Causal Inference Is Not Just a Statistics Problem

Figures & data

Fig. 1 Example DAG. Here, there are three nodes representing three factors: cause, other cause, and effect. The arrows demonstrate the causal relationships between these factors such that cause causes effect and other cause causes both cause and effect.

Fig. 2 Directed acyclic graphs describing the four data generating mechanisms: (1) Collider (2) Confounder (3) Mediator (4) M-Bias.

Table 1 Causal terminology along with the data generating mechanism for each of the four datasets included in the causal quartet.

Technical term	Explanation	Data generating mechanism
(1) Collider	The exposure, X, causes a factor, Z, and the outcome, Y, causes a factor, Z. Adjusting for Z when estimating the effect of X on Y would yield a biased result.	$X\sim N(0,1),Y=X+{\epsilon }_y,{\epsilon }_y\sim N(0,1)$ $Z=0.45X+0.77Y+{\epsilon }_z,{\epsilon }_z\sim N(0,1)$
(2) Confounder	A factor, Z, causes both the exposure, X, and the outcome, Y. Failing to adjust for Z when estimating the effect of X on Y would yield a biased result.	$Z\sim N(0,1),X=Z+{\epsilon }_x,{\epsilon }_x\sim N(0,1)$ $Y=0.5X+Z+{\epsilon }_y,{\epsilon }_y\sim N(0,1)$
(3) Mediator	An exposure, X, causes a factor, Z, which causes the outcome, Y. Adjusting for Z when estimating the effect of X on Y would yield the direct effect, not adjusting for Z would yield the total effect of X on Y. The direct effect represents the relationship between X and Y independent of any mediator, while the total effect includes both the direct effect and any indirect effects mediated by the potential mediator.	$X\sim N(0,1)$, $Z=X+{\epsilon }_z,{\epsilon }_z\sim N(0,1)$ $Y=Z+{\epsilon }_y,{\epsilon }_y\sim N(0,1)$
(4) M-Bias	There are two additional factors, U1 and U2. Both cause Z, U1 causes the exposure, X, and U2 causes the outcome, Y. Adjusting for Z when estimating the effect of X on Y will yield a biased result.	$U_1\sim N(0,1),U_2\sim N(0,1)$ $Z=8U_1+U_2+{\epsilon }_z,{\epsilon }_z\sim N(0,1)$ $X=U_1+{\epsilon }_x,{\epsilon }_x\sim N(0,1)$ $Y=X+U_2+{\epsilon }_y,{\epsilon }_y~N(0,1)$

Fig. 3 100 points generated using the data generating mechanisms specified (1) Collider (2) Confounder (3) Mediator (4) M-Bias. The blue line displays a linear regression fit estimating the relationship between X and Y; in each case, the slope is 1.

Table 2 Correct causal models and average causal effects for each data-generating mechanism.

Data generating mechanism	Correct causal model	Correct average causal effect
(1) Collider	Y ˜ X	1
(2) Confounder	Y ˜ X ; Z	0.5
(3) Mediator	Direct effect: Y ˜ X ; Z Total Effect: Y ˜ X	Direct effect: 0 Total effect: 1
(4) M-Bias	Y ˜ X	1

NOTE: The notation $X;Z$ implies that we should adjust for Z when estimating the causal effect. In other words, for the confounder data generating mechanism and direct effect mediator model, the potential outcomes are independent of exposure given the observed factor Z.

Table 3 Estimated average treatment effects under each data generating mechanism with and without adjustment for Z as well as the correlation between X and Z.

Data generating	$\widehat{\text{ATE}}$ not adjusting	$\widehat{\text{ATE}}$ adjusting	Correlation
mechanism	for Z	for Z	of X and Z
(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

Fig. 4 Time-ordered collider DAG (with time increasing from left to right) where each factor is measured twice. X is the exposure, Y is the outcome, and Z is the measured factor. The highlighted Z node indicates which time point is being adjusted for when estimating the average treatment effect of the highlighted X on the highlighted Y.

Table 4 Coefficients for the exposure under each data generating mechanism depending on the model fit as well as the correlation between X and Z.

Data generating mechanism	$\widehat{\text{ATE}}$ not adjusting for pre-exposure Z	$\widehat{\text{ATE}}$ adjusting for pre-exposure Z	Correct average causal effect
(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

Data Availability Statement

The causal quartet datasets presented in this article are available in an R package titled quartets (D’Agostino McGowan 2023). https://r-causal.github.io/quartets/