Fixed-priority preemption-threshold scheduling (FPTS) has been proposed as a generalization of fixed-priority preemptive scheduling (FPPS) and fixed-priority non-preemptive scheduling (FPNS) with the aim to improve schedulability and reduce run-time overheads. In this paper, we show that the existing worst-case response time (WCRT) analysis for FPTS is pessimistic and present an exact WCRT analysis. Moreover, we refine the task model for FPTS, making FPTS also a generalization of fixed-priority scheduling with deferred preemption (FPDS). Finally, we present exact analysis for FPTS for this refined task model and an example showing that FPTS can improve on FPPS and FPDS