Value Assumption

---

## Stable Unit Treatment Value Assumption

.box-inv-1[
The potential outcomes for any unit do not vary with the exposures assigned to other units, and, for **each unit, there are no different forms or versions of each exposure level, which lead to different potential outcomes.**]

---

# Exposure assignment

.box-1.medium[
`$Y_{me}^{obs}$`(🍦) = 😀
]

.box-inv-1[I decided to eat ice cream because I was feeling quite grumpy and thought it might make me feel better! It did!]

---
class: title title-1

# Exposure assignment

.box-1.medium[
`$Y_{you}^{obs}$`(no 🍦) = 😀
]

---

# <svg style="height:0.8em;top:.04em;position:relative;" viewBox="0 0 640 512"><path d="M624 416H381.54c-.74 19.81-14.71 32-32.74 32H288c-18.69 0-33.02-17.47-32.77-32H16c-8.8 0-16 7.2-16 16v16c0 35.2 28.8 64 64 64h512c35.2 0 64-28.8 64-64v-16c0-8.8-7.2-16-16-16zM576 48c0-26.4-21.6-48-48-48H112C85.6 0 64 21.6 64 48v336h512V48zm-64 272H128V64h384v256z"/></svg> Application Exercise

# Exposure assignment

.box-inv-1[Understanding the **assignment mechanism** is crucial for understanding the causal effect]

.box-1[Assignment probability must be **individualistic** (your covariates / potential outcomes can't influence my assignment)]

---

# Exposure assignment

.box-inv-1[Understanding the **assignment mechanism** is crucial for understanding the causal effect]

.box-1[Assignment probability must be **individualistic** (your covariates / potential outcomes can't influence my assignment)]

---

# Exposure assignment

.box-inv-1[Understanding the **assignment mechanism** is crucial for understanding the causal effect]

.box-2[Assignment probability must be **individualistic** (your covariates / potential outcomes can't influence my assignment)]

---

# Exposure assignment

.box-inv-1[Understanding the **assignment mechanism** is crucial for understanding the causal effect]

.box-2[Assignment probability must be **individualistic** (your covariates / potential outcomes can't influence my assignment)]

# Notation

.box-inv-1.medium[
.huge[
`$\mathbf{Y}(0), \mathbf{Y}(1)$`
]
]

---

# Notation

.box-inv-1.medium[
.huge[
`$\mathbf{X}$`
]
]

---

# Causal inference is fundamentally a missing data problem

---

# Outcomes for an individual

.box-inv-1[
`$Y_i^{obs}= Y_i(X_i)=\begin{cases}Y_i(0)&\textrm{if }X_i=0\\Y_i(1)&\textrm{if }X_i = 1\end{cases}$`
]

--

.box-1[
`$Y_i^{mis}= Y_i(1-X_i)=\begin{cases}Y_i(1)&\textrm{if }X_i=0\\Y_i(0)&\textrm{if }X_i = 1\end{cases}$`
]

---

# Outcomes for an individual

.box-inv-1[
`$Y_i(0)= \begin{cases}Y_i^{mis}&\textrm{if }X_i=1\\Y_i^{obs}&\textrm{if }X_i = 0\end{cases}$`
]

.box-1[
`$Y_i(1)= \begin{cases}Y_i^{mis}&\textrm{if }X_i=0\\Y_i^{obs}&\textrm{if }X_i = 1\end{cases}$`
]
---
class: section-title-1 title-1 middle

# Causal inference is fundamentally a missing data problem

# Assignment mechanism

.box-inv-1.medium[

`$P(\mathbf{X}|\mathbf{Z}, \mathbf{Y}(0), \mathbf{Y}(1))$`
]

---

# Assignment mechanism

.box-inv-1.medium[

`$P(\mathbf{X}|\mathbf{Z}, \mathbf{Y}(0), \mathbf{Y}(1))$`
]

.box-inv-1[
`$$\sum_{X\in\{0,1\}^N}P(\mathbf{X}|\mathbf{Z}, \mathbf{Y}(0), \mathbf{Y}(1)) =1$$`
for all `$\mathbf{Z}$`, `$\mathbf{Y}(0)$`, and `$\mathbf{Y}(1)$`
]

---

# Assignment mechanism

.box-inv-1[
In words: this is the probability that a particular value for the full assignment will occur -- like person 1 gets assigned to the exposure, 2 & 3 to control, 4 to exposure, and so on. The probabilities across the full set of `$2^N$` possible assignment vectors must sum to 1.
]

.box-1.medium[This is **not** the probability that a particular person will be assigned a particular exposure]

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# Unit Assignment Probability

`$P_i(\mathbf{Z}, \mathbf{Y}(0), \mathbf{Y}(1))=\sum_{\mathbf{X}:X_i=1}P(\mathbf{X}|\mathbf{Z}, \mathbf{Y}(0),\mathbf{Y}(1))$`
]

.box-1[
You can get this by summing across all possible assignment vectors, `$\mathbf{X}$` where `$X_i=1$`
]

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# Propensity score: The average unit assignment probability for units where `$X_i=x$`

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# Propensity Score

.box-inv-1.medium[

`$e_i(x)=\frac{1}{N(x)}\sum_{i: X_i=x}P_i(\mathbf{Z}, \mathbf{Y}(0), \mathbf{Y}(1))$`
]

---