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Recently Lundberg, 2021 [1] emphasized the necessity to define a unit-specific quantity, target population, and causal diagram, to clarify the theoretical and empirical estimands of any quantitative study. The state "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." However, almost all the examples I see compare the aggregate (usually average) unit-specific quantity across two groups of a dichotomous variable, such as males vs. females, treated vs. control, ... I found no single example of analyzing the correlation between two continuous variables or the causal effect of one continuous variable on another one.

Please give me such an example or explain how one can formulate the unit-specific quantity for the causal effect of one continuous variable on another one.

References:

[1] 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.

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  • $\begingroup$ Can you describe what you mean by "unit-specific"? Do you mean individual treatment effects? We generally can't estimate those with any certainty regardless of the treatment type. Average treatment effects may be identified but individual treatment effects are not. There are average treatment effects for continuous treatments, though; are those what you are asking about? $\endgroup$
    – Noah
    Jan 9, 2022 at 0:03
  • $\begingroup$ Please check out the cited paper. They've fully explained it. $\endgroup$ Jan 9, 2022 at 16:34

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Unless you are in the rare situation where X vs Y is linear, the most sensible estimand is the continuous function relating X to a property of Y such as the median or mean. If the relationship were linear then the (constant) slope would be sufficient. The estimator of the regression function would be for example a regression spline.

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  • $\begingroup$ Please check out the cited paper for the definitions of theoretical and empirical estimands. $\endgroup$ Jan 9, 2022 at 16:34
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    $\begingroup$ I did check it. The paper does not show how to deal with the most common situation where X has a nonlinear effect. It only covers an arbitrary "select two values of X and see how Y differs at those two values" approach. $\endgroup$ Jan 9, 2022 at 19:18

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