The risk analysis for the occurrence of recurrent events is a major concern in many clinical research studies or epidemiological studies. In the field of oncology, therapeutic strategies are evaluated in randomised clinical trials in which efficacy is assessed through the occurrence of sequential events that define the progression of the disease. In HIV-infected patients, the infection evolves in several stages that have been defined by the occurrence of successive clinical events. The frame of this work is the regression models for the risk of multiple successive events. In practice, the hypothesis of existing correlations between the inter-event times cannot be a priori discarded. The aim of this work is to develop a regression model that would assess such correlations. In this setting, the most common method is to assume that correlations between inter-event times are induced by a random, unobserved heterogeneity across individuals. The corresponding model defines the individual hazard as a function of a random variable, or " frailty ", assumed to be gamma-distributed with a variance that quantifies the heterogeneity across individuals and incidentally the correlations between inter-event times. However, the use of this model when evaluating the correlations has the drawback that it tends to underestimate the variance of the frailty.A first approach was proposed for two sequential events in a "gap-timescale", in which the risk is defined as a function of the time elapsed since the previous event. The proposed method was derived from an approximation of the risk of second event given the first time-to-event in a frailty model for various frailty distributions. Another approach was defined in "calendar-time", in which the risk is expressed as a function of the time elapsed since the beginning of the subject's follow-up. The proposed method was derived from an approximation of the intensity conditional on the past in a frailty model. In both timescales, the method that was developed consists in including in the model an internal covariate, that is calculated on the history of the process, and that corresponds to the difference between the observed number of events and the expected number of events in the past period given the individual's other covariates.A review of the literature involving simulation studies showed that when defining the generation processes, most authors considered the case of heterogeneity in the population. However, in many simulation studies, only constant hazards are considered, and no event-dependence is introduced. Simulations studies showed that in both timescales, the test of the effect of the internal covariate in the proposed model proved more powerful that the usual test of homogeneity in the gamma frailty model. This gain of power is more noticeable in gap-time. Additionally, in this timescale, the proposed model provides a better estimation of the variance of the frailty when heterogeneity is low or moderate, more particularly in small samples.The method developed in gap-time was used to analyse data from a cohort of HIV-infected patients. It showed a negative correlation between the time from infection to first minor manifestation of immunodeficiency and the time from first minor manifestation of immunodeficiency to AIDS. The method developed in calendar-time was used to study the occurrence of repeated progressions and severe toxicities in a clinical trial for patients with advanced colorectal cancer. In this example, the method corroborated the results obtained with a gamma frailty model which showed a significant heterogeneity.
