# How can I design a staircase function that converges on a given hit-rate?

I'm in the process of creating a simple contrast staircase function where subjects are initially presented with high-contrast Gabor patches and are subsequently queried as to their orientation in a forced-choice task.

I only understand staircase functions superficially, which is to say that I conceive them as follows:

1. present a stimulus
2. query some feature of the stimulus
3. If correct, reduce the intensity of the stimulus by n, else increase its intensity by m where m < n.

Intuitively, it seems like this should converge on a hit-rate of .5, but I know for a fact that a staircase can be designed to converge on an arbitrary hit-rate. How would I modify my simplistic algorithm to converge on, say, .8 ?

Thank you!

If n != m then it will not home in on the 50 % threshold. In these simple N-up/N-down staircases, you can modify either the stepsize (as you proposed) or the number of successes/failures to act as a criterion for upgrade/downgrade.

A comprehensive introduction to these staircases and the effect of changing these properties can be found in this paper. The 80 % threshold is simple because it's simply nDown=1 and nUp=3 with the same stepsize in either direction.

Several staircase methods are implemented in the PsychoPy python package/software and it actually has the nDown=1 and nUp=3 as default value if you don't specify anything else.

Note: If you can assume a nice mathematical relationship between stimulus parameter and perception (i.e. linear, logistic, power etc.), then it'll probably be more efficient to use regression because even though your parameter is way off the desired threshold, it's still informative about the mathematical relationship. Thus you can use it to infer where the threshold should be and home in much more efficiently than the "scanning-like" approach of N-up/N-down methods.

• Aha! Thanks for pointing out my flawed assumption and thank you for the reference! Jun 7, 2014 at 15:07
• You're welcome. I just added a bit of info on staircasing using regression. Jun 9, 2014 at 7:26
• You can gain even more flexibility in the desired thresholds by varying the size of the up- and down-steps. Have a look at Kaernbach (1991).
– crsh
Sep 18, 2014 at 8:22
• And adding to the comment of @crash Zwislocki 2001 describes how to obtain a 75% threshold by a 1-up 3-down rule. This can be modified to obtain other thresholds.
– AliceD
Oct 31, 2014 at 3:45