# Appropriate link function for 2AFC data?

Contrary to some here, others (e.g. Brian Ripley, the authors of sensR, and the authors of psyphy) appear to think that using a standard binomial link function when analyzing two alternative forced choice data in which the minimum expected proportion correct is .5 is incorrect. However, their approach as to what the link function should be varies.

1.The sensR library uses:

function (mu)
{
tres <- mu
for (i in 1:length(mu)) {
if (mu[i] > 0.5)
tres[i] <- sqrt(2) * qnorm(mu[i])
if (mu[i] <= 0.5)
tres[i] <- 0
}
tres
}


2.The psyphy library uses:

function (mu)
{
m <- 2
mu <- pmax(mu, 1/m + .Machine\$double.eps)
qlogis((m * mu - 1)/(m - 1))
}


3.Gabriel Baud-Bovy implicitly recommends (1+exp(x)/(1+exp(x)))/2.

The approach selected seems like it may have some consequences for the result. Is there a "correct" link function to be using with these sorts of problems, or so long as the link, inverse link, mu.eta, and variance functions all agree is everything going to be all right? Is there a single source material that provides any authoritative guidance on this issue?

Following John's advice I plotted these functions...

alt text http://psychlab2.variablesolutions.org/~russell/ForInternet/2AFCFunctionPlots.jpg

The black line is a standard logistic function. The red line is the function from sensR. The blue line is from psyphy and the cyan line is from Gabriel Baud-Bovy, but given the oddness of the shape it provides, perhaps I misinterpreted him. The psyphy function line looks like what I'd expect a logistic function to look like in a psychophysics 2AFC experiment.

It doesn't just seem like it will have consequences, it will have large consequences. Fit that second function to the data you put in your last question on this. It goes dramatically negative as it approaches 0.5.

Perhaps more importantly, you also need to consider what the different equations mean for how one interprets the functioning of the mind.

There is no known function that is just best for all 2AFC*. Such a function would be tantamount to proving a universal law of the operations of the mind. You have to model your data if you want the very best fit.

*OK, some models like splines will just fit most anything but you'd have to justify why you have all the extra parameters theoretically.

(ASIDE: you were opposed to clipping when difficulty achieved maximum (or minimum). Consider, if you were modelling a robotic arm at the maximum point of travel you would just clip the results at the maximum point of travel (something your first equation does). Just because you didn't know what that point was before you found it doesn't mean anything. You found it when performance reached chance.)

• It is an interesting idea to fit the previous dataset here, it would tell me the extent to which the "problem" I'm worried about is solved by using these functions. Also, I may fit existing psychometric data individually by subject using glm and produce the output curves on the probability scale. Perhaps one will pass the eyeball test and the other won't. I'll update the question when I know more. – russellpierce Aug 12 '10 at 14:49
• ASIDE: I'll grant that the model should match the process. In the case of modeling successes and failures of the robot arm you propose, values past its maximum movement are non-nonsensical, consequently they are past the point of observed values; so if I did that I'd be committing another error, attempting to extrapolate beyond the range of observed values. – russellpierce Aug 12 '10 at 14:57
• ASIDE continued: However, in psychophysics we have a system that starts off not working at all, gradually starts being able to process the information provided to it, and then asymptotes near 100% when the information is clearly available. Here I may have presented stimuli that are too hard to discriminate (beyond its range on the lower bound) or presented stimuli that are too easy to discriminate (beyond its range on the upper bound). Here it is reasonable ask what the system will do at those extreme values. – russellpierce Aug 12 '10 at 15:03