In this section you can find some more information about the different shapes your psychometric function can take. Which one you go for is mainly dictated by your data but you should also take theoretical aspects into account.
A variety of different parametric shapes for psychometric functions have been used. In probit analysis for example, the data are essentially fit by a cumulative gaussian; visual contrast detection data have been reported to be well fit by a weibull distribution function. Fitting visual contrast detection with a weibull function is also theoretically appealing because it corresponds to the quick pooling model ([Graham_1989] p. 165).
Psignifit supports a relatively large number of psychometric function shapes. These are selected using two keywords: ‘sigmoid’ and ‘core’ (this is independent of whether you are using bootstrap or Bayes). To understand the meaning of these two keywords, let us take a look at the model that psignifit tries to fit:
Here, is a parameter vector (in forced choice tasks is fixed). The critical term that determines the shape of the psychometric function is . We decompose in two functions, a scalar function and a higherdimensional function , such that
In many cases (but not all), will be a simple linear transformation, while will inject a nonlinearity. We will call the ‘sigmoid’ and the ‘core’.
The figure illustrates how sigmoid and core are related to each other. A sigmoid does not have any parameters. Thus, fitting a psychometric function with only a sigmoid would always result in the same psychometric function. Two such sigmoids are shown in the left column of the figure: The first is a logistic sigmoid and the second is the cumulative distribution function of the standard exponential distribution. In order to have parameters that describe the shape of the psychometric function, we use a core object. The top row of the figure illustrates two core objects: the first is an abCore that can be requested with the keyword ‘ab’. We can see that the output of this core is simply a linear function of . However, the slope and intercept of this linear function depends on the two parameters and . The second plot in the first row illustrates a polyCore, as requested with the keyword ‘poly’. Note that the poly core is a nonlinear function of . Again, the two parameters and determine the precise form of the nonlinear function. In order to illustrate the fact that each core object represents a large number of different functions in , four different combinations of and have been plotted.
The four plots in the lower right of the figure demonstrate how sigmoids and cores can be combined to allow for a large number of possible psychometric function shapes. For instance, the lower right plot is a combination of the Exponential sigmoid and the poly core. The resulting function is the cumulative distribution function of the weibull distribution. The combination of logistic sigmoid and ab core corresponds to the logistic function that was the default setting in earlier versions of psignifit. The advantage of separating sigmoid and core is that we can now use a different core object, to specify that a function should be fitted on different axes (e.g. logarithmic instead of linear) or in a different parameterization. Also note, that the figure only presents two sigmoids and two cores. This results in two different function families for the psychometric function. Psignifit includes 6 different sigmoids and 5 different cores, resulting in 30 different function families.
The following two sections describe the sigmoids and cores in more detail. Then finally, there is a section about common combinations of sigmoids and cores.
Six different sigmoids can be selected. All of them correspond to cumulative distributions functions.
There are also six different cores to be selected. The first three are simply linear transformations of the stimulus intensities. The remaining three cores are nonlinear transformations. Typically, these will be needed to define a weibull function.
As already mentioned above, combinations of ‘sigmoid’ and ‘core’ determine the shape of the nonlinear function . There are some shapes that are particularly interesting in psychophysical applications. This section explains how to obtain these typical shapes.
In this case, we combine the ‘logistic’ sigmoid with one of the linear cores (ab,mw,linear). Depending on the core used, this results in different parameterizations.
where . This allows to be interpreted as the 75% threshold and as the width of the interval in which rises from to . A typical choice for is 0.1.
This parameterization does not allow a psychophysically meaningful interpretation of the parameters.
The cumulative gaussian is obtained by combining the gauss sigmoid with one of the linear cores (ab,mw,linear). The parameterizations are precisely the same as for the logistic function with one exception: The scaling factor z(alpha) for the mw parameterization is , where is the inverse of the the cumulative gaussian.
Also for the cumulative Gumbel sigmoids, the parameterizations are similar to the logistic function. However, the Gumbel distribution is skewed. This implies that the alpha parameter of the ab parameterization can not be interpreted as a 75% threshold. For the mw parameterization this is solved in a different way. The lgumbel + mw function is parametrized as follows:
There are a number of ways to parametrize the Weibull function.
which is implemented using the combination of an exponential-sigmoid and a poly-core.