How the Staircase Works

Xianji uses a 3-down-1-up adaptive staircase, the standard psychophysical method for finding a threshold without running hundreds of trials at every difficulty level.

The rule

Three correct in a row → the next trial is harder (occlusion moves earlier)
One incorrect → the next trial is easier (occlusion moves later)

A timeout counts as incorrect for this rule, and is recorded separately.

This rule converges on the point where the athlete is correct about 79.4% of the time. That figure is not arbitrary — it is where a 3-down-1-up rule settles mathematically, since the level at which the probability of three consecutive correct responses equals 0.5 is 0.5^1/3 ≈ 0.794.

What a session looks like

offset
 (ms)
 200│                                    ╭─╮   ╭─╮
    │                            ╭───╮  ╱   ╲ ╱   ╲
 150│                    ╭───╮  ╱     ╲╱     V     ╲
    │            ╭───╮  ╱     ╲╱                    ╲
 100│    ╭───╮  ╱     ╲╱        ← reversals: the peaks
    │   ╱     ╲╱                  and troughs
  50│  ╱
    │ ╱  ← coarse steps        ← fine steps →
   0│╱
    └────────────────────────────────────────────── trial
     start          converging          settled

The offset climbs while the athlete succeeds, drops when they fail, and settles into oscillation around their threshold. Each direction change is a reversal.

Step sizes

The staircase starts with coarse steps to reach the right region quickly, then switches to fine steps to resolve the threshold precisely. For badminton the coarse step is 33 ms (two display refreshes at 60 Hz) and the fine step is 17 ms (one refresh); the switch happens after 2 reversals.

The deadline tightens

The response deadline starts at 800 ms and shortens by 50 ms at each reversal, down to a floor of 400 ms.

This prevents a strategy where the athlete compensates for a hard occlusion by deliberating longer. Anticipation that takes 900 ms to produce is not anticipation as it functions in play.

Threshold estimation

threshold = mean(last 6 reversals)
SE        = standard deviation(last 6 reversals) / √6

Early reversals are discarded because they reflect the staircase travelling toward the threshold, not oscillating around it.

When a session is rejected

A session reports no threshold if either:

Condition	Meaning
Fewer than 6 reversals	The staircase never settled. Usually too few trials
SE > 25 ms	The reversals are too scattered to support an estimate. Usually drifting performance

The session terminates at 12 reversals or 60 trials, whichever comes first.

Why not just measure accuracy at a fixed offset?

A fixed offset measures the wrong thing for most athletes. Set it too early and everyone is at chance; too late and everyone is at ceiling. Neither discriminates, and neither is comparable between athletes of different ability.

The staircase instead finds each athlete’s own operating point, expressed on a physical scale (milliseconds before contact) that means the same thing for everyone.

References

Levitt, H. (1971). Transformed up-down methods in psychoacoustics. Journal of the Acoustical Society of America, 49(2), 467–477.
García-Pérez, M.A. (1998). Forced-choice staircases with fixed step sizes: asymptotic and small-sample properties. Vision Research, 38(12), 1861–1881.
Abernethy, B. & Russell, D.G. (1987). The relationship between expertise and visual search strategy in a racquet sport. Human Movement Science, 6(4), 283–319.
Müller, S. & Abernethy, B. (2012). Expert anticipatory skill in striking sports. Research Quarterly for Exercise and Sport, 83(2), 175–187.