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Define $\pi_i$ as the probability that person $i$ will be missing from your sample and $Y_i = 1$ denotes that a subject is missing. Say we're in a missing at random (MAR) scenario where $\pi_i$ depends on two known continuous variables $X_1$ and $X_2$:

$$logit(\pi_i) = \beta_1 x_1 + \beta_2 x_2$$

Let's say that I introduce a "baseline" rate of missing values, $\beta_0$, that is independent of all covariates. For a specific example, let's say there is a 5% chance that an observation is missing. In this case, $\pi_i$ is described by:

$$\text{logit}(\pi_i) = \beta_0 + \beta_1 x_1 + \beta_2 x_2$$.

In our specific example of 5% baseline missing, this would be: $$\text{logit}(\pi_i) = \text{logit}^{-1}(0.05) + \beta_1 x_1 + \beta_2 x_2$$

In this scenario, there parts of $\pi$ that depend on known covariates, i.e. $\beta_1$ and $\beta_2$. There is also a part of $\pi$ that doesn't depend on any covariates: $\beta_0$.

If we observe missing values through this second scenario, is this considered MAR? My guess it that it is because the distribution of $\pi$ depends on $X_1$ and $X_2$ even if not all parts of $\pi$ do.

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Yes, the second scenario would be considered MAR. The usual definition of MCAR is something like $P(Y=1|data) = P(Y=1)$; the probability of missingness is independent of the data. That doesn't apply in this case unless $\beta_1, \beta_2 = 0$. Therefore this scenario is not MCAR.

I suppose technically one could imagine the other direction. If we have MCAR data, can we say it is also MAR? I think the answer is yes, depending on how exactly MAR is defined.

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