Our particular problem arises from a survey of acute respiratory tract infection (ARI), in which 1,312 kindergarten children in Beijing have been followed day by day for 1 year. Six types of infection are considered: upper respiratory tract infection (including influenza), pyogenic tonsillitis, laryngitis, otitis media, bronchitis, and pneumonia. The record of each child consists of not only the types of diseases and the times of illness, but also age, sex, some indices of health, family history, and personal history of respiratory diseases. During the data analysis, in addition, we considered several basic meteorological variables kindly provided by the Beijing Meteorological Observatory. A parametric statistical model based on a renewal process with competing risks and time-dependent covariates is used to analyse these data. Similar models have been considered by several authors, including Kalbfleisch and Prentice ( 1980), Prentice, Williams, and Peterson ( 1981), and Hsieh, Crowley, and Tormey (1983). These authors use reordering of the time scale in order to modify the partial likelihood techniques of the (semiparametric) proportional hazards model. In contrast to this, our parametric model allows a direct approach, avoiding the complicated conditioning argument in the model with time-dependent covariates, enjoying full efficiency (since inference is based on the full likelihood), and performing the analysis directly in chronological time as opposed to the reordered time scales of the above authors. The asymptotic statistical properties are standard for our fully parametric model. The statistical model was proposed in J.-Q. Fang's unpublished Ph.D. dissertation (University of California at Berkeley, 1985). The history of each individual can make several contributions to the full likelihood in the sense of a renewal process; the proportional hazards model is assumed with a general parametric underlying hazard (log-polynomial);