Confounding

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# Confounding
## What If: Chapter 7
### Elena Dudukina
### 2021-02-04

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# Confounding
## Lack of exchangeability
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🧀
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**↙️**   **↘️**
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# 7.1 The structure of confounding
## Bias due to common causes
- A: treatment
- Y: outcome
- L: confounder (a common cause of A and Y)

## Paths between A and Y
- A ➡️ Y: causal effect (directed path)
- A ⬅️ L ➡️ Y: biasing backdoor path (starts with a head pointing into A)

*  The associational risk ratio `\(\frac{Pr[Y=1|A=1]}{Pr[Y=1|A=0]}\)` is not causal risk ratio `\(\frac{Pr[Y^a=1]}{Pr[Y^a=0]}\)`

![:scale 20%](Screenshot 2021-01-31 at 17.46.49.png)
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## 7.1 Examples
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Occupational factors (healthy worker bias)
- A: working as a firefighter
- Y: mortality
- L: being fit to work as a firefighter

Confounding by indication (channeling bias)
- A: drug (aspirin)
- Y: mortality
- L: indication for the drug (heart disease)
- U: (unmeasured) atherosclerosis
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![:scale 40%](Screenshot 2021-01-31 at 17.46.49.png)

![:scale 40%](Screenshot 2021-01-31 at 18.12.59.png)
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Lifestyle
- A: behavior (exercise)
- Y: mortality
- L: lifestyle (smoking)
- U: (unmeasured) personality/socioeconomic determinants

Reverse causation
- A: behavior
- Y: clinical disease
- L: lifestyle (smoking)
- U: (unmeasured) subclinical disease
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![:scale 40%](Screenshot 2021-01-31 at 18.14.29.png)

![:scale 40%](Screenshot 2021-01-31 at 18.14.29.png)
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Genetic factors (linkage disequilibrium)
- A: DNA sequence
- Y: developing a trait
- L: DNA sequence confounding effect of A if L and A are often being inherited together and L causes Y
- U: ethnicity

Social factors
- A: income at age 65
- Y: disability at age 75
- L: disability at age 55 
- Time-varying flavor

Environmental exposures
- A: airborne particle
- Y: coronary heart disease
- L: other pollutants, which levels co-vary together with A
- U: (unmeasured) weather (controls the level of all types of airborne pollutants)
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## Assumptions for the remainder of the chapter
- No selection bias (Chapter 8)

- No measurement bias (Chapter 9)

- No random variabiity (Chapter 10)
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# 7.2 Confounding and exchangeability
## Assuming consistency and positivity hold
- Exchangeability holds `\(Y^a \perp\perp A\)`: potential outcome under the treatment regime is independent from actually observed treatment for all treatment levels (perfect randomization)

- When `\(Y^a \perp\perp A\)` is true and treatment is binary
    - the average causal effect `\(E[Y^{a=1}] - E[Y^{a=0}]\)` equals the observed associational effect `\(E[Y|A=1] - E[Y|A=0]\)`
    
--

- When *conditional* exchangeability holds `\(Y^a \perp\perp A|L\)`, the verage caussal effect is identifiable
    -  `\(E[Y^{a=1}] - E[Y^{a=0}] = \Sigma_{l}E[Y|L=l, A=1]*Pr[L=l] - \Sigma_{l}E[Y|L=l, A=0]*Pr[L=l]\)`
    - G-methods (IPTW, standardization) for marginal estimates
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# Is there a set of variables L that guarantees conditional exchangeability holds?
## No guarantees (unless well-conducted RCT)
* Use of DAGs to show the data-generating mechanism and assumptions
* No residual confounding (conditional exchangeability) assumption
* Backdoor criterion (J. Pearl, 1995)
    - No node in adjustment set is a descendant (child) of exposure (not a mediator)
    - The adjustment set blocks all paths pointing into exposure (blocks all backdoor paths)
    - Consider magnitude and te direction of bias by unmeasured confounding
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# 7.3 Confounding and the backdoor criterion
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Adjustment sets sufficient to eliminate **confounding**:
- 7.2:  {L}, {U}

- 7.3: {L}, {U}
- 7.4: {} (empty set)
    - adjusting for L opens a biasing path
    - A ⬅️ U2 ➡️ [L] ⬅️ U1 ➡️ Y

- think **confounding**, not **confounders** 
- role of the variable changes when other variables are adjusted
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![:scale 40%](Screenshot 2021-01-31 at 18.12.59.png)

![:scale 40%](Screenshot 2021-01-31 at 18.14.29.png)

![:scale 40%](Screenshot 2021-01-31 at 21.02.16.png)
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Fig 7.5
-  A ⬅️ L ⬅️ U1 ➡️ Y (open biasing path, L is not a collider)
-  A ⬅️ U2 ➡️ L ⬅️ U1 ➡️ Y (biasing path is closed with the collider L)
- {U1}, {L, U2}, {L, U1}

Fig 7.6
- {L1}, {U1}, {L, L2}, {L, U2}, {L, L1}, {L, U1}
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![:scale 40%](Screenshot 2021-01-31 at 21.16.10.png)

![:scale 40%](Screenshot 2021-01-31 at 21.21.04.png)

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# 7.4 Confounding and confounders
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Traditional criteria
- Association with the exposure
- Association with the outcome
- Not on a causal pathway

Change of the estimate after adjustment may occur for the reasons other than adjusting confounding (selection bias, non-collapsibility of effect measures)
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![:scale 30%](Screenshot 2021-01-31 at 21.02.16.png)

![:scale 30%](Screenshot 2021-01-31 at 21.36.18.png)

Is L a confounder?
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# 7.6 Confounding adjustment
- Sufficient set for confounding adjustment
- Methods
    - G-methods (marginal and conditional effects)
        - Standardization
        - IP weighting (deleting an arrow L ➡️ A)
        - g-estimation
    - Stratification-based methods (conditional effects only)
        - Stratification
        - Restriction
        - Matching
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# References
Hernán MA, Robins JM (2020). Causal Inference: What If. Boca Raton: Chapman & Hall/CRC (v. 31jan21)