Signal Detection Booth

The model

On every trial the observer forms one internal number x. If only noise is present, x ~ N(0, 1). If a signal is present, x ~ N(d', s^2): the whole distribution is shifted by the sensitivity d' (measured in noise standard deviations) and may be wider (s > 1). The observer answers YES when x is above a fixed criterion lambda.

Four outcomes

Hit rate H = 1 - Phi((lambda - d')/s), false-alarm rate F = 1 - Phi(lambda), where Phi is the normal cumulative distribution. Misses are 1 - H and correct rejections 1 - F. Moving the criterion trades hits for false alarms but does not change how well the two distributions are separated.

Measuring d' and bias

From counts, d' = z(H) - z(F) and bias c = -(z(H) + z(F))/2, with z = Phi^-1. With zero or perfect rates a log-linear correction adds 0.5 to each count. For equal spreads the estimate is unbiased; for s != 1 it is not, which is why the readout shows a different value.

The ROC curve

Plotting H against F for every criterion gives the receiver operating characteristic. Its area is the probability that a random signal trial feels stronger than a random noise trial: AUC = Phi(d' / sqrt(1 + s^2)). The diagonal is chance (d' = 0).

Ideal observer and payoffs

With signal probability p, hit reward V_H, miss cost 1, correct-rejection reward 1 and false-alarm cost C_FA, expected payoff per trial is p (H V_H - (1 - H)) + (1 - p)((1 - F) - F C_FA). It is maximised where the likelihood ratio equals beta = (1 - p)/p * (1 + C_FA)/(V_H + 1); for equal spreads that is lambda = d'/2 + ln(beta)/d'. Rare signals and costly false alarms push the best criterion to the right, big hit rewards pull it to the left.

Simplifications

Gaussian internal responses, a fixed criterion with no trial-to-trial noise or learning, independent trials, and a single sensory channel. The picture on the monitor is illustrative: the model works on the internal number x, not on the pixels. Real observers drift, have lapses and unequal-variance curves; this lab shows the idealised theory.

1. Set d' = 3 and move the criterion: hits stay high while false alarms collapse. Now d' = 0.8 and see how little any criterion can do.
2. Press the preset "Liberal observer" and then "Conservative observer": the orange dot slides along the same ROC curve, so sensitivity did not change.
3. Set signal probability to 10 % and press "Move criterion to the ideal observer": the best criterion moves right and the payoff rises.
4. Give hits a reward of 4 and compare the ideal criterion with the neutral one.
5. Watch the yellow d' estimate converge to the true value as trials accumulate (graph 3).