Missing not at random (MNAR)

Data are missing not at random when the probability of missingness still depends on an unobserved value after conditioning on the observed data.

Model the joint distribution by multiplying a response model by a sequential dropout model. For example, the logit of dropout before visit $j$, conditional on remaining observed, can depend on the currently unobserved $Y_{ij}$ as well as previous responses and baseline covariates. The observed-data likelihood sums over the missing binary responses. Such a dependence parameter is usually weakly identified or unidentified from observed responses alone; sensitivity analysis must vary it over scientifically justified values.

A response model and a dropout model share latent random effects. Conditional on those effects and observed predictors, the two processes may be independent, yet marginally dropout is missing not at random because the shared effects also predict unseen outcomes. Joint likelihood fitting integrates the product of both models over the shared effects; integrating the outcome model alone is insufficient. This is an explicit identifying model, and its latent-dependence assumptions should be assessed by sensitivity analysis.