The unit-specific quantity might be:

- "**Descriptive**: the factual value that some variable actually would take for unit *i* in the absence of intervention;" e.g., $Y_{i}$: "whether person *i* is employed ...
- **Causal**: the value some variable would take if a treatment variable *D* was set to a particular value *d*, producing the potential outcome;" e.g., $Y_{i}(d)$: "whether person *i* would be employed if they received job training)"
- **Causal Difference** ("A difference in potential outcomes): the effect of job training on the employment of person *i*"
- **Caused by Multipel Variables**: "An outcome that would be realized under intervention of more than one variable;" e.g.,
   - "A **mediation** claim about the outcome $Y_{i}(d, m)$ that unit *i* would realize if the treatment were set to $D = d$ and some mediator were set to $M = m$."
   - An **interaction** claim of "whether person *i* would be employed if they received job training and childcare."

Different Forms of Unit-specific Quantity of Theoretical Estimand

"The unit-specific quantity clarifies whether the research goal is causal, and if so, what counterfactual intervention is being considered." It "involves a substantive question: what factual or counterfactual thing would we like to know for each unit in the population?"

University of Michigan - Ann Arbor

- Unit-specific quantity
- Target population

"The goal is often a quantity involving two components: a unit-specific quantity subscripted by i aggregated over a target population of units."

Components of Theoretical Estimand

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

Unit-specific Quantity of Theoretical Estimand

The target Population specifies "over whom or what do we aggregate that unit-specific quantity?"

Target Population of Theoretical Estimand

1- **Estimated Rate**:
$$\frac{1}{n} \sum_{i=1}^n Y_{i}$$
- **Target population**: Mean over every *i* among U.S. adults
- **Unit-specific quantity ($Y_{i}$)**: Whether each *i* is employed

2- **Causal question**: How would the probability of employment differ if we enrolled a randomly chosen individual in job training or not?
$$\frac{1}{n} \sum_{i=1}^n \left( Y_{i}(1) - Y_{i}(0) \right)$$
- **Target population**: Mean over every *i* among U.S. adults
- **Unit-specific quantity**: Whether each *i* is employed
   - **$Y_{i}(1)$**: Employment if enrolled in job training
   - **$Y_{i}(0)$**: Employment if not enrolled in job training


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