In this article we consider the case that each patient in a longitudinal study may experience two or more distinct failures. The corresponding failure times, which are possible censored, are recorded for each patient. The logarithm of each marginal failure time is assumed to be linearly related to its covariates; however, the distributional form of the error term in the model does not have to be specified in the analysis. Furthermore, no specific structure of dependence among the distinct failure times on each subject has to be imposed. Various linear regression methods for analyzing multivariate failure time observations are proposed. Our procedures do not involve the unstable nonparametric hazard function estimation. Extensive numerical studies are conducted to evaluate the new proposals. Recommendations are also made for their practical usage.