How to fit averaged data to obtain a single psychometric function?

Short answer
Averaging psychometric curves may not be the preferred way to pool psychophysical data.

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Typically, extracted gold-standard outcome measures from the psychometric curves will be pooled and averaged to perform statistical analyses and so on.

For example, in the visual sciences a much-used outcome measure is the visual acuity where, for example, gratings in four orientations are shown using standard psychophysical tests. Since it's a 4AFC task, the 62.5% correct score will be taken as a measure of the threshold, where the gratings will be varied in their width, measures in cycles per degree (cpd) or related measure of visual angle.

To pool and average individual subject scores the acuity scores will be averaged, measured in cpd, and not the psychometric curves. For examples see Nau et al. (2013) and Bach et al. (1996).

If you would really insist, you could pool every measurement in case the method of constant stimuli or related paradigm was used. Then one single psychometric fit could be performed on the congregate data. Problem with this approach, as opposed to the preferred method described above, is that the fit will yield awesome outcomes in terms of superbly small variances in the fitted outcome parameters, as well as favorable descriptive statistical parameters, such as the correlation coefficient, simply because there are so many data points and hence many degrees of freedom. Further, random errors occurring in one, or a few subjects will now affect the overall fit and 'weird' data points will tend to be obscured by the multitude of data points per x value.

A better approach in terms of statistical descriptive outcomes would be to first average every data point and then do the fit. However, also here outliers will be obscured because of the averaging procedure. The power of psychometric fits is that individual subjects can be analyzed.

In case an adaptive method is used above procedure won't hold as each subject will have different x values. Adaptive procedures in general do not lend themselves very well for curve fitting as the data points around threshold are dense, but the trials targeting chance level or 100% correct are sparse or nonexistent altogether. Hence the asymptotes are ill-defined. You could average these data, if you insist, by averaging each of the fitted parameters and generate a 'master' fit out of those. Beware of logarithmic values, though, as averaging those is not arbitrary. Again, descriptive statistical parameters become obscure in such a master fit, and statistical analyses become difficult.

In all, my advice is to stick with pooling the single outcome measures obtained from the psychometric curve, i.e, stick to pooling your PSE outcomes.

<sub>References
- Nau et al., Transl Vis Sci Technol (2013); 2(3): 1
- Bach et al., optom Vis Sci (1996); 73(1): 49-53</sub>