The survival probability in a group of individuals may evolve in time due to the influence of a time-varying covariate. In this paper we present a model-based approach allowing the estimation of the functional link between the survival probability and a time-varying covariate when data are grouped and time-period censored. The approach is based on an underlying model consisting in non-stationary Markov processes and describing the survival of individuals. The underlying model is aggregated in time and at the group level to handle the group structure of data and the censoring. The aggregation yields a generalized non-linear mixed model. Then, a Bayesian procedure allows the estimation of the model parameters and the description of the link between the survival probability and the time-varying covariate. This approach is applied in order to explore the relationship between the daily survival probability of mosquitofish (Gambusia holbrooki) and their time-varying lengths (small mosquitofish die with a higher rate than large ones because they are more affected by predation, cannibalism and environmental stress).