The psychometric function relates the performance of an observer to some quantitative stimulus characteristic (e.g., Kingdom & Prins,
2010). I will refer to the latter in the remainder simply as the
intensity of the stimulus, symbolized by
x. In models of observer performance, a specific shape of the function is typically assumed and is almost invariably some sigmoidal function such as the cumulative Gaussian distribution or the Weibull function. In the remainder, I will use
F(
x;
α,
β), or simply
F, to refer to the function that characterizes performance of an underlying sensory mechanism. Two parameters characterize the function, its threshold and its slope. The threshold parameter (symbolized by
alpha:
α) specifies the function's location and is typically defined as that stimulus intensity at which a specific level of performance is achieved. The slope parameter (symbolized by
beta:
β) determines the function's rate of change. In this paper I use the Gumbel (or log-Weibull) function throughout as the model of
F and use the subscript “
G” to specify this:
In practice, we cannot measure
F directly. In psychophysical studies, we instead infer
F from the pattern of responses (e.g., proportion correct in a multiple alternative forced-choice [mAFC] task or proportion endorsement [e.g., “I perceived the stimulus”] in a “yes/no” task) observed across trials in which stimuli of varying intensity
x are used. The probability of observing a positive response as a function of stimulus intensity
x is usually modeled as:
FG,
x,
α, and
β are as defined above. The parameter
γ corresponds to the lower asymptote of
ψ, while the parameter
λ determines the upper asymptote. Neither
γ nor
λ characterizes the performance of the underlying sensory mechanism, which is our primary interest. In a “yes/no” task, the lower asymptote,
γ, corresponds to the false alarm rate and characterizes the decision process. In the more common mAFC task,
γ is determined by the task and is assumed to equal 1/
m in a task with
m response categories. Parameter
λ determines the upper asymptote of
ψ. The upper asymptote often deviates from unity due to so called lapses, which are negative responses (“no” or incorrect) that are stimulus independent (for example, “finger errors”). Thus, parameter
λ also does not characterize the underlying sensory mechanism but rather is a function of observer characteristics such as attention or vigilance. Since
γ and
λ do not describe the underlying sensory mechanism but nevertheless do affect our observations (which we must model to recover the underlying function
F), they are considered to be “nuisance parameters”.