How the SDT confidence model reads a Space Observer trial
On every trial, a planet briefly appears covered in orange and green aliens. The brain gets a noisy impression of which colour is more common, picks one, and reports how sure it is. Signal detection theory turns that story into five steps and a handful of parameters.
1The stimulus
Each display shows a single planet with 68 aliens scattered over its surface, some orange and some green, shown only briefly. The task is to say which colour is more abundant. We code each trial by the difference between the two colours, scaled by the total:
st = (ngreen − norange) / 68
Positive values mean more green, negative values more orange. Which colour gets the plus sign is an arbitrary convention and the model is symmetric. A staircase adjusts the difference so that each person gets about 73% correct, so most trials are close calls.
Try the flash at a difference of 6 or 8. That is roughly where the staircase holds most people.
2The internal evidence, and β
The observer doesn't register the colour difference perfectly. On each trial they get a noisy internal sample of how much greener than orange the display looked:
xt = β · st + ε, ε ~ Normal(0, 1)
β is sensitivity. It converts the physical colour difference into internal signal, measured in units of the noise. Someone with β = 8 gets twice as much signal from the same display as someone with β = 4. The noise is fixed at 1 because it sets the unit; only the ratio of signal to noise can be estimated.
The classic SDT measure d′ is the distance between the internal distributions for a "more orange" and a "more green" display of the same difficulty. Here that's d′t = 2 · β · |st|. Shekhar and Rahnev fitted a separate d′ for each difficulty level. However, with a staircase it could be the case that there are too few trials per level, so we assume d′ grows linearly with the colour difference and β is the slope.
3The choice
The decision rule is simple: answer "green" if xt > c0, otherwise answer "orange". At c0 = 0 the observer is unbiased; a positive c0 means they need more evidence before saying green, so they say orange more often. Combining steps 2 and 3 gives a probit psychometric function:
P(green | s) = Φ(β · s − c0)
β sets how steep the curve is, and c0 shifts it sideways. Φ is the standard normal cumulative distribution function.
4Confidence
This is the central SDT assumption. Confidence uses the same internal sample that drove the choice, and reads off how far it landed from the criterion. A sample just past the boundary feels like a guess; a sample far out feels certain.
With four confidence levels there are three criteria, mirrored on both sides of c0. They're fitted as a starting point plus positive gaps, so they can never cross: k1 = d1, k2 = k1 + d2_inc, k3 = k2 + d3_inc.
| Distance of x from c0 | Confidence |
|---|---|
| less than k1 | 1 (lowest) |
| k1 to k2 | 2 |
| k2 to k3 | 3 |
| more than k3 | 4 (highest) |
So on any single trial the evidence distribution is cut into eight regions, four confidence levels for each answer. The area of each region is the model's probability of that exact response. Those eight numbers are the whole model.
Sample an internal reading and see which response it produces.
Fitting works by looking up, on each of a participant's roughly 80 trials, the height of the bar for the response they actually gave, and multiplying these across trials. A fixed lapse of 0.1% per trial keeps one odd response from driving the likelihood to zero. The best parameters are the ones that make the bars tall where this person's responses were.
5What the model predicts across difficulty
Two classic predictions follow. Accuracy rises with the colour difference along the probit curve. Confidence shows an X pattern: it rises with easier displays on correct trials but falls on error trials, because on an easy trial the only way to be wrong is for the sample to land just barely on the wrong side.
The staircase adds something specific to this task. It targets 73% correct, which happens when β·|s| ≈ 0.61. So the colour difference a person ends up seeing is about 68 × 0.61 / β.
What this means for β
Move the β slider above and the d′ readout never changes. The staircase gives everyone a different display but the same internal difficulty; a sensitive observer just ends up seeing displays where the two colours are closer to even. Two consequences follow.
First, β in this task works like an inverse threshold. It says how small a colour difference someone can handle, not how accurate they are, since everyone is held near 73%.
Second, the internal evidence looks much the same for everyone once the staircase has settled. If confidence is also binned into per-person quartiles, the fitted criteria end up nearly identical across participants, so they carry little information about individual differences.
What SDT leaves out
SDT assumes confidence is a noise-free readout of the very sample that drove the choice: perfect metacognition relative to the evidence. Every other model in the Shekhar and Rahnev comparison relaxes this in some way. Some add noise to the confidence readout, some add information arriving after the decision, and some let the choice draw on evidence that confidence cannot see. When SDT wins a model comparison on 80 trials, it means those departures can't be detected with this much data, not that confidence is truly noise-free.