Positivity assumption and Conditional exchangeability

Question

Let's imagine a simple-traditional scenario: A <--- L ---> Y and A ---> Y. Thus, we have exposure A (treatment and no treatment) and outcome Y. L represents confounder(s). For my specific question, it entails a multidimensional L (i.e., multiple variables). Moreover, we can assume that these covariates L are sufficient for adjusting for all confounders between A and Y (conditional exchangeability). My objective is to estimate the ATE {E(Y^1 - Y^0)}.

I proceed to use any type of debiased machine learning that allows me to estimate E(Y^1) and E(Y^0). Once that is done, I take a simple difference. Until now, everything is good.

However, after looking at the results, I decided to investigate the inverse probability weights (1/probability of being treated) and realized that some people have very little chance of receiving treatment. Hence, the positivity assumption is (almost surely) violated. This assumption, from my understanding, means that every individual, regardless of their covariates, has a nonzero probability of receiving each treatment level. It ensures that for each level of L, there is data on both the treatment and control conditions.

However, this scenario seems to be weird given that we could assume conditional exchangeability. In other words, for conditional exchangeability to be a meaningful and testable condition, it presupposes that each level of L observed in the data must have both treated and untreated individuals. If this were not the case (i.e., positivity is violated), then there would be subsets of the covariate space L where it's impossible to observe and thus verify whether potential outcomes are indeed independent of treatment assignment. If this is not the case, what am I not understanding?

Answer 1

So, I take the perspective that we should phrase the assumptions like, "exchangeability with positivity". My reasoning follows a similar line to what you state above: exchangeability is only well-defined mathetmatically if positivity is met. We can see this in the expression for exchangeability (in terms of probability to make things a little simpler to write than expectations, but same logic applies to both) $$ \Pr(Y^a = 1 | L=l) = \Pr(Y^a = 1 | A=a,L=l) \; \forall \; a\in\{0,1\}, l \in \mathcal{L} $$ where $\mathcal{L}$ is the support of $L$. We can rewrite the right-hand side as $$ \frac{\Pr(Y^a =1, A=a, L=l)}{\Pr(A=a, L=l)} $$ following the definition of conditional probabilities.

Recall that positivity is $$ \Pr(A=a | L=l) = \frac{\Pr(A=a,L=l)}{\Pr(L=l)} > 0 \; \forall \; a\in\{0,1\}, l \in \mathcal{L} $$ This condition ensures that $\Pr(A=a, L=l)$ is non-zero wherever $\Pr(L=l) > 0$. If $\Pr(A=a, L=l)$ were zero, then exchangeability is not well-defined since it involves a division by zero. So, this should give some intuition as to why exchangeability and positivity assumptions are linked together.

One item I will note is that when you look at the inverse probability weights, or propensity scores, they do not actually check the positivity assumption above. We can distinguish between structural positivity (assumption above) and random positivity (positivity in finite data samples). You can have violations of the random positivity assumption without structural positivity. Further, even if using debiased machine learning, you still are assuming the statistical model is correctly specified. Incorrect model specification can look like non-positivity. So, you cannot really say that the "positivity assumption is (almost surely) violated" in that context.

Finally, exchangeability is not a testable condition. Neither is the positivity assumption above. These are assumptions we make about the world. They themselves are not testable, unless you make other assumptions.

You can read a more formal write-up of my thoughts on positivity in this preprint.