Week 4: Clearing the Path from Cause to Effect

DSAN 5650: Causal Inference for Computational Social Science
Summer 2026, Georgetown University

Class Sessions

Author

Affiliation

Jeff Jacobs

jj1088@georgetown.edu

Published

Wednesday, June 10, 2026

Open slides in new window →

Schedule

Today’s Planned Schedule:

	Start	End	Topic
Lecture	6:30pm	7:10pm	PGM as Modeling Language →
	7:10pm	7:30pm	The Ladder of Causal Inference →
	7:30pm	7:50pm	Elemental Confounds I: Forks and Chains →
Break!	7:50pm	8:00pm
	8:00pm	8:50pm	Elemental Confounds II: ⚠️Colliders⚠️ →
	8:50pm	9:00pm	Elemental Confounds III: Proxies →

Roadmap

5300 → Now

In e.g. 5300, you learned a bunch of ad hoc models: Linear Regression, Decision Trees, SVMs
PGMs provide a formalized modeling language for “writing out” models unambiguously in a way your computer understands: specifying exactly how to estimate parameters from data

Now → August: Class splits into two themes, running in parallel!

What kinds of cool comp social sci models are unlocked, that we can now implement in this language? [HW2]
How can we expand PGM vocabulary to incorporate causality? [Midterm]

From PGMs to SWIGs (Bezuidenhout et al. 2024)

Why Take the Time to Learn a Modeling Language (vs. Individual Models)?

My answer: allows you to adapt to specifics/idiosyncrasies of your problem!
Language metaphor: Learning models vs. learning modeling language $\Leftrightarrow$ Learning phrases in a language vs. learning to speak the language
“Hello, one hamburger please” is good, but what if you…
Are allergic to ketchup and need to make sure it’s removed
Want to replace sesame seed bun with poppy seed bun, if they have it
Prefer spicy, but not too spicy, mustard Bun only Animal style …

Languages give us a syntax…

S	$\rightarrow$	NP VP
NP	$\rightarrow$	DetP N \| AdjP NP
VP	$\rightarrow$	V NP
AdjP	$\rightarrow$	Adj \| Adv AdjP
N	$\rightarrow$	`frog` \| `tadpole`
V	$\rightarrow$	`sees` \| `likes`
Adj	$\rightarrow$	`big` \| `small`
Adv	$\rightarrow$	`very`
DetP	$\rightarrow$	`a` \| `the`

…For expressing arbitrary (infinitely many!) sentences

Example 1: Multilevel Tadpoles (McElreath, Ch. 13)

Need a language that can communicate the following info to estimation algorithm:

Unit of observation is tadpole, but unit of analysis is tank
Ultimately, I care about $Y =$ survival rate (dependent var), as function of $X =$ tank properties (independent var)
…But the $n_i = 48$ tanks actually come in $n_j = 3$ types: small (10 bois), medium (25), large (35) (Bonus: What if there are different numbers of tanks per type?)
I need it to account for impact of tank size, then pool info across tank sizes

Example 2: Dissertation Nightmare

*Above*: Data from Soviet archives; *Above Right*: US Military archives; *Below Right*: NATO archives

Nightmarish Without a Modeling Language!

Modeling language $\Rightarrow$ Unambiguously “encode” idiosyncratic domain knowledge
Dissertation: Cold War $\times$ “Third World” $\leadsto$ Cuban 🇨🇺 trans-continental operations¹
Main narrative (for estimation): 1975 (South Africa invades Angola, 14 Oct → 🇨🇺 intervention, 4 Nov) to 1979 (USSR requests 🇨🇺 troops to Ethiopia for Ogaden War)
[Ontology] Fix 1979 geographic entities at National level (as modeling choice, like fixing 2000 USD to measure inflation): $\textsf{Cuba}_{1979}$, $\textsf{Angola}_{1979}$, $\textsf{PDRY}_{1979}$, $\textsf{YAR}_{1979}$
Different tokens (Think NLP: "Congo", "DRC", "Republic of Congo") can then be contextualized: can “track” and link data appropriately despite splits, merges, name changes
Say we have data on “Number of Communist Militants in $X$” (Hoover Yearbook)…

Entity	Data from 1947-1971 at...		Data from 1971-Present at...
$\textsf{Pakistan}_{1979}$	National Level:	$\frac{62}{62+70} \times$ “Pakistan”	National Level:	“Pakistan”
$\textsf{Pakistan}_{1979}$	Subnational Level:	“West Pakistan”	Subnational Level:	$\sum_{i \in \text{Regions}}\text{data}_i$
$\textsf{Bangladesh}_{1979}$	National Level:	$\frac{70}{62+70} \times$ “Pakistan”	National Level:	“Bangladesh”
$\textsf{Bangladesh}_{1979}$	Subnational Level:	“East Pakistan”	Subnational Level:	$\sum_{i \in \text{Regions}}\text{data}_i$

The Ladder of Causal Inference

		Counterfactuals: What would have happened, if history was slightly different… $\Pr(Y_{M=M_0} \mid \textsf{do}(X)) - \Pr(Y_{M=M_0} \mid \textsf{do}(\neg X))$
	Intervention: What happens if I… $\Pr(Y \mid \textsf{do}(X)) - \Pr(Y \mid \textsf{do}(\neg X))$
Association: What happened? $\Pr(Y \mid X) - \Pr(Y \mid \neg X)$

$\leadsto$ Stuff we add to probability theory in 5650 is to combat confounding: to “fix” whatever is making $\Pr(Y \mid X) \neq \Pr(Y \mid \textsf{do}(X))$!

Recap: The Four Elemental Confounds

From Richard McElreath’s *Statistical Rethinking* Lectures

Code

library(tidyverse) # For ggplot

── Attaching core tidyverse packages ──────────────────────── tidyverse 2.0.0 ──
✔ dplyr     1.2.1     ✔ readr     2.2.0
✔ forcats   1.0.1     ✔ stringr   1.6.0
✔ lubridate 1.9.5     ✔ tibble    3.3.1
✔ purrr     1.2.1     ✔ tidyr     1.3.2
── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
✖ dplyr::filter() masks stats::filter()
✖ dplyr::lag()    masks stats::lag()
ℹ Use the conflicted package (<http://conflicted.r-lib.org/>) to force all conflicts to become errors

Code

library(extraDistr) # For rbern()


Attaching package: 'extraDistr'

The following object is masked from 'package:purrr':

    rdunif

Code

library(patchwork) # For side-by-side plotting
library(ggtext) # For colors in titles
library(rethinking)

Loading required package: cmdstanr
This is cmdstanr version 0.9.0
- CmdStanR documentation and vignettes: mc-stan.org/cmdstanr
- CmdStan path: /Users/jpj/.cmdstan/cmdstan-2.36.0
- CmdStan version: 2.36.0

A newer version of CmdStan is available. See ?install_cmdstan() to install it.
To disable this check set option or environment variable cmdstanr_no_ver_check=TRUE.
Loading required package: posterior
This is posterior version 1.7.0

Attaching package: 'posterior'

The following objects are masked from 'package:stats':

    mad, sd, var

The following objects are masked from 'package:base':

    %in%, match

Loading required package: parallel
rethinking (Version 2.42)

Attaching package: 'rethinking'

The following objects are masked from 'package:extraDistr':

    dbern, dlaplace, dpareto, rbern, rlaplace, rpareto

The following object is masked from 'package:purrr':

    map

The following object is masked from 'package:stats':

    rstudent

Code

library(dagitty)
n_d <- 10000 # For discrete RVs
n_c <- 300 # For continuous RVs

Pipes $X \rightarrow Z \rightarrow Y$: Conditioning = Blocking

Code

library(rethinking)
library(dagitty)
library(ggdag)
pipe_dag <-dagitty("dag{
X[exposure]
Y[outcome]
X -> Y
X -> Z
Z -> Y
}")
coordinates(pipe_dag) <- list(
    x=c(X=0, Z=0.5, Y=1),
    y=c(X=1, Z=0.5, Y=1)
)
drawdag(pipe_dag, cex=4, lwd=5, radius=10)
drawopenpaths(pipe_dag, lwd=5)
adj_sets <- adjustmentSets(
    pipe_dag, effect="direct"
)
writeLines("Adjustment sets (direct effect):")
adj_sets
set.seed(5650)
cpipe_df <- tibble(
    X = rnorm(n_c),
    Z = rbern(n_c, plogis(X)),
    Y = rnorm(n_c, 2 * Z - 1)
)
cpipe_lm <- lm(Y ~ X, data=cpipe_df)
cpipe_slope <- round(cpipe_lm$coef['X'], 3)
cpipe_z0_lm <- lm(Y ~ X, data=cpipe_df |> filter(Z == 0))
cpipe_z0_slope <- round(cpipe_z0_lm$coef['X'], 2)
cpipe_z0_label <- paste0("<span style='color: #e69f00;'>Slope<sub>Z=0</sub> = ",cpipe_z0_slope,"</span>")
cpipe_z1_lm <- lm(Y ~ X, data=cpipe_df |> filter(Z == 1))
cpipe_z1_slope <- round(cpipe_z1_lm$coef['X'], 2)
cpipe_z1_label <- paste0("<span style='color: #56b4e9;'>Slope<sub>Z=1</sub> = ",cpipe_z1_slope,"</span>")
cpipe_z_texlabel <- paste0(cpipe_z0_label, " | ", cpipe_z1_label)
cpipe_xmin <- min(cpipe_df$X)
cpipe_xmax <- max(cpipe_df$X)
ggplot() +
  # Points
  geom_point(
    data=cpipe_df |> filter(Y > -3),
    aes(x=X, y=Y, color=factor(Z)),
    size=0.4*g_pointsize,
    alpha=0.8
  ) +
  # Overall lm
  geom_smooth(
    data=cpipe_df, aes(x=X, y=Y),
    method = lm, se = FALSE,
    linewidth = 3, color='white'
  ) +
  geom_smooth(
    data=cpipe_df, aes(x=X, y=Y),
    method = lm, se = FALSE,
    linewidth = 2.5, color='black'
  ) +
  theme_dsan(base_size=18) +
  theme(
    plot.title = element_text(size=18),
    plot.subtitle = element_markdown(size=16)
  ) +
  coord_equal() +
  labs(
    title = paste0(
      "Unstratified Slope = ",cpipe_slope
    ),
    x = "X", y = "Y", color = "Z"
  )


Attaching package: 'ggdag'

The following object is masked from 'package:stats':

    filter

Adjustment sets (direct effect):
{ Z }

`geom_smooth()` using formula = 'y ~ x'
`geom_smooth()` using formula = 'y ~ x'

Code

library(rethinking)
library(dagitty)
library(ggdag)
pipe_dag_closed <-dagitty("dag{
X[exposure]
Y[outcome]
Z[adjustedNode]
X -> Y
X -> Z
Z -> Y
}")
coordinates(pipe_dag_closed) <- list(
    x=c(X=0, Z=0.5, Y=1),
    y=c(X=1, Z=0.5, Y=1)
)
drawdag(pipe_dag_closed, cex=4, lwd=5, radius=10)
drawopenpaths(pipe_dag_closed, Z="Z", lwd=5)
adj_sets_closed <- adjustmentSets(
    pipe_dag_closed
)
writeLines("Adjustment sets (direct effect):")
adj_sets_closed
set.seed(5650)
cpipe_df <- tibble(
    X = rnorm(n_c),
    Z = rbern(n_c, plogis(X)),
    Y = rnorm(n_c, 2 * Z - 1)
)
cpipe_lm <- lm(Y ~ X, data=cpipe_df)
cpipe_slope <- round(cpipe_lm$coef['X'], 3)
cpipe_z0_lm <- lm(Y ~ X, data=cpipe_df |> filter(Z == 0))
cpipe_z0_slope <- round(cpipe_z0_lm$coef['X'], 2)
cpipe_z0_label <- paste0("<span style='color: #e69f00;'>Slope<sub>Z=0</sub> = ",cpipe_z0_slope,"</span>")
cpipe_z1_lm <- lm(Y ~ X, data=cpipe_df |> filter(Z == 1))
cpipe_z1_slope <- round(cpipe_z1_lm$coef['X'], 2)
cpipe_z1_label <- paste0("<span style='color: #56b4e9;'>Slope<sub>Z=1</sub> = ",cpipe_z1_slope,"</span>")
cpipe_z_texlabel <- paste0(cpipe_z0_label, " | ", cpipe_z1_label)
cpipe_xmin <- min(cpipe_df$X)
cpipe_xmax <- max(cpipe_df$X)
ggplot() +
  # Points
  geom_point(
    data=cpipe_df |> filter(Y > -3),
    aes(x=X, y=Y, color=factor(Z)),
    size=0.6*g_pointsize,
    alpha=0.8
  ) +
  # Stratified lm
  # (slightly larger black lines)
  geom_smooth(
    data=cpipe_df,
    aes(x=X, y=Y, group=factor(Z)),
    method=lm, se=FALSE, fullrange=TRUE,
    linewidth=2.75, color='black'
  ) +
  # (Colored lines)
  geom_smooth(
    data=cpipe_df,
    aes(x=X, y=Y, color=factor(Z)),
    method=lm, se=FALSE, fullrange=TRUE,
    linewidth=2
  ) +
  theme_dsan(base_size=18) +
  theme(
    plot.title = element_markdown(size=18),
    plot.subtitle = element_markdown(size=16)
  ) +
  coord_equal() +
  labs(
    title=cpipe_z_texlabel,
    x = "X", y = "Y", color = "Z"
  )

Adjustment sets (direct effect):
{}

`geom_smooth()` using formula = 'y ~ x'
`geom_smooth()` using formula = 'y ~ x'

Forks $X \leftarrow Z \rightarrow Y$: Conditioning = Blocking

Code

library(rethinking)
library(dagitty)
library(ggdag)
pipe_dag <-dagitty("dag{
X[exposure]
Y[outcome]
X -> Y
Z -> X
Z -> Y
}")
coordinates(pipe_dag) <- list(
    x=c(X=0, Z=0.5, Y=1),
    y=c(X=1, Z=0.5, Y=1)
)
drawdag(pipe_dag, cex=4, lwd=5, radius=10)
drawopenpaths(pipe_dag, lwd=5)
adj_sets <- adjustmentSets(
    pipe_dag, effect="direct"
)
writeLines("Adjustment sets (direct effect):")
adj_sets
library(ggtext)
set.seed(5650)
cfork_df <- tibble(
    Z = rbern(n_c),
    X = rnorm(n_c, 2 * Z - 1),
    Y = rnorm(n_c, 2 * Z - 1)
)
library(latex2exp)
overall_lm <- lm(Y ~ X, data=cfork_df)
overall_slope <- round(overall_lm$coef['X'], 3)
z0_lm <- lm(Y ~ X, data=cfork_df |> filter(Z == 0))
z0_slope <- round(z0_lm$coef['X'], 2)
z0_label <- paste0("<span style='color: #e69f00;'>Slope<sub>Z=0</sub> = ",z0_slope,"</span>")
z1_lm <- lm(Y ~ X, data=cfork_df |> filter(Z == 1))
z1_slope <- round(z1_lm$coef['X'], 2)
z1_label <- paste0("<span style='color: #56b4e9;'>Slope<sub>Z=1</sub> = ",z1_slope,"</span>")
z_texlabel <- paste0(z0_label, " | ", z1_label)
cfork_xmin <- min(cfork_df$X)
cfork_xmax <- max(cfork_df$X)
ggplot() +
  # Points
  geom_point(
    data=cfork_df,
    aes(x=X, y=Y, color=factor(Z)),
    size=0.6*g_pointsize,
    alpha=0.8
  ) +
  # Overall lm
  geom_smooth(
    data=cfork_df, aes(x=X, y=Y),
    method = lm, se = FALSE,
    linewidth = 2.5, color='black'
  ) +
  theme_dsan(base_size=18) +
  theme(
    plot.title = element_text(size=18),
    plot.subtitle = element_markdown(size=16)
  ) +
  coord_equal() +
  labs(
    title = paste0(
      "Unstratified Slope = ",overall_slope
    ),
    x = "X", y = "Y", color = "Z"
  )

Adjustment sets (direct effect):
{ Z }

`geom_smooth()` using formula = 'y ~ x'

Code

library(rethinking)
library(dagitty)
library(ggdag)
fork_dag_closed <-dagitty("dag{
X[exposure]
Y[outcome]
Z[adjustedNode]
X -> Y
Z -> X
Z -> Y
}")
coordinates(fork_dag_closed) <- list(
    x=c(X=0, Z=0.5, Y=1),
    y=c(X=1, Z=0.5, Y=1)
)
fork_dag_closed <- setVariableStatus(fork_dag_closed, "adjustedNode", "Z")
drawdag(fork_dag_closed, cex=4, lwd=5, radius=10)
drawopenpaths(fork_dag_closed, Z="Z", lwd=5)
library(ggtext)
set.seed(5650)
cfork_df <- tibble(
    Z = rbern(n_c),
    X = rnorm(n_c, 2 * Z - 1),
    Y = rnorm(n_c, 2 * Z - 1)
)
library(latex2exp)
overall_lm <- lm(Y ~ X, data=cfork_df)
overall_slope <- round(overall_lm$coef['X'], 3)
z0_lm <- lm(Y ~ X, data=cfork_df |> filter(Z == 0))
z0_slope <- round(z0_lm$coef['X'], 2)
z0_label <- paste0("<span style='color: #e69f00;'>Slope<sub>Z=0</sub> = ",z0_slope,"</span>")
z1_lm <- lm(Y ~ X, data=cfork_df |> filter(Z == 1))
z1_slope <- round(z1_lm$coef['X'], 2)
z1_label <- paste0("<span style='color: #56b4e9;'>Slope<sub>Z=1</sub> = ",z1_slope,"</span>")
z_texlabel <- paste0(z0_label, " | ", z1_label)
cfork_xmin <- min(cfork_df$X)
cfork_xmax <- max(cfork_df$X)
ggplot() +
  # Points
  geom_point(
    data=cfork_df,
    aes(x=X, y=Y, color=factor(Z)),
    size=0.6*g_pointsize,
    alpha=0.8
  ) +
  # Overall lm
  geom_smooth(
    data=cfork_df, aes(x=X, y=Y),
    method = lm, se = FALSE,
    linewidth = 2.5, color='black'
  ) +
  # Stratified lm
  # (slightly larger black lines)
  geom_smooth(
    data=cfork_df,
    aes(x=X, y=Y, group=factor(Z)),
    method=lm, se=FALSE, fullrange=TRUE,
    linewidth=2.75, color='black'
  ) +
  # (Colored lines)
  geom_smooth(
    data=cfork_df,
    aes(x=X, y=Y, color=factor(Z)),
    method=lm, se=FALSE, fullrange=TRUE,
    linewidth=2
  ) +
  theme_dsan(base_size=18) +
  theme(
    plot.title = element_text(size=18),
    plot.subtitle = element_markdown(size=16)
  ) +
  coord_equal() +
  labs(
    title = paste0(
      "Unstratified Slope = ",overall_slope
    ),
    subtitle=z_texlabel,
    x = "X", y = "Y", color = "Z"
  )

`geom_smooth()` using formula = 'y ~ x'
`geom_smooth()` using formula = 'y ~ x'
`geom_smooth()` using formula = 'y ~ x'

Conditioning on a Proxy for $Z$

With just $X \rightarrow Z \rightarrow Y$, we’d have a pipe
Observing $A \Rightarrow$ some (not all!) info about $Z$

Code

library(tidyverse)
library(extraDistr)
library(latex2exp)
set.seed(5650)
cprox_df <- tibble(
    X = rnorm(n_c),
    Z = rbern(n_c, plogis(X)),
    Y = rnorm(n_c, 2 * Z - 1),
    A = rbern(n_c, (1-Z)*0.86 + Z*0.14)
)
cprox_lm <- lm(Y ~ X, data=cprox_df)
cprox_slope <- round(cprox_lm$coef['X'], 3)
cprox_a0_lm <- lm(Y ~ X, data=cprox_df |> filter(A == 0))
cprox_a0_slope <- round(cprox_a0_lm$coef['X'], 2)
cprox_a0_label <- paste0("<span style='color: #e69f00;'>Slope<sub>Z=0</sub> = ",cprox_a0_slope,"</span>")
# A == 1 lm
cprox_a1_lm <- lm(Y ~ X, data=cprox_df |> filter(A == 1))
cprox_a1_slope <- round(cprox_a1_lm$coef['X'], 2)
cprox_a1_label <- paste0("<span style='color: #56b4e9;'>Slope<sub>Z=1</sub> = ",cprox_a1_slope,"</span>")
cprox_a_texlabel <- paste0(cprox_a0_label, " | ", cprox_a1_label)
cprox_xmin <- min(cprox_df$X)
cprox_xmax <- max(cprox_df$X)
ggplot() +
  # Points
  geom_point(
    data=cprox_df |> filter(Y > -3),
    aes(x=X, y=Y, color=factor(A)),
    size=0.5*g_pointsize,
    alpha=0.8
  ) +
  # Overall lm
  geom_smooth(
    data=cprox_df, aes(x=X, y=Y),
    method = lm, se = FALSE,
    linewidth = 2.5, color='black'
  ) +
  # Stratified lm
  # (slightly larger black lines)
  geom_smooth(
    data=cprox_df,
    aes(x=X, y=Y, group=factor(A)),
    method=lm, se=FALSE, fullrange=TRUE,
    linewidth=2.75, color='black'
  ) +
  # (Colored lines)
  geom_smooth(
    data=cprox_df,
    aes(x=X, y=Y, color=factor(A)),
    method=lm, se=FALSE, fullrange=TRUE,
    linewidth=2
  ) +
  theme_dsan(base_size=22) +
  theme(
    plot.title = element_text(size=22),
    plot.subtitle = element_markdown(size=20)
  ) +
  coord_equal() +
  labs(
    title = paste0(
      "Unstratified Slope = ",cprox_slope
    ),
    subtitle=cprox_a_texlabel,
    x = "X", y = "Y", color = "A"
  )

`geom_smooth()` using formula = 'y ~ x'
`geom_smooth()` using formula = 'y ~ x'
`geom_smooth()` using formula = 'y ~ x'

⚠️Colliders⚠️ $X \rightarrow Z \leftarrow Y$: Conditioning = Opening

Code

library(rethinking)
library(dagitty)
library(ggdag)
coll_dag <-dagitty("dag{
X[exposure]
Y[outcome]
X -> Y
X -> Z
Y -> Z
}")
coordinates(coll_dag) <- list(
    x=c(X=0, Z=0.5, Y=1),
    y=c(X=1, Z=0.5, Y=1)
)
drawdag(coll_dag, cex=4, lwd=5, radius=10)
drawopenpaths(coll_dag, lwd=5)
adj_sets_coll <- adjustmentSets(
    coll_dag, effect="direct"
)
writeLines("Adjustment sets (direct effect):")
adj_sets_coll
set.seed(5650)
ccoll_df <- tibble(
    X = rnorm(n_c),
    Y = rnorm(n_c),
    Z = rbern(n_c, plogis(2 * (X + Y - 1)))
)
ccoll_lm <- lm(Y ~ X, data=ccoll_df)
ccoll_slope <- round(ccoll_lm$coef['X'], 3)
ccoll_z0_lm <- lm(Y ~ X, data=ccoll_df |> filter(Z == 0))
ccoll_z0_slope <- round(ccoll_z0_lm$coef['X'], 2)
ccoll_z0_label <- paste0("<span style='color: #e69f00;'>Slope<sub>Z=0</sub> = ",ccoll_z0_slope,"</span>")
ccoll_z1_lm <- lm(Y ~ X, data=ccoll_df |> filter(Z == 1))
ccoll_z1_slope <- round(ccoll_z1_lm$coef['X'], 2)
ccoll_z1_label <- paste0("<span style='color: #56b4e9;'>Slope<sub>Z=1</sub> = ",ccoll_z1_slope,"</span>")
ccoll_z_texlabel <- paste0(ccoll_z0_label, " | ", ccoll_z1_label)
ccoll_xmin <- min(ccoll_df$X)
ccoll_xmax <- max(ccoll_df$X)
ggplot() +
  # Points
  geom_point(
    data=ccoll_df |> filter(Y > -3),
    aes(x=X, y=Y, color=factor(Z)),
    size=0.4*g_pointsize,
    alpha=0.8
  ) +
  # Overall lm
  geom_smooth(
    data=ccoll_df, aes(x=X, y=Y),
    method = lm, se = FALSE,
    linewidth = 3, color='white'
  ) +
  geom_smooth(
    data=ccoll_df, aes(x=X, y=Y),
    method = lm, se = FALSE,
    linewidth = 2.5, color='black'
  ) +
  theme_dsan(base_size=18) +
  theme(
    plot.title = element_markdown(size=18),
    plot.subtitle = element_markdown(size=16)
  ) +
  coord_equal() +
  labs(
    title=ccoll_z_texlabel,
    x = "X", y = "Y", color = "Z"
  )

Adjustment sets (direct effect):
{}

`geom_smooth()` using formula = 'y ~ x'
`geom_smooth()` using formula = 'y ~ x'

Code

library(rethinking)
library(dagitty)
library(ggdag)
fork_dag_closed <-dagitty("dag{
X[exposure]
Y[outcome]
Z[adjustedNode]
X -> Y
X -> Z
Y -> Z
}")
coordinates(fork_dag_closed) <- list(
    x=c(X=0, Z=0.5, Y=1),
    y=c(X=1, Z=0.5, Y=1)
)
fork_dag_closed <- setVariableStatus(fork_dag_closed, "adjustedNode", "Z")
drawdag(fork_dag_closed, cex=4, lwd=5, radius=10)
drawopenpaths(fork_dag_closed, Z="Z", lwd=5)
set.seed(5650)
ccoll_df <- tibble(
    X = rnorm(n_c),
    Y = rnorm(n_c),
    Z = rbern(n_c, plogis(2 * (X + Y - 1)))
)
ccoll_lm <- lm(Y ~ X, data=ccoll_df)
ccoll_slope <- round(ccoll_lm$coef['X'], 3)
ccoll_z0_lm <- lm(Y ~ X, data=ccoll_df |> filter(Z == 0))
ccoll_z0_slope <- round(ccoll_z0_lm$coef['X'], 2)
ccoll_z0_label <- paste0("<span style='color: #e69f00;'>Slope<sub>Z=0</sub> = ",ccoll_z0_slope,"</span>")
ccoll_z1_lm <- lm(Y ~ X, data=ccoll_df |> filter(Z == 1))
ccoll_z1_slope <- round(ccoll_z1_lm$coef['X'], 2)
ccoll_z1_label <- paste0("<span style='color: #56b4e9;'>Slope<sub>Z=1</sub> = ",ccoll_z1_slope,"</span>")
ccoll_z_texlabel <- paste0(ccoll_z0_label, " | ", ccoll_z1_label)
ccoll_xmin <- min(ccoll_df$X)
ccoll_xmax <- max(ccoll_df$X)
ggplot() +
  # Points
  geom_point(
    data=ccoll_df |> filter(Y > -3),
    aes(x=X, y=Y, color=factor(Z)),
    size=0.4*g_pointsize,
    alpha=0.8
  ) +
  # Stratified lm
  # (slightly larger black lines)
  geom_smooth(
    data=ccoll_df,
    aes(x=X, y=Y, group=factor(Z)),
    method=lm, se=FALSE, fullrange=TRUE,
    linewidth=2.75, color='black'
  ) +
  # (Colored lines)
  geom_smooth(
    data=ccoll_df,
    aes(x=X, y=Y, color=factor(Z)),
    method=lm, se=FALSE, fullrange=TRUE,
    linewidth=2
  ) +
  theme_dsan(base_size=18) +
  theme(
    plot.title = element_markdown(size=18),
    plot.subtitle = element_markdown(size=16)
  ) +
  coord_equal() +
  labs(
    title=ccoll_z_texlabel,
    x = "X", y = "Y", color = "Z"
  )

`geom_smooth()` using formula = 'y ~ x'
`geom_smooth()` using formula = 'y ~ x'

References

Bezuidenhout, Dana, Sarah Forthal, Kara Rudolph, and Matthew R Lamb. 2024. “Single World Intervention Graphs (SWIGs): A Practical Guide.” American Journal of Epidemiology, September 11, kwae353. https://doi.org/10.1093/aje/kwae353.

Gleijeses, Piero. 2013. Visions of Freedom: Havana, Washington, Pretoria, and the Struggle for Southern Africa, 1976-1991. UNC Press. https://books.google.com?id=eY74AAAAQBAJ.

McElreath, Richard. 2020. Statistical Rethinking: A Bayesian Course with Examples in R and STAN. CRC Press. https://books.google.com?id=FuLWDwAAQBAJ.

Footnotes

Helpful metaphor (Gleijeses 2013): Cuba $\approx$ Forward-deployed “3rd World Outpost” for USSR (Soviet $ but Cuban training of PAIGC → MPLA), as Israel $\approx$ Forward-deployed “3rd World Outpost” for US (US $ but Israeli training of SAVAK → SADF)↩︎

S	\(\rightarrow\)	NP VP
NP	\(\rightarrow\)	DetP N \| AdjP NP
VP	\(\rightarrow\)	V NP
AdjP	\(\rightarrow\)	Adj \| Adv AdjP
N	\(\rightarrow\)	`frog` \| `tadpole`
V	\(\rightarrow\)	`sees` \| `likes`
Adj	\(\rightarrow\)	`big` \| `small`
Adv	\(\rightarrow\)	`very`
DetP	\(\rightarrow\)	`a` \| `the`

Week 4: Clearing the Path from Cause to Effect

Schedule

Roadmap

Why Take the Time to Learn a Modeling Language (vs. Individual Models)?

Example 1: Multilevel Tadpoles (McElreath, Ch. 13)

Example 2: Dissertation Nightmare

Nightmarish Without a Modeling Language!

The Ladder of Causal Inference

Recap: The Four Elemental Confounds

Pipes \(X \rightarrow Z \rightarrow Y\): Conditioning = Blocking

Forks \(X \leftarrow Z \rightarrow Y\): Conditioning = Blocking

Conditioning on a Proxy for \(Z\)

⚠️Colliders⚠️ \(X \rightarrow Z \leftarrow Y\): Conditioning = Opening

References

Footnotes