Notation for independence of potential outcomes

Question

If we say (binary) treatment status, t is independent of potential outcomes, $\{y_1,y_o\}$, it is usually writing as

$t_i \perp \!\!\! \perp \{y_{i1},y_{io}\}$ .

I take this to mean intuitively, for one example, that there is no systematic relationship where those with higher potential outcomes from treatment are more likely to receive treatment, or the same for lower and so forth. i.e. one arbitrary example where this is not true is all the people with $y_1$ say, above the median value of the distribution of potential outcomes from treatment, are very likely to be in the treatment group.

I do not get how this is encapsulated in the above formulation- it seems to be saying that treatment status for individual i is independent of i's potential outcome with treatment and i's potential outcome without treatment? This to me looks like its saying that i's potential outcome in both states of the word constitute two different random variables? and nothing about a broader population of all people (and just this one individual i)?

Basically I do not understand this notation and how it implies what I believe it is supposed to imply.

Answer 1

This notation, as for my understanding, says: random variable $t_i$ is independent (or ortogonal) with the vector of random variables $\{y_{i0}, y_{i1}\}$. This probably means it should be independent with both of them separately, and group independence should also be true.

Indeed Rubin Causal Model can be interpreted the way, that both potential outcomes exist as random variables, but only one of them can be realised, so we can check only one value. But, again, both of them exist as separate beings - random variables.

Therefore realisation of random variable describing treatment $t_i$ do not correlate with values of nor $y_{i1}$ nor $y_{i0}$, however it does affect which one of them we do observe.

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To answer the second question - how it relates to the rest of population.

It is common laziness for econometricians to write a model as:

$$ y_i = \beta_0 + \beta_1 x_i + \varepsilon_i$$

without specifying $i=1,...,N$.

I think that this is often 'assumed' addition in such iterated statements. It seems that it is also in mind here, and in fact such independence assumption should hold for every single observation (you could write them separately for every observed unit).

Answer 2

The problem with the notation is that without additional definitions it is ambiguous if $i$ is being treatment as a random variable or a fixed variable.

When I see $t_i\perp\{y_{i1},y_{i0}\}$

without any additional definitions or clarifications I think that means conditional on $i$, so I then think that is shorthand for this condition holding for all subjects $i=1,\ldots,n$ which could be written as

When I see $t_1\perp\{y_{11},y_{10}\},\ldots,t_n\perp\{y_{n1},y_{n0}\}$

And I am pretty sure there are plenty of other readers struggling to understand the notation think the same thing. But this makes no sense because for each subject the potential outcomes $y_{i1}$ and $y_{i0}$ are (by definition) fixed and any fixed random variable is always independent of any other random variable which would mean the condition ALWAYS holds. So then we are forced (by realization that this makes no sense) to decide we are supposed to be treating $i$ as random. Which would mean for a RANDOMLY selected subject $i$ treatment is independent of the potential outcomes.

Note that if we change the setup to make each individual person's potential outcomes random that does not help. We would still have the independence conditional on $i$.