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A recorded variable included in imputation models to help predict missing values or explain response patterns, even if it is absent from the final substantive model. Useful auxiliaries can improve precision and make missing at random more plausible; they cannot establish that assumption from the observed data.

Let $R$ indicate observation of a regression response $Y$, with fully observed predictors $X$. If $Y\perp R\mid X$ and $\mathbb{P}(R=1\mid X)>0$ on the target support, the observed conditional density equals the full conditional density. Thus a correctly specified regression model can be fitted to observed responses under conditional missing at random. This does not justify the unadjusted complete-case mean when observation rates and outcome means both vary with $X$, nor does it imply missing completely at random.

For monotone missing data, let $R_{ij}$ indicate observation and let ${\rho }_{ij}$ be the product of the conditional retention probabilities through visit $j$. Under sequential missing at random, conditioning on the complete response history makes these retention probabilities depend only on the history already observed at each decision. Therefore $\mathbb{E}(R_{ij}/{\rho }_{ij}\mid Y_i,X_i)=1$. Weighting a working-independence generalized estimating equation by $R_{ij}/{\rho }_{ij}$ restores its mean-zero property. This needs correct retention probabilities, positivity and regularity; uncertainty must account for estimating the weights.

A subject has a random intercept $B_i\sim N(0,{\tau }^2)$, with responses conditionally independent given $B_i$ and their covariates. Integrating the product of Bernoulli likelihoods over $B_i$ defines the joint subject likelihood and induces within-subject dependence. Its coefficients describe conditional odds ratios for a fixed latent subject effect. Under missing at random with distinct parameters, the observed-response likelihood integrates over the same random effect and sums out unobserved responses. Dependence of dropout on $B_i$ is not generally ignorable merely because conditional independence holds given $B_i$.

When a response $Y$ is missing at random conditional on a fully observed category $X$, estimate its conditional probabilities using respondents within each category and average them with the category frequencies from all patients. For a binary response, $\hat{P}(Y=1)={\sum }_x\hat{P}(X=x)\hat{P}(Y=1\mid X=x,R=1)$. The weighting uses the full cohort rather than only complete cases.

The mechanism assigns probabilities to observation patterns $R$ conditional on complete measurements $D$: $p(R\mid D)$. Whether it depends on missing values distinguishes missing completely at random, missing at random and missing not at random.

Specify a conditional distribution for each incomplete variable given the others, cycle through parameter draws and missing-value draws, and retain multiple completed datasets after suitable convergence checks. Preserve observed entries. Adequate models, appropriate support and the intended analysis structure matter; arbitrary conditional models need not be compatible with one joint distribution. Standard applications assume missing at random given included predictors. Use Rubin's rules to pool estimates and uncertainty.

If a regression response $Y$ is observed but covariate $X$ is sometimes missing, likelihood inference requires a joint model: for example $f_{\beta }(y\mid x)g_{\eta }(x)$. An incomplete record contributes $\int f_{\beta }(y\mid x)g_{\eta }(x)dx$ to the observed-data likelihood. Missing at random with distinct parameters allows the missingness mechanism to be omitted; it does not remove the need to integrate over $X$.

For a population question use a population-averaged logistic model for repeated binary outcomes. Code $z_i=1$ for the new treatment and $0$ for the control, with visit times $t_j=2j$ months for $j=1,2,3$. Let $w_{ij}=(1,z_i,x_i^T,t_j{)}^T$, ${\gamma }_M=({\alpha }_M,{\varphi }_M,{\beta }_M^T,{\delta }_M{)}^T$, andDifferent subjects are independent. Each response has a Bernoulli distribution with this marginal mean, but no independence of a subject's three responses is assumed. The linear time effect and absence of treatment-by-time interaction are substantive mean assumptions; if inappropriate they can be replaced by time indicators and interaction terms.

For complete data, set $m_i=(m_{i1},m_{i2},m_{i3}{)}^T$, $A_i=\mathrm{diag}(m_{ij}(1-m_{ij}))$, and $D_i=\partial m_i/\partial {\gamma }_M^T=A_i{W}_i$, where $W_i$ has rows $w_{ij}^T$. A working exchangeable correlation matrix has unit diagonal and off-diagonal value $\rho \in (-1/2,1)$; set $V_i=A_i^{1/2}C(\rho )A_i^{1/2}$. The generalized estimating equation isWith correct means and independent subjects, a positive-definite working covariance matrix and regularity give consistency even if the working correlation is wrong. Write $H={\sum }_i{D}_i^T{V}_i^{-1}{D}_i$ and $u_i=D_i^T{V}_i^{-1}(Y_i-m_i)$. The fitted subject-level sandwich covariance matrix isevaluated at the estimates. This inference is asymptotic in the number of independent subjects, not the number of visits.

For the actual incomplete data, define $R_{ij}=1$ if visit $j$ is observed, with $R_{i0}=1$ and monotone dropout. Let ${\mathcal{H}}_{i,j-1}$ contain treatment, baseline covariates and responses observed before visit $j$. Under sequential missing at random, specifyThe assumption is that the retention decision, conditional on this history, does not additionally depend on unseen current or future outcomes. Fit the $q_{ij}$, for example by visit-specific logistic regressions. A simple valid choice uses working independence and the inverse-observation-weighted estimating equations for longitudinal dropout:Terms with $R_{ij}=0$ are zero and require no missing response. Given the full response vector, sequential missing at random makes $\mathbb{E}(R_{ij}/{\rho }_{ij}\mid Y_i,z_i,x_i)=1$. Consequently the weighted score has the same expectation as the complete-data score, proving its mean-zero property. Correct retention probabilities, positivity and a correctly specified marginal mean are required. Use cluster-robust uncertainty, accounting for fitted retention-model coefficients, or resample entire subjects and refit both models. Under missing completely at random, the simpler unweighted observed-response GEE can suffice; ordinary unweighted GEE is not valid under arbitrary history-dependent missing at random.

The treatment coefficient gives a population conditional-on-baseline odds ratio $e^{{\varphi }_M}$ at each visit. To communicate an absolute population treatment benefit, standardize predicted probabilities over the full baseline population, rather than only completers:A causal reading additionally uses the randomized controlled trial assignment, consistency of the treatment definition, no interference and the dropout assumptions; the statistical model alone does not supply those conditions.

For conditional individual response profiles use a logistic random-intercept model for repeated binary outcomes. Introduce a subject effect $B_i\sim N(0,{\tau }^2)$ independently across subjects and independently of treatment and baseline covariates, under the specified model. Conditional on $B_i,z_i,x_i$, assume the three responses are independent Bernoulli random variables withHere $B_i$ is persistent unmeasured subject heterogeneity and ${\tau }^2$ its variance. It induces dependence after integration: for distinct visits, the response covariance is ${\mathrm{Cov}}_B(p_{ij}(B),p_{ik}(B))$, generally positive. This is a complete joint generalized linear mixed model, unlike a mean-only GEE specification.

Under missing at random conditional on the observed responses and baseline variables, with distinct parameters for missingness and outcomes, the missingness model is ignorable for likelihood inference. Let ${\mathcal{O}}_i=\{j:R_{ij}=1\}$. The outcome observed-data likelihood isThe omitted Bernoulli factors sum to one over missing outcomes. Fit the fixed coefficients and ${\tau }^2$ by maximizing this integrated likelihood, with suitable quadrature or another justified integration method. If $\tau =0$, use the limiting point-mass model. Correct response and random-effect distributions, identifiable coefficients and regularity are needed; the latent distribution assumption is not merely a convenient working covariance.

For patient $i$, combine the fitted model with the posterior density of $B_i$ proportional to its normal density times that patient's observed Bernoulli factors. Integrating $p_{ij}(b)$ over this posterior gives a patient-specific predicted response profile. The treatment odds ratio for a fixed $b$ and the same baseline variables is $e^{{\varphi }_C}$. This conditional contrast does not by itself identify each person's unobserved causal treatment effect: a parallel-arm trial observes only one treatment per person. Random slopes or heterogeneous treatment effects require further modelling and sufficient information; they are not implied by fitting a random intercept.

The population-averaged logistic regression describes $P(Y_{ij}=1\mid z_i,x_i)$ after averaging over unmeasured individual heterogeneity. Its intercept is a marginal log odds at reference baseline values and time zero, its treatment coefficient is a population conditional-on-baseline log odds ratio, and its time coefficient is the change in that marginal log odds per month. Time zero is an extrapolation here because the recorded visits begin later; centring time at the first visit would give a more interpretable intercept.

In the random-intercept logistic model, the intercept is instead the conditional log odds for a subject with $B_i=0$. The treatment and time coefficients apply at fixed $B_i$; the subject's actual intercept is ${\alpha }_C+B_i$. Even with a treatment-independent random intercept, the conditional and marginal coefficients generally differ through noncollapsibility of the odds ratio. Indeed, if $\eta ={\alpha }_C+{\varphi }_{C}z+{\beta }_C^{T}x+{\delta }_{C}t$, marginalization giveswhich is not generally a logistic curve with the same linear coefficients. To quantify random-intercept attenuation of marginal logistic slopes, let $p_B={\mathrm{logit}}^{-1}(\eta +B)$. ThenFor a nondegenerate finite random intercept this derivative lies strictly between zero and one and usually varies with $\eta$. Thus the marginal treatment contrast is an integrated, attenuated version of the conditional contrast, and the marginal time slope need not be constant. One should not equate the two sets of coefficients, even in a randomized controlled trial without treatment confounding.

For missing completely at random, an unweighted observed-response GEE with cluster-robust standard errors can consistently fit the marginal mean, and a correctly specified integrated GLMM likelihood is valid as well. Under missing at random depending on previous observed outcomes, ordinary unweighted GEE is generally biased because the remaining responses are selected by informative observed history. Correct inverse-observation-weighted estimating equations for longitudinal dropout recover the marginal target under the sequential MAR and positivity assumptions already stated. Covariate-only observation mechanisms are a simpler special case in which correctly conditioned unweighted mean equations can remain valid.

A correctly specified joint response GLMM, fitted by observed-data likelihood, is valid under ignorable missing at random with distinct parameters, including dependence of missingness on observed response history. This is a likelihood property, not a claim that every mixed model automatically fixes missingness. Under missing not at random, neither ordinary GEE, MAR-based weights nor a response-only GLMM is generally valid; the unseen-outcome dependence requires further modelling and sensitivity assumptions.

First investigate what predicts dropout among variables actually observed, record reasons where possible, and seek additional outcome follow-up. Dependence on observed history alone can be missing at random; informative dependence on unseen responses or latent heterogeneity requires a missing not at random analysis. Observed data do not ordinarily distinguish MAR from all MNAR explanations.

An explicit selection model for informative longitudinal dropout can augment a joint response model $f_{\vartheta }(Y_i\mid z_i,x_i)$ by dropout hazards. Define $R_{ij}$ as above and, among subjects retained through visit $j-1$, putOmit the previous-response term at the first visit. A nonzero $\kappa$ makes dropout depend on the currently missing response. For monotone observation indicators $r_i$, letThe appropriate observed-data likelihood isThus a patient's dropout pattern contributes information in the model; dropping the $g_{\eta }$ factor is not justified under this informative mechanism.

Another option is a shared-parameter model for informative dropout: retain the conditional Bernoulli response model, but let the dropout logit contain the same latent effect, for example $a_j+c^T{x}_i+d{z}_i+\lambda Y_{i,j-1}+{\kappa }_B{B}_i$. Conditional on the shared effect and observed history the processes can be independent, while marginally dropout remains associated with unseen responses. The joint likelihood must integrate the response factors and dropout factors together over $B_i$. A response-only mixed-model fit does not generally account for this dependence.

Because $\kappa$ or ${\kappa }_B$ can be weakly identified from the observed outcomes, report sensitivity of the treatment conclusion to plausible informative-dropout assumptions. One can fix a range of these dependence parameters and refit, or use pattern-mixture sensitivity analysis that shifts the imputed post-dropout log odds by a specified amount from the MAR prediction. Present population risks and treatment contrasts across these scenarios with their uncertainty. These are explicit identifying assumptions, not an empirical test proving which missingness mechanism is true. Neither a complete-case analysis nor carrying forward the last response supplies such a justification.

Let $R_i=1$ indicate an observed outcome. Fitting the conditional outcome model to responders requireswith a positive observation probability over the relevant predictor support. This is conditional missing at random for the predictors used in the regression: after conditioning on them, observation must not further select participants by their unobserved outcome. It gives $f(Y_i\mid x_i,Z_i,R_i=1)=f(Y_i\mid x_i,Z_i)$, the complete-case regression under conditional missing at random condition. Correct specification of the conditional mean model is also required. Missing completely at random is sufficient but stronger; simply assuming missingness depends only on a post-randomization variable excluded from this model would not suffice.

Within the displayed sequential response selection model, $\theta =0$ makes every attempt's nonresponse probability independent of $Y$, and hence gives the outcome-independence needed for complete-case analysis. The estimate $\hat{\theta }=0.38$ suggests that larger outcomes are associated with greater odds of an unsuccessful attempt. Per one-unit outcome increase, the fitted odds ratio is $e^{0.38}\approx 1.46$, with the reported interval transformed to $(e^{-0.06},e^{0.82})\approx (0.94,2.27)$.

The confidence interval for $\theta$ includes zero, so there is no rejection of outcome-independent missingness at the five-percent level within this model. This does not establish that the missing at random assumption is true: the interval also allows appreciable outcome dependence, and inference depends on the chosen selection model. The intercepts describe response levels, not independence from the outcome. Nor can the respondents' mean and standard deviation establish the missing outcomes' distribution. There are $300-211=89$ unobserved outcomes, about $29.7\%$, so conclusions about the full trial depend materially on the missingness assumptions.

Let $Y$ denote the full data and $R$ the pattern of indicators, with $R_j=1$ for an observed component and $R_j=0$ for a missing one. For each fixed pattern $r$, split the data as $y=(y_{\mathrm{obs}}(r),y_{\mathrm{mis}}(r))$. The missing at random condition isHere $\psi$ parameterizes the missingness mechanism. Equivalently its conditional probability, with the observed data fixed, is constant over all possible completions of the missing data. The restriction is pattern-specific because the observed components depend on $r$. Missing at random allows missingness to depend on observed values; missing completely at random imposes independence from all the data.

Under missing completely at random, the observed second-year outcomes form a random subsample, so the direct pooled complete-case analysis estimate isIt is valid under missing completely at random because dropout no longer changes the marginal distribution of second-year use. Pooling is simpler than reporting separate conditional probabilities, but MCAR alone does not make this estimate more efficient than the estimate in the previous part. The fully observed first-year variable carries useful outcome information and can improve precision even when dropout is completely random.

To make the qualification explicit, write $X=Y_1$, $m(X)=P(Y_2=1\mid X)$, $\theta =E\{m(X)\}$ and $\rho =P(R=1)$ for the constant response probability. For $N$ patients the leading variance of the pooled estimator is $\mathrm{Var}(Y_2)/(N\rho )$. The standardization estimator in the preceding part has leading varianceIndeed its first-order centered contribution is $m(X)-\theta +R\{Y_2-m(X)\}/\rho$: the two terms have zero covariance, and their variances give that expression. The law of total variance therefore yieldsThe inequality is strict when there is dropout and first-year use predicts second-year use, as the distinct conditional rates suggest here. Thus the requested universal efficiency claim needs qualification. In the unrestricted joint binary-outcome model, maximizing the observed-data likelihood under either missing at random or missing completely at random gives the same standardization estimate $0.384889$: the all-patient first-year proportion and the two observed conditional second-year proportions maximize its factored outcome likelihood. Restricting the distinct missingness parameters to a common response probability affects their factor, not this estimate. The pooled value is the simple valid MCAR answer; retaining the first-year information gives the efficient MCAR answer and does not require replacing the previous estimate.