Longitudinal observation patterns and missingness mechanisms

Let $m={\sum }_j\mathrm{\mid }{S}_j\mathrm{\mid }$ and let $Y^{\ast }\in {\mathbb{R}}^m$ be the complete scheduled response vector for subjects $j$ and visits $i$. Let the declared observed covariates take values in a measurable space $(\mathcal{X},\mathcal{A})$. The finite set of observation patterns is $\mathcal{R}=\{0,1{\}}^m$; $r_{ji}=1$ means visit $(j,i)$ is observed. A missingness mechanism is a function $g:\mathcal{R}\times {\mathbb{R}}^m\times \mathcal{X}\to [0,1]$ such that, for every $(y^{\ast },x)$, ${\sum }_{r\in \mathcal{R}}g(r\mid y^{\ast },x)=1$, and, for each fixed $r$, $(y^{\ast },x)\mapsto g(r\mid y^{\ast },x)$ is measurable for $\mathcal{B}({\mathbb{R}}^m)\otimes \mathcal{A}$. Thus $g(\cdot \mid y^{\ast },x)$ is a probability mass function on $\mathcal{R}$, and its dependence on the full data state is the finite-output kernel specified by the observation-mechanism field in Scientific statistical model contracts, which requires a probability measure at each state and measurable state dependence. For a pattern $r$, write $Y_{\mathrm{o}\mathrm{b}\mathrm{s}}(r)$ for the subvector of $Y^{\ast }$ at coordinates with $r_{ji}=1$.

The mechanism is missing completely at random (MCAR) if, for each pattern $r$, $g(r\mid y^{\ast },x)$ does not depend on $y^{\ast }$. It is missing at random (MAR) if, for each $r$, it depends on the full response vector only through $Y_{\mathrm{o}\mathrm{b}\mathrm{s}}(r)$ and $x$. It is missing not at random (MNAR) relative to this partition if the MAR condition fails for at least one pattern on the declared support. Monotone dropout means that, within each subject, observed visits form an initial segment of the scheduled visits, possibly empty or complete.

These are properties of the declared observation kernel. A mixed-model mean or covariance equation does not specify the kernel or imply MCAR, MAR, or MNAR.

For an LMM written on observed rows, $O_j=\{i:r_{ji}=1\}$ determines which coordinates and design rows are retained. If $r$ depends on the full scheduled outcomes, the distribution of the selected rows conditional on that pattern need not equal the corresponding unselected marginal subvector law.