Chapter 5: The Race in the Dark

Ground floor

Two children stand at a jar of marbles in a dim room, arguing about whether there are more red ones or more blue. They cannot count them; they can only glimpse. Each glimpse is a smudge of impression: reddish, then maybe blue, then reddish again. Nothing decisive, ever. So each child keeps a private running tally, tipping a little toward red with this glimpse, back toward blue with that one, and at some point one of them says a word out loud.

Now ask the only interesting question. What determines when the child speaks?

Two things, and they have nothing to do with each other. The first is how good the glimpses are: in a brighter room, each glimpse tips the tally further in the right direction, and the child arrives at the truth quickly. The second is how far the tally has to travel before the child is willing to speak at all. One child speaks the moment the tally leans; another waits until it is decisively over. The first is fast and wrong a lot. The second is slow and right a lot.

Here is what matters, and it is the whole chapter: those two children can produce identical scores. The hasty child in a bright room and the careful child in a dim room take the same time and make the same number of mistakes. Any measurement that reports speed and accuracy, and mixes them into one number, cannot tell them apart. It will call them equally able. One of them is extracting far more from every glimpse.

You have met this before, in the village with the bell. Sensitivity and criterion, eyes and trigger. This chapter is that same fork, but now the observer is not a snapshot judge. He is a process, running in time, and the time is data.

The stairs

The model is called the drift diffusion model, and it is, quietly, one of the most successful models in the history of psychology: it explains speed, accuracy, and the shape of the entire reaction-time distribution, correct and incorrect, from a handful of parameters, and its parameters have since been found in the firing rates of actual neurons in monkeys doing exactly this task. It is worth knowing properly, because it is the answer to a question you may already be answering by hand.

Picture a particle starting between two boundaries. Evidence arrives continuously and pushes it: on average toward the correct boundary, but with noise, so it staggers. When it touches a boundary, the decision is made and the response is emitted. Three parameters do almost all the work.

Drift rate, v. The average pull per unit time toward the correct answer: the quality of the evidence entering the system, which is to say the ability of this person on this item. Higher v means faster and more accurate, together. This is the parameter you actually want when you claim to measure a mind.

Boundary separation, a. How far the particle must travel before committing: caution. Wider boundaries mean slower and more accurate; narrower mean faster and sloppier. Crucially, a is under strategic control, it moves with instructions, with mood, with coffee, with how much the user cares today, and it moves without any change in ability whatsoever.

Non-decision time, Ter. Everything that is not deliberation: light hitting the retina, the finger moving, the browser noticing. Typically two to three hundred milliseconds of pure overhead, sitting inside every reaction time you have ever recorded, having nothing to do with cognition.

Look at what falls out for free. The speed-accuracy tradeoff, that ancient annoyance, is not a nuisance to be controlled away; it is simply a being moved, and the model predicts the exact curve. The awkward fact that errors are sometimes faster and sometimes slower than correct responses, which no simple theory could ever explain, falls out of where the starting point sits and how variable the drift is. And the skew of the reaction-time distribution, that long right tail from the last chapter, is not a defect of the data needing a fix; it is precisely the shape a first-passage time has. The model explains why the data was never bell-shaped in the first place.

Now, the part that makes this immediately usable rather than admirable. Fitting the full model is a research undertaking. But there is a shortcut, EZ-diffusion, that recovers the three parameters in closed form from three summary numbers you almost certainly already store: the proportion correct (Pc), the mean reaction time of correct responses (MRT), and the variance of the reaction times of correct responses (VRT), the last in seconds squared. Fix a scaling constant s = 0.1 by convention, because the model has one degree of freedom too many and someone has to nail it down. Then:

Let L = ln(Pc / (1 − Pc)), the log-odds of being correct.

The drift rate is

v = sign(Pc − 0.5) · s · [ L · (L·Pc² − L·Pc + Pc − 0.5) / VRT ]^(1/4)

The boundary is

a = s² · L / v

And the mean decision time is MDT = (a / 2v) · (1 − e^(−v·a/s²)) / (1 + e^(−v·a/s²)), from which the non-decision time is simply Ter = MRT − MDT.

Do not be intimidated by the fourth root; it is arithmetic, and it runs in ten lines of code on data you already have. What you get back is three numbers where you previously had two, and the third one is the one that was ruining the other two.

Because here is the practical detonation, and it is aimed at a specific, extremely common design. Suppose an instrument scores a person by combining accuracy with pace, awarding more for fast-correct than for slow-correct, using constants chosen by hand, a floor for slow-but-right, a bonus for quick-and-right, an expected median time per item. That design is a sincere attempt to respect the tradeoff, and it is measurably better than scoring accuracy alone. But look at what it does to our two children. Every such formula reads caution as ability. The user who is tired, or careful, or has just been burned by a wrong answer, widens a; their score falls; the instrument reports a cognitive decline that did not happen. The user who learns that the system rewards speed narrows a; their score rises; the instrument reports improvement that did not happen either. The hand-tuned constants are, in effect, an unstated and unexamined guess about the exchange rate between speed and accuracy, applied identically to every person and every state, when the whole point of the last fifty years of this literature is that the exchange rate is a free parameter the user sets, moment to moment, and can be measured rather than assumed.

The repair is not to throw the instrument away. It is that the same trials, with no change to the interface and no additional burden on the user, already contain the decomposition. Report v as the ability estimate. Report a as a state indicator, and notice, this is the gift, that a is not noise to be regressed out but a genuine readout of how the person is approaching the task today: caution, fatigue, engagement. An instrument that wanted to answer “how am I functioning today?” rather than only “how good am I?” has been sitting on the answer the whole time, filed under nuisance.

One honesty clause, because this book would not survive its own chapter three otherwise. EZ-diffusion is the simplified version: it assumes an unbiased starting point and no trial-to-trial variability in the parameters, and it will misbehave on very high accuracy (Pc near 1 makes L explode) and on tiny trial counts. It is a diagnostic instrument, not a final word. But it converts a hand-tuned guess into a principled estimate for the price of one afternoon, and it names its own assumptions out loud, which is more than the hand-tuned constants ever did.

The locked door

Behind the door: the full diffusion model with across-trial variability in drift, starting point, and non-decision time, fit by hierarchical Bayesian methods that pool across users without pretending they are the same user; collapsing boundaries, where caution decreases as time passes, because real deadlines exist; the linear ballistic accumulator and the racing-accumulator family for choices among more than two options; and the neuroscience, where the accumulating particle turns out to be visible as firing rates in parietal cortex, one of the very few times a psychological model has been caught red-handed being implemented. The names are Ratcliff, whose 1978 paper started it and who has been refining it for five decades, and Wagenmakers, van der Maas and Grasman for the EZ shortcut above.