When the junction $A \leftarrow B \rightarrow C$ appears in a Causal Diagram, we say that B is a confounder for both A and C. It might appear that there is a causal relationship between A and C, because there is an observable relationship between A and C when B is varied. However, if we condition on B ("hold B constant") we observe that A and C become conditionally independent. 

$A \leftarrow B \rightarrow C$: The Fork and Confouders

When a single node is caused by two other nodes we encounter something called "collider-bias" where one condition explains-away another. 

Since A and C both effect the outcome of B, by conditioning on B, A and C become **dependent**.

$A \rightarrow B \leftarrow C$: The Collider

A proxy is used as a substitute measure when a variable that is not directly measurable is needed for analysis.  For a variable to be a proxy for another variable, the proxy variable needs to be highly correlated with the variable of interest.  The proxy can be used for any variables such as a mediator, confounder, collider.  

Proxy

When the junction $$A \rightarrow B \rightarrow C$$ appears in a causal diagram, we note that the presence of $$B$$ "screens-off" the value of $$A$$. This means that $$C$$ and $$A$$ are conditionally independent given the value of $$B$$.

Chain $$A \rightarrow B \rightarrow C$$: The Screening-Off Effect

Pearl outlines 3 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)

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

As defined in "The Book of Why", causal diagrams are Bayesian networks where every arrow indicates either a direct causal relation or the possibility of a direct causal relation in the direction that the arrow is facing. Unlike causal diagrams, Bayesian networks do not have to be based on causal relations.

The meaning of the arrows is a major difference between Bayesian networks and causal diagrams. 

In a probabilistic Bayesian network, the arrows into $Y$ mean that the probability of Y is governed by the conditional probability tables for $Y$, given observations of its parent variables. The same is true for causal Bayesian networks, except that the conditional probability tables specify the probability of $Y$ given interventions on the parent variables. Both models specify probabilities for $Y$, not a specific value of $Y$. 

In a structural causal model, there are no conditional probability tables. The arrows simply mean $Y$ is a function of its parents, as well as the exogenous variable $U_Y$:
$Y = f_Y(X, A, B, C,…, U_Y)$


Relation between Bayesian Networks and Causal Diagrams

Simplified causal diagram describing a firing squad 

Causal Diagram Example

The 3 Junctions in Causal Diagram

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.

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)."

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