Assumptions A1–A8

Every method in longmi is valid only under stated assumptions. This file is the canonical enumeration; imputers and pooled results reference these labels in their ValidityDeclaration. None of A2, A5, A8 is fully verifiable from observed data. The package distinguishes what it enforces, what it declares, and what must be argued substantively.

At a glance

Assumption Plain-language meaning Software checks it? Analyst must justify it? Sensitivity / recourse
A1 Participants or clusters are independent Partly (id column is the unit) Yes Cluster-level resampling
A2 Given the observed data, missingness does not depend on the missing outcomes No — untestable Yes Delta adjustment
A3 Outcome and missingness parameters are distinct No Yes Shared-parameter model needed if violated
A4 Every observed history has some chance of follow-up Partly Yes Inspect follow-up probabilities
A5 The imputation distribution is adequate for the estimand No Yes Predictive checks; simulation
A6 Parameter and outcome uncertainty are propagated Yes (backend-verified) No Backend declaration
A7 The complete-data estimator and covariance are valid Partly (adapter returns the robust covariance) Yes Alternative analysis
A8 Imputation and analysis models are compatible Partly (nesting by construction where documented) Yes Bootstrap-then-impute (planned); richer imputation model

A1. Independent clusters

\[ (Y_i, R_i, X_i) \perp (Y_k, R_k, X_k), \qquad i \neq k. \]

Dependence within participants is allowed and must be explicitly modeled. The participant (or cluster) is the sampling unit for all resampling-based inference.

Practical check: confirm the id column identifies the independent unit; if participants are nested in sites, the site may be the correct unit.

A2. Missing at random (MAR)

\[ p_\psi(R \mid Y^{\mathrm{obs}}, Y^{\mathrm{mis}}, X) = p_\psi(R \mid Y^{\mathrm{obs}}, X). \]

Missingness may depend on observed baseline and longitudinal outcomes, but not additionally on the missing outcomes after conditioning on the observed data. MCAR is the stronger special case \(p(R \mid Y, X) = p(R)\).

Practical check: MAR is not empirically testable against MNAR from the observed data alone. The package therefore treats MNAR sensitivity analysis (delta adjustment) as the appropriate response to doubt about A2, not a hypothesis test.

A3. Distinct parameters

The response-model parameter \(\theta\) and missingness-model parameter \(\psi\) are variation independent: the joint parameter space is \(\Theta \times \Psi\), and the prior or likelihood imposes no shared restrictions that would break ignorability.

Practical check: rarely violated in applied settings; violated when, e.g., a shared random effect drives both outcome and dropout (a shared-parameter MNAR model).

A4. Positivity

For histories represented in the target population,

\[ P(R_{ij} = 1 \mid H_{i,j-1}) > 0. \]

No observed type of participant can have a structurally zero probability of follow-up.

Practical check: inspect follow-up rates within strata of predictors and observed history; structural zeros (e.g., a site that never administered wave 3) must be handled by design, not imputation.

A5. Correct imputation model

\(p_\theta(Y^{\mathrm{mis}} \mid Y^{\mathrm{obs}}, X)\) is correctly specified for the estimand of interest — or at least sufficiently rich that the relevant conditional means, interactions, and variances are correctly represented.

Practical check: posterior-predictive checks on observed margins; simulation under known data-generating processes (the simulation suite deliberately includes misspecified imputation models to demonstrate failure).

A6. Proper parameter and outcome draws

Each imputation includes uncertainty in both the model parameters and the missing outcomes (Proposition 2). Fixing parameters at point estimates produces improper, overconfident imputations.

Enforced: every imputer's ValidityDeclaration must state parameter_uncertainty_propagated; the reference imputers draw \(\theta^{(m)}\) per imputation.

A7. Valid complete-data estimator

The estimator applied to each completed dataset is consistent and asymptotically normal when complete data are observed, with a valid covariance estimator (for GEE, the robust sandwich covariance).

Practical check: the analysis adapter must return the covariance actually recommended for the complete-data method, not a model-based shortcut.

A8. Congeniality / compatibility

The imputation model preserves the relationships needed by the analysis model. In particular it must include:

  • every analysis variable;
  • the exposure-by-time interaction;
  • the relevant time functional form;
  • clustering / within-participant dependence;
  • auxiliary variables used to explain missingness.

A categorical-wave imputation model contains a linear-time analysis as a restricted special case, so imputing with categorical wave effects (and the corresponding exposure-by-wave interactions) is the safe default.

Special care with GEE: standard Rubin variance is not universally robust to uncongeniality when the completed-data estimator is method-of-moments (Meng 1994; Yang & Kim 2016). See gee_after_imputation.md for the qualification and robust alternatives.