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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.

A generalized linear model is overdispersed when the conditional variance exceeds the variance prescribed by its response family and mean, for example $\mathrm{Var}(Y_i\mid X_i)>V({\mu }_i)$ for a unit-dispersion Poisson or binomial model. Unobserved heterogeneity or dependence among repeated observations can cause it. A generalized linear mixed model adds random effects for the corresponding clusters or subjects, thereby representing this heterogeneity and induced dependence.

The generalized linear mixed model tries to explain overdispersion by replacing the fixed minority coefficient with a Gaussian random intercept. Conditional on the group effect $b_m$,

This is a poor use of a random effect because minority has only two levels. Two realized intercepts contain almost no information about a random-effects distribution or its variance, and the two levels are substantively fixed categories rather than a sample from a population of groups. The model also has worse Akaike information criterion than model 1, $1132.8>1122.3$, and its fit does not establish that the original overdispersion has disappeared.

For response $j$ from subject $s$, model2 is the generalized linear mixed modelwith conditional independence given the random intercepts. The fitted values are ${\hat{\beta }}_0=8.9966$, ${\hat{\beta }}_1=-1.2317$, ${\hat{\tau }}^2=1.7533$, and fitted residual dispersion $0.7591$.

The random intercept models persistent between-subject differences and the resulting within-subject dependence among repeated measurements. That is the main feature absent from model1.