The psychometric function model

A short informal introduction to psychometric functions.

Psychometric functions are a common way to summarize behaviour in a task with two possible outcomes which depend on continuous stimulus level set by the experimenter. We assume that one outcome is dominant at low stimulus levels and the other outcome is dominant at high stimulus levels, such that the probability of the latter outcome is a sigmoid function of the stimulus level as plotted in the following figure:

As psychometric functions are most commonly used for performance data we call the outcome we converge to at high stimulus levels correct and model the percent or proportion of correct responses at each stimulus level. To summarize this function we use 4 parameters as illustrated in A:

• the threshold m: The level where moved half-way from guessing to maximum performance
• the width w: The difference between the 5% and the 95% level of the function
• the lapse rate lambda: How many incorrect responses are given at maximum performance
• the guess rate gamma: How many correct responses are given at infinitely low or 0 stimulus level

Of these the threshold m and width w are usually of much higher interest. Lambda and gamma are usually given by the experimental design or expected to be close to 0.

As illustrated in B & C, there are many possible choices for the exact shape of the sigmoid function used in the model. All of them can be parametrized using the threshold and width, which have comparable meaning for all of them, such that we can handle them all the same. The different function shapes each have their own standard parameters, which can be obtained with helper functions from the threshold and width.

Classically, the proportion of correct trials is modelled as a binomial distribution. Measured average performances sometimes deviate farther from their mean than expected from the binomial model. To compensate for this overdispersion we use a beta-binomial distribution which can allow for higher variance.

For more details and the exact mathematical treatment refer to our paper (sections 2.1 & 2.2).