Mixed Missingness Mechanisms in Data Analysis

• Mixed-missingness mechanisms are scenarios where multiple missing data processes (MCAR, MAR, MNAR) coexist and interact across variables.
• They require integrated modeling approaches such as pattern-mixture models, copula models, and Bayesian methods to address non-ignorable missingness.
• Advancements in these methods improve bias reduction, variance estimation, and causal inference in fields like biomedicine, survey analysis, and semi-supervised learning.

A mixed-missingness mechanism refers to any data situation in which multiple types of missingness mechanisms (e.g., MCAR, MAR, MNAR) or multiple causes of missingness are present—potentially acting simultaneously on different variables, data blocks, entry positions, or across repeated measurements. In statistical inference and machine learning, these mixed or composite mechanisms necessitate inference strategies, identifiability analysis, and imputation methods that go beyond the classic single-mechanism frameworks, such as Rubin’s MCAR/MAR/MNAR taxonomy. Modern modeling integrates multicausal indicators, distinguishes structured dependence on observed or unobserved data, and frequently incorporates the presence of non-ignorable (MNAR) components that cannot be marginalized out or ignored without bias. Mixed-missingness frameworks are critical in biomedicine, causal inference, survey science, and semi-supervised learning, where practical missingness processes rarely conform to a single, ignorable type.

1. Formal Definitions and Taxonomies

The mixed-missingness paradigm encompasses both pattern-mix (different missingness rules across variables or entry patterns), cause-mix (different mechanisms by reason), and structural-mix (missingness indicators depending on other missingness indicators). Morikawa and Kano formalized this by introducing a categorical missingness indicator $M_t \in \{0, 1, ..., C\}$ at each time or data location, encoding both the presence and cause of missingness (Morikawa et al., 2014). Each cause-specific mechanism may independently be “MAR” (missing at random, depending only on observed values) or “NMAR” (missing not at random, depending on unobserved values):

• MAR for cause $c$: $$P(M_t=c|Y_{obs}, Y_{mis}; \phi_c) = P(M_t=c|Y_{obs}; \phi_c)$$.
• NMAR for cause $c$: $P(M_t=c|Y_{obs}, Y_{mis}; \phi_c)$ depends on $Y_{mis}$.

In the multivariate context, structured mixed-missingness mechanisms as in Jackson et al. (Jackson et al., 2023) allow each missingness indicator $M_j$ to depend not only on $X$ but also on the vector of other missingness indicators $M_{-j}$: $$p(M | X, \gamma) = \prod_{j=1}^p p(M_j | X, M_{-j}, \gamma_j)$$. The classification of mixed-missingness thus spans:

• Unstructured MCAR, MAR, MNAR: Each column/entry has a purely independent (by data and/or missingness values) mechanism.
• Structured MCAR/MAR/MNAR: Each mechanism may be functionally or probabilistically dependent on other missingness indicators.
• Strong or weak structures: Missingness in one variable may logically determine (strong) or affect probabilistically (weak) the missingness in another.

Hierarchical mixed-missingness mechanisms—with strict cause priority—permit certain ignorability results even in the presence of NMAR components (Morikawa et al., 2014).

2. Likelihood Factorization and Ignorability in Mixed Contexts

Under general mixed-missingness, the observed-data likelihood cannot always be factorized into a product of the data and missingness models (i.e., $L(\theta, \phi) = L^Y(\theta)L^M(\phi)$) unless all mechanisms are ignorable (fully MAR/MCAR). In presence of any NMAR component, the missingness mechanism parameters $\phi$ and data parameters $\theta$ are typically entangled in the observed-data likelihood:

$$L_n(\theta, \phi) = \prod_{i=1}^n \int f(Y_{obs,i}, y_{mis,i};\theta) P(M_i | Y_{obs,i}, y_{mis,i};\phi) dy_{mis,i}$$

Precisely, as shown by Morikawa and Kano (Morikawa et al., 2014):

• All-MAR/MAR-combinations: Each can be ignored for inference on $\theta$.
• Any NMAR present: None are ignorable, and even MAR components coupled in the same pattern with NMAR cannot be omitted unless a strict hierarchical structure and full knowledge of causes is available.
• Hierarchical structure: MAR causes with lower or equal priority to the least NMAR cause remain ignorable, simplifying inference under known-order causality.

Structured missingness (Jackson et al., 2023) further complicates ignorability: when strong dependencies exist between missingness indicators (e.g., file-matching or block-wise missingness), standard multiple imputation under MAR fails, even if each variable is MCAR or MAR univariately.

3. Model-based Approaches for Mixed-Missingness

A variety of likelihood-based, Bayesian, and algorithmic approaches directly address mixed-missingness. Prominent frameworks include:

• Pattern-Set Mixture Models: Deep generative models clustering missingness patterns and interpolating between ignorable and non-ignorable regimes (Ghalebikesabi et al., 2021).
• Probit/Logit Selection Models: For each data block or variable, a parametric selection model (often a link function applied to the unobserved variable) is jointly coupled with a data model (e.g., M5 for proteomics (O'Brien et al., 2015)).
• Nonparametric Copula and Mixture Models: Flexible modeling of multivariate mixed-type data and missingness, where missingness is MAR overall but can accommodate complex nonlinear dependencies (Feldman et al., 2022).
• Gaussian Process Latent Variable Models: For NMAR in mixed data, missingness indicators are modeled as probit/logistic GPs sharing the same latent space as the substantive variables (Mitsuhiro et al., 2021).
• Finite Mixture Models with MCAR+MAR: As in SSLfmm (McLachlan et al., 3 Dec 2025), label-missingness in semi-supervised learning is modeled as a mixture of MCAR (random dropping) and entropy-based MAR (ambiguity-driven), with an explicit mixture structure in the missing indicator likelihood.
• Maximum Mean Discrepancy (MMD) Estimation: Robust M-estimation under MCAR with explicit bias quantification under arbitrary contamination of the missingness mechanism (Chérief-Abdellatif et al., 1 Mar 2025).

Algorithmically, inference proceeds via EM/Gibbs samplers (for pattern-mixture or selection models), fast convex optimization (for two-stage matrix completion in survey data (Mao et al., 6 Feb 2024)), and stochastic gradient (for MMD-based M-estimators).

4. Causal, Semi-supervised, and Survey Applications

Mixed-missingness mechanisms are central in:

• Causal inference with mechanism shifts: $lm$-graphs generalize $m$-graphs to encode mechanism changes induced by missing entry values, yielding estimands such as the Full Average Treatment Effect (FATE) and Natural Average Treatment Effect (NATE) (Aguas et al., 18 Jun 2025). These frameworks account for missingness-induced shifts in downstream mechanisms by context-sensitive graphical labeling.
• Semi-supervised learning (SSL): In SSLfmm, the class-label missingness mechanism is explicitly modeled as a mixture of MCAR and entropy-based MAR. The resulting expectation-conditional-maximization algorithm leverages both labeled and unlabeled data, and the model can outperform fully supervised training with only labeled data available (McLachlan et al., 3 Dec 2025).
• Survey analysis under heterogeneous and stratified informative missingness: Two-stage matrix completion approaches fit separate logistic models for each block/question/covariate, then perform IPW-regularized likelihood estimation, yielding error bounds robust to entry-wise heterogeneity (Mao et al., 6 Feb 2024).
• Multiple imputation for mediation and TMLE: In scenarios with multiple MNAR points—e.g., each mediator, confounder, and outcome influencing its own missingness (Dashti et al., 26 Mar 2024, Dashti et al., 2021)—empirical studies show that MI performance deteriorates unless imputation models are tailored to the full m-DAG, include substantial interactions/nonlinearities, or enforce outcome-model compatibility.

5. Practical Implications, Identifiability Results, and Empirical Findings

Mixed-missingness mechanisms fundamentally challenge ignorability, identifiability, and effective use of standard imputation/inference pipelines:

• Ignorability and identifiability: Only in special cases (all-MAR, hierarchical structure with known causes) is complete-data likelihood inference possible without modeling missingness intricacies (Morikawa et al., 2014). Under general “mixed” MNAR, recovery of parameters typically requires (i) explicit modeling of each cause/variable’s missingness and (ii) incorporating missing data indicators or offsets in prediction/imputation models (Beesley et al., 2021, Dashti et al., 2021).
• Multiple imputation bias: Simulations in mixed MNAR settings consistently show that ignoring self-driven missingness (or omitting critical interactions) produces substantial bias in both mediation and causal effect estimation (Dashti et al., 26 Mar 2024, Dashti et al., 2021). The best-performing MI strategies either enforce strict compatibility between imputation and substantive models (SMCFCS), or employ tree-based/fully nonparametric imputation to capture nonlinearity-driven missingness.
• Empirical gains from mixed-modeling: Explicit mixed-missingness models—such as midpoint mixed models with probit selection (O'Brien et al., 2015), entropy-driven MAR label dropout in SSL (McLachlan et al., 3 Dec 2025), or NMAR-aware data fusion via GPs (Mitsuhiro et al., 2021)—yield lower bias, improved coverage, and sometimes even lower prediction error than complete-data baselines.
• Variance estimation and algorithmic considerations: BootMI (bootstrap-then-impute) provides less biased interval coverage than MI-then-bootstrap (MIBoot) under multivariable mixed MNAR mechanisms (Dashti et al., 26 Mar 2024). Algorithmic efficiency with structured missingness can require innovative optimization routines (e.g., nuclear norm penalization with fast singular value thresholding (Mao et al., 6 Feb 2024)).

These observations underscore the necessity for mechanism-aware modeling, flexible specification (allowing hierarchical, block, or multicausal dependence), and careful inferential design whenever mixed-missingness is present or cannot be robustly ruled out.

6. Extensions and Current Directions

Active research emphasizes several directions in mixed-missingness:

• Structured missingness detection: Systematic characterization via graphs, tree-based analyses, and association metrics can help uncover and model deterministic or probabilistic cross-indicator dependencies (Jackson et al., 2023).
• Causal effect recovery with mechanism shifts: $lm$-SCMs and their identification theory differentiate full-data and natural (observed-mechanism) effects, and motivate the development of estimator classes robust to context-specific independence shifts (Aguas et al., 18 Jun 2025).
• Contamination-robust estimators: Approaches such as parametric MMD estimation provide explicit quantification of bias due to both model and mechanism misspecification, offering finite-sample and asymptotic control under “mildly” adversarial missingness (Chérief-Abdellatif et al., 1 Mar 2025).
• Flexible and nonparametric imputation: Bayesian copula models for mixed data (Feldman et al., 2022), as well as modified chained equations with missingness indicators/interactions/offsets (Beesley et al., 2021), enable robust inference in complex, structured, and high-dimensional missingness regimes.
• Software and reproducibility: Implementation packages such as SSLfmm (for SSL under mixed label-missingness) (McLachlan et al., 3 Dec 2025) and BootImpute (for valid MI-based variance estimation (Dashti et al., 26 Mar 2024)) encapsulate state-of-the-art methods fitting modern mixed-missingness settings.

The consensus across recent literature is that recognizing, correctly specifying, and computationally leveraging mixed-missingness mechanisms are essential for unbiased inference, valid uncertainty quantification, and efficient learning in contemporary, real-world data environments.