Deriving the functional form to fit confidence ratings in psychophysical experiments

Introduction: Measuring subjects’ confidence, or metacognition, permits insight into subjects’ decision making processes. Subjects can report their confidence in each decision they make during a psychophysical experiment. Typically, researchers fit an arbitrary function, if anything, to describe these data. Here I describe a simple procedure to derive a functional form for confidence ratings, based on the psychometric curve. Methods: A subject’s confidence during a psychophysical task should be proportional to the total amount of information available for each decision. If the psychometric function is known, fitting their decision data, then a subject’s uncertainty is proportional to the slope of that function, plus any intrinsic information in the stimuli. For example, if a subject must discriminate between two stimuli of different contrasts, then the total amount of information would be proportional to the derivative of the psychometric function plus the sum of the log contrast of each stimulus; i.e., subjects are more confident for suprathreshold stimuli. Results: I demonstrate this using two experiments in which subjects discriminated between Gabors. In one experiment, subjects judged which stimulus had the higher contrast (comparative judgement), and in the second, subjects judged whether the two stimuli were equal (equality judgment). These decisions have different functional forms. The comparative judgement is monotonic, with a maximum slope at the point of subjective equality. The equality judgment is a negative convex function with an absolute maximum—therefore its derivative has two humps. In both experiments, a linear term fits the sum of the stimuli log contrasts. Conclusion: As an application, I show that, even when the psychometric functions are identical, subjects’ uncertainty can be used to discriminate intrinsically biased decisions, of which subjects are unaware, from explicit bias, e.g. instructions to make a given choice unless subjects are certain that another choice is correct.