24 Causal inference

He is wise who bases causal inference
on an explicit causal structure
that is defensible on scientific grounds.

Aristotle (384–322 BC)

The previous chapter on prediction (Chapter 23) showed how we can anticipate outcomes from data — but a good prediction need not tell us why an outcome occurs, nor what would happen if we intervened. We learn that “correlation does not imply causation”, but not what positively would imply causation. More importantly, we really would like to understand why things are what they are. For ultimately, we need a causal understanding of the mechanisms governing our environment in order to gain control over them.

When studying correlational data, the human mind often intuitively imposes structure — which we embrace as a feature, rather than a bug. However, EDA and statistics lack the tools for dealing with causal structures. We need a language and methodology to ask and answer causal questions.

Understanding causal concepts and corresponding models will allow us to think beyond data. Essentially, this addresses and uncovers the data generating process. It also assigns a clear role to our own interpretation and understanding of data, rather than relying on a vague hope that data will eventually reveal itself. Especially when analyzing large sets of data (which may often be observational in nature), we will see that qualitative assumptions are important for the interpretation of results and the conclusions that can be drawn from data.

Please note

This chapter currently is in an early drafting stage — mostly a collection of notes and ideas, and a placeholder for future content. However, it contains useful links to better resources.

Preparation

Recommended readings for this chapter include:

• The Causal inference in R blog, the Causal inference in R workshop and the corresponding book Causal Inference in R (by Malcolm Barrett, Lucy D’Agostino McGowan, and Travis Gerke)

• Introductory articles by Lübke et al. (2020) and McGowan et al. (2024)

• Background articles and books by Judea Pearl (e.g., Pearl, 2009; Pearl et al., 2016; Pearl & Mackenzie, 2018)

Preflections

• What is the case? Which associations can we see in our data?
• What happens if we intervene and actively change something?
• What would have happened otherwise — in a counterfactual world?

24.1 Introduction

Having discussed “data”, “science”, and “statistics”, is “causal modeling” just another confusing term? On the contrary,

it will anchor the elusive notions of science, knowledge, and data
in a concrete and meaningful setting

(Pearl & Mackenzie, 2018, p. 11)

Main reason: Data does not reveal itself.

Playing a round of “good cop, bad cop”:

• The bad news is:

Data are profoundly dumb

(Pearl & Mackenzie, 2018, p. 6)

We need to organize observations and provide structure — go beyond data.

• The good news is:

You are smarter than your data

(Pearl & Mackenzie, 2018, p. 21)

24.1.1 Data and tools

Using functionality from the ggdag package (Barrett, 2024) and data from the quartets package (D’Agostino McGowan, 2023):

library(ggdag)     # drawing causal diagrams
library(quartets)  # data

See the online documentation for details on the quartets datasets.

The R packages used in the book Causal Inference in R (by Malcolm Barrett, Lucy D’Agostino McGowan, and Travis Gerke) can be installed as follows:

# install.packages("pak")  # utility tool for installing sets of R pkgs
pak::pak(c(
  "r-causal/causalworkshop",
  "r-causal/ggdag",
  "r-causal/halfmoon",
  "r-causal/propensity",
  "r-causal/tipr",
  "LucyMcGowan/touringplans"
))

# Note that this installed 48 packages.

Color settings:

library(unikn)
col_node <- pal_seeblau[[1]]
col_lbl  <- "black" # "white"

24.2 Essentials

Detecting patterns is the realm of statistics. Causal models go beyond the data by imposing a structure.

24.2.1 Causal inference

The ladder of causation distinguishes 3 rungs/levels of analysis and insight (Pearl & Mackenzie, 2018):

1. see: Detecting associations/Observing patterns in data: What corresponds to seeing \(x\)?

2. do: Performing interventions: What if I do \(x\)? How?

3. imagine: Counterfactuals enable understanding: What if I had done \(y\) instead? Why?

Examples

Examples from Pearl & Mackenzie (2018):

Causal diagrams provide a representation of scenarios, but also allow for reasoning: Working out the effects of observations, interventions, and counterfactuals.

Diagram with deterministic edges:

Reasoning through effects of:

1. observation: If prisoner is observed to be dead, was the court order present?
2. intervention: What if A decided to shoot?
3. counterfactual: What if A refused?

Diagram with probabilistic edges:

Value of counterfactual reasoning:

• If a vast majority of the population is vaccinated, it seems that vaccination kills more people than the disease.
• However, if nobody was vaccinated, the disease would cause far more deaths.

24.2.2 A causal quartet

Guiding question:

• Can we infer the causal mechanism (i.e., the interplay between causes and effects) from the data (values of variables)?

An introductory example from McGowan et al. (2024):

Table 24.1: Table 3 of D’Agostino et al. (2023).

dataset	ate_x	ate_xz	cor
(1) Collider	1	0.55	0.7
(2) Confounder	1	0.50	0.7
(3) Mediator	1	0.00	0.7
(4) M-Bias	1	0.88	0.7

Table 24.2: Table 4 of D’Agostino et al. (2023).

dataset	ate_x	ate_xz	truth
(1) Collider	1	1.00	1.0
(2) Confounder	1	0.50	0.5
(3) Mediator	1	1.00	1.0
(4) M-Bias	1	0.88	1.0

Despite some differences in the raw data, the linear relation between exposure and outcome are identical for all four sets of data within a quartet.

24.2.3 Causal diagrams

On the methodology of using graphs:

Using graphs as “reasoning engines,” namely, bringing to light the logical ramifications of the information used in their construction.

Blog of Judea Pearl (2023-01-04)

Causal diagrams are a tool for visualizing our assumptions about the causal structure of a question we aim to answer. They both anchor our thinking, as well as allow us to communicate our hypotheses. And when trained to read them, they even inform us about possible ways of estimating unbiased effects in a causal network of variables.

A popular form of causal diagram is called directed acyclic graphs (DAGs). Visually, DAGs depict the causal structure between variables as edges and nodes. The variables are shown as nodes (aka. points or vertices), while the arrows going from one variable to another are edges (aka. arcs or arrows). DAGs are

• directed because their arrows point in one direction
• acyclic because variables must not cause themselves (i.e., no circles)

The causal DAGs we introduce in this section are also known as structural causal models (SCMs, Pearl et al., 2016).

The following examples are based on Section 4.1 Visualizing causal assumptions of the textbook Causal inference in R. The introductory article by Lübke et al. (2020) provides similar examples.

In Figure 24.1, there are 2 nodes, x and y, and one edge going from x to y. This represents that “x causes y” or “y listens to x”:

Figure 24.1: A causal diagram (or DAG) for “x causes y” or “y listens to x”.

Measuring the causal effect of x on y essentially asks for a numeric estimate of this arrow.

• Typical DAGs involve more than two variables and one arrow. A series of arrows form a path. The three types of paths in DAGs are known as forks, chains, and colliders (aka. inverse forks).

Figure 24.2 shows these three types of causal relationships (i.e., chains, forks, and colliders):

Figure 24.2: Three basic causal structures or DAGs.

In these DAGs, the direction of the arrows and the relationships of interest determine which type of path a series of variables represents:

1. chains represent direct causes. In chain paths, a series of arrows points in the same direction. The variable or node q is called a mediator: it lies on the causal path from x to y. In our diagram, the only path from x to y is mediated through q.

2. forks represent a common or mutual cause of two variables. In fork paths, the arrows from x to y point in different directions. In our diagram, q causes both x and y, so that q is a confounder.

3. colliders represent a mutual descendant of two variables. In collider paths, two arrowheads meet at one variable. As we have seen for forks, the arrows from x to y point in different directions, but in the opposite direction than in forks (which is why colliders are also called reverse forks). This means that the collider variable q is caused by two other variables (and q itself often called a collider). Here, x and y both cause q.

Additionally, we can also categorize DAGs into “open” vs. “closed” paths:

• Paths that transmit association are open paths;

• Paths that do not transmit association are closed paths.

Thus, chains and forks are open paths, while colliders are closed paths.

Quick demo structures

The ggdag package (Barrett, 2024) provides utility functions for quickly creating DAGs for demonstrating basic causal structures, including mediation_triangle(), confounder_triangle(), collider_triangle(), m_bias(), and butterfly_bias(). For instance:

library(ggdag)
m_bias() |> ggdag() + theme_dag()

24.2.4 Viewing DAGs through a statistical lens

When analyzing quantitative data, our interpretations and conclusions still rely on qualitative assumptions. This is especially true when dealing with observational data (i.e., data without randomized exposure to experimental conditions). The following section uses simulated data and either visualizations or simple linear regressions to illustrate the effects of bias, confounding, and of controlling for covariates.

When viewing causal structures through the lens of statistics, the simplest possible question concerns the relationship between two variables x and y. If we only consider their correlation, we can characterize our three key DAGs from above as follows:

1. In the chain, x and y are associated, but their relationship is mediated by q.

2. In the fork, x and y are associated, but there is no arrow pointing from x to y. As the mutual cause q causes both x and y, x and y are confounded by q and show a spurious association. Statistically adjusting for q (or “controlling” for q) will block the bias from confounding and reveal the true relationship between x and y.

3. In the collider, x and y are not associated. However, controlling for q has the opposite effect than with confounding: It introduces bias.

What happens to the relation between x and y, when we statistically control for q? The following examples illustrate the effects of controlling for q on the relation between x and y for each of the three causal paths (see DAGs above).

1. Chains: Controlling for a mediator reveals direct effect

For chains, whether or not we adjust for mediators depends on the research question. Here, adjusting for a mediator q results in a null estimate of the effect of x on y. Because the only effect of x on y is via q, no other effect remains.

The effect of x on y mediated by q is called the indirect effect, while the effect of x on y directly is called the direct effect. If we only care about a direct effect, controlling for q might be what we want. But if we want to know about both effects, we should not adjust for q.

Figure 24.3 illustrates the difference between total and direct effects by adjusting for a mediator q:

Figure 24.3: When an effect of x on y is mediated by q, adjusting for q reveals the direct effect, rather than the total effect.

Interpretation of Figure 24.3:

• The unadjusted effect of x on y (in Panel A) represents the total (direct and indirect) effect.

• Since the total effect is entirely due to the path mediated by q, no relationship remains when we adjust for q (in Panel B). This null effect is the direct effect.

Linear regression

The following code shows the difference between the total effect (of x on y, without adjustment for the mediator q) and the direct effect (with adjustment for the mediator q) through a linear regression lens:

Table 24.3: Chain A: Not adjusting for mediator q (total effect)

	Estimate	Std. Error	t value	Pr(>|t|)
(Intercept)	1.043	0.041	25.253	0
x	0.624	0.040	15.621	0

Table 24.4: Chain B: Adjusting for mediator q (direct effect)

	Estimate	Std. Error	t value	Pr(>|t|)
(Intercept)	0.023	0.054	0.423	0.673
x	-0.028	0.042	-0.674	0.500
q	2.066	0.087	23.651	0.000

Note that both the total effect and the direct effect is real — and neither is due to bias. However, they address and answer different research questions.

2. Fork: Controlling for a confounder removes bias

In a fork, two variables x and y are not causing each other, but are both affected by a common cause q. If we naively measured the correlation between x and y, we may think that both variables are related. But this relationship can change, when controlling for the confounding influence of q.

Figure 24.4 illustrates the control/removal of bias by adjusting for a confounder q:

Figure 24.4: When an effect of x on y is confounded by q, adjusting for q removes the bias.

Interpretation of Figure 24.4:

• x and y are not causing each other, but both affected by a mutual cause q. Statistically measuring the unadjusted effect of y ~ x (in Panel A) yields a biased result, as it includes information about q.

• When controlling for q, however (in Panel B), the bias disappears: Within each level of q, x and y are unrelated.

Linear regression

The following code shows the difference between the biased effect (of x on y, without adjustment for the confounder q) and the unbiased effect (with adjustment for the confounder q) through a linear regression lens:

Table 24.5: Fork A: Not adjusting for confounder q (biased)

	Estimate	Std. Error	t value	Pr(>|t|)
(Intercept)	1.188	0.058	20.403	0
x	0.644	0.047	13.663	0

Table 24.6: Fork B: Adjusting for confounder q (unbiased)

	Estimate	Std. Error	t value	Pr(>|t|)
(Intercept)	0.067	0.046	1.457	0.145
x	0.026	0.033	0.775	0.439
q	2.932	0.074	39.769	0.000

3. Collider: Controlling for a collider introduces bias

Colliders differ from forks. In a collider, x and y are not associated, but both cause q. Adjusting for q has the opposite effect than with confounding: It opens a biasing pathway. Sometimes, people draw the path opened up by conditioning on a collider that connects x and y.

Figure 24.5 illustrates the introduction of bias by adjusting the continuous relationship between x and y for a (binary) collider q:

Figure 24.5: When x and y both cause / collide into q, adjusting for q introduces bias.

Interpretation of Figure 24.5:

• When we do not include q (in Panel A), we find no relationship between x and y. That’s the correct result (given the data-generating mechanism here):

• However, when we include q (in Panel B), we can detect information about both x and y, and they appear correlated (although they do not directly cause each other): Across levels of x, those with q = 0 have lower levels of y. Paradoxically, this association seemingly flows back in time. Of course, that can’t happen from a causal perspective, so controlling for q is the wrong thing to do here. We end up with a biased effect of x on y.

Linear regression

The following code shows the difference between the unbiased effect (of x on y, without adjustment for the collider q) and the biased effect (with adjustment for the collider q) through a linear regression lens:

Table 24.7: Collider A: Not adjusting for collider q (unbiased)

	Estimate	Std. Error	t value	Pr(>|t|)
(Intercept)	-0.020	0.032	-0.649	0.516
x	-0.028	0.032	-0.885	0.376

Table 24.8: Collider B: Adjusting for collider q (biased)

	Estimate	Std. Error	t value	Pr(>|t|)
(Intercept)	-0.693	0.035	-19.748	0
x	-0.281	0.026	-10.768	0
q	1.376	0.052	26.454	0

In all three paths, the interpretation of the effect of x on y drastically changes if we adjust for a third variable q.

Conclusion

Real causal structures will typically involve more than three variables. But before we can start tackling more complicated scenarios, we ought to understand these basic ABC cases.

Importantly, our artificial and fictitious examples used known data-generating mechanisms. Thus, they jointly illustrate that the effects and interpretation of controlling for a variable q differ depending on the data-generating mechanism. Thus, we cannot determine the correct analysis or interpretation without considering the causal relations between the variables.

Continue with:

• Section 4.2 DAGs in R.

24.3 Conclusion

Felix, qui potuit rerum cognoscere causas
(Lucky is he who has been able to understand the causes of things)

Virgil (29 BC) (from Pearl et al., 2016, p. xii)

24.3.1 Summary

Key points

Causal inference asks not just what goes with what, but why — and what would happen if we intervened:

Why causal inference

• Correlation does not imply causation: To explain outcomes, anticipate interventions, or assign credit, we need a model of the data-generating process, not just patterns in the data.
• Data alone are “profoundly dumb” (Pearl & Mackenzie, 2018) — drawing causal conclusions requires explicit, qualitative assumptions that we bring to the data.

The ladder of causation

• Pearl’s three rungs (Pearl & Mackenzie, 2018) distinguish levels of causal insight: seeing (associations: what goes with \(x\)?), doing (interventions: what if I do \(x\)?), and imagining (counterfactuals: what if I had done \(y\) instead?).
• Statistics and EDA mostly live on the first rung; causal models let us climb to the second and third.

Causal diagrams (DAGs)

• A directed acyclic graph (DAG) encodes our causal assumptions as nodes (variables) and edges (causal arrows) — directed because arrows point one way, acyclic because nothing causes itself.
• Paths take three basic shapes: chains (x -> q -> y, where q is a mediator), forks (x <- q -> y, where q is a confounder), and colliders (x -> q <- y).

Adjusting for a third variable

• Whether “controlling for” a variable q helps or hurts depends entirely on the causal structure: it reveals a direct effect in a chain, removes confounding bias in a fork, but introduces bias in a collider.
• Hence we cannot choose the correct analysis from the data alone — the causal assumptions (the DAG) must come first.

24.3.2 Resources

Add pointers to cheatsheets and additional links here.

Background readings

• Foundational articles and books by Judea Pearl (e.g., Pearl, 2009; Pearl et al., 2016; Pearl & Mackenzie, 2018)

• Introductory articles by Lübke et al. (2020) and McGowan et al. (2024)

R resources

• The Causal inference in R blog, the Causal inference in R workshop and the corresponding textbook Causal Inference in R (by Malcolm Barrett, Lucy D’Agostino McGowan, and Travis Gerke)

• Tutorial on Statistical Modeling and Causal Inference (by Simon Munzert, Lisa Oswald, and Sebastian Ramirez Ruiz, 2021)

24.3.3 Preview

The remaining chapters of Part 6 turn from analyzing data to creating with it: generative art (Chapter 25) and interactive applications with Shiny (Chapter 26).

24.4 Exercises

24.4.1 Adjusting causes bias

This exercise is based on 3.1 Example 1: Adjusting causes bias of Lübke et al. (2020) (p. 134f., but with different labels):

Data (and data-generating mechanism):

1. we hear: \(h = E_h\), with \(E_h ∼ \mathcal{N}(0, 1)\)
2. we know: \(k = 5h + E_k\), with \(E_k ∼ \mathcal{N}(0, 1)\)
3. we tell: \(t = 3k + E_t\), with \(E_t ∼ \mathcal{N}(0, 1)\)

where \(\mathcal{N}(\mu = 0, \sigma = 1)\) stands for the Normal distribution.

DAG

Linear regression analysis

# A. Total effect:
lm_1a <- stats::lm(tell ~ hear)
summary(lm_1a)$coefficients |> knitr::kable(digits = 3, label = NA, caption = "(\\#tab:causal-ex01-3a) A. Not adjusting for `know`")

Table 24.9: A. Not adjusting for know

	Estimate	Std. Error	t value	Pr(>|t|)
(Intercept)	-0.022	0.097	-0.231	0.818
hear	15.121	0.098	154.585	0.000

# B. Adjust for mediator "know" (as covariate):
lm_1b <- lm(tell ~ hear + know)
summary(lm_1b)$coefficients |> knitr::kable(digits = 3, label = NA, caption = "(\\#tab:causal-ex01-3b) B. Adjusting for `know`")

Table 24.10: B. Adjusting for know

	Estimate	Std. Error	t value	Pr(>|t|)
(Intercept)	-0.005	0.031	-0.161	0.872
hear	0.122	0.163	0.748	0.455
know	2.982	0.032	94.026	0.000

24.4.2 Adjusting removes bias

This exercise is based on 3.2 Example 2: Adjusting removes bias of Lübke et al. (2020) (p. 135f., but with different labels):

Data (and data-generating mechanism):

1. IQ: Intelligence score \(I\) is noisy: \(I = U_I\), with \(U_I ∼ \mathcal{N}(100, 15)\)
2. learn: Learning time \(L\) listens to \(I\): \(L = 200 − I + U_L\), with \(U_L ∼ \mathcal{N}(0, 1)\)
3. test: Test score \(T\) listens to \(I\) and \(L\): \(T = (.50 \cdot I) + (.10 \cdot L) + U_T\), with \(U_T ∼ \mathcal{N}(0, 1)\)

DAG

• Note that IQ is a common cause of both learn and test.

Linear regression analysis

# A. Biased effect (not adjusting for 'IQ'):  
lm_2a <- lm(test ~ learn)
summary(lm_2a)$coefficients |> knitr::kable(digits = 3, label = NA, caption = "(\\#tab:causal-ex02-3a) A. Not adjusting for IQ")

Table 24.11: A. Not adjusting for IQ

	Estimate	Std. Error	t value	Pr(>|t|)
(Intercept)	100.083	0.234	427.866	0
learn	-0.401	0.002	-173.920	0

# => A seemingly negative effect of 'learn' on 'test'!

# B. Adjust for common cause 'IQ' (as covariate):
lm_2b <- lm(test ~ learn + IQ)
summary(lm_2b)$coefficients |> knitr::kable(digits = 3, label = NA, caption = "(\\#tab:causal-ex02-3b) B. Adjusting for IQ")

Table 24.12: B. Adjusting for IQ

	Estimate	Std. Error	t value	Pr(>|t|)
(Intercept)	3.446	6.339	0.544	0.587
learn	0.082	0.032	2.577	0.010
IQ	0.484	0.032	15.253	0.000

# => True effect of 'learn' on 'test' is positive!

24.4.3 Randomized experiments

This exercise is based on 3.4 Example 4: Randomized experiments of Lübke et al. (2020) (p. 136f., but with different labels):

Data (and data-generating mechanism):

1. IQ: Intelligence score \(I\) is noisy: \(I = U_I\), with \(U_I ∼ \mathcal{N}(100, 15)\)

2. learn: Learning time is randomly set to be either 80 or 120: \(L = (U_L \cdot 80) + (1− U_L) \cdot 120\), with \(U_L ∼ \mathcal{B}(.50)\)

3. test: Test score \(T\) listens to \(I\) and \(L\): \(T = (.50 \cdot I) + (.10 \cdot L) + U_T\), with \(U_T ∼ \mathcal{N}(0, 1)\)

DAG

Descriptives

#> [1] -0.998
#> [1] 0.015

Table 24.13: Descriptive summary of experimental results

learn_exp	n	mn_IQ	mn_test_exp
80	504	99.39	57.66
120	496	99.83	61.95

Linear regression analysis

Table 24.14: A. Exp. effect, not adjusting for IQ

	Estimate	Std. Error	t value	Pr(>|t|)
(Intercept)	49.079	1.200	40.884	0
learn_exp	0.107	0.012	9.100	0

Table 24.15: B. Exp. effect, adjusting for IQ

	Estimate	Std. Error	t value	Pr(>|t|)
(Intercept)	-0.029	0.262	-0.109	0.913
learn_exp	0.102	0.002	65.034	0.000
IQ	0.499	0.002	235.985	0.000

Table 24.16: C. Effect of IQ (unconditional)

	Estimate	Std. Error	t value	Pr(>|t|)
(Intercept)	9.924	0.487	20.386	0
IQ	0.501	0.005	103.559	0

Conclusion

No causation without manipulation

(Holland, 1986)

Experimental manipulation (of learn_exp) makes the effects interpretable.

24.4.4 More causal quartets

1. Variation quartets The data variation_causal_quartet demonstrates that we can get the same average treatment effect despite variability across some pre-treatment characteristic (here called covariate):

Table 24.17: 1. Using variation_causal_quartet: Average treatment effect by dataset

dataset	ATE
(1) Constant effect	0.1
(2) Low variation	0.1
(3) High variation	0.1
(4) Occasional large effects	0.1

2. Heterogeneity quartets The data heterogeneous_causal_quartet demonstrates how we can observe the same causal effect under different patterns of treatment heterogeneity:

Table 24.18: 2. Using heterogeneous_causal_quartet: Average treatment effect by dataset

dataset	ATE
(1) Linear interaction	0.1
(2) No effect then steady increase	0.1
(3) Plateau	0.1
(4) Intermediate zone with large effects	0.1

In both cases, create a visualization that illustrates what is really going on between covariate and outcome (grouped by exposure) in each dataset.

Sources: This exercise is based on Gelman et al. (2024) and data from the quartets package (D’Agostino McGowan, 2023).