Causal Inference

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# Causal Inference
### CWAN| 王　超辰
### 2020-07-31 (Fri) <br> <span class="citation">@HOME</span> 15:00-16:00

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# Chapter 1 <br> Definition

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## Outline

.large[
- Purpose
1. Individual causal effect
2. Average causal effects
3. Measures of causal effect
4. Random variability
5. Causation versus association
]

---
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## Purpose

.full-width[.content-box-red[.bold[
To introduce the **mathematical notation** <br> that could formalize the causal intuition.
]]]

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## Individual causal effect (1)

.full-width[.content-box-red[.bold[
We compare (mentally) the outcome <br> **when an action `$A$` is taken** <br> with the outcome <br> **when the action `$A$` is withheld.**
]]]

.med[
- Zeus is waiting for a heart transplant. 
  - He dies five days later after receiving a new heart
  - He survives five days later if he did not had transplant
  - We say **the heart transplant caused** Zeus's death.
]

---
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## Individual causal effect (2)

.full-width[.content-box-red[.bold[
We compare (mentally) the outcome <br> **when an action `$A$` is taken** <br> with the outcome <br> **when the action `$A$` is withheld.**
]]]

.med[
- Hera is also waiting for a heart transplant. 
  - She survives five days later after heart transplant
  - She also survives five days later if she did not had transplant
  - We say **the heart transplant has no causal effect**.
]

---
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## Individual causal effect (3)

.full-width[.content-box-red[.bold[
We compare (mentally) the outcome 
<br> **when an action `$A$` is taken** 
<br> with the outcome 
<br> **when the action `$A$` is withheld.** 
<br>  `$A$` could be an intervention, an exposure, 
<br>  or a treatment.
]]]

---
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## Individual causal effect (4)

- Consider a dichotomous treatment variable `$A$`
    - 1 = treated
    - 0 = untreated
- And a dichotomous outcome variable `$Y$`
    - 1 = death
    - 0 = survival
- We define: 
    - `$Y^{a = 1}$`: the outcome that would be observed under treatment `$a = 1$`
    - `$Y^{a = 0}$`: `$Y$` under treatment `$a = 0$`
 
---
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## Individual causal effect (5)

.pull-left[
- Zeus has: 
    - `$Y^{a = 1} = 1$`
    - `$Y^{a = 0} = 0$`
- Hera has: 
    - `$Y^{a = 1} = 0$`
    - `$Y^{a = 0} = 0$`
]

.pull-right[
- `$Y^{a = 1} \neq Y^{a = 0}$` <br> then there is a <br> **causal effect**
- `$Y^{a = 1}, Y^{a = 0}$` are <br> **potential or counterfactual outcomes**
]

???
- "Potential" because either of these two outcomes can be potentially observed
- "Counterfactual" because these outcomes represent situations that may not actually occur

---
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## Individual causal effect (6)

.full-width[.content-box-red[.bold[
**Consistency:**
<br> For each individual, one of the potential 
<br> outcomes will actually occur. 
<br> For example, Zeus was actually treated 
<br> `$(A = 1)$`, his counterfactual outcome 
<br> `$Y^{a = 1} = 1$` is equal to what was observed. 
<br> `$Y = Y^{a = 1} = 1$` is called **consistency**.
]]]

---
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## Average causal effect (1)

.full-width[.content-box-red[.bold[
We need three pieces of information: 
- an outcome of interest
- the action `$a = 1, a= 0$` to be compared
- the counterfactual outcomes `$Y^{a = 1}, Y^{a = 0}$`
]]]

.med[Normally it is not possible to identify individual causal effects. We turn to an **aggregated causal effect**.]

---
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## Average causal effect (2)

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  <tr>
    <th class="tg-0lax"></th>
    <th class="tg-1wig">\[\mathbf{Y^{a = 0}}\] </th>
    <th class="tg-1wig"> \[\mathbf{Y^{a = 1}}\] </th>
    <th class="tg-0lax"></th>
    <th class="tg-1wig"> \[\mathbf{Y^{a = 0}}\] </th>
    <th class="tg-1wig"> \[\mathbf{Y^{a = 1}}\] </th>
  </tr>
</thead>
<tbody>
  <tr>
    <td class="tg-bhpp">Rheia</td>
    <td class="tg-cyim">0</td>
    <td class="tg-cyim">1</td>
    <td class="tg-bhpp">Leto</td>
    <td class="tg-cyim">0</td>
    <td class="tg-cyim">1</td>
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    <td class="tg-bhpp">Kronos</td>
    <td class="tg-cyim">1</td>
    <td class="tg-cyim">0</td>
    <td class="tg-bhpp">Ares</td>
    <td class="tg-cyim">1</td>
    <td class="tg-cyim">1</td>
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  <tr>
    <td class="tg-bhpp">Demeter</td>
    <td class="tg-cyim">0</td>
    <td class="tg-cyim">0</td>
    <td class="tg-bhpp">Athena</td>
    <td class="tg-cyim">1</td>
    <td class="tg-cyim">1</td>
  </tr>
  <tr>
    <td class="tg-bhpp">Hades</td>
    <td class="tg-cyim">0</td>
    <td class="tg-cyim">0</td>
    <td class="tg-bhpp">Hephaestus</td>
    <td class="tg-cyim">0</td>
    <td class="tg-cyim">1</td>
  </tr>
  <tr>
    <td class="tg-bhpp">Hestia</td>
    <td class="tg-cyim">0</td>
    <td class="tg-cyim">0</td>
    <td class="tg-bhpp">Aphrodite</td>
    <td class="tg-cyim">0</td>
    <td class="tg-cyim">1</td>
  </tr>
  <tr>
    <td class="tg-bhpp">Poseidon</td>
    <td class="tg-cyim">1</td>
    <td class="tg-cyim">0</td>
    <td class="tg-bhpp">Cyclope</td>
    <td class="tg-cyim">0</td>
    <td class="tg-cyim">1</td>
  </tr>
  <tr>
    <td class="tg-bhpp">Hera</td>
    <td class="tg-cyim">0</td>
    <td class="tg-cyim">0</td>
    <td class="tg-bhpp">Persephone</td>
    <td class="tg-cyim">1</td>
    <td class="tg-cyim">1</td>
  </tr>
  <tr>
    <td class="tg-bhpp">Zeus</td>
    <td class="tg-cyim">0</td>
    <td class="tg-cyim">1</td>
    <td class="tg-bhpp">Hermes</td>
    <td class="tg-cyim">1</td>
    <td class="tg-cyim">0</td>
  </tr>
  <tr>
    <td class="tg-bhpp">Artemis</td>
    <td class="tg-cyim">1</td>
    <td class="tg-cyim">1</td>
    <td class="tg-bhpp">Hebe</td>
    <td class="tg-cyim">1</td>
    <td class="tg-cyim">0</td>
  </tr>
  <tr>
    <td class="tg-bhpp">Apollo</td>
    <td class="tg-cyim">1</td>
    <td class="tg-cyim">0</td>
    <td class="tg-bhpp">Dionysus</td>
    <td class="tg-cyim">1</td>
    <td class="tg-cyim">0</td>
  </tr>
</tbody>
</table>
]

.pull-right[.small[
- 10 out of 20 would have died if they had received the treatment
`$\text{Pr}[\mathbf{Y}^{a = 1} = 1] = 10/20 \\ = 0.5$`

- 10 out of 20 would have died if they if they had not received the treatment
`$\text{Pr}[\mathbf{Y}^{a = 0} = 1] = 10/20 \\ = 0.5$`
]]

---
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## Average causal effect (3)

.full-width[.content-box-red[.bold[
An **average causal effect** of treatment `$A$` on outcome `$Y$` is present if

$$
\text{Pr}[\mathbf{Y}^{a = 1} = 1] \neq \text{Pr}[\mathbf{Y}^{a = 0} = 1]
$$
]]]

In the previous table, because `$\text{Pr}[\mathbf{Y}^{a = 1} = 1] = \text{Pr}[\mathbf{Y}^{a = 0} = 1] = 0.5$`, 
<br> we say that the average causal effect is null, or the
<br> **null hypothesis of no average causal effect is true**.

---
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## Measures of causal effect (1)

.full-width[.content-box-red[.bold[
These are equivalent ways of representing the causal null:

- **Risk difference**: `$\text{Pr}[\mathbf{Y}^{a = 1} = 1] - \text{Pr}[\mathbf{Y}^{a = 0} = 1] = 0$`
- **Risk ratio**: 
`$\frac{\text{Pr}[\mathbf{Y}^{a = 1} = 1]}{\text{Pr}[\mathbf{Y}^{a = 0} = 1]} = 1$`
- **Odds ratio**: 
`$\frac{\text{Pr}[\mathbf{Y}^{a = 1} = 1] / \text{Pr}[\mathbf{Y}^{a = 1} = 0]}{\text{Pr}[\mathbf{Y}^{a = 0} = 1]/\text{Pr}[\mathbf{Y}^{a = 0} = 0]} = 1$`
]]]

---
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## Measures of causal effect (2)

.med[
- Causal risk difference, risk ratio, and odds ratio
    - quantify the strength of the same causal effect on **different scales**
    - also called **causal effects**
- Consider use different measures for different purposes:
    - 3 in a million vs. 1 in a million deaths
        - causal risk difference: 0.000002
        <br> the absolute number of cases of disease attributable to the treatment.
        - causal risk ratio: 3
        <br> how many times treatment relative to no treatment increases the risk
]

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## Measures of causal effect (3) - **Number needed to treat, NNT**

.full-width[.content-box-red[.bold[
.small[
Consider 100 million patients in which 
<br> 20 million would die within 5 years if treated `$(a = 1)$`
<br> 30 million would die within 5 years if untreated `$(a = 0)$`
<br> the causal risk difference is 
<br> `$\text{Pr}[\mathbf{Y}^{a = 1} = 1] - \text{Pr}[\mathbf{Y}^{a = 0} = 1] = 0.2 - 0.3 = -0.1$`
- If one treats the 100 million patients, there will be 10 million fewer deaths 
- One needs to treat 100 million patients to save 10 million lives, i.e. **treat 10 patients to save 1 life (= NNT)**
- NNT is equal to the inverse of the causal risk difference
]

]]]

---
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## Random variability

.med[
- The populations of interest are typically much larger. In practice, we only collect information on a sample of the population of interest.

- The "hat" over `$\text{Pr}$` indicates that the sample proportion `$\widehat{\text{Pr}}[\mathbf{Y}^{a} = 1]$` is an **estimator** of the corresponding population quantity `$\text{Pr}[\mathbf{Y}^{a} = 1]$`.

- **The larger** the number of individuals in the sample, **the smaller** the difference between `$\widehat{\text{Pr}}[\mathbf{Y}^{a} = 1]$` and `$\text{Pr}[\mathbf{Y}^{a} = 1]$` is expected to be.

- The error due to sampling variability is random.
]

---
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## Causation versus association (1)

.pull-left[
.med[In the real world, we only can observe one of these (potential) outcomes because the individual is either treated or untreated.
- 7 died in 13 who were treated `$(A=1)$` 
`$\text{Pr}[Y = 1|A = 1] \\ = 7/13$`
- `$\text{Pr}[Y = 1|A = a]$` conditional probability
]
]

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    <th class="tg-1wig">A</th>
    <th class="tg-1wig">Y</th>
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</thead>
<tbody>
  <tr>
    <td class="tg-1wig">Rheia</td>
    <td class="tg-0lax">0</td>
    <td class="tg-0lax">0</td>
    <td class="tg-1wig">Leto</td>
    <td class="tg-0lax">0</td>
    <td class="tg-0lax">0</td>
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    <td class="tg-1wig">Kronos</td>
    <td class="tg-0lax">0</td>
    <td class="tg-0lax">1</td>
    <td class="tg-1wig">Ares</td>
    <td class="tg-0lax">1</td>
    <td class="tg-0lax">1</td>
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  <tr>
    <td class="tg-1wig">Demeter</td>
    <td class="tg-0lax">0</td>
    <td class="tg-0lax">0</td>
    <td class="tg-1wig">Athena</td>
    <td class="tg-0lax">1</td>
    <td class="tg-0lax">1</td>
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    <td class="tg-1wig">Hades</td>
    <td class="tg-0lax">0</td>
    <td class="tg-0lax">0</td>
    <td class="tg-1wig">Hephaestus</td>
    <td class="tg-0lax">1</td>
    <td class="tg-0lax">1</td>
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    <td class="tg-1wig">Hestia</td>
    <td class="tg-0lax">1</td>
    <td class="tg-0lax">0</td>
    <td class="tg-1wig">Aphrodite</td>
    <td class="tg-0lax">1</td>
    <td class="tg-0lax">1</td>
  </tr>
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    <td class="tg-1wig">Poseidon</td>
    <td class="tg-0lax">1</td>
    <td class="tg-0lax">0</td>
    <td class="tg-1wig">Cyclope</td>
    <td class="tg-0lax">1</td>
    <td class="tg-0lax">1</td>
  </tr>
  <tr>
    <td class="tg-1wig">Hera</td>
    <td class="tg-0lax">1</td>
    <td class="tg-0lax">0</td>
    <td class="tg-1wig">Persephone</td>
    <td class="tg-0lax">1</td>
    <td class="tg-0lax">1</td>
  </tr>
  <tr>
    <td class="tg-1wig">Zeus</td>
    <td class="tg-0lax">1</td>
    <td class="tg-0lax">1</td>
    <td class="tg-1wig">Hermes</td>
    <td class="tg-0lax">1</td>
    <td class="tg-0lax">0</td>
  </tr>
  <tr>
    <td class="tg-1wig">Artemis</td>
    <td class="tg-0lax">0</td>
    <td class="tg-0lax">1</td>
    <td class="tg-1wig">Hebe</td>
    <td class="tg-0lax">1</td>
    <td class="tg-0lax">0</td>
  </tr>
  <tr>
    <td class="tg-1wig">Apollo</td>
    <td class="tg-0lax">0</td>
    <td class="tg-0lax">1</td>
    <td class="tg-1wig">Dionysus</td>
    <td class="tg-0lax">1</td>
    <td class="tg-0lax">0</td>
  </tr>
</tbody>
</table>
]

---
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## Causation versus association (2)

- When `$\text{Pr}[Y = 1|A = 1] = \text{Pr}[Y = 1|A = 0]$` <br> we say that treatment `$A$` and outcome `$Y$` are **independent**: `$A \perp \!\!\! \perp Y$`

- We say that treatment `$A$` is not associated with `$Y$` or, `$A$` does not predict `$Y$`.

- We also say that `$A$` is associated with `$Y$` if `$\text{Pr}[Y = 1|A = 1] \neq \text{Pr}[Y = 1|A = 0]$`

---
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## Association measures

Similarly, associational risk differences, risk ratio, and odds ratio can be defined as:

.full-width[.content-box-red[.bold[
- **Risk difference**: `$\text{Pr}[Y =1 | A= 1] - \text{Pr}[Y = 1| A= 0]$` 
- **Risk ratio**: `$\frac{\text{Pr}[Y =1 | A= 1]}{\text{Pr}[Y = 1| A= 0]}$`
- **Odds ratio**: `$\frac{\text{Pr}[Y =1 | A= 1] / \text{Pr}[Y = 0 | A= 1]}{\text{Pr}[Y = 1| A= 0] / \text{Pr}[Y = 0| A= 0]}$`
]]]

---
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## Causation versus association (3)

In our sample of 20 individuals, we found

1. no causal effect, because <br> `$\text{Pr}[\mathbf{Y}^{a = 1} = 1] - \text{Pr}[\mathbf{Y}^{a = 0} = 1] = 0.5 - 0.5 = 0$`
2. an association because <br> `$\text{Pr}[Y =1 | A= 1] - \text{Pr}[Y = 1| A= 0] \\ =7/13 - 3/7 = 0.11$`

---
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background-image: url("./fig/causalassociation.png")
background-position: 50% 50%
background-size: contain

---
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## Causation versus association (4)

1. **Causation**:a contrast between the whole white diamond (**all individuals treated**) and the whole grey diamond (**all individuals untreated**). <br> .blue[*"What would be the risk if everybody had been treated/untreated?"*]

2. Whereas **association**: a contrast between the white (**the treated**) and the grey (**the untreated**) <br> .blue[*"What is the risk in the treated/untreated?"*]

---
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## Association is not causation

In our example, 
- There was association because risk in the treated (= 7/13) was greater than risk in the untreated (3/7).
- There was no causation because risk if everybody treated was the same as the risk if everybody was untreated.

---
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## Association is not causation

Suppose that 5-year mortality for aspirin vs. no aspirin,

- causal risk ratio is 0.5 
- associational risk ratio is 1.5

This is possible because individuals at high risk of CVD death are preferentially prescribed aspirin. 
If a physician decides to withhold aspirin because those who treated had a greater risk of dying compared with untreated. This is a malpractice.

---
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background-image: url("./fig/causalnotassociation.mp4")
background-position: 50% 50%
background-size: contain

---
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# Chapter 2 <br> Randomized Experiments

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## Outline

- Randomization

- Conditional randomization

- Standardization

- Inverse probability weighting

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### Only one of the counterfactual outcome is known

???
The data in this table can only good to compute association measures.

---
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## Randomization

- Randomized studies also can only generate data with missing values of the counterfactual outcomes

- However, randomization ensures that these missing values **occurred by chance**.

As a result, effect measures can be computed - **consistently estimated** - in randomized experiments despite the missing data.

---
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## Exchangeability (1)

.full-width[.content-box-red[.bold[
An ideal randomized experiments ensures that the treatment groups are **exchangeable**.
]]]

.pull-halfleft[.med[
- Treatment in white 
![](fig/Randomization0.png)
- associational risk ratio: 0.3/0.6 = 0.5
- associational risk difference: 0.3 - 0.6 = -0.3

]]

.pull-halfright[.med[
- Treatment in grey
![](fig/Randomization1.png)
- associational risk ratio: 0.3/0.6 = 0.5
- associational risk difference: 0.3 - 0.6 = -0.3
]]

???

- If we randomly assign group labels. 
- We will have the same results of association measures both ways.
- Because we believe that the proportion of deaths among treated or untreated no matter who was treated are the same. 
- So in the study, who was actually treated or untreated does not change the results of our observations `$\text{Pr}[Y =1 | A = 0]$`
- We call this feature as "exchangeability".

---
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## Exchangeability (2)

.pull-halfleft[.med[
- Treatment in white 
![](fig/Randomization0.png)
- associational risk ratio: 0.3/0.6 = 0.5
- associational risk difference: 0.3 - 0.6 = -0.3
]]

.pull-halfright[.med[
- Treatment in grey
![](fig/Randomization1.png)
- associational risk ratio: 0.3/0.6 = 0.5
- associational risk difference: 0.3 - 0.6 = -0.3
]]

???

- This exchangeability means that the risk of death in the white would have been the same as the risk of death in the grey had individuals in the white received the treatment given to the grey.

---
class: midde
## Exchangeability (3)

.full-width[.content-box-red[.bold[
.small[
- The risk under potential treatment `$a$` **among the treated** <br> `$\text{Pr}[Y^a = 1 | A =1]$` equals
- The risk under potential treatment `$a$` **among the untreated** <br>  `$\text{Pr}[Y^a = 1 | A =0]$` equals 
- The risk under potential treatment `$a$` **in the whole population** <br> `$\text{Pr}[Y^a = 1]$`

`$$Y^a \perp \!\!\! \perp A \forall \text{ (= for all) } a$$`

- The actual treatment `$A$` **does not predict** the counterfactual outcome `$Y^a$`.

- Randomization is so highly valued since it is expected to produce this exchangeability.
]]]]

???

- Which means that the risk (here is the proportion of death) under potential treatment does not depend on whether individuals were actually treated or not.

---
class: midde
## Why exchangeability is so great?

.med[
In the presence of exchangeability. **Association is Causation.**

- The counterfactual risk in the treatment group is equal to the counterfactual risk under treatment in the entire population.

`$$\text{Pr}[Y^{a = 1} = 1 | A =1] = \text{Pr}[Y^{a = 1} = 1]$$`

- And the counterfactual risk in the treatment group is equal to their observed actual outcome (by consistency).

`$$\text{Pr}[Y^{a = 1} = 1 | A =1] = \text{Pr}[Y = 1 | A = 1]$$`

- **The risk in the treated is the same as the risk as if everybody had been treated.**
]

---
class: middle,

## Conditional randomization

.pull-left[.med[
- `$L$` is a prognostic factor  <br> (1 if critical condition,  <br>  0 otherwise)
- Conditional randomization design: 
    - Randomly select 50% from those who is in noncritical condition
    - Randomly select 75% from those who is in critical condition

]]

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    <th class="tg-0pky"></th>
    <th class="tg-7btt">L</th>
    <th class="tg-7btt">A</th>
    <th class="tg-7btt">Y</th>
    <th class="tg-0pky"></th>
    <th class="tg-7btt">L</th>
    <th class="tg-7btt">A</th>
    <th class="tg-7btt">Y</th>
  </tr>
</thead>
<tbody>
  <tr>
    <td class="tg-fymr">Rheia</td>
    <td class="tg-c3ow">0</td>
    <td class="tg-c3ow">0</td>
    <td class="tg-c3ow">0</td>
    <td class="tg-fymr">Leto</td>
    <td class="tg-c3ow">1</td>
    <td class="tg-c3ow">0</td>
    <td class="tg-c3ow">0</td>
  </tr>
  <tr>
    <td class="tg-fymr">Kronos</td>
    <td class="tg-c3ow">0</td>
    <td class="tg-c3ow">0</td>
    <td class="tg-c3ow">1</td>
    <td class="tg-fymr">Ares</td>
    <td class="tg-c3ow">1</td>
    <td class="tg-c3ow">1</td>
    <td class="tg-c3ow">1</td>
  </tr>
  <tr>
    <td class="tg-fymr">Demeter</td>
    <td class="tg-c3ow">0</td>
    <td class="tg-c3ow">0</td>
    <td class="tg-c3ow">0</td>
    <td class="tg-fymr">Athena</td>
    <td class="tg-c3ow">1</td>
    <td class="tg-c3ow">1</td>
    <td class="tg-c3ow">1</td>
  </tr>
  <tr>
    <td class="tg-fymr">Hades</td>
    <td class="tg-c3ow">0</td>
    <td class="tg-c3ow">0</td>
    <td class="tg-c3ow">0</td>
    <td class="tg-fymr">Hephaestus</td>
    <td class="tg-c3ow">1</td>
    <td class="tg-c3ow">1</td>
    <td class="tg-c3ow">1</td>
  </tr>
  <tr>
    <td class="tg-fymr">Hestia</td>
    <td class="tg-c3ow">0</td>
    <td class="tg-c3ow">1</td>
    <td class="tg-c3ow">0</td>
    <td class="tg-fymr">Aphrodite</td>
    <td class="tg-c3ow">1</td>
    <td class="tg-c3ow">1</td>
    <td class="tg-c3ow">1</td>
  </tr>
  <tr>
    <td class="tg-fymr">Poseidon</td>
    <td class="tg-c3ow">0</td>
    <td class="tg-c3ow">1</td>
    <td class="tg-c3ow">0</td>
    <td class="tg-fymr">Cyclope</td>
    <td class="tg-c3ow">1</td>
    <td class="tg-c3ow">1</td>
    <td class="tg-c3ow">1</td>
  </tr>
  <tr>
    <td class="tg-fymr">Hera</td>
    <td class="tg-c3ow">0</td>
    <td class="tg-c3ow">1</td>
    <td class="tg-c3ow">0</td>
    <td class="tg-fymr">Persephone</td>
    <td class="tg-c3ow">1</td>
    <td class="tg-c3ow">1</td>
    <td class="tg-c3ow">1</td>
  </tr>
  <tr>
    <td class="tg-fymr">Zeus</td>
    <td class="tg-c3ow">0</td>
    <td class="tg-c3ow">1</td>
    <td class="tg-c3ow">1</td>
    <td class="tg-fymr">Hermes</td>
    <td class="tg-c3ow">1</td>
    <td class="tg-c3ow">1</td>
    <td class="tg-c3ow">0</td>
  </tr>
  <tr>
    <td class="tg-fymr">Artemis</td>
    <td class="tg-c3ow">1</td>
    <td class="tg-c3ow">0</td>
    <td class="tg-c3ow">1</td>
    <td class="tg-fymr">Hebe</td>
    <td class="tg-c3ow">1</td>
    <td class="tg-c3ow">1</td>
    <td class="tg-c3ow">0</td>
  </tr>
  <tr>
    <td class="tg-fymr">Apollo</td>
    <td class="tg-c3ow">1</td>
    <td class="tg-c3ow">0</td>
    <td class="tg-c3ow">1</td>
    <td class="tg-fymr">Dionysus</td>
    <td class="tg-c3ow">1</td>
    <td class="tg-c3ow">1</td>
    <td class="tg-c3ow">0</td>
  </tr>
</tbody>
</table>
]

---
class: middle

## Conditional exchangeability (1)

- Under the randomization conditional on `$L$` design, we only have exchangeability property within each strata of the condition.
- Because it is two randomized experiment combined: 
    - one conducted in the subset of individuals in critical condition.
    - the other in the subset of individuals in noncritical condition.

---
class: middle

## Conditional exchangeability (2)

Among those who in critical condition `$(L = 1)$`,

`$$\text{Pr}[Y^a = 1 | A = 1, L = 1] = \text{Pr}[Y^a = 1 | A = 0, L = 1]$$`

Or,

`$$Y^a \perp \!\!\! \perp A |L = 1 \forall a$$`

Read as: <br> " `$Y^a$` (the potential outcomes) and `$A$` (the actual treatment groups) are independent given `$L = 1$`."

---
class: middle

## Conditional causal risk ratio

- Because association is causation within each subset
- We can compute stratum-specific causal effect 
    - `$\frac{\text{Pr}[Y = 1 | L = 1, A = 1]}{\text{Pr}[Y = 1 | L = 1, A = 0]}$` in critical condition.
    - `$\frac{\text{Pr}[Y = 1 | L = 0, A = 1]}{\text{Pr}[Y = 1 | L = 0, A = 0]}$` in noncritical condition.
    - If stratum-specific RRs are different between subsets, we say that the effect of treatment is **modified** by `$L$`, or that there is *effect modification* by `$L$`

---
class: middle

## Standardization (1)

.pull-left[.between[
Among those in noncritical condition `$(L = 0)$`:
- The risk of death among the treated: 
<br> `$\text{Pr}[Y = 1 | L = 0, A = 1] = 1/4$`
- The risk of death among the untreated:
<br> `$\text{Pr}[Y = 1 | L = 0, A = 0] = 1/4$`

Among those in critical condition `$(L = 1)$`:
- The risk of death among the treated: 
<br> `$\text{Pr}[Y = 1 | L = 1, A = 1] = 6/9 = \frac{2}{3}$`
- The risk of death among the untreated:
<br> `$\text{Pr}[Y = 1 | L = 1, A = 0] = 2/3$`

]]

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  <tr>
    <th class="tg-de2y"></th>
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    <th class="tg-9ydz">A</th>
    <th class="tg-9ydz">Y</th>
    <th class="tg-de2y"></th>
    <th class="tg-9ydz">L</th>
    <th class="tg-9ydz">A</th>
    <th class="tg-9ydz">Y</th>
  </tr>
</thead>
<tbody>
  <tr>
    <td class="tg-sn4r">Rheia</td>
    <td class="tg-2jdy">0</td>
    <td class="tg-2jdy">0</td>
    <td class="tg-2jdy">0</td>
    <td class="tg-sn4r">Leto</td>
    <td class="tg-02vm">1</td>
    <td class="tg-02vm">0</td>
    <td class="tg-02vm">0</td>
  </tr>
  <tr>
    <td class="tg-sn4r">Kronos</td>
    <td class="tg-2jdy">0</td>
    <td class="tg-2jdy">0</td>
    <td class="tg-2jdy">1</td>
    <td class="tg-sn4r">Ares</td>
    <td class="tg-amug">1</td>
    <td class="tg-amug">1</td>
    <td class="tg-amug">1</td>
  </tr>
  <tr>
    <td class="tg-sn4r">Demeter</td>
    <td class="tg-2jdy">0</td>
    <td class="tg-2jdy">0</td>
    <td class="tg-2jdy">0</td>
    <td class="tg-sn4r">Athena</td>
    <td class="tg-amug">1</td>
    <td class="tg-amug">1</td>
    <td class="tg-amug">1</td>
  </tr>
  <tr>
    <td class="tg-sn4r">Hades</td>
    <td class="tg-2jdy">0</td>
    <td class="tg-2jdy">0</td>
    <td class="tg-2jdy">0</td>
    <td class="tg-sn4r">Hephaestus</td>
    <td class="tg-amug">1</td>
    <td class="tg-amug">1</td>
    <td class="tg-amug">1</td>
  </tr>
  <tr>
    <td class="tg-sn4r">Hestia</td>
    <td class="tg-2ral">0</td>
    <td class="tg-2ral">1</td>
    <td class="tg-2ral">0</td>
    <td class="tg-sn4r">Aphrodite</td>
    <td class="tg-amug">1</td>
    <td class="tg-amug">1</td>
    <td class="tg-amug">1</td>
  </tr>
  <tr>
    <td class="tg-sn4r">Poseidon</td>
    <td class="tg-2ral">0</td>
    <td class="tg-2ral">1</td>
    <td class="tg-2ral">0</td>
    <td class="tg-sn4r">Cyclope</td>
    <td class="tg-amug">1</td>
    <td class="tg-amug">1</td>
    <td class="tg-amug">1</td>
  </tr>
  <tr>
    <td class="tg-sn4r">Hera</td>
    <td class="tg-2ral">0</td>
    <td class="tg-2ral">1</td>
    <td class="tg-2ral">0</td>
    <td class="tg-sn4r">Persephone</td>
    <td class="tg-amug">1</td>
    <td class="tg-amug">1</td>
    <td class="tg-amug">1</td>
  </tr>
  <tr>
    <td class="tg-sn4r">Zeus</td>
    <td class="tg-2ral">0</td>
    <td class="tg-2ral">1</td>
    <td class="tg-2ral">1</td>
    <td class="tg-sn4r">Hermes</td>
    <td class="tg-amug">1</td>
    <td class="tg-amug">1</td>
    <td class="tg-amug">0</td>
  </tr>
  <tr>
    <td class="tg-sn4r">Artemis</td>
    <td class="tg-02vm">1</td>
    <td class="tg-02vm">0</td>
    <td class="tg-02vm">1</td>
    <td class="tg-sn4r">Hebe</td>
    <td class="tg-amug">1</td>
    <td class="tg-amug">1</td>
    <td class="tg-amug">0</td>
  </tr>
  <tr>
    <td class="tg-sn4r">Apollo</td>
    <td class="tg-02vm">1</td>
    <td class="tg-02vm">0</td>
    <td class="tg-02vm">1</td>
    <td class="tg-sn4r">Dionysus</td>
    <td class="tg-amug">1</td>
    <td class="tg-amug">1</td>
    <td class="tg-amug">0</td>
  </tr>
</tbody>
</table>
]

???

- As one example, you may be interested in the average causal effect in the entire population, rather than in the stratum-specific average causal effects. 
- IF you do not expect to have information about L for making a decision in the future (for example, the variable L is expensive to measure on everybody), we can use two ways to have the same results. The first way is called standardization.

---
class: middle

## Standardization (2) - the combined causal risk ratio

$$
\frac{\text{Pr}[Y^{a = 1} = 1]}{\text{Pr}[Y^{a = 0} = 1]}
$$

.med[
- The numerator is the risk if all 20 individuals had been treated
    - `$\frac{1}{4}$` in 8 non-critical; and `$\frac{2}{3}$` in 12 critical:
      - `$\frac{1}{4} \times \frac{8}{20} + \frac{2}{3} \times \frac{12}{20} =0.5$`
- The denominatior is the risk if all 20 individuals had been untreated
    - `$\frac{1}{4}$` in 8 non-critical; and `$\frac{2}{3}$` in 12 critical:
      - `$\frac{1}{4} \times \frac{8}{20} + \frac{2}{3} \times \frac{12}{20} =0.5$`
]

---
class: middle

## Standardization (3) - the combined causal risk ratio

$$
\frac{\text{Pr}[Y^{a = 1} = 1]}{\text{Pr}[Y^{a = 0} = 1]} = 0.5/0.5 = 1
$$

.full-width[.content-box-red[.bold[
.small[
More formally, <br>
`$\text{Pr}[Y^{a} = 1]$` is **the weighted average of the stratum-specific risks**, the weights is the proportion of individuals in each subset: 
<br> `$$\text{Pr}[Y^{a} = 1] = \text{Pr}[Y^a = 1|L = 0]\text{Pr}[L = 0] \\+ \text{Pr}[Y^a = 1|L = 1]\text{Pr}[L = 1]$$`
This is known in epidemiology as **standardization**.

]]]]

---
class: middle, center, inverse

## Inverse probability weighting

---
class: middle

##  Inverse probability weighting (1)

So far,

- the causal risk ratio was computed in a conditionally randomized experiment via standardization.

From this slide,

- we compute this same causal risk ratio via inverse probability weighting.

---
class: middle

##  Data as a probability tree:

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  <tr>
    <th class="tg-de2y"></th>
    <th class="tg-9ydz">L</th>
    <th class="tg-9ydz">A</th>
    <th class="tg-9ydz">Y</th>
    <th class="tg-de2y"></th>
    <th class="tg-9ydz">L</th>
    <th class="tg-9ydz">A</th>
    <th class="tg-9ydz">Y</th>
  </tr>
</thead>
<tbody>
  <tr>
    <td class="tg-sn4r">Rheia</td>
    <td class="tg-2jdy">0</td>
    <td class="tg-2jdy">0</td>
    <td class="tg-2jdy">0</td>
    <td class="tg-sn4r">Leto</td>
    <td class="tg-02vm">1</td>
    <td class="tg-02vm">0</td>
    <td class="tg-02vm">0</td>
  </tr>
  <tr>
    <td class="tg-sn4r">Kronos</td>
    <td class="tg-2jdy">0</td>
    <td class="tg-2jdy">0</td>
    <td class="tg-2jdy">1</td>
    <td class="tg-sn4r">Ares</td>
    <td class="tg-amug">1</td>
    <td class="tg-amug">1</td>
    <td class="tg-amug">1</td>
  </tr>
  <tr>
    <td class="tg-sn4r">Demeter</td>
    <td class="tg-2jdy">0</td>
    <td class="tg-2jdy">0</td>
    <td class="tg-2jdy">0</td>
    <td class="tg-sn4r">Athena</td>
    <td class="tg-amug">1</td>
    <td class="tg-amug">1</td>
    <td class="tg-amug">1</td>
  </tr>
  <tr>
    <td class="tg-sn4r">Hades</td>
    <td class="tg-2jdy">0</td>
    <td class="tg-2jdy">0</td>
    <td class="tg-2jdy">0</td>
    <td class="tg-sn4r">Hephaestus</td>
    <td class="tg-amug">1</td>
    <td class="tg-amug">1</td>
    <td class="tg-amug">1</td>
  </tr>
  <tr>
    <td class="tg-sn4r">Hestia</td>
    <td class="tg-2ral">0</td>
    <td class="tg-2ral">1</td>
    <td class="tg-2ral">0</td>
    <td class="tg-sn4r">Aphrodite</td>
    <td class="tg-amug">1</td>
    <td class="tg-amug">1</td>
    <td class="tg-amug">1</td>
  </tr>
  <tr>
    <td class="tg-sn4r">Poseidon</td>
    <td class="tg-2ral">0</td>
    <td class="tg-2ral">1</td>
    <td class="tg-2ral">0</td>
    <td class="tg-sn4r">Cyclope</td>
    <td class="tg-amug">1</td>
    <td class="tg-amug">1</td>
    <td class="tg-amug">1</td>
  </tr>
  <tr>
    <td class="tg-sn4r">Hera</td>
    <td class="tg-2ral">0</td>
    <td class="tg-2ral">1</td>
    <td class="tg-2ral">0</td>
    <td class="tg-sn4r">Persephone</td>
    <td class="tg-amug">1</td>
    <td class="tg-amug">1</td>
    <td class="tg-amug">1</td>
  </tr>
  <tr>
    <td class="tg-sn4r">Zeus</td>
    <td class="tg-2ral">0</td>
    <td class="tg-2ral">1</td>
    <td class="tg-2ral">1</td>
    <td class="tg-sn4r">Hermes</td>
    <td class="tg-amug">1</td>
    <td class="tg-amug">1</td>
    <td class="tg-amug">0</td>
  </tr>
  <tr>
    <td class="tg-sn4r">Artemis</td>
    <td class="tg-02vm">1</td>
    <td class="tg-02vm">0</td>
    <td class="tg-02vm">1</td>
    <td class="tg-sn4r">Hebe</td>
    <td class="tg-amug">1</td>
    <td class="tg-amug">1</td>
    <td class="tg-amug">0</td>
  </tr>
  <tr>
    <td class="tg-sn4r">Apollo</td>
    <td class="tg-02vm">1</td>
    <td class="tg-02vm">0</td>
    <td class="tg-02vm">1</td>
    <td class="tg-sn4r">Dionysus</td>
    <td class="tg-amug">1</td>
    <td class="tg-amug">1</td>
    <td class="tg-amug">0</td>
  </tr>
</tbody>
</table>
]

---
class: middle

### Use the prob tree to compute the causal risk ratio (1)

.pull-left[.med[
- The denominator of the causal risk ratio, `$\text{Pr}[Y^{a = 0} = 1]$` 
<br> is the counterfactual risk 
<br> of death **had everybody remained untreated**.
- `$\text{Pr}[Y^{a = 0} = 1] \\= (2 + 8)/20 = 0.5$`
]]

---
class: middle

### Use the prob tree to compute the causal risk ratio (2)

.pull-left[.med[
- The numerator of the causal risk ratio, `$\text{Pr}[Y^{a = 1} = 1]$` 
<br> is the counterfactual risk 
<br> of death **had everybody remained treated**.
- `$\text{Pr}[Y^{a = 1} = 1] \\= (2 + 8)/20 = 0.5$`
- The causal risk ratio is
<br> `$0.5 / 0.5 = 1$`
]]

---
class: top, left
background-image: url("./fig/Tree weights.png")
background-position: 60% 50%
background-size: contain

## How did this work?

???

- The two trees in the previous figures can be combined together as shown here. 
- This pooled population is twice as large as the original study sample, is called pseudo-population.

- If we look carefully that in the first arm of treatment, four untreated individuals with non-critical condition are now used to create a 8 members of the psudo-population. That means each of them receives a weight of 2, which is equal to the inverse of 0.5.

- Similarly, the 9 treated individuals with critical condition are used to create a 12 member of the pseudo-population. Each of them receives a weight of 1.33 = 1/0.75

---
class: middle

##  Inverse probability weighting (3)

.full-width[.content-box-red[.bold[.med[
This is called **inverse probability weighting (IPW)**.

- The pseudo-population is created by weighting each individual in the population by the **inverse of the conditional probability of receiving treatments**

- IPW yielded the same result as standardization. 
<br>In fact, they are mathematically equivalent.

]]]]

---
class: middle

# Summary

- Both standardization and IP weighting simulate what would have been observed if the variable `$L$` had not been used to decide the probability of treatment, we often say that these methods **adjust/control for `$L$`**.
- However, we cannot always do randomized control study. Frequently, we turn to observational study for answers.

---
class: middle, inverse, center

# Chapter 3 <br> Observational Studies