In this article we consider cases where data consist of many small and independent groups of correlated failure time observations. For each failure time, which may be censored, some important covariates are also recorded. Our goal is to examine the covariate effects on the individual failure time observations. We assume that the logarithm of each failure time is linearly related to its covariates. We then take a population-averaged model approach to obtain inference procedures for the regression parameters without specifying the joint distribution of the observations within the group. The new proposals do not need complicated and unstable nonparametric estimates for the hazard function of the error term. Their properties are extensively examined for practical sample sizes. Comparisons among various procedures are also performed. All the methods studied in this article are illustrated with examples.