Multifactor software reliability modeling with software test metrics data is well known to be useful for predicting the software reliability with higher accuracy, because it utilizes not only software fault count data but also software testing metrics data observed in the development process. In this paper we generalize the existing Cox proportional hazards regression-based software reliability model by introducing more generalized hazards representation, and improve the goodness-of-fit and predictive performances. In numerical examples with real software development project data, we show that our generalized model can significantly outperform several logistic regression-based models as well as the existing Cox proportional hazards regression-based model.