1- "Set a **theoretical estimand**, clearly connecting this quantity to theory"

2- "Link to an **empirical estimand**, which is informative about the theoretical estimand under some identification assumptions"

3- Use "an **estimation strategy** to learn the empirical estimand (e.g., a regression model) ... from data."

Three Critical Choices in Causal Inference

1- "The former may involve unobservable quantities such as counterfactuals, whereas the latter involves only observable data."

2- "The choice of theoretical estimands requires substantive argument about the theory and goals; the choice of empirical estimands requires conceptual argument about unobserved data."

Theoretical Estimand vs. Empirical Estimand

"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)."

University of Michigan - Ann Arbor

"The estimand is the target quantity—the purpose of the statistical analysis ... the thing we are estimating" " that connects statistical evidence to theory."

It refers "to both the theoretical and the empirical estimands."

Estimand

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)

Lundberg, I., Johnson, R., & Stewart, B. M. (2021). What is your estimand? Defining the target quantity connects statistical evidence to theory. American Sociological Review, 86(3), 532-565.
https://journals.sagepub.com/doi/pdf/10.1177/00031224211004187?casa_token=kd-8FImHZ90AAAAA:jzdKy6ZOAbd6PmxW2sXQG3CwYeSScDGeNKRJkPI0yGRzTBr2j7dUw384Bgg3Fihk9TiVLnOOUL81L4E

What Is Your Estimand? Defining the Target Quantity Connects Statistical Evidence to Theory

Theoretical estimand "sets the targets of inference" and defines the estimand "outside of any statistical model ... like an average causal effect or a population mean..., clearly connecting this quantity to theory" through argument.

"The unit-specific quantity and target population combine to define the theoretical estimand: the thing we would like to know if we had data for the full
population in all factual or counterfactual worlds of interest. A paper may have multiple theoretical estimands."

Theoretical Estimand

"We make only one point in this article. Every quantitative study must be able to answer the question: what is your estimand? The estimand is the target quantity—the purpose of the statistical analysis. Much attention is already placed on how to do estimation; a similar degree of care should be given to defining the thing we are estimating. We advocate that authors state the central quantity of each analysis—the theoretical estimand—in precise terms that exist outside of any statistical model. In our framework, researchers do three things: (1) set a theoretical estimand, clearly connecting this quantity to theory; (2) link to an empirical estimand, which is informative about the theoretical estimand under some identification assumptions; and (3) learn from data. Adding precise estimands to research practice expands the space of theoretical questions, clarifies how evidence can speak to those questions, and unlocks new tools for estimation. By grounding all three steps in a precise statement of the target quantity, our framework connects statistical evidence to theory."

Empirical Estimand

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

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)

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

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