Simplified causal diagram describing a firing squad 

University of Michigan - Ann Arbor

A unidirectional graph of nodes and arrows, where each node represents a variable (measure) under the study and the arrows represent how the nodes causally affect each other. "The Book of Why" calls them the language of knowledge.

Instead of causal diagrams, econometricians use mathematical equations and statisticians use a set of assumptions that characterize the arrows in causal diagrams. 

A causal diagram, regardless of its method of demonstration, should illustrate the cause-effect relations in the environment that have affected the data generation process.

Causal diagram (for Path analysis)

Pearl, J., & Mackenzie, D. (2018). $$\mathit{The \ Book \ of \ Why: \ The \ New \ Science \ of \ Cause \ and \ Effect.}$$ Basic Books.

The Book of Why

Staplin, N., Herrington, W. G., Judge, P. K., Reith, C. A., Haynes, R., Landray, M. J., ... & Emberson, J. (2017). Use of causal diagrams to inform the design and interpretation of observational studies: an example from the Study of Heart and Renal Protection (SHARP). Clinical Journal of the American Society of Nephrology, 12(3), 546-552. https://doi.org/10.2215/cjn.02430316

Use of Causal Diagrams to Inform the Design and Interpretation of Observational Studies: An Example from the Study of Heart and Renal Protection (SHARP)

- Ancestors
- Descendants
- Children
- Parents

Types of relations in a Causal Diagram

A conventional measure of a treatment's effectiveness. It averages treatment efficacy over a population and can be found using a randomized control trial (RCT). It measures how much an increase in a single unit of treatment variable X will affect the outcome variable Y. When you have properly eliminated all confounding variables, you can obtain an estimate through running a regression.

Average Causal Effect (ACE)

Path analysis is the use of linear regression and multiple regression techniques to evaluate relationships between multiple dependent variables and multiple independent variables. Lines are drawn unidirectionally based on correlation/association between two variables. Path analysis is often thought of as the earliest form of statistical analysis closest to causal inference despite still being association-based and can be thought of as a predecessor to Bayesian networks.

Path Analysis 

Streiner, D. L. (2005). Finding Our Way: An Introduction to Path Analysis. The Canadian Journal of Psychiatry, 50(2), 115-122. doi:10.1177/070674370505000207
Link: https://journals.sagepub.com/doi/pdf/10.1177/070674370505000207

Introduction to Path Analysis 

"Each theoretical estimand is linked to an empirical estimand involving only observable quantities (e.g., a difference in means in a population) by assumptions about the relationship between the data we observe and the data we do not. These identification assumptions can be conveyed in a Directed
Acyclic Graph (DAG. i.e., a causal diagram)."

Empirical Estimand

Pearl outlines three basic types of junctions in causal diagrams which appear commonly and help explain the data. The three junctions take the forms: 1. $$A \rightarrow B \rightarrow C$$ (Chain) 2. $$A \leftarrow B \rightarrow C$$ (Fork) 3. $$A \rightarrow B \leftarrow C$$ (Collider)

Three Basic Junctions in Causal Diagrams

While causal diagrams are a specific type of Bayesian network where arrows indicate direct causal relations (or the possibility of one), general Bayesian networks do not require causal relationships. The primary difference lies in the meaning of the arrows:

- **Probabilistic Bayesian Networks:** Arrows directed into $$Y$$ mean that the probability of $$Y$$ is governed by conditional probability tables for $$Y$$, given observations of its parent variables.
- **Causal Bayesian Networks:** Conditional probability tables specify the probability of $$Y$$ given interventions on the parent variables.
- **Structural Causal Models:** Lacking conditional probability tables, arrows mean $$Y$$ is a deterministic function of its parents and an exogenous variable $$U_Y$$: $$Y = f_Y(X, A, B, C,…, U_Y)$$.

Both probabilistic and causal Bayesian networks specify probabilities for $$Y$$, not specific values.

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