A fundamental issue in the design and the interpretation of experimental studies of perception relates to the question of whether the participants in these experiments could perform the perceptual task assigned to them using another feature, or cue, than that intended by the experimenter. An approach frequently used by auditory- and visual-perception researchers to guard against this possibility involves applying random variations to the stimuli across presentations or trials so as to make the "unwanted" cue unreliable for the participants. However, the theoretical basis of this widespread practice is not well developed. In this article, we describe a 2-channel model based on general principles of psychophysical signal detection theory, which can be used to assess the respective contributions of the unwanted cue and of the primary cue to performance or thresholds measured in perceptual discrimination experiments involving stimulus randomization. Example applications of the model to the analysis of results obtained in representative studies from the auditory- and visual-perception literature are provided. In several cases, the results of the model-based analyses indicate that the effectiveness of the randomization procedure was less than originally assumed by the authors of these studies. These findings underscore the importance of quantifying the potential influence of unwanted cues on the results of psychophysical experiments, even when stimulus randomization is used. (Contains 6 figures, 1 table and 3 footnotes.)