We present a new decision-making model that can account for trial-by-trial variability induced by a process ("pre-process") that occurs before an explicit sensory signal specifying a later motor response. A process after explicit sensory signals, referred to herein as the "post-process", has been investigated by a variety of so-called rise-to-threshold models including the LATER model. The LATER model formulates post-process variability but treats the pre-process as fixed within a block of an experiment. We propose an extension of the LATER model, which we call the extended LATER (ELATER) model, to account for trial-by-trial variability of both pre- and post-processes together. We present the mathematical formulation of the ELATER model and analyze its characteristics, including numerical examples and an example of saccade latency data in reward-manipulated conditions with caudate activity. The ELATER model is useful for investigating decision making by taking account of trial-by-trial variability of both pre- and post-processes.