The diffusion model (Ratcliff, 1978) and the leaky competing accumulator model (LCA, Usher & McClelland, 2001) were tested against two-choice data collected from the same subjects with the standard response time procedure and the response signal procedure. In the response signal procedure, a stimulus is presented and then, at one of a number of experimenter-determined times, a signal to respond is presented. The models were fit to the data from the two procedures simultaneously under the assumption that responses in the response signal procedure were based on a mixture of decision processes that had already terminated at response boundaries before the signal and decision processes that had not yet terminated. In the latter case, decisions were based on partial information in one variant of each model or on guessing in a second variant. Both variants of the diffusion model fit the data well and both fit better than either variant of the LCA model, although the differences in numerical goodness-of-fit measures were not large enough to allow decisive selection between the models.