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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 $B_i$.

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.

Ordinary multiple imputation by chained equations relies on missing at random, conditional on the variables included in its models, together with adequate model specification and suitable conditional distributions. For Bayesian inference that ignores the response mechanism, the usual sufficient condition also includes distinct parameters for data and missingness models, with an appropriately separate prior.

The suspected dependence on withheld responses makes the usual missing-at-random assumption doubtful. An accurately fitted observed-data model cannot, on its own, establish what the nonresponders would have reported. Use substantive judgement and sensitivity analysis under plausible missing not at random departures, for example a pattern-mixture sensitivity analysis shifting the imputed condom log odds or partner-count distribution relative to the observed-data prediction. Report how the final association changes. Additional informative measurements may make missing at random more credible, but cannot verify it.

A missingness mechanism is ignorable for likelihood-based inference about the outcome parameter when the observed-data likelihood can be formed from the outcome model alone. The standard sufficient conditions are MAR and distinct parameters: the outcome-model parameter and missingness-model parameter have a product parameter space. The missingness indicators then carry no additional likelihood information about the outcome parameter once the observed outcomes are given.