Missingness Has a Universe

A Typed and Compositional Foundation for Missing-Data Research

Huayin Wang datumwise · independent open-source research project Version 2.0 · 3 August 2026

Central claim. Missingness is not merely a stochastic relation between a response indicator and variables in a pre-given data matrix. It is a typed, compositional law of a member inside a declared universe. Before asking why a value is missing, an analysis must establish which points exist, at what anchor the value is defined, which observed determinants are lawfully available at those points, and how missingness propagates through transformation.

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Abstract

Modern missing-data theory has developed powerful accounts of missingness mechanisms, ignorability, recoverability, testability, imputation, sensitivity analysis, and nonresponse. Yet most of this work begins after a data foundation has already been fixed. Units, variables, observation opportunities, coordinate alignment, population membership, and the relationship between a value and its covariates are usually treated as given by the complete-data model. In real analytical systems, these are not harmless preliminaries. They are among the main sources of ambiguity and error.

This paper proposes a data-foundational account of missingness derived from the Theory of Data. A typed value lives at a typed anchor point inside a universe whose existence law determines which points belong to the population. An event universe is generated by occurrences; a spine universe establishes expected points independently of value presence. Missingness becomes defined only after an eligible universe point has been established.

For a target member $v$, the paper introduces an M-contract with four determinant classes: coordinates functionally reachable from the value anchor, other observed members lawfully transported to the target point, the unobserved target value itself, and latent determinants absent from the observed governed model. The M-anchor is the codomain of the functional coordinate-determinant map from the V-anchor. Nonfunctional context cannot enter the M-anchor directly; it must first be converted into a single-valued observed determinant member by a declared transformation. MCAR, MAR, self-dependent MNAR, and latent-dependent MNAR are then derived properties of the complete M-contract rather than primitive labels attached to a rectangular dataset.

The deeper contribution is compositional. Missingness is treated as a law that must transform with the member under mapping, reduction, alignment, filtering, expansion, and universe crossing. This creates a research program for missingness-preserving analytical systems: typed eligibility, determinant transport, support propagation, coverage semantics, proof-carrying transformation, and evidence-ranked mechanism declarations. Part of that program is already formal: the population, support, and coverage rules used here specialize transformation rules proved in the accompanying Contract Calculus, and the paper states which contract components are mechanically checkable and which remain statistical.

The framework does not solve the classical identification problem, nor does it claim that all semantic premises are testable from observed data. It instead exposes the premises that conventional analyses must already make, gives them typed locations, and makes their consequences available to analytical computation.

Keywords: missing data; MCAR; MAR; MNAR; missingness mechanisms; data foundations; typed coordinates; analytical grain; universe; member; M-anchor; functional dependency; coarsening; incomplete databases; compositional semantics

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1. The problem begins before the missingness mechanism

The standard missing-data question is often written in terms of a complete-data variable $Y$, an observed part $Y_{\mathrm{obs}}$, a missing part $Y_{\mathrm{mis}}$, and a missingness indicator $R$. The analyst asks whether the distribution of $R$ depends on observed data, missing values, or latent causes, and what that dependence permits for inference.

That question is indispensable. It is not always the first question.

Before a missingness mechanism can be defined, a real analysis must answer:

1. Which population is under study?
2. Which points belong to that population?
3. At which coordinates is the variable defined?
4. Is the absence of a row the absence of an event, the absence of an expected value, or exclusion from the population?
5. Which observed covariates are available at the same observational opportunity?
6. If a covariate lives at another grain, by what operation is it made available?
7. If several related records exist, which one—or which reduction of them—determines missingness?
8. How does missingness change when the target variable is aggregated, aligned, lagged, filtered, expanded, or crossed into another population?

Conventional notation often suppresses these questions by treating a complete data matrix, vector, or sample space as already formed. That is a reasonable abstraction when the data-producing design has fixed the units, variables, and observation schedule. It is much less safe in analytical data systems assembled from transactions, snapshots, registries, classifications, forecasts, sensors, and operational joins.

The main thesis of this paper is therefore not that established missing-data theory is wrong. It is that the theory generally begins after several data-constituting decisions have been made.

Missing-data theory studies the observation process over a complete-data object. A data foundation must also explain how that complete-data object acquires its population, coordinates, grain, and lawful relationships.

The distinction matters because an incorrect complete-data object cannot be repaired by a correct missingness model.

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2. What existing research already establishes

This proposal depends on, rather than replaces, several major traditions.

2.1 Rubin’s mechanism and ignorability framework

Rubin formalized conditions under which the process causing missing data may be ignored for particular forms of inference. The distinction among missing completely at random, missing at random, and nonignorable missingness concerns how the response process depends on complete and observed data, together with parameter-distinctness and inferential conditions (Rubin, 1976).

The present paper inherits:

• the distinction between the substantive data process and the observation process;
• the missingness indicator;
• mechanism factorization;
• MCAR, MAR, and nonignorable missingness;
• the warning that MAR alone is not identical to universal ignorability.

Later clarification has shown that MAR terminology must be stated with care: realized versus everywhere conditions, the conditioning information, and the inferential paradigm all matter (Seaman et al., 2013).

2.2 Coarsening at random

Heitjan and Rubin generalized missingness to coarse data: instead of observing an exact element of the ideal sample space, one observes a set containing it. Coarsening includes missingness, censoring, rounding, heaping, and grouping (Heitjan and Rubin, 1991). Gill, van der Laan, and Robins further developed the structure and limitations of coarsening-at-random models (Gill et al., 1997).

Coarsening is the closest statistical predecessor to the current proposal because it asks what was observed relative to an ideal data space. The difference is that the present work does not treat the ideal data space as the final foundation. It asks how the relevant population and typed coordinate space were constituted and how they behave under analytical transformation.

2.3 Missingness graphs

Mohan, Pearl, and Tian introduced graphical representations that explicitly encode dependencies between substantive variables and missingness indicators and developed results on recoverability under MAR and MNAR structures (Mohan et al., 2013). Subsequent work studied testability and graphical definitions of MAR (Mohan and Pearl, 2014; Tian, 2015).

The current proposal inherits the insight that missingness mechanisms should be explicit structural objects, not labels inferred from null patterns alone. Its additional question is how each node or determinant in a missingness graph is typed, located, aligned, and made available at the target variable’s observational opportunity.

2.4 Incomplete-information databases

Database theory has developed formal semantics for nulls, possible worlds, marked nulls, conditional tables, integrity constraints, and safe query answering. Imieliński and Lipski gave foundational conditions for semantically meaningful relational operations over incomplete information (Imieliński and Lipski, 1984). Other work examined functional dependencies in the presence of nulls and incomplete information.

This tradition addresses a problem statistical missing-data work often leaves to implementation: what queries over incomplete records are semantically justified. A recent paper directly combines relational querying with missingness mechanisms represented by Bayesian networks and missingness graphs (Bertossi et al., 2026).

The present proposal lies at the intersection of these traditions but changes the primitive data object. Rather than beginning with rows, relations, or an already aligned vector of variables, it begins with typed values at typed anchor points inside explicit universes.

2.5 Estimands, survey nonresponse, and planned missingness

Statistical practice has converged on the priority of data-constituting decisions from two further directions.

The ICH E9(R1) addendum on estimands requires a trial analysis to fix the target population, the variable, the handling of intercurrent events, and the population-level summary before an estimator is chosen (ICH, 2019). This is a regulatory expression of the principle defended here: the object of inference must be constituted before the observation process over it is modeled.

Survey methodology distinguishes frame undercoverage, unit nonresponse, and item nonresponse: a unit may be absent from the sampling frame, present but not responding, or responding with particular items missing (Little and Rubin, 2019). Register-based statistics develops the frame side explicitly, treating the register of expected units as a governed object in its own right (Wallgren and Wallgren, 2014). These distinctions are the survey-world form of the population, eligibility, and support chain of Section 4, and a sampling frame is the survey-world form of a spine universe.

Planned missingness designs deliberately schedule items to be unobserved for sampled units (Graham et al., 2006). In the present vocabulary, such a design declares a spine universe whose eligibility law makes some absences design-intended rather than nonresponse. The design works precisely because the distinction between an ineligible point and an eligible unobserved point has been fixed in advance.

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3. The silent fixed-data premise

The conventional complete-data object usually carries several assumptions that are rarely written as part of the missingness mechanism.

3.1 Population is already fixed

The notation assumes that the relevant units exist and are members of the analysis population.

In practice, absence can mean:

• no event occurred;
• an event occurred but was not recorded;
• the entity was not eligible;
• an expected observation was not collected;
• the unit failed to map into a classification;
• a join removed the unit;
• the source system had incomplete coverage;
• the value was not applicable.

These are not equivalent forms of missingness.

3.2 Variables are already aligned

A model such as

$$ P(R_Y\mid X,Y) $$

assumes that $X$, $Y$, and $R_Y$ can be evaluated for the same observational unit.

In real data, $X$ may live at:

• the same unit and time;
• a coarser organizational level;
• a finer event level;
• a prior time;
• a rolling window;
• another universe connected by a many-to-many relationship.

The expression is not operationally meaningful until the transport from $X$‘s location to $Y$‘s observation point has been declared.

3.3 Grain is already fixed

The word “variable” can hide multiple grains. Monthly income, annual income, person-level income, household income, and an income event are not interchangeable merely because they share a name and type.

A missingness model at one grain does not automatically induce a missingness model at another.

3.4 Observation opportunity is already fixed

A response indicator $R_Y$ presumes that there was an opportunity for $Y$ to be observed.

For an event-generated population, the absence of a point may mean that no opportunity existed. For a scheduled or spine-generated population, the point may exist while the value is absent.

3.5 Transformations are treated as external preprocessing

Analysts often build complete cases, aggregate records, create lags, align sources, or impute before fitting the final model. Each operation changes the support and can change the missingness mechanism. Yet the transformation is commonly documented procedurally rather than incorporated into the formal missingness contract.

These assumptions are not trivial metadata. They define the object to which MCAR, MAR, or MNAR is being attributed.

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4. Data foundations: datums, members, anchors, and universes

The Theory of Data supplies the underlying vocabulary.

4.1 Anchor and anchor point

An anchor $A$ is a typed coordinate schema assembled from selected levels of declared axes. Its point space is:

$$ \operatorname{Pts}(A)

\prod_{\alpha\in\operatorname{Axes}(U)} |A(\alpha)|. $$

An anchor point is:

$$ a\in\operatorname{Pts}(A). $$

For example:

$$ a=(\text{Patient 17},\text{Visit 4},\text{Assay A}) $$

is not an arbitrary database key. It is one typed location.

4.2 Datum

A datum is one typed value at one anchor point:

$$ p=(a,x), \qquad x\in |X|. $$

The key is the anchor point. The value is typed.

4.3 Universe

An anchor space describes structurally possible coordinates. It does not say which points exist in one data population.

A universe is:

$$ U=(\kappa,A_U,P_U), $$

where:

• $\kappa$ is the universe type;
• $A_U$ is the native anchor;
• $P_U\subseteq\operatorname{Pts}(A_U)$ is the inhabited population.

Membership in $P_U$ is determined by the universe’s existence law, written $\lambda_U$ in the Theory’s canonical form: an event universe is generated by occurrences; a spine universe by declared expectation. The existence law is what licenses statements about points that carry no values.

The complete grain is:

$$ \operatorname{Grain}=(U,A). $$

Two variables can share the same displayed anchor while belonging to different universes.

4.4 Event and spine universes

In an event universe, occurrence generates membership.

A transaction point exists because a transaction occurred. If no transaction occurred, the point may not exist. Its absence is not automatically a missing value.

In a spine universe, membership is established independently of value presence.

A patient-visit-assay point may exist because a protocol scheduled the assay. The assay result may then be absent at an existing eligible point.

Events generate coordinates. Spines await values.

This distinction precedes MCAR, MAR, and MNAR. It establishes where a missingness indicator can first be defined.

4.5 Member

The Theory of Data distinguishes a measure family from its anchored realizations. A measure is the stable governed family, such as revenue or hemoglobin level. A member is one governed realization of that family at a particular anchor in a particular universe. Earlier drafts of the Theory used the word measure for the realization itself; the canonical vocabulary now reserves it for the family. This paper concerns one target member and, where the family is not at issue, calls it simply the member $v$.

Let $E_v\subseteq P_{U,A_V}$ be the set of eligible universe points for the target member $v$.

The conceptual complete member is:

$$ v^\star:E_v\rightarrow X. $$

Define the response indicator:

$$ R_v:E_v\rightarrow{0,1}, $$

with:

$$ R_v(a)= \begin{cases} 1,&\text{if }v^\star(a)\text{ is observed},\ 0,&\text{if }v^\star(a)\text{ is missing}. \end{cases} $$

The observed support is:

$$ S_v={a\in E_v:R_v(a)=1}, $$

and the observed member is:

$$ v:S_v\rightarrow X, \qquad v(a)=v^\star(a)\text{ for }a\in S_v. $$

This separates three objects that are frequently conflated:

$$ \operatorname{Pts}(A_V) \supseteq P_{U,A_V} \supseteq E_v \supseteq S_v. $$

They mean, respectively:

1. structurally possible points;
2. universe members;
3. eligible observation opportunities for $v$;
4. points at which $v$ was observed.

Missingness is defined on $E_v$, not on every conceivable coordinate and not merely on the rows that happened to be stored.

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5. The M-contract

The target member’s missingness behavior is represented by:

$$ \mathcal M_v

(\mu_v,Z_v,s_v,L_v,K_v,\mathcal E_v). $$

The components are:

Component	Role
$\mu_v:E_v\to\operatorname{Pts}(A_M)$	Functional coordinate-determinant map
$A_M$	M-anchor: the coarsest typed coordinate resolution at which missingness may vary — the coarsest $A_M$ for which $R_v$ is conditionally independent of finer coordinates given $\mu_v$ and $Z_v$ (§9)
$Z_v$	Lawfully transported frame of other observed determinants
$s_v\in{0,1}$	Whether self-dependence is admitted
$L_v$	Declared latent determinant class
$K_v$	Missingness probability kernel
$\mathcal E_v$	Evidential status of the declarations

The mechanism factors as:

$$ R_v(a) \sim K_v!\left( \mu_v(a), Z_v(a), v^\star(a)\mathbf 1[s_v=1], L_v(a) \right). $$

The notation does not claim that every component is identifiable. It records what the model asserts, what the system can operationalize, and what evidence supports each assertion.

The components divide by the kind of adjudication they admit. Functionality of $\mu_v$, the lawfulness of each declared determinant transport, eligibility containment, and support propagation are mechanical properties: a governed system can check them against declarations and data. The kernel $K_v$ is not. It is a statistical object; no calculus adjudicates it, and the framework does not claim otherwise. The division is recorded in the evidence component: mechanical clauses may reach verified status; kernel-level clauses are at best corroborated, and are commonly assumed or unidentifiable.

Two further clauses belong to a joint contract over several members rather than to any single one: the coupling premise that assembles per-member kernels into one mechanism, and the distinctness of mechanism parameters from the data-model parameter. Both are statistical rather than mechanical, both carry evidence statuses, and both are defined in §8.

The contract is written in the selection-model direction: it declares determinants of the response indicator given the complete data. Pattern-mixture parameterizations condition the data model on response patterns instead (Little, 1993). The contract does not privilege either factorization for estimation. It records what is asserted about the observation process, at which typed locations, with what evidence; either estimation strategy may consume the record.

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6. Four determinant classes

6.1 Coordinate determinants

Coordinate determinants are obtained through:

$$ \mu_v:E_v\rightarrow\operatorname{Pts}(A_M). $$

They divide into two kinds.

Native coordinate determinants

These are projections or lawful hierarchy movements from the V-anchor.

If:

$$ A_V={\mathrm{Patient},\mathrm{Clinic},\mathrm{Day}} $$

and:

$$ A_M={\mathrm{ClinicType},\mathrm{Month}}, $$

the M-map may be:

$$ (\mathrm{patient},\mathrm{clinic},\mathrm{day}) \mapsto (\mathrm{type}(\mathrm{clinic}),\mathrm{month}(\mathrm{day})). $$

The selected hierarchy level matters. Missingness may vary by clinic type without varying by individual clinic.

Contextual coordinate determinants

A determinant may lie outside the axes displayed in $A_V$ while being functionally determined by each V-anchor point.

For example:

$$ (\mathrm{Store},\mathrm{Day}) \rightarrow \mathrm{CollectionTeam}. $$

If every store-day has exactly one responsible team, collection team may enter $A_M$.

The admission rule is:

A coordinate can enter the M-anchor only when each eligible V-anchor point determines one value of that coordinate under declared universe structure.

Equivalently, $\mu_v$ must be functional.

If a patient visit is associated with several clinicians, clinician cannot enter the M-anchor without a further rule. The model must first define primary clinician, last clinician, clinician count, weighted exposure, or another single-valued construction.

Functionality is declared relative to the universe and may require a time coordinate. If responsibility for collection is reassigned over time, the functional edge is $(\mathrm{store},\mathrm{day})\rightarrow\mathrm{CollectionTeam}$, not $\mathrm{store}\rightarrow\mathrm{CollectionTeam}$. A mapping that drifts is functional only with its validity period in the domain. Declaring the edge at the wrong resolution is a contract error, and one the system can detect when observed data contradict the declared map.

This is a key distinction between an M-anchor and an unrestricted set of predictors.

6.2 Other observed member determinants

Missingness may depend on another observed member $z_j$. The source member need not share the target’s V-anchor. It must be lawfully evaluable at each target observation point.

Let:

$$ z_j:S_j\rightarrow Y_j. $$

A determinant transport is:

$$ \tau_j:z_j\mapsto\widetilde z_j, $$

where:

$$ \widetilde z_j:E_v\rightarrow Y’_j. $$

The transport may be:

• identity at the same anchor;
• broadcast from a coarser anchor through a functional map;
• reduction from a finer anchor;
• lag or window construction;
• alignment within a universe;
• declared crossing from another universe.

The observed determinant frame is:

$$ Z_v= \langle \widetilde z_1,\ldots,\widetilde z_k \rangle. $$

The transport is part of the mechanism. “Missingness depends on prior disease severity” is incomplete until the theory states which severity member, at which prior anchor, under which lag rule, with which missingness of its own.

Other response indicators may also enter $Z_v$. The missingness of one member can depend on whether another member was observed, even when the other value is unavailable.

6.3 Self-dependence

Self-dependence means:

$$ R_v(a)\leftarrow v^\star(a). $$

This is the familiar self-masking form of MNAR. It deserves an explicit flag because the unobserved determinant is the target value itself, not an unspecified latent variable.

When:

$$ s_v=1, $$

the kernel may depend on $v^\star(a)$.

6.4 Latent determinants

A latent determinant is a missingness determinant not represented in the observed governed model.

It may be:

• an unrecorded process state;
• latent disease severity;
• respondent motivation;
• device failure;
• an undocumented operational policy;
• an external event;
• an omitted variable.

Write:

$$ L_v= \langle\ell_1,\ldots,\ell_r\rangle. $$

A latent determinant need not be “outside the real-world universe.” It is outside the observed model available for operational evaluation.

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7. MCAR, MAR, and MNAR as derived contract properties

The classical categories are retained but derived from the M-contract.

7.1 MCAR relative to the declared model

MCAR holds when:

$$ \mu_v(a)=* $$

is constant at the terminal M-anchor,

$$ Z_v=\varnothing, \qquad s_v=0, \qquad L_v=\varnothing. $$

Then:

$$ P(R_v(a)=1\mid\mathcal D^\star)=c. $$

This is MCAR relative to the declared data model and universe. It does not prove independence from every unknown external cause.

7.2 MAR

MAR holds when missingness may depend on observed coordinate and member determinants:

$$ \mu_v\text{ nonconstant} \quad\text{and/or}\quad Z_v\neq\varnothing, $$

while:

$$ s_v=0, \qquad L_v=\varnothing. $$

Thus:

$$ P(R_v\mid\mathcal D^\star)

P(R_v\mid\mu_v,Z_v). $$

This includes:

• coordinate-driven MAR;
• observed-member-driven MAR;
• mixed coordinate-and-member MAR.

7.3 MNAR-self

MNAR-self holds when:

$$ s_v=1. $$

The probability of observation may depend on the missing target value.

7.4 MNAR-latent

MNAR-latent holds when:

$$ L_v\neq\varnothing. $$

The mechanism may depend on an unobserved determinant other than the target value.

7.5 General MNAR

The mechanism is MNAR when:

$$ s_v=1 \quad\text{or}\quad L_v\neq\varnothing. $$

This decomposition makes MNAR less opaque. It distinguishes self-selection from dependence on other unobserved context.

7.6 Everywhere and realized forms

The definitions above are everywhere-style statements: they constrain the mechanism over the eligible set, not only over the realized missingness pattern. Realized-data variants exist and matter for likelihood arguments (Seaman et al., 2013). A complete contract should state which form is asserted, because the two support different inferential claims.

The properties above belong to a single member’s contract. Their aggregation to a joint, ignorable mechanism is a separate question, characterized in §8.

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8. Aggregation: from per-member contracts to joint ignorability

Sections 5 to 7 define the M-contract of a single target member and read MCAR, MAR, and MNAR off that one contract. The inferential payoff of MAR, however, is ignorability (Rubin, 1976): under MAR together with a parameter-distinctness condition, the missingness mechanism may be dropped from the likelihood or posterior for the data-model parameter. Ignorability is a property of the joint law of the whole response pattern, and it does not decompose member by member. Per-member contracts that are each MAR can aggregate to a non-ignorable joint mechanism.

This section supplies the joint object, states when aggregation preserves MAR, and names the two premises the aggregation requires. No new statistical theory is introduced; the classical results (Rubin, 1976; Robins and Gill, 1997; Seaman et al., 2013) are specialized to the typed framework, and the contribution is the location of the premises and the checkability of the obstruction.

Throughout, the everywhere form of MAR (§7.6) is used, and each per-member contract is assumed to have excluded self- and latent-dependence:

$$ s_{v_j}=0,\qquad L_{v_j}=\varnothing\qquad (j=1,\dots,k). \tag{A} $$

If (A) fails for any member, the joint mechanism is MNAR by inspection.

8.1 The joint frame

Target members $v_1,\dots,v_k$ can be said to have a pattern only after they are aligned to a common eligible frame.

Definition (joint eligible frame). A joint eligible frame is a declared point set $E$ together with, for each $j$, an eligibility inclusion $E\subseteq E_{v_j}$ and a transported member $v_j^\star:E\to\mathcal X_j$ obtained by the alignment operations of §11.4.

The construction is itself a contract obligation, and §11.3’s distinction applies with full force: if $E$ is formed as an intersection of observed supports it is a carve, not a restriction, and the joint object then denotes a selected subpopulation with a mechanism of its own. A lawful joint frame is built from eligibility sets.

At each $a\in E$ write the complete vector, the response vector, and the observed part:

$$ D^\star(a)=\bigl(v_1^\star(a),\dots,v_k^\star(a)\bigr), \qquad R(a)=\bigl(R_{v_1}(a),\dots,R_{v_k}(a)\bigr)\in{0,1}^k, $$

$$ D_{\mathrm{obs}}(a)=\Bigl(\text{coordinates of }a;\ R(a);\ {v_j^\star(a):R_{v_j}(a)=1}\Bigr). $$

The pattern $R(a)$ is always part of $D_{\mathrm{obs}}(a)$: one always knows whether a value was observed. This is why dependence on other indicators is harmless below, while dependence on other values is the constrained case.

Each member’s determinant frame $Z_{v_j}$ splits accordingly:

• $W_j(a)$ — the coordinate determinants $\mu_{v_j}(a)$ and any indicator determinants $R_{v_l}(a)$ appearing in $Z_{v_j}$. These are functions of the anchor point and the pattern, hence always in $D_{\mathrm{obs}}(a)$.
• ${v_l^\star(a):l\in\operatorname{pa}(j)}$ — the value determinants: members whose starred value enters $Z_{v_j}$. The index set $\operatorname{pa}(j)\subseteq{1,\dots,k}\setminus{j}$ is the set of value-parents of $j$.

8.2 The coupling premise

The per-member kernels are conditionals of individual indicators. They do not by themselves determine a joint mechanism $g_\phi\bigl(R(a)\mid D^\star(a)\bigr)$; a coupling premise is required. Fix a declared order on the members and write the joint mechanism in sequential form:

$$ g_\phi\bigl(R(a)\mid D^\star(a)\bigr) =\prod_{j=1}^{k} K_{v_j}\Bigl(R_{v_j}(a)\ \Big|\ R_{v_1}(a),\dots,R_{v_{j-1}}(a),\ W_j(a),\ {v_l^\star(a)}_{l\in\operatorname{pa}(j)}\Bigr). \tag{SF} $$

Two special cases matter. When no factor conditions on earlier indicators, (SF) reduces to conditional independence of the indicators given the complete data — the simplest coupling, appropriate for item nonresponse arising independently across items. When missingness is absorbing, as in dropout, the later factors are deterministic given an earlier indicator and consult no values at all; this case is not conditionally independent, and a formulation that assumed independence would misdescribe it.

(SF) is a located premise that the per-member contracts leave implicit. It should be a declared clause of the joint contract with its own evidence status, because without it the labels of §7 have no joint meaning. It can fail substantively: a shared device outage that drops several items together induces indicator dependence not mediated by $D^\star$, which must be modeled as a common latent determinant, returning the joint object to MNAR-latent.

Definition (joint M-contract). The joint M-contract over $E$ is

$$ \mathbf M=\bigl(E,\ {\mathcal M_{v_j}}_{j=1}^{k},\ \text{(SF)},\ \Delta\bigr), $$

with $\Delta$ the distinctness clause of §8.6.

8.3 Per-member MAR does not imply joint MAR

A minimal example shows the failure.

Let $k=2$, both contracts satisfying (A), with $\operatorname{pa}(1)={2}$ and $\operatorname{pa}(2)={1}$: each member’s response depends on the other’s value, through the observed-member determinant class of §6.2. Per member, both contracts have empty self- and latent-classes, so §7.2 classifies each as MAR.

Consider a point $a$ with the non-monotone realization $R_{v_1}(a)=1$, $R_{v_2}(a)=0$. The factor for $R_{v_1}$ reads $v_2^\star(a)$, which is missing. The joint mechanism therefore varies with a missing value,

$$ g_\phi\bigl(R(a)\mid D_{\mathrm{obs}}(a),D_{\mathrm{mis}}(a)\bigr)\neq g_\phi\bigl(R(a)\mid D_{\mathrm{obs}}(a)\bigr), $$

and is MNAR. Two MAR members compose to a non-ignorable joint mechanism, and the failure is specific: a response indicator was routed through a co-missing value.

8.4 The value-dependence digraph

Definition. The value-dependence digraph $\mathcal G$ of a joint M-contract has nodes $v_1,\dots,v_k$ and a directed edge $v_l\to v_j$ exactly when $l\in\operatorname{pa}(j)$. Coordinate determinants and indicator determinants induce no edges.

The digraph is declared, not estimated, and the condition below is read off it together with the realized pattern.

8.5 When aggregation preserves MAR

Proposition (aggregation). Let $\mathbf M$ satisfy (A) and (SF), with each kernel non-degenerate in its value arguments. Then the joint mechanism is everywhere-MAR if and only if, at every $a\in E$, every factor that is non-degenerate at the realized pattern has its value-parents observed:

$$ \forall a\in E,\ \forall j\ \text{with $K_{v_j}$ non-degenerate at }R(a),\ \forall l\in\operatorname{pa}(j):\quad R_{v_l}(a)=1. \tag{$\ast$} $$

Proof. By (SF) the joint mechanism is a product of factors, each of which depends on $D_{\mathrm{mis}}(a)$ only through its missing value-parents; the terms $W_j(a)$ and the conditioning indicators lie in $D_{\mathrm{obs}}(a)$. If $(\ast)$ holds, every non-degenerate factor is a function of $D_{\mathrm{obs}}(a)$ alone, so the product is constant as the missing coordinates range over their space and equals $g_\phi(R(a)\mid D_{\mathrm{obs}}(a))$. If $(\ast)$ fails, some non-degenerate factor reads a missing value-parent and, by non-degeneracy, varies with it. $\blacksquare$

Two structural regimes force $(\ast)$.

• (C1) Complete parents. Every value-parent is fully observed on the frame: $S_{v_l}=E$ for all $l\in\operatorname{pa}(j)$, all $j$. Missingness is driven only by completely observed covariates.
• (C2) Monotone coherence. There is a total order $\prec$ with every value-edge pointing backward ($l\in\operatorname{pa}(j)\Rightarrow l\prec j$), and the pattern is monotone with respect to $\prec$: the observed set is a $\prec$-downward-closed prefix. Under (SF), once a member is missing the subsequent factors are deterministic and consult no values, so the only non-degenerate factors are those at or before the frontier, whose value-parents are $\prec$-earlier and therefore observed.

These are the two settings in which multivariate MAR is usable in practice — missingness driven by fully observed covariates, and monotone dropout with history dependence — and the proposition recovers them. The obstruction in all other cases is the same one: non-monotone internal missingness with mutual value-dependence, where $(\ast)$ cannot be guaranteed structurally.

8.6 The distinctness clause

Ignorability requires, beyond MAR, that the mechanism parameters be separated from the data-model parameter. This is a relation between the kernel family and the data model, so its natural home is the joint contract:

$$ \Delta:\quad (\theta;\ \phi_1,\dots,\phi_k)\ \text{are distinct,} $$

meaning variation-independent for likelihood ignorability and a priori independent for Bayesian ignorability, where $\phi_j$ parameterizes $K_{v_j}$.

With (SF) and $\Delta$ recorded, the three premises of ignorability — MAR structure, mechanism-model parameter separation, and the coupling that makes “the mechanism” one object — each occupy a declared position with an evidence status. Like the kernel $K_v$ of §5, $\Delta$ is a statistical clause: no calculus adjudicates parameter distinctness. It enters the evidence component as assumed, or as verified by design where a design guarantees it, as in planned missingness (§2.5).

Corollary (ignorability). Under (A), (SF), $(\ast)$, and $\Delta$, the joint mechanism is ignorable for likelihood and Bayesian inference on $\theta$, and observed-data inference may proceed from the observed-data model alone.

8.7 A short instance

Three members on a customer-month frame: plan_tier ($v_1$), usage ($v_2$), and satisfaction_score ($v_3$), the last collected by survey.

Ignorable case. plan_tier is complete on the frame ($S_{v_1}=E$, it is a billing attribute). Survey response depends on plan tier and on month: $\operatorname{pa}(3)={1}$, with the month entering as a coordinate determinant. Usage is complete. The digraph has one edge, $v_1\to v_3$, and its source is fully observed: (C1) holds, $(\ast)$ holds, and with $\Delta$ the survey mechanism may be dropped from inference on the usage-satisfaction model.

Non-ignorable case. One edge is added: heavy users are billed on a metered plan whose tier is recorded only when usage is captured, so $\operatorname{pa}(1)={2}$ as well, and usage itself is sometimes missing. Now $v_2\to v_1\to v_3$, and at any point where usage is missing the factor for plan_tier reads a missing value-parent. $(\ast)$ fails, the joint mechanism is MNAR, and the labels attached to each member separately do not license dropping the mechanism.

The difference between the two cases is one declared edge and one support fact. Both are recorded in the contract, so the distinction is checkable before any estimation is attempted.

8.8 Scope of this section

$(\ast)$ is a characterization under (A) and (SF); (C1) and (C2) are sufficient structural guarantees, not necessary ones. Non-monotone mechanisms may satisfy $(\ast)$ pointwise without either regime holding, but this is fragile and generally not certifiable in advance; Robins and Gill (1997) develop randomized monotone missingness as the well-behaved non-monotone family.

The mechanical and statistical division of §5 carries to the joint object cleanly. Whether $(\ast)$ is structurally guaranteed — whether (C1) or (C2) holds — is mechanically checkable: it is a property of the declared digraph, the declared supports, and the declared monotonicity of the pattern, all of which a governed system holds. Whether the kernels are correctly specified, and whether (SF) and $\Delta$ hold, remain statistical.

The proposition is stated for everywhere-MAR. The realized-pattern variant (Seaman et al., 2013) weakens $(\ast)$ to the observed pattern and supports likelihood factorization but not every Bayesian claim; a complete joint contract should declare which form it asserts, exactly as §7.6 requires per member.

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9. M-irrelevance, not “MCAR along an axis”

Suppose:

$$ A_V= {\mathrm{Store},\mathrm{Product},\mathrm{Month}} $$

and:

$$ A_M={\mathrm{Store}}. $$

It is tempting to say that missingness is “MCAR along product and month.” That wording confuses a local conditional-independence statement with the global MCAR classification.

The precise assertion is:

$$ R_v \perp (\mathrm{Product},\mathrm{Month}) \mid \mathrm{Store}, Z_v, v^\star\mathbf 1[s_v=1], L_v. $$

We call an omitted axis M-irrelevant when its coordinates provide no additional information about missingness after conditioning on the complete M-contract.

This formulation also handles hierarchy levels. If $A_M$ contains Region rather than Store, missingness may vary between regions but not between stores within the same region, conditional on the remaining determinants.

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10. Why universe type changes the missing-data problem

10.1 Transaction example

Consider transaction revenue at:

$$ A_V={\mathrm{Transaction}}. $$

If no transaction occurred, there is no transaction point. Defining $R_{\mathrm{revenue}}=0$ for every nonexistent transaction would manufacture an infinite missing population.

The relevant missingness problem begins only after a transaction point exists and revenue is eligible.

10.2 Scheduled laboratory example

Consider a protocol spine:

$$ A_V= {\mathrm{Patient},\mathrm{Visit},\mathrm{Assay}}. $$

The protocol establishes eligible assay points. An absent assay result at an eligible point is a missing observation.

10.3 Operational reporting example

A store-day report may be expected only when a store is active. The eligibility set is therefore not the full Cartesian product of stores and dates:

$$ E_v

{(\mathrm{store},\mathrm{day}): \mathrm{active}(\mathrm{store},\mathrm{day})=1}. $$

If store activity is defined in another source, its transport and evidential status are part of the missingness foundation.

10.4 Population loss through transformation

An inner join may remove units without a mapping. The resulting dataset does not merely have “some missing rows.” It may represent a different universe:

$$ U_{\mathrm{all\ customers}} \neq U_{\mathrm{region\ mapped\ customers}}. $$

A missingness model fitted after the join can be internally correct while answering a question about the wrong population.

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11. The deepest claim: missingness is compositional

Most missing-data theories specify a mechanism for a given data object. Analytical systems repeatedly transform that object.

A foundational theory must say how the M-contract changes.

Missingness is not only a property of source data. It is a law that must propagate through analytical transformation.

11.1 Mapping

Let:

$$ w=f\circ v. $$

If $f:X\to Y$ is total, the observed support is inherited:

$$ R_w(a)=R_v(a). $$

If $f$ is partial—for example, logarithm on nonpositive values—then the transformation introduces a new definedness mechanism:

$$ R_w(a)

R_v(a)\cdot \mathbf 1[f(v(a))\text{ is defined}]. $$

This second absence is not source missingness. It is transformation-induced undefinedness. The output contract should preserve the distinction.

11.2 Reduction

Let:

$$ q:A_V\rightarrow B $$

and:

$$ w(b)

g{v(a):q(a)=b}. $$

The output observation indicator cannot be inferred from $R_v$ without a coverage rule.

Possible policies include:

• observed if any source value is observed;
• observed only if every eligible source value is observed;
• observed if weighted coverage exceeds a threshold;
• observed with an explicit coverage state;
• unavailable when the aggregator’s missingness requirements are violated.

Let the eligible fiber be:

$$ E_b={a\in E_v:q(a)=b}. $$

A reduction missingness policy is:

$$ R_w(b)

H_g\bigl( {R_v(a):a\in E_b}, E_b \bigr). $$

The definition of $H_g$ is part of the reducer family, not a generic null-handling preference.

These policies are not hypothetical. The Contract Calculus proves transformation rules for a finite population fragment in which eligible counts, observed counts, and a declared coverage mode travel as sufficient state through staged reduction (Wang, 2026b). $H_g$ is the M-contract reading of that machinery.

The output M-contract may also require the pushforward of coordinate determinants, summaries of observed determinant members, and new latent assumptions induced by aggregation.

11.3 Restriction and carving

A filter can have two meanings.

A restriction narrows the represented points while retaining the same universe interpretation.

A carve defines a new population.

The difference matters for missingness. Excluding units because a determinant is unavailable may change the estimand, not merely the support.

The distinction is formal. In the Contract Calculus the two operations produce different certified contracts, and their non-equivalence is a proved theorem of the population fragment rather than a modeling preference.

11.4 Alignment

To form a frame from members $v_1,\ldots,v_k$, the system needs a common eligible population and support policy.

Complete-case alignment uses:

$$ S_{\mathrm{cc}}

\bigcap_j S_{v_j}. $$

But that intersection may define a selected subpopulation with its own missingness mechanism. It should not silently inherit the universe name of the original members.

11.5 Lag and windows

If:

$$ \widetilde z(a_t)=z(a_{t-1}), $$

the determinant is available only where a predecessor exists and was observed.

The missingness of the lagged determinant combines:

• boundary absence;
• source missingness;
• temporal alignment;
• possible changes in universe membership.

11.6 Expansion and universe crossing

A one-to-many expansion may copy, allocate, weight, assign, or refuse a value.

Missingness must follow the same passage law. If one source point maps to several target points, an absent source may propagate to every target, while missing target mappings may create a different support failure.

A many-to-many crossing requires an explicit face and conservation law. Otherwise both values and missingness indicators can be duplicated or selectively dropped without a declared meaning.

11.7 A proof-backed core, and the G_M program

Part of the propagation calculus this section calls for already exists. For finite fragments, the Contract Calculus proves: inheritance of support under total mapping, with transformation-induced undefinedness kept separate; coverage-aware reduction whose eligible and observed counts stage correctly as sufficient state; the non-equivalence of restriction and carving; and relation expansion under declared replication, assignment, and exact allocation, with refusal of undeclared fan-out (Wang, 2026b; Wang, 2026c).

Those results govern populations, supports, and values. They do not yet carry the full M-contract. The natural formal companion to this paper is therefore a fragment—call it $G_M$—that extends the proved contracts with the M-contract and establishes, rule by rule, that the declared determinant classes, transports, and coverage policies are preserved or explicitly transformed by every admitted operation. Sections 10.1 through 10.6 are the specification of that fragment. It requires no new statistical theory. It requires the discipline already applied to values and populations, extended to the observation process.

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12. Three-anchor placement

The Theory of Data describes a governed member through three jurisdictions.

V-anchor

The V-anchor says where values live:

$$ A_V. $$

M-contract and M-anchor

The M-contract governs observation. Its M-anchor says at which coordinate resolution missingness may vary, while its observed, self, and latent clauses specify the other determinant classes.

B-law

B-law—the contract-inheritance boundary in the Theory’s current vocabulary—states where a particular transformation must stop. It is operator- and movement-specific.

Together:

V locates values. M governs their observation process. B restricts their transformation.

The M-contract is not merely an annotation on the V-anchor. It interacts with it through functional reachability and with B-law through transformation. A reduction that is permitted for observed values may still be unavailable because the output missingness policy is undefined or its coverage obligations fail.

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13. What is inherited and what is contributed

13.1 Established foundations

The following are established:

• response indicators and complete-data variables;
• MCAR, MAR, and MNAR;
• self-masking and latent-variable mechanisms;
• missingness graphs;
• recoverability and testability questions;
• coarsening-at-random;
• estimand frameworks and survey nonresponse structure;
• null and incomplete-database semantics;
• functional dependencies in incomplete relations.

13.2 Proposed contributions

The candidate contributions of this paper are:

1. Universe-before-mechanism principle Missingness is defined only after population membership and observation eligibility are established.

2. Event–spine distinction Occurrence-generated and expectation-generated universes induce different meanings of absence.

3. M-anchor as a typed coordinate determinant The coordinate part of missingness factors through a hierarchy-sensitive map from the V-anchor.

4. Functional-reachability admission rule A coordinate may enter the M-anchor only if each eligible V-anchor point determines one M-coordinate.

5. Boundary between coordinates and observed members Nonfunctional context must first be converted into a single-valued observed determinant member by a declared operation.

6. Lawful determinant transport Observed covariates at other grains or universes require explicit broadcast, reduction, lag, alignment, or crossing.

7. Four-class M-contract Coordinate, observed-member, self, and latent determinants are represented in one typed contract.

8. M-irrelevance Omitted axes receive a conditional-independence interpretation relative to the full contract.

9. Compositional missingness calculus M-contracts must transform under mapping, reduction, alignment, restriction, lag, expansion, and crossing. A population-and-support core of the required rules is already proved in the accompanying Contract Calculus; the M-contract extension is specified here as the $G_M$ program.

10. Evidence-ranked mechanism declarations Operationally testable claims, substantive assumptions, and unidentifiable components are separated rather than uniformly asserted.

11. Aggregation of contracts to joint ignorability Per-member M-contracts, coupled by a declared sequential mechanism, preserve MAR exactly when no active response factor depends on a co-missing value; the two structural guarantees recover the classical tractable regimes, and whether they hold is checkable from the declared digraph, supports, and pattern.

The novelty claim is provisional pending a systematic literature review. The contribution should not be described as the discovery of new missingness categories. It is a proposed foundation that makes the existing categories typed, population-aware, and compositional.

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14. Evidence and identifiability

The framework must not confuse declaration with proof.

Each M-contract clause should carry an evidential status:

$$ \mathcal E_v \in { \mathrm{verified}, \mathrm{corroborated}, \mathrm{assumed}, \mathrm{unidentifiable}, \mathrm{contradicted} }. $$

Examples:

• A declared functional map from clinic to region may be verified against data.
• Variation in response rates by clinic may corroborate a clinic-level M-anchor.
• MCAR can be contradicted by observed coordinate patterns.
• Absence of detected association does not verify MCAR.
• Exclusion of self-dependence is commonly an assumption.
• A latent determinant may be substantively motivated but unidentifiable.
• A transport from event-level history to patient-month may be verified as a computation while its causal interpretation remains assumed.

The framework does not make MNAR identifiable. It makes the MNAR assumption explicit, located, and consequential for computation.

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15. Implications for analysis and software

15.1 Analysis specification

A real-world missing-data analysis should declare:

• target universe;
• universe type;
• eligibility set;
• V-anchor;
• complete and observed member definitions;
• M-map and M-anchor;
• observed determinant transports;
• self and latent clauses;
• transformation history;
• evidence statuses;
• target estimand and inferential assumptions.

15.2 Imputation

An imputation model should operate over the declared eligible universe and target anchor.

A predictor can be included only after its transport to the target point is defined. This prevents accidental leakage across time, duplication across many-to-many relations, and arbitrary mixing of grains.

15.3 Complete-case analysis

Complete-case selection should be represented as a carve with an explicit resulting universe, not as a neutral deletion step.

15.4 Data pipelines

Pipelines should propagate:

• eligibility;
• support;
• missingness indicators;
• coverage;
• transformation-induced undefinedness;
• contract versions;
• evidence state.

15.5 Query systems

A query engine should distinguish:

• no universe point;
• eligible point with missing value;
• inapplicable value;
• inaccessible value;
• transformation-induced undefinedness;
• population excluded by a carve;
• unmatched crossing point.

Collapsing all of these to SQL NULL loses the distinctions required for missing-data analysis.

A serving layer should also expose the status of each returned field—governed, provisional, terminal, coordinate, or annotation—so that a downstream analysis can distinguish a governed member from a displayed calculation.

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16. Falsifiable research program

The proposal should be tested through claims that can fail.

16.1 Representational necessity

For stated classes of real analyses, removing universe, V-anchor, M-anchor, or determinant transport should create distinguishable accepted errors.

16.2 Independent reproducibility

Independent analysts should assign compatible universes, eligibility sets, anchors, and determinant transports at an acceptable rate.

16.3 Mechanism transport soundness

The proposed mapping, reduction, alignment, and crossing rules should preserve the declared missingness semantics. For the population and support components, this standard is already met by machine-checkable rules in the proved fragments. The $G_M$ extension should be held to the same standard: checkable rules, not narrative claims.

16.4 Failure prediction

The framework should predict errors caused by:

• event/spine confusion;
• population substitution;
• grain mismatch;
• illegal determinant broadcast;
• many-to-many predictor duplication;
• complete-case universe drift;
• aggregation coverage loss;
• lag boundary leakage.

16.5 Comparative intervention

Typed M-contracts should prevent held-out analytical failures more effectively than conventional metadata, null checks, or prose documentation alone.

16.6 Recoverability integration

The framework should connect its typed contracts to existing graphical recoverability criteria without changing their statistical validity.

16.7 Query semantics

Certified queries over governed missing data should be safe with respect to both incomplete-information semantics and missingness-mechanism assumptions.

16.8 Refusal correctness

When a system refuses because eligibility, transport, or missingness propagation is undefined, no equivalent plan should exist under the same declared contract.

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17. Limitations

This paper has several clear limits.

1. It is a foundations proposal, not an estimator.
2. It does not replace Rubin-style likelihood theory or graphical recoverability.
3. It does not identify MNAR mechanisms from observed data.
4. It does not yet provide a complete algebra for propagating M-contracts. A population-and-support core is proved; the determinant-transport extension is specified as the $G_M$ program but not yet formalized. Joint ignorability is characterized in §8 rather than left open; the residual item there is the treatment of non-monotone mechanisms that satisfy the condition pointwise without a structural guarantee.
5. The event–spine distinction may require additional universe types.
6. Functional reachability may be too restrictive for set-valued or distribution-valued coordinate contexts; those cases need explicit typed extensions.
7. The boundary between a contextual coordinate and an observed member may admit more than one valid modeling choice.
8. Evidence ranks require operational criteria.
9. Novelty claims require a systematic review across statistics, databases, survey methodology, longitudinal analysis, measurement theory, and formal data systems.
10. A rich contract can be wrong. The framework can enforce an incorrect declaration consistently.

These are research obligations, not reasons to hide the foundation.

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18. Conclusion

Missing-data research has made deep progress by formalizing the observation process, ignorability, recoverability, and nonresponse. Yet its usual complete-data object often arrives with population, coordinate, grain, and alignment decisions already settled.

In real analytical systems, those decisions are not background. They determine whether an absence is missingness, whether a covariate is available, whether a response indicator is defined, and whether a transformed dataset still denotes the original population.

This paper proposes a different starting point:

Missingness is a typed, compositional law of a member inside a universe.

The universe establishes which points exist. The V-anchor locates the values. The M-anchor locates the coordinate determinants of observation. Other observed determinants must be lawfully transported to the target point. Self and latent clauses distinguish forms of MNAR. Transformations must carry the missingness contract forward rather than treating it as preprocessing residue.

MCAR, MAR, and MNAR remain essential. They become more precise when attached to a complete data contract.

The paper is one layer of a larger program. The same closure discipline that governs values and populations in analytical engineering is extended here to the observation process. A defect introduced when the data object is constituted cannot be repaired by correct statistical methodology applied above it; a refusal at the lower layer is what protects the layer above.

The deepest research direction is therefore not another imputation method. It is a compositional missingness calculus capable of answering, for every analytical transformation:

• Which population now exists?
• Which values were eligible?
• Which values were observed?
• Which determinants were available?
• Which missingness claims remain valid?
• Which new absence was introduced?
• Which inference is still justified?

A mature theory of missing data should not begin with a matrix and ask only why some cells are empty.

It should first explain why those cells exist.

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