Notation

We begin by reviewing notation for unintentional missing data.
Let $y$ represent the complete data that would be observed with no missing values.
The structure of $y$ may be a vector of univariate measurements or a matrix with multivariate responses for each unit.
The complete data $y$ may also include covariates, such as in a multivariate normal model for predictors and outcomes jointly.

Write
$y = (y_{\text{obs}},, y_{\text{mis}})$,
where $y_{\text{obs}}$ denotes observed values and $y_{\text{mis}}$ denotes missing values.

An indicator variable $I$ has the same dimension as $y$.
Each element equals $1$ if the corresponding value of $y$ is observed and $0$ if missing.
The indicator $I$ is fully observed.
Item nonresponse corresponds to $I_{ij} = 0$ for unit $i$, variable $j$; unit nonresponse corresponds to $I_{ij} = 0$ for all $j$ for a given unit $i$.

The joint distribution of $(y, I)$ given parameters $(\theta, \phi)$ is$$p(y, I \mid \theta, \phi) = p(y \mid \theta)\, p(I \mid y, \phi).$$

The observed information is $(y_{\text{obs}}, I)$.
The observed-data distribution integrates over missing components:$$p(y_{\text{obs}}, I \mid \theta, \phi) = \int p(y_{\text{obs}}, y_{\text{mis}} \mid \theta)\, p(I \mid y_{\text{obs}}, y_{\text{mis}}, \phi)\, dy_{\text{mis}}.$$

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Missing at Random (MAR)

Data are missing at random (MAR) if the missingness mechanism does not depend on the missing values:$$p(I \mid y_{\text{obs}}, y_{\text{mis}}, \phi) = p(I \mid y_{\text{obs}}, \phi).$$

Under MAR, the observed-data distribution becomes$$p(y_{\text{obs}}, I \mid \theta, \phi) = p(I \mid y_{\text{obs}}, \phi)\, p(y_{\text{obs}} \mid \theta),$$

where$$p(y_{\text{obs}} \mid \theta) = \int p(y_{\text{obs}}, y_{\text{mis}} \mid \theta)\, dy_{\text{mis}}.$$

If $\theta$ (data-model parameters) and $\phi$ (missingness parameters) are distinct, meaning independent in the prior, then inference for $\theta$ may proceed using only the observed-data likelihood $p(y_{\text{obs}} \mid \theta)$.
In this case, the missing-data mechanism is ignorable.

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Missing Completely at Random (MCAR)

Data are missing completely at random (MCAR) if$$p(I \mid y_{\text{obs}}, y_{\text{mis}}, \phi) = p(I \mid \phi).$$

MCAR is stronger than MAR and usually implausible.
MAR + distinct parameters is generally sufficient to avoid modeling the missingness mechanism explicitly.

Including more relevant observed variables in $y$ increases the plausibility of MAR by reducing dependence of missingness on unobserved information.

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Examples

Consider two variables $y = (\text{age}, \text{income})$, with age fully observed and income partially missing.
Assume a bivariate normal distribution for $(\text{age}, \text{income})$.

1. MCAR:
   Missingness of income does not depend on age or income.
2. MAR but not MCAR:
   Missingness depends on age but not on income given age.
   The mechanism is ignorable if $\theta$ and $\phi$ are distinct.
3. Not MAR (nonignorable):
   Missingness depends on age and also on income within age groups.

The relevance of the missing-data mechanism depends on analytic goals.
For inference about age only, income data may be ignored and the mechanism becomes irrelevant.
For inference about income, the missing-data process must be modeled.