Week 4: Clearing the Path from Cause to Effect

DSAN 5650: Causal Inference for Computational Social Science
Summer 2025, Georgetown University

Author

Affiliation

Jeff Jacobs

jj1088@georgetown.edu

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 →

\[ \DeclareMathOperator*{\argmax}{argmax} \DeclareMathOperator*{\argmin}{argmin} \newcommand{\bigexp}[1]{\exp\mkern-4mu\left[ #1 \right]} \newcommand{\bigexpect}[1]{\mathbb{E}\mkern-4mu \left[ #1 \right]} \newcommand{\definedas}{\overset{\small\text{def}}{=}} \newcommand{\definedalign}{\overset{\phantom{\text{defn}}}{=}} \newcommand{\eqeventual}{\overset{\text{eventually}}{=}} \newcommand{\Err}{\text{Err}} \newcommand{\expect}[1]{\mathbb{E}[#1]} \newcommand{\expectsq}[1]{\mathbb{E}^2[#1]} \newcommand{\fw}[1]{\texttt{#1}} \newcommand{\given}{\mid} \newcommand{\green}[1]{\color{green}{#1}} \newcommand{\heads}{\outcome{heads}} \newcommand{\iid}{\overset{\text{\small{iid}}}{\sim}} \newcommand{\lik}{\mathcal{L}} \newcommand{\loglik}{\ell} \DeclareMathOperator*{\maximize}{maximize} \DeclareMathOperator*{\minimize}{minimize} \newcommand{\mle}{\textsf{ML}} \newcommand{\nimplies}{\;\not\!\!\!\!\implies} \newcommand{\orange}[1]{\color{orange}{#1}} \newcommand{\outcome}[1]{\textsf{#1}} \newcommand{\param}[1]{{\color{purple} #1}} \newcommand{\pgsamplespace}{\{\green{1},\green{2},\green{3},\purp{4},\purp{5},\purp{6}\}} \newcommand{\pedge}[2]{\require{enclose}\enclose{circle}{~{#1}~} \rightarrow \; \enclose{circle}{\kern.01em {#2}~\kern.01em}} \newcommand{\pnode}[1]{\require{enclose}\enclose{circle}{\kern.1em {#1} \kern.1em}} \newcommand{\ponode}[1]{\require{enclose}\enclose{box}[background=lightgray]{{#1}}} \newcommand{\pnodesp}[1]{\require{enclose}\enclose{circle}{~{#1}~}} \newcommand{\purp}[1]{\color{purple}{#1}} \newcommand{\sign}{\text{Sign}} \newcommand{\spacecap}{\; \cap \;} \newcommand{\spacewedge}{\; \wedge \;} \newcommand{\tails}{\outcome{tails}} \newcommand{\Var}[1]{\text{Var}[#1]} \newcommand{\bigVar}[1]{\text{Var}\mkern-4mu \left[ #1 \right]} \]

source("../dsan-globals/_globals.r")

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

Linear Regression as a PGM (Source)

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

From McElreath (2020)

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”
	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”
	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 \(\Pr(Y \mid X) \neq \Pr(Y \mid \textsf{do}(X))\)!

The Four Elemental Confounds

From Richard McElreath’s Statistical Rethinking Lectures

library(tidyverse) # For ggplot
library(extraDistr) # For rbern()
library(patchwork) # For side-by-side plotting
n_d <- 10000 # For discrete RVs
n_c <- 300 # For continuous RVs

The Fork: \(X \leftarrow Z \rightarrow Y\)

set.seed(5650)
fork_df <- tibble(
    Z = rbern(n_d),
    X = rbern(n_d, (1-Z)*0.1 + Z*0.9),
    Y = rbern(n_d, (1-Z)*0.1 + Z*0.9),
)

plot_freqs <- function(df, plot_title, y_lab=TRUE) {
  df_cor <- cor(df$X, df$Y)
  df_label <- paste0("Cor(X,Y) = ",round(df_cor,3))
  freq_df <- df |>
    group_by(X, Y) |>
    summarize(count=n())
  freq_plot <- freq_df |>
    ggplot(
      aes(x=factor(X), y=factor(Y), fill=count)
    ) +
    geom_tile() +
    coord_equal() +
    scale_fill_distiller(
      palette="Greens", direction=1,
      limits=c(0,5000)
    ) +
    geom_label(
      aes(label=count),
      fill="white", color="black", size=7
    ) +
    labs(
      title = plot_title,
      subtitle = df_label,
      x="X", y="Y"
    ) +
    theme_dsan(base_size=24) +
    theme(
      plot.title = element_text(size=21),
      plot.subtitle = element_text(size=18)
    ) +
    remove_legend()
  if (!y_lab) {
    freq_plot <- freq_plot + theme(
      axis.title.y = element_blank()
    )
  }
  return(freq_plot)
}
# The full df
full_label <- paste0("Raw Data (n = 10K)")
full_plot <- plot_freqs(fork_df, full_label)

`summarise()` has grouped output by 'X'. You can override using the `.groups`
argument.

# Conditioning on Z = 0
z0_df <- fork_df |> filter(Z == 0)
z0_n <- nrow(z0_df)
z0_label <- paste0("Z == 0 (",z0_n," obs)")
z0_plot <- plot_freqs(z0_df, z0_label, y_lab=FALSE)

`summarise()` has grouped output by 'X'. You can override using the `.groups`
argument.

# Conditioning on Z = 1
z1_df <- fork_df |> filter(Z == 1)
z1_n <- nrow(z1_df)
z1_label <- paste0("Z == 1 (",z1_n," obs)")
z1_plot <- plot_freqs(z1_df, z1_label, y_lab=FALSE)

`summarise()` has grouped output by 'X'. You can override using the `.groups`
argument.

full_plot | z0_plot | z1_plot

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("$Slope_{Z=0} = ",z0_slope,"$")
z0_leg_label <- TeX(paste0("0 $(m=",z0_slope,")$"))
z1_lm <- lm(Y ~ X, data=cfork_df |> filter(Z == 1))
z1_slope <- round(z1_lm$coef['X'], 2)
z1_label <- paste0("$Slope_{Z=1} = ",z1_slope,"$")
z_texlabel <- TeX(paste0(z0_label, " | ", z1_label))
cfork_xmin <- min(cfork_df$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=24) +
  theme(
    plot.title = element_text(size=24),
    plot.subtitle = element_text(size=20)
  ) +
  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'

The Pipe: \(X \rightarrow Z \rightarrow Y\)

set.seed(5650)
pipe_df <- tibble(
    X = rbern(n_d),
    Z = rbern(n_d, (1-X)*0.1 + X*0.9),
    Y = rbern(n_d, (1-Z)*0.1 + Z*0.9),
)

# The full df
pipe_full_label <- paste0("Raw Data (n = 10K)")
pipe_full_plot <- plot_freqs(pipe_df, pipe_full_label)

`summarise()` has grouped output by 'X'. You can override using the `.groups`
argument.

# Conditioning on Z = 0
pipe_z0_df <- pipe_df |> filter(Z == 0)
pipe_z0_n <- nrow(pipe_z0_df)
pipe_z0_label <- paste0("Z == 0 (",pipe_z0_n," obs)")
pipe_z0_plot <- plot_freqs(pipe_z0_df, pipe_z0_label, y_lab=FALSE)

`summarise()` has grouped output by 'X'. You can override using the `.groups`
argument.

# Conditioning on Z = 1
pipe_z1_df <- pipe_df |> filter(Z == 1)
pipe_z1_n <- nrow(pipe_z1_df)
pipe_z1_label <- paste0("Z == 1 (",pipe_z1_n," obs)")
pipe_z1_plot <- plot_freqs(pipe_z1_df, pipe_z1_label, y_lab=FALSE)

`summarise()` has grouped output by 'X'. You can override using the `.groups`
argument.

pipe_full_plot | pipe_z0_plot | pipe_z1_plot

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("$Slope_{Z=0} = ",cpipe_z0_slope,"$")
cpipe_z1_lm <- lm(Y ~ X, data=cpipe_df |> filter(Z == 1))
cpipe_z1_slope <- round(cpipe_z1_lm$coef['X'], 2)
cpipe_z1_label <- paste0("$Slope_{Z=1} = ",cpipe_z1_slope,"$")
cpipe_z_texlabel <- TeX(paste0(cpipe_z0_label, " | ", cpipe_z1_label))
cpipe_xmin <- min(cpipe_df$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
  ) +
  # Overall lm
  geom_smooth(
    data=cpipe_df, aes(x=X, y=Y),
    method = lm, se = FALSE,
    linewidth = 2.5, color='black'
  ) +
  # 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=24) +
  theme(
    plot.title = element_text(size=24),
    plot.subtitle = element_text(size=20)
  ) +
  coord_equal() +
  labs(
    title = paste0(
      "Unstratified Slope = ",cpipe_slope
    ),
    subtitle=cpipe_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'

⚠️The Collider⚠️: \(X \rightarrow Z \leftarrow Y\)

set.seed(5650)
coll_df <- tibble(
    X = rbern(n_d),
    Y = rbern(n_d),
    Z = rbern(n_d, ifelse(X + Y > 0, 0.9, 0.2)),
)

# The full df
coll_full_label <- paste0("Raw Data (n = 10K)")
coll_full_plot <- plot_freqs(coll_df, coll_full_label)

`summarise()` has grouped output by 'X'. You can override using the `.groups`
argument.

# Conditioning on Z = 0
coll_z0_df <- coll_df |> filter(Z == 0)
coll_z0_n <- nrow(coll_z0_df)
coll_z0_label <- paste0("Z == 0 (",coll_z0_n," obs)")
coll_z0_plot <- plot_freqs(coll_z0_df, coll_z0_label, y_lab=FALSE)

`summarise()` has grouped output by 'X'. You can override using the `.groups`
argument.

# Conditioning on Z = 1
coll_z1_df <- coll_df |> filter(Z == 1)
coll_z1_n <- nrow(coll_z1_df)
coll_z1_label <- paste0("Z == 1 (",coll_z1_n," obs)")
coll_z1_plot <- plot_freqs(coll_z1_df, coll_z1_label, y_lab=FALSE)

`summarise()` has grouped output by 'X'. You can override using the `.groups`
argument.

coll_full_plot | coll_z0_plot | coll_z1_plot

• Conditioning on colliders induces correlation where there previously was none ☠️

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("$Slope_{Z=0} = ",ccoll_z0_slope,"$")
ccoll_z1_lm <- lm(Y ~ X, data=ccoll_df |> filter(Z == 1))
ccoll_z1_slope <- round(ccoll_z1_lm$coef['X'], 2)
ccoll_z1_label <- paste0("$Slope_{Z=1} = ",ccoll_z1_slope,"$")
ccoll_z_texlabel <- TeX(paste0(ccoll_z0_label, " | ", ccoll_z1_label))
ccoll_xmin <- min(ccoll_df$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=0.6*g_pointsize,
    alpha=0.8
  ) +
  # Overall lm
  geom_smooth(
    data=ccoll_df, aes(x=X, y=Y),
    method = lm, se = FALSE,
    linewidth = 2.5, color='black'
  ) +
  # 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=24) +
  theme(
    plot.title = element_text(size=24),
    plot.subtitle = element_text(size=20)
  ) +
  coord_equal() +
  labs(
    title = paste0(
      "Unstratified Slope = ",ccoll_slope
    ),
    subtitle=ccoll_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'

• …This is why we have to think, rather than just “control for everything”! 😭

Proxies for \(Z\)

set.seed(5650)
prox_df <- tibble(
  X = rbern(n_d),
  Z = rbern(n_d, (1-X)*0.1 + X*0.9),
  Y = rbern(n_d, (1-Z)*0.1 + Z*0.9),
  A = rbern(n_d, (1-Z)*0.1 + Z*0.9)
)

# The full df
prox_full_label <- paste0("Raw Data (n = 10K)")
prox_full_plot <- plot_freqs(prox_df, prox_full_label)

`summarise()` has grouped output by 'X'. You can override using the `.groups`
argument.

# Conditioning on A == 0
prox_a0_df <- prox_df |> filter(A == 0)
prox_a0_n <- nrow(prox_a0_df)
prox_a0_label <- paste0("A == 0 (",prox_a0_n," obs)")
prox_a0_plot <- plot_freqs(prox_a0_df, prox_a0_label, y_lab=FALSE)

`summarise()` has grouped output by 'X'. You can override using the `.groups`
argument.

# Conditioning on A == 1
prox_a1_df <- prox_df |> filter(A == 1)
prox_a1_n <- nrow(prox_a1_df)
prox_a1_label <- paste0("A == 1 (",prox_a1_n," obs)")
prox_a1_plot <- plot_freqs(prox_a1_df, prox_a1_label, y_lab=FALSE)

`summarise()` has grouped output by 'X'. You can override using the `.groups`
argument.

prox_full_plot | prox_a0_plot | prox_a1_plot

• With just \(X \rightarrow Z \rightarrow Y\), we’d have a pipe
• Observing \(A\) gives us some (not all!) information about \(Z\)

library(tidyverse)
library(extraDistr)
library(latex2exp)
set.seed(5650)
n_c <- 300
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("$Slope_{A=0} = ",cprox_a0_slope,"$")
# A == 1 lm
cprox_a1_lm <- lm(Y ~ X, data=cprox_df |> filter(A == 1))
cprox_a1_slope <- round(cprox_a1_lm$coef['X'], 2)
cprox_a1_label <- paste0("$Slope_{A=1} = ",cprox_a1_slope,"$")
cprox_a_texlabel <- TeX(paste0(cprox_a0_label, " | ", cprox_a1_label))
cprox_xmin <- min(cprox_df$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=0.6*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_text(size=18)
  ) +
  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'

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, 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 Books.

McElreath, Richard. 2020. Statistical Rethinking: A Bayesian Course with Examples in R and STAN. CRC Press.

Footnotes

1. 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)