Mathematical foundations and validity conditions¶

longmi implements established missing-data theory; it does not introduce a new proof of multiple imputation. This chapter states the framework, the assumptions (detailed in assumptions.md), and four cited propositions. Each proposition records what is proved, under which assumptions, and where the original proof lives. Full citations are in references.md.

What these propositions establish¶

1. Under MAR and parameter distinctness, likelihood-based outcome inference may ignore the missingness model (Proposition 1).
2. Drawing parameters and then missing outcomes generates posterior-predictive imputations (Proposition 2).
3. Repeated completed-data estimates can be pooled under proper-imputation and compatibility conditions (Proposition 3).
4. GEE and other estimating-equation analyses need additional care, because Rubin variance is not universally robust to uncongeniality (Proposition 4).

Plain-language background is in why impute the response? and MCAR, MAR, and MNAR.

1. Full and observed data¶

Participant \(i = 1, \dots, n\) has longitudinal response vector

at known observation times \(t_1, \dots, t_J\), and fully observed predictors \(X_i\). The response indicator is

and the response partitions as \(Y = (Y^{\mathrm{obs}}, Y^{\mathrm{mis}})\).

Two models are in play:

• the longitudinal response model \(p_\theta(Y \mid X)\), parameterized by \(\theta\);
• the missingness model \(p_\psi(R \mid Y, X)\), parameterized by \(\psi\).

The estimand \(Q\) is a functional of the response model (in the motivating application, coefficients of a marginal count-outcome regression).

2. Assumptions¶

Stated in full in assumptions.md; in brief:

• A1 (independent clusters) — \((Y_i, R_i, X_i)\) are independent across \(i\); dependence within a participant is allowed and explicitly modeled.
• A2 (missing at random) — \(p_\psi(R \mid Y^{\mathrm{obs}}, Y^{\mathrm{mis}}, X) = p_\psi(R \mid Y^{\mathrm{obs}}, X)\).
• A3 (distinct parameters) — \(\theta\) and \(\psi\) are variation independent.
• A4 (positivity) — observed histories have nonzero follow-up probability.
• A5 (correct imputation model) — \(p_\theta(Y^{\mathrm{mis}} \mid Y^{\mathrm{obs}}, X)\) is correctly specified for the estimand of interest.
• A6 (proper parameter and outcome draws) — each imputation propagates uncertainty in both the parameters and the missing outcomes.
• A7 (valid complete-data estimator) — the completed-data estimator is consistent and asymptotically normal with a valid covariance estimator.
• A8 (congeniality / compatibility) — the imputation model preserves the relationships needed by the analysis model.

3. Proposition 1 — Ignorability under MAR¶

Claim. Under A2 (MAR) and A3 (distinctness), the missingness mechanism may be ignored for likelihood-based or Bayesian inference about \(\theta\).

Source. Rubin DB. Inference and Missing Data. Biometrika. 1976;63(3):581–592.

Derivation. The observed-data likelihood is

Under MAR the missingness density does not depend on \(Y^{\mathrm{mis}}\), so it moves outside the integral:

If \(\theta\) and \(\psi\) are distinct (A3), the factorization implies inference about \(\theta\) can be based on \(p_\theta(Y^{\mathrm{obs}} \mid X)\) alone, without fitting the missingness model.

What this does and does not establish. This is ignorability under MAR, not a proof that MAR is true. The theorem establishes the consequence of MAR if it is assumed; MAR itself is not empirically testable from the observed data (see A2 in assumptions.md).

4. Proposition 2 — Posterior-predictive imputation¶

Claim. Drawing parameters from their observed-data posterior and then missing responses from their conditional distribution generates draws from the posterior-predictive distribution \(p(Y^{\mathrm{mis}} \mid Y^{\mathrm{obs}}, X)\).

Sources. Rubin 1978; Rubin 1987; Tanner & Wong 1987; Schafer 1997.

Derivation. For imputation \(m\), draw

Marginalizing over \(\theta\),

which is ordinary Bayesian marginalization: the two-stage scheme samples exactly this mixture.

Consequence for implementation. Every longmi imputer must draw both model parameters and missing outcomes (A6). Fixing \(\theta = \widehat\theta\) and sampling only outcomes omits parameter uncertainty and generally yields overconfident inference.

Terminological care. The displayed integral proves that the two-stage algorithm generates posterior-predictive draws. "Proper imputation" has a more technical repeated-sampling definition in Rubin (1987, ch. 4). Posterior-predictive draws from a correctly specified Bayesian model are the canonical route to proper MI, but the two statements are distinguished throughout this documentation.

Count outcomes. For a negative-binomial response model, one posterior-predictive draw uses the gamma–Poisson representation:

which guarantees \(Y_{ij}^{\mathrm{mis},(m)} \in \{0, 1, 2, \dots\}\).

5. Proposition 3 — Repeated-imputation inference¶

Claim. Under proper imputation, valid complete-data inference, regularity conditions, and congeniality, completed-data estimates may be pooled with Rubin's repeated-imputation rules; the pooled estimator is consistent for \(Q_0\) as \(n \to \infty\).

Sources. Rubin 1987; Meng 1994; Wang & Robins 1998.

Point estimate. For completed dataset \(m\), compute \(\widehat Q^{(m)}\); the MI point estimate is

Conditional on the observed data, the completed datasets are posterior-predictive draws, so by the conditional law of large numbers

That much is elementary Monte Carlo. The stronger claim — that under identification, correct imputation-model specification (A5), regularity, a consistent complete-data estimator (A7), and compatibility (A8),

— is the subject of Wang & Robins (1998), who derive the asymptotic behavior and variance structure of parametric multiple-imputation estimators, including procedures beyond Rubin's original class of proper imputations.

Variance. With within-imputation variance \(U^{(m)} = \widehat{\mathrm{Var}}(\widehat Q^{(m)} \mid Y^{\mathrm{complete},(m)})\), define

The motivation is the law of total variance,

with \(\overline U\) estimating the first term, \(B\) the second, and \(B/M\) accounting for the Monte Carlo error in \(\overline Q_M\) itself.

This one-line decomposition is motivation, not a complete proof. Exact repeated-imputation validity additionally requires proper imputations (A6), valid complete-data inference (A7), and congeniality between the imputation and analysis procedures (A8; Meng 1994). The same formula applies with covariance matrices for vector \(Q \in \mathbb{R}^p\). Degrees of freedom and fraction of missing information follow Barnard & Rubin (1999); the exact formulas as implemented are in ../algorithms/rubin_pooling.md.

6. Proposition 4 — Estimating-equation qualification¶

Claim. When the substantive analysis is a method-of-moments or GEE estimator, standard Rubin variance requires additional compatibility conditions; robust alternatives include Robins–Wang variance, over-imputation, or bootstrap-then-impute.

Sources. Robins & Wang 2000; Yang & Kim 2016; Bartlett & Hughes 2020; see also Meng 1994.

A GEE estimator solves \(\sum_i U_i(\beta) = 0\) and is not derived from a likelihood. Meng (1994) formalizes congeniality — whether the imputation and analysis procedures can be viewed as arising from a coherent common Bayesian model — and shows Rubin's variance can be conservative or anticonservative without it. Robins & Wang (2000) derive variance estimators that remain consistent when the imputation and analysis models are incompatible or misspecified. Yang & Kim (2016) show Rubin's variance estimator can be asymptotically biased when the complete-data estimator is method-of-moments and congeniality fails.

longmi therefore does not claim that any imputation model plus any GEE plus Rubin pooling is automatically valid. The defensible claim, stated in gee_after_imputation.md, is:

Standard Rubin pooling is supported when imputations are proper, the imputation model is compatible or congenial with the GEE estimand, and the complete-data sandwich covariance is valid.

Bootstrap-then-impute (Bartlett & Hughes 2020) is the planned robust fallback: resample participants, refit the imputation model within each replicate, impute, refit the analysis, and take inference from the bootstrap distribution. The interface anticipates this mode now; implementation is scheduled for 0.2.

7. What the software enforces¶

Proofs establish validity under stated assumptions; the package enforces what is mechanically checkable and declares the rest:

• checkable, and enforced by LongitudinalData / CompletedDatasetCollection: observed values never overwritten; every eligible missing value imputed; count outcomes remain nonnegative integers; predictors fully observed; identical term ordering across completed-data fits;
• declared, via each imputer's ValidityDeclaration and surfaced by result.validity_report(): the assumed missingness mechanism, whether parameter and outcome uncertainty are propagated, congeniality status, and the recommended pooling method;
• assessed empirically, via the simulation suite (bias, coverage, widening with missing information, deliberate misspecification) and cross-language parity with R.