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I am wondering, within the context of causal inference, what it means to "non-parametrically" identify a causal effect within the super-population perspective. For example, in Hernan/Robins Causal Inference Book Draft:

https://cdn1.sph.harvard.edu/wp-content/uploads/sites/1268/2019/02/hernanrobins_v1.10.38.pdf

It defines non-parametric identification on pg. 43 and 123 as:

...identification that does not require any modeling assumptions when the size of the study population is quasi-infinite. By acting as if we could obtain an unlimited number of individuals for our studies, we could ignore random fluctuations and could focus our attention on systematic biases due to confounding, selection, and measurement. Statisticians have a name for problems in which we can assume the size of the study population is effectively infinite: identification problems.

I understand the identification part to mean that under the strong ignorability assumption, there is only ONE way for the observed data to correspond to a causal effect estimand. What confuses me is why we need to assume the size of the study is quasi-infinite.

For example, in the book it gives an example of a 20 person study where each subject was representative of 1 billion identical subjects, and to view the hypothetical super-population as that of 20 billion people. Specifically, on pg. 13 it states that:

... we will assume that counterfactual outcomes are deterministic and that we have recorded data on every subject in a very large (perhaps hypothetical) super-population. This is equivalent to viewing our population of 20 subjects as a population of 20 billion subjects in which 1 billion subjects are identical to the 1st subject, 1 billion subjects are identical to the 2nd subject, and so on.

My confusion here is what it means to assume a single person is representative of 1 billion identical individuals. Is it assuming that each of the 1 billion are identical with respect to their outcomes and treatment only, but differ with respect to the covariates? Or is it assuming the individual is a summary measure of the 1 billion? My instinct is that the notion of the 1 billion is entertaining the fact we may draw many times without having a case where we have a lack of samples. I.e., small sample sizes result in more unstable estimates.

Essentially, what is so crucial about assuming there are many identical individuals in the "background", if they are just going to be the same as a patient you observe? What happens or breaks down if instead of the 1 billion, we only had 2 identical individuals?

Thank you for any insight.

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Thank you for bringing this interesting book to our attention. Below are my two cents.

...we will assume that counterfactual outcomes are deterministic and that we have recorded data on every subject in a very large (perhaps hypothetical) super-population.

The above seems to merely be referring to the Central Limit Theorem and the Law of Large Numbers. In other words, as the sample size N increases, from a frequentist perspective, your standard error around an effect modifier (or equally causal risk ratio, risk difference, etc.) estimate is shrinking to virtually zero; or from a Bayesian perspective, your credible interval is collapsing to a single point. In other words, the posterior distribution becomes a point estimate. So in theory a deterministic value as opposed to a random variable.

Is it assuming that each of the 1 billion are identical with respect to their outcomes and treatment only, but differ with respect to the covariates? Or is it assuming the individual is a summary measure of the 1 billion? My instinct is that the notion of the 1 billion is entertaining the fact we may draw many times without having a case where we have a lack of samples. I.e., small sample sizes result in more unstable estimates.

I agree with your intuition. See also the side note on Pg. 9:

Technically, when $i$ refers to a specific individual, such as Zeus, $Y_i^a$ is not a random variable because we are assuming that individual counterfactual outcomes are deterministic.

In addition on Pg. 14 they note:

However, for pedagogic reasons, we will continue to largely ignore random error until Chapter 10. Specifically, we will assume that counterfactual outcomes are deterministic

So all of the slightly confusing language (around 1 observation representing 1 billion, etc.) is for simplification to ignore "random error" in these estimates and focus the attention on systematic biases (errors) due to confounding, selection, and measurement.

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  • $\begingroup$ Thanks for your input. Just to clarify, would you suggest that their definition of "identical" refers to 1 billion individuals who have the same covariates, but different observed outcomes as assigned by the treatment assignment? In other words, the potential outcomes and the covariates are the same, but the observed outcome will be possibly different as assigned by the treatment assignment? $\endgroup$ – user321627 May 1 '19 at 18:25
  • $\begingroup$ To me it reads as 1 billion individuals who have the identical co-variate values and observed outcomes. If the observed outcomes could vary (randomness), your effect modifiers, etc. would no longer be deterministic. $\endgroup$ – Zhubarb May 2 '19 at 7:25
  • $\begingroup$ Thanks, I am trying to think of a counterexample to the above where instead of 1 billion, we had say only 5 identical individuals. I am doing this to try to understand why the 1 billion is needed. In the case of 5 identical individuals, do you have any insight where things could go wrong? It seems if we already have a fixed sample from the superpopulation, why would we worry about the other billion identical possibilities that we do not actually consider in our sample? Thank you for any input! $\endgroup$ – user321627 May 2 '19 at 19:35
  • $\begingroup$ Hello, nothing is wrong with 5 samples in reality, it is just that the estimate (e.g. of the causal risk ratios, etc.) will have randomness. The higher the sample size, the smaller the (standard) error on your estimate.They suppose very large numbers just to be able to say, "assume there is no random error" so they can fully focus on systematic error. $\endgroup$ – Zhubarb May 7 '19 at 8:19

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