Introduction to causal inference

# Introduction to causal inference

**Session 11**

.light[MATH 80667A: Experimental Design and Statistical Methods for Quantitative Research in Management 
HEC Montréal
]

]

---
name: outline
class: title title-inv-1

# Outline
--

.box-2.large.sp-after-half[Directed acyclic graphs]

--
.box-3.large.sp-after-half[Causal mediation]

---

# Directed acyclic graphs

## .color-light-1[Slides by Dr. Andrew Heiss, CC BY-NC 4.0 License.]

---

---

# Types of data

.box-inv-2.medium.sp-after-half[Experimental]

.box-2.sp-after-half[You have control over which units get treatment]

]

.box-inv-2.medium.sp-after-half[Observational]

.box-2.sp-after-half[You don't have control over which units get treatment]

]

---

# Causal diagrams

.box-inv-2.medium.sp-after-half[Directed acyclic graphs (DAGs)]

.box-2.SMALL[**Directed**: Each node has an arrow that points to another node]

.box-2.SMALL[**Acyclic**: You can't cycle back to a node (and arrows only have one direction)]

]

]

---

# Causal diagrams

.box-inv-2.medium.sp-after-half[Directed acyclic graphs (DAGs)]

.box-2.SMALL[Graphical model of the process that generates the data]

.box-2.SMALL[Maps your philosophical model]

.box-2.SMALL[Fancy math ("*do*-calculus") tells you what to control for to isolate and identify causation]

]

![](11-slides_files/figure-html/simple-dag-1.png)
]
---

# How to draw a DAG
.box-inv-2.medium.sp-after-half[What is the causal effect of an additional year of education on earnings?]
--
.box-2.sp-after-half[Step 1: List variables]
--
.box-2.sp-after-half[Step 2: Simplify]
--
.box-2.sp-after-half[Step 3: Connect arrows]
--
.box-2.sp-after-half[Step 4: Use logic and math to determine which nodes and arrows to measure]
---

# 1. List variables
.box-2.sp-after[Education (treatment) → Earnings (outcome)]
.center.float-left.sp-after-half[
.box-inv-2.sp-after-half[Location]&ensp;.box-inv-2.sp-after-half[Ability]&ensp;.box-inv-2.sp-after-half[Demographics]
]
.center.float-left.sp-after-half[
.box-inv-2.sp-after-half[Socioeconomic status]&ensp;.box-inv-2.sp-after-half[Year of birth]
]
.center.float-left[
.box-inv-2.sp-after-half[Compulsory schooling laws]&ensp;.box-inv-2.sp-after-half[Job connections]
]
---

# 2. Simplify
.box-2.sp-after[Education (treatment) → Earnings (outcome)]
.center.float-left.sp-after-half[
.box-inv-2.sp-after-half[Location]&ensp;.box-inv-3[Ability]&ensp;.box-inv-3[Demographics]
]
.center.float-left.sp-after-half[
.box-inv-3[Socioeonomic status]&ensp;.box-inv-2.sp-after-half[Year of birth]
]
.center.float-left[
.box-inv-2.sp-after-half[Compulsory schooling laws]&ensp;.box-inv-2.sp-after-half[Job connections]
]
.box-inv-4[Background]
---

# 3. Draw arrows
.pull-left-narrow[
.box-inv-2.sp-after-half[Education causes earnings]
]
.pull-right-wide[
<img src="11-slides_files/figure-html/edu-earn-simple-1.png" width="100%" style="display: block; margin: auto;" />

]

---

# 3. Draw arrows

.box-inv-2.sp-after-half[Background, year of birth, location, job connections, and school requirements all cause education]

]

]

---

# 3. Draw arrows

.box-inv-2.sp-after-half[Background, year of birth, and location all cause earnings too]

]

]

---

# 3. Draw arrows

.box-inv-2.sp-after-half[Education causes job earnings]

]

]

---

# 3. Draw arrows

.box-inv-2.sp-after-half[Location and background are probably related, but neither causes the other. Something unobservable (U1) does that.]

]

]

---

# Causal identification
.box-inv-2.medium.sp-after-half[A causal effect is *identified* if the association between treatment and outcome is propertly stripped and isolated]
---

# Paths and associations
.box-inv-2.medium.sp-after-half[Arrows in a DAG transmit associations]
.box-inv-2.medium.sp-after-half[You can redirect and control those paths by "adjusting" or "conditioning"]
---

# Three types of associations
.pull-left-3[
.box-2.medium[Confounding]
<img src="11-slides_files/figure-html/confounding-dag-1.png" width="100%" style="display: block; margin: auto;" />

.box-inv-2.small[Common cause]
]

.pull-middle-3.center[
.box-2.medium[Causation]

.box-inv-2.small[Mediation]
]

.box-inv-2.small[Selection / endogeneity]
]

---

# Confounding

.pull-left-wide[
<img src="11-slides_files/figure-html/confounding-dag-big-1.png" width="100%" style="display: block; margin: auto;" />
]

.box-inv-2.medium.sp-after-half[But **Z** causes both **X** and **Y**]

.box-inv-2.medium.sp-after-half[**Z** * confounds* the **X** → **Y** association]
]

---

# Paths

.pull-left-wide[
![](11-slides_files/figure-html/confounding-dag-big-1.png)
]
.pull-right-narrow[
.box-inv-2.sp-after-half[Paths between **X** and **Y**?]
.box-2.sp-after-half[**X** → **Y**]
.box-2.sp-after-half[**X** ← **Z** → **Y**]
.box-inv-2.medium.sp-after-half[**Z** is a *backdoor*]
]
---

# *d*-connection
.pull-left-wide[
![](11-slides_files/figure-html/confounding-dag-big-1.png)
]
.pull-right-narrow[
.box-inv-2.sp-after-half[**X** and **Y** are "*d*-connected" because associations can pass through **Z**]
.box-2.sp-after-half[The relationship between **X** and **Y** is not identified / isolated]
]
---

# Effect of money on elections
.box-inv-2.medium.sp-after-half[What are the paths between **money** and **win margin**?]
.pull-left[
<img src="11-slides_files/figure-html/money-elections-1.png" width="100%" style="display: block; margin: auto;" />
]

.box-2.sp-after-half[Money ← Quality → Margin]

.box-inv-2.sp-after-half[Quality is a *backdoor*]
]

---

# Closing doors

.pull-left[
<img src="11-slides_files/figure-html/confounding-dag-adjusted-1.png" width="100%" style="display: block; margin: auto;" />
]

---

# Closing doors

.pull-left[
.box-inv-2.small.sp-after-half[Find the part of campaign money that is explained by quality, remove it. This is the residual part of money.]

.box-inv-2.small[Find the part of win margin that is explained by quality, remove it. This is the residual part of win margin.]

.box-inv-2.small[Find the relationship between the residual part of money and residual part of win margin. **This is the causal effect**.]
]

.pull-right[
<img src="11-slides_files/figure-html/money-elections-adjusted-1.png" width="100%" style="display: block; margin: auto;" />
]

---

# Closing doors

# How to adjust
.box-inv-2.medium[Include term in regression]
$$
`\begin{aligned}
\text{Win margin} =& \beta_0 + \beta_1 \text{Campaign money} +\\
& \beta_2 \text{Candidate quality} + \varepsilon
\end{aligned}`
$$
.center.float-left[
.box-inv-2.medium[Matching] .box-inv-2.medium[Stratifying]
.box-inv-2.medium.sp-before-half[Inverse probability weighting]
]
---

# *d*-separation
.pull-left[
![](11-slides_files/figure-html/confounding-dag-adjusted-1.png)
]
.pull-right[
.box-inv-2.medium[If we control for **Z**, **X** and **Y** are now "*d*-separated" and the association is isolated!]
]
---

# Closing backdoors
.pull-left-narrow[
.box-inv-2[Block all backdoor paths to identify the main pathway you care about]
]
.pull-right-wide[
![](11-slides_files/figure-html/edu-earn-full-1.png)
]
---

# All paths
.pull-left.left[
.box-2.smaller.sp-after-half[Education → Earnings]
.box-2.smaller.sp-after-half[Education → Job connections → Earnings]
.box-3.smaller.sp-after-half[Education ← Background → Earnings]
.box-3.smaller.sp-after-half[Education ← Background ← U1 → Location → Earnings]
.box-3.smaller.sp-after-half[Education ← Location → Earnings]
.box-3.smaller.sp-after-half[Education ← Location ← U1 → Background → Earnings]
.box-3.smaller.sp-after-half[Education ← Year → Earnings]
]
.pull-right[
![](11-slides_files/figure-html/edu-earn-full-1.png)
]
---

# All paths
.pull-left-narrow[
.box-inv-2[Adjust for **Location**, **Background** and **Year** to isolate the **Education → Earnings** causal effect]
]
.pull-right-wide[
<img src="11-slides_files/figure-html/edu-earn-adjust-1.png" width="100%" style="display: block; margin: auto;" />
]

---

# How do you know if this is right?

.pull-left-narrow[
.box-inv-2[You can test the implications of the model to see if they're right in your data]

$$
X \perp Y\ |\ Z
$$

.box-2.small[X is independent of Y, given Z]
]

.pull-right-wide[
<figure>
 <img src="img/12/dagitty-implications.png" alt="Dagitty implications" title="Dagitty implications" width="100%">
</figure>
]

---

# Causation

.pull-left-wide[
<img src="11-slides_files/figure-html/causation-dag-big-1.png" width="100%" style="display: block; margin: auto;" />
]

.box-inv-2.medium.sp-after-half[**X** causes **Z** which causes **Y**]

.box-2.medium.sp-after-half[**Z** is a mediator]
]

---

# Colliders
.pull-left-wide[
<img src="11-slides_files/figure-html/collider-dag-big-1.png" width="100%" style="display: block; margin: auto;" />
]

.box-inv-2.medium.sp-after-half[**Y** causes **Z**]

.box-2.medium.sp-after-half[Should you control for **Z**?]
]

---

# Programming and social skills

.box-inv-2.medium[Do programming skills reduce social skills?]

.pull-left[
<img src="11-slides_files/figure-html/programming-social-skills-1.png" width="100%" style="display: block; margin: auto;" />
]

.pull-right[
.box-2[You go to a tech company and conduct a survey. You find a negative relationship! Is it real?]
]

---

# Programming and social skills

.box-inv-2.medium[Do programming skills reduce social skills?]

.pull-left[
![](11-slides_files/figure-html/programming-social-skills-1.png)
]
.pull-right[
.box-2.sp-after-half[No! **Hired by a tech company** is a collider and we controlled for it.]
.box-2.sp-after-half[This inadvertently connected the two.]
]
---
layout: false
.pull-left[
.box-2.medium[Colliders can create fake causal effects]
]
.pull-right[
.box-2.medium[Colliders can hide real causal effects]
]
<img src="11-slides_files/figure-html/bulls-scores-1.png" width="50%" style="display: block; margin: auto;" />

---

---

# Life is inherently complex

Source: Andrew Heiss (?), likely from

McQuire, C., Daniel, R., Hurt, L. et al. The causal web of foetal alcohol spectrum disorders: a review and causal diagram. Eur Child Adolesc Psychiatry 29, 575–594 (2020). https://doi.org/10.1007/s00787-018-1264-3

---
class: title title-2
# Three types of associations

![](11-slides_files/figure-html/confounding-dag-1.png)
.box-inv-2.small.sp-after-half[Common cause]
.box-inv-2.small[Causal forks **X** ← **Z** → **Y**]
]
.pull-middle-3[
.box-2.medium[Causation]
![](11-slides_files/figure-html/mediation-dag-1.png)
.box-inv-2.small.sp-after-half[Mediation]
.box-inv-2.small[Causal chain **X** → **Z** → **Y**]
]
.pull-right-3[
.box-2.medium[Collision]
![](11-slides_files/figure-html/collision-dag-1.png)
.box-inv-2.small.sp-after-half[Selection / endogeneity]
.box-inv-2.small[inverted fork **X** → **Z** ← **Y**]
]

---

layout: false
name: causal-mediation
class: center middle section-title section-title-3

# Causal mediation

---

---
# Key references

- Imai, Keele and Tingley (2010), [A General Approach to Causal Mediation Analysis](https://doi.org/10.1037/a0020761), *Psychological Methods*.
- Pearl (2014), [Interpretation and Identification of Causal Mediation](http://dx.doi.org/10.1037/a0036434), *Psychological Methods*.
- Baron and Kenny (1986), [The Moderator-Mediator  Variable  Distinction in Social Psychological Research: Conceptual,  Strategic, and Statistical Considerations](https://doi.org/10.1037/0022-3514.51.6.1173), *Journal of Personality and Social Psychology*

Limitations:

- Bullock, Green, and Ha (2010), [Yes, but what’s the mechanism? (don’t expect an easy answer)](https://doi.org/10.1037/a0018933)
- Uri Simonsohn (2022) [Mediation Analysis is Counterintuitively Invalid](http://datacolada.org/103)

---

# Fundamental problem of causal inference

.box-3.medium[Observe outcome for a single treatment]

With binary treatment `$X_i$`, I observed either `$Y_i \mid \text{do}(X_i=1)$` or `$Y_i \mid \text{do}(X_i=0)$` given intervention **do**.

.box-3.medium[Thus define causal effect as an **average** treatment]

`$$\mathsf{E}[Y_i \mid \text{do}(X_i=1)] - \mathsf{E}[Y_i \mid \text{do}(X_i=0)]$$`

# Sequential ignorability assumption

Define

- treatment of individual `$i$` as `$X_i$`, 
- potential mediation given treatment `$x$` as `$M_i(x)$` and 
- potential outcome for treatment `$x$` and mediator `$m$` as `$Y_i(x, m)$`.

Given pre-treatment covariates `$W$`, potential outcomes for mediation and treatment are conditionally independent of treatment assignment.
$$ Y_i(x', m), M_i(x) \perp\mkern-10mu\perp X_i \mid W_i = w$$
Given pre-treatment covariates and observed treatment, potential outcomes are independent of mediation.
$$ Y_i(x', m) \perp\mkern-10mu\perp  M_i(x) \mid X_i =x, W_i = w$$

Formulation in terms of potential outcomes (what if?) with intervention.

---
# Total effect

**Total effect**: overall impact of `$X$` (both through `$M$` and directly)

`$$\begin{align*}\mathsf{TE}(x, x^*) = \mathsf{E}[ Y \mid \text{do}(X=x)] - \mathsf{E}[ Y \mid \text{do}(X=x^*)]\end{align*}$$`

This can be generalized for continuous `$X$` to any pair of values `$(x_1, x_2)$`.

.pull-left[
.box-inv-3[
**X** → **M** → **Y** plus **X** → **Y**
]
]
.pull-right[
<img src="11-slides_files/figure-html/moderation-1.png" width="80%" style="display: block; margin: auto;" />
]
---
# Average controlled direct effect

`$$\begin{align*}\textsf{CDE}(m, x, x^*) &= \mathsf{E}[Y \mid \text{do}(X=x, m=m)] - \mathsf{E}[Y \mid \text{do}(X=x^*, m=m) \\&= \mathsf{E}\{Y(x, m) - Y(x^*, m)\} 
\end{align*}$$`
Expected population change in response when the experimental factor changes from `$x$` to `$x^*$` and the mediator is set to a fixed value `$m$`.

---
# Direct and indirect effects

**Natural direct effect**: `$\textsf{NDE}(x, x^*) = \mathsf{E}[Y\{x, M(x^*)\} - Y\{x^*,  M({x^*})\}]$`    
   - expected change in `$Y$` under treatment `$x$` if `$M$` is set to whatever value it would take under control `$x^*$`

**Natural indirect effect**: `$\textsf{NIE}(x, x^*) = \mathsf{E}[Y\{x^*, M(x)\} - Y\{x^*,  M(x^*)\}]$`  
   - expected change in `$Y$` if we set `$X$` to its control value and change the mediator value which it would attain under `$x$`

.small[
Counterfactual conditioning reflects a physical intervention, not mere (probabilistic) conditioning.
]

Total effect is `$\mathsf{TE}(x, x^*) = \textsf{NDE}(x, x^*) - \textsf{NIE}(x^*, x)$`
???

# Necessary and sufficiency of mediation

From Pearl (2014):

> The difference `$\textsf{TE}-\textsf{NDE}$` quantifies the extent to which the response of `$Y$` is owed to mediation, while `$\textsf{NIE}$` quantifies the extent to which it is explained by mediation. These two components of mediation, the necessary and the sufficient, coincide into one in models void of interactions (e.g., linear) but differ substantially under moderation

- In linear systems, changing the order og arguments amounts to flipping signs
- This definition works under temporal reversal and gives the correct answer (the regression-slope approach of the linear structural equation model does not).