Correlated geometric models of order k and its application to intensive care unit and leprosy data

Geometric models are used to analyse the discrete time until the occurrence of an event of interest (success or consecutive successes). In two real data sets, named leprosy and intensive care unit (ICU), the events correspond, respectively, to abandoning the clinical treatment of leprosy, where abandonment corresponds to four consecutive patient absences from treatment, and the patient's discharge from the ICU. The distribution proposed in this article, called the correlated geometric distribution of order k (or correlated k-order geometric distribution), $$ , consists of including a correlation parameter in the geometric distribution of order k, thus considering the dependence between patient responses until the occurrence of the event. This model proves to be a better option for real data analysis where the effect of individual correlation is considered. The model is applied to real leprosy data to estimate the treatment abandonment probability. Bayesian methods are used to determine the parameter estimators of the models and to evaluate regression models. The covariates are related to the probability of the event by an appropriate link function chosen by Bayesian selection criteria. A diagnostic analysis evaluates the models fit by posterior randomized quantile residuals and influential observations by $$ -divergence measures. This methodology is illustrated by simulation studies and real ICU admission data analysis. Studies show a good fit of the proposed model. Real data analyses also find that the probabilities of the event of interest can be overestimated or underestimated when modeled without considering the effect of dependency on the model.