Causal Inference

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

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# Chapter 1, 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)

.pull-left[
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<thead>
  <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>
  </tr>
  <tr>
    <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>
  </tr>
  <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-0lax"></th>
    <th class="tg-1wig">A</th>
    <th class="tg-1wig">Y</th>
  </tr>
</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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    <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>
  <tr>
    <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>
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    <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")
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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, Definition