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I'm wondering when it is possible to distinguish between a data generating process that is missing at random (MAR) vs one that is not missing at random (NMAR) by analyzing a directed acyclic graph (DAG). For example, suppose we have the following:

$$ \begin{matrix} T& \longrightarrow & Y \\ \downarrow & & \\ R_{Y} \end{matrix} $$

where $R$ is the indicator for missingness on the outcome variable $Y$ and $T$ is treatment and always observed. In this case it's clear that $R$ is conditionally independent of $Y$ given $T$, which is what's required for making a determination of MAR (at least according to most definitions I've seen).

But suppose we had a slightly different graph:

$$ \begin{matrix} T& \longrightarrow & Y \\ \downarrow & & \\ L & & \\ \downarrow \\ R_{Y} \end{matrix} $$

where $L$ is an unobserved/latent variable. Suppose that $R_Y=0$ ($Y$ is missing) if $L$ is greater than some threshold value. Like in the first graph, we still have a situation in which missingness on $Y$ is conditionally independent of $Y$ given $T$ (which is always observed). But now missingness on $Y$ does depend on the unobserved value of $L$.

  • Does that make this NMAR?
  • Or is it still MAR since even though we don't observe $L$ we know that it depends on the always observed variable $T$?
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