Software effort estimation in the early stages of the software life cycle is one of the most essential and daunting tasks for project managers. In this research, a new model based on non-linear regression analysis is proposed to predict software effort from use case diagrams. It is concluded that, where software size is classified from small to very large, one linear or non-linear equation for effort estimation cannot be applied. Our model with three different non-linear regression equations can incorporate the different ranges in software size.