Graphical representation of causal effects

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# Graphical representation of causal effects
## What If: Chapter 6
### Elena Dudukina
### 2021-01-11

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# Draw your assumptions before your conclusions
- Causal diagrams
- Directed acyclic graphs (DAGs)

![:scale 30%](Screenshot 2021-01-06 at 13.07.36.png)
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# DAGs

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![:scale 60%](Screenshot 2021-01-06 at 13.13.05.png)

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![:scale 60%](Screenshot 2021-01-06 at 13.14.17.png)

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.footnote[Arah, O. A. 2008. “The Role of Causal Reasoning in Understanding Simpson’s Paradox, Lord’s Paradox, and the Suppression Effect: Covariate Selection in the Analysis of Observational Studies.” Emerg Themes Epidemiol 5 (February): 5. https://doi.org/10.1186/1742-7622-5-5]
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# DAGs
![:scale 30%](Screenshot 2021-01-06 at 13.14.29.png)

.footnote[Arah, O. A. 2008. “The Role of Causal Reasoning in Understanding Simpson’s Paradox, Lord’s Paradox, and the Suppression Effect: Covariate Selection in the Analysis of Observational Studies.” Emerg Themes Epidemiol 5 (February): 5. https://doi.org/10.1186/1742-7622-5-5]
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# When we observe a statistical association between two variables
1) Two variables are a cause and the consequence

![:scale 30%](Screenshot 2021-01-06 at 13.36.06.png)

2) Two variables have a common cause (a confounder)
- A and Y are associated even though A does not cause Y
- Example: L - smoking, A - carrying a lighter, Y - lung cancer
    
![:scale 30%](Screenshot 2021-01-06 at 13.34.51.png)
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# When we observe a statistical association between two variables
3) Two variables share a common consequence or a child of a common consequence, which was conditioned on
- E and D are associated even though E does not cause D (or vice versa)

![:scale 50%](Screenshot 2021-01-06 at 13.41.23.png)

![:scale 50%](Screenshot 2021-01-06 at 13.44.15.png)

.footnote[Hernán, Miguel A., Sonia Hernández-Díaz, and James M. Robins. 2004. “A Structural Approach to Selection Bias.” Epidemiology 15 (5): 615–25]
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# Associatioin vs Causation
- "Association, unlike causation, is a symmetric relationship between two variables; thus, when present, association flows between two variables regardless of the direction of the causal arrows"

- Association is **not** causation
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# Marginal vs conditional probabilities
- Conditional
    - The probability that an event Y occurs, given  we know some other event A has occurred, is the conditional probability of Y given A
    - `\(Pr(Y|A)\)`
- Marginal(unconditional)
    - `\(Pr(Y)\)`
    - Remembering standardization/adjustment formula
        - `\(Pr(Y^a)=\Sigma_lPr(Y=1|A=a, L=l)*Pr(L=l)\)`
        - Probability of PO `\(Y^a\)` marginal over confounder `\(L\)`
        
.footnote[Causal Inference in Statistics: A Primer. Judea Pearl, Madelyn Glymour, Nicholas P. Jewell]
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![:scale 30%](Screenshot 2021-01-11 at 19.59.20.png)

- `\(A \perp\perp Y|B\)`
- `\(A \not\perp\not\perp Y\)`
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# Conditioning
- `\(A \perp\perp Y|L\)`

![:scale 20%](Screenshot 2021-01-11 at 20.05.44.png)

- `\(A \not\perp\not\perp Y|L\)`

![:scale 20%](Screenshot 2021-01-11 at 20.08.05.png)

![:scale 20%](Screenshot 2021-01-11 at 20.16.05.png)

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# Biasing path
**For the association between A and Y**
- A --> [L] <--Y

![:scale 30%](Screenshot 2021-01-11 at 20.14.42.png)

![:scale 30%](Screenshot 2021-01-11 at 20.16.49.png)
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# Backdoor path
- A <-- L --> Y

![:scale 30%](Screenshot 2021-01-11 at 20.18.04.png)
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# Exchangeability, positivity, consistency

## inference is never assumption-free

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# Systematic bias
- Infinite sample size won't help eliminating systematic bias
- The magnitude of the effect is off
- Structural definition of bias 
    * Common causes --> **confounding**
    
  ![:scale 20%](Screenshot 2021-01-06 at 13.34.51.png)

* Conditioning on common effects --> **selection bias (collider stratification bias)**
    
  ![:scale 15%](Screenshot 2021-01-11 at 20.08.05.png)

![:scale 20%](Screenshot 2021-01-11 at 20.16.05.png)
  
  * Information (measurement) error --> later in the book
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# Effect modification structure
- Causal vs surrogate efect modifiers

.pull-left[![:scale 60%](Screenshot 2021-01-11 at 20.38.11.png) ![:scale 60%](Screenshot 2021-01-11 at 20.44.48.png)]

.pull-right[![:scale 40%](Screenshot 2021-01-11 at 20.46.29.png) ![:scale 40%](Screenshot 2021-01-11 at 20.47.02.png)

![:scale 40%](Screenshot 2021-01-11 at 20.48.25.png)]

- Structure of the association of the effect modifier with the outcome matters
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## Interaction in DAGs
- Augmented DAGs

<blockquote class="twitter-tweet"><p lang="en" dir="ltr">Right. Personally, I’ve found it useful to augment the DAG for heterogeneity as in these from my SER 2015 conference presentation: <a href="https://t.co/8QJ26BTx6Y">pic.twitter.com/8QJ26BTx6Y</a></p>&mdash; Onyi Arah🎄waiting for vaccine🌍 MD, DSc, PhD (@oacarah) <a href="https://twitter.com/oacarah/status/1169561608642863104?ref_src=twsrc%5Etfw">September 5, 2019</a></blockquote> <script async src="https://platform.twitter.com/widgets.js" charset="utf-8"></script>