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The whole idea around ignorability is still leaving me a bit confused. I did read this post from the site: Strong ignorability: confusion on the relationship between outcomes and treatment. It had made things a bit clearer. But I'm trying to figure out what is the difference between ignorability and strong ignorability. It seems all the literature discusses strong ignorability as an assumption that we check in theory to determine the "quality" (there is probably a better word to describe it that is more poignant, but I'm not sure what) of our experiment, but I don't see anything on just ignorability. Does it have a mathematical formulation like strong ignorability's $(Y_0,Y_1) \perp \!\!\! \perp T|X$? This is all new to me and feeling a bit overwhelming....

D.C. the III
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2 Answers2

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Unless there is a specific reference in which you saw "ignorability" defined as something different from "strong ignorability," these terms are conventionally used interchangeably. Both are referring to the assumption that $(Y_0,Y_1) \perp \!\!\! \perp T|X$.

For what it's worth, the term "unconfoundedness" is also typically used interchangeably with these terms: all three terms typically all refer to the same assumption that treatment is independent of potential outcomes (conditional on $X$).

doubled
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    In Rosenbaum & Rubin (1983), strong ignorability is defined to contrast with ignorability. I personally am unable to parse the difference between the two. – Noah Mar 04 '22 at 15:08
  • I found the difference out as per my professor. At least "symbolically" strong ignorability is characterized as $P(T| y^0, y^1, X) = P(T|X)$. Whereas ignorability is $P(T| y^0, y^1, X) = P(T)$. Where $T$ is treatment, $y^i$ are the potential outcomes, $X$ are the covariates. – D.C. the III Mar 05 '22 at 02:10
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I will give a response from the epidemiological side of causal inference, which uses a slightly different terminology ('exchangeable' rather than 'ignorable') with some subtle differences. Citing from What If (Hernán & Robins, 2023, p. 28, emphasis mine):

Rubin (1974, 1978) extended Neyman’s theory for randomized experiments to observational studies. Rosenbaum and Rubin (1983) referred to the combination of exchangeability and positivity as weak ignorability, and to the combination of full exchangeability (see Technical Point 2.1) and positivity as strong ignorability

Then, from Technical point 2.1 (p. 15, emphasis mine), we can see that:

Formally, let $\mathcal{A} = \{a, a', a'', \dots \}$ denote the set of all treatment values present in the population, and $Y^{\mathcal{A}} = \{Y^a, Y^{a'}, Y^{a''}, \dots\}$ the set of all counterfactual outcomes. Randomization makes $Y^{\mathcal{A}}⊥A$. We refer to this joint independence as full exchangeability. [...] For a continuous outcome, exchangeability $Y^{\mathcal{a}}⊥A$ implies mean exchangeability $\operatorname{E}[Y^a | A = a'] = \operatorname{E}[Y^a]$ but mean exchangeability does not imply exchangeability because distributional parameters other than the mean (e.g., variance) may not be independent of treatment.

In other words, the distinction between weak ignorability and strong ignorability boils down to whether we are assuming mean exchangeability or full exchangeability (which I deem slightly more informative terms).

It is no surprise this distinction is hard to grasp since the bulk of most applications and expositions of causal inference problems is only concerned with average causal effects, so we end up erasing the distinction and just talking about ignorability or exchangeability per se. At the same time, causal assumptions from Rubin's terminology involve more than a single property (especially in the case of SUTVA), so that also does not help with clarity.

Kuku
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