Figure 1
Illustration of the Signal Detection Theory (SDT) framework. A) The observer makes a measurement, x, of the stimulus, either stimulus A or stimulus B. The probability of a measurement, conditioned on the stimulus, is assumed to be Gaussian distributed due to the effect of sensory noise. The observer selects a decision criterion, c, that provides a mapping from x to a response (e.g., say “A”). Perceptual sensitivity, d′, is reflected in the separation of the curves. B) Each pair of d′ and c predict a unique combination of Hit and False Alarm rates. Curves reflect all the possible Hit rate and False Alarm rate combinations for a given perceptual sensitivity (e.g., d′ = 1). Two example criterion placements (green and gold) are shown on the curves and the SDT model insets. The green criterion setting leads to more Hits and False alarms (shaded regions) compared to the gold criterion setting. C) When priors are not equal for the two choice alternatives, the optimal criterion, copt, is no longer at the neutral location centred between the two measurement distributions. As shown, stimulus A is more probable than stimulus B, causing a rightward shift in copt. Consequently, stronger evidence of B is needed to report “B”. D) Similarly for unequal payoffs. Correctly guessing A receives 2 units of reward, whereas correctly guessing B receives 4 units. This shifts copt leftward so that stronger evidence of A is needed to report poorly rewarded “A”. In an unequal reward context, affective bias is an increased preference to report the low-reward option (here “A”). This can be modelled as a shift in c towards the high-reward distribution. An example of a criterion influenced by affective bias is shown by the dashed line.
Figure 2
The tone-discrimination task of Aylward, Hales, et al. (2019). A) Participants reported if they heard the low-frequency tone (red) or the high-frequency tone (blue). The reward for correctly guessing the high-frequency tone was four times larger than for the low-frequency tone. On some trials, an ambiguous tone (purple) was played and was rewarded randomly. The graph illustrates the SDT model with the expected probability distributions of the decision variable for each of the tones. Example shown for an observer with a perceptual sensitivity of d′ = 3.5 (i.e., the sensitivity of the average anxiety-group participant). The optimal decision boundary (dashed) provides the decision rule that maximises expected reward: choose “low tone” if the decision variable is lower than this value, and “high tone” if it is above. Critically, this boundary does not align with the mid-point between the red and blue distributions (peak of the purple distribution) because it is optimal to have a bias for responding with the more highly rewarded “high tone”. Average criterion for control group (green) and anxiety group (gold) are also shown, demonstrating a conservative shift away from optimal, and an even greater shift for the clinical group interpreted as affective bias. B) Boxplots of the empirical distributions of the proportion of “high-tone” judgements for the ambiguous stimulus, split by test population. The lower proportion of high-tone responses of the anxiety group is the affective bias effect (p = 0.003, BF10 = 12.51; Aylward, Hales, et al., 2019). Horizontal dashed lines indicate the expected proportion of responses for the ideal observer who uses the optimal decision boundary. The expected proportions are shown for several perceptual sensitivities ranging from no sensitivity (d′ = 0) to near-perfect sensitivity (d′ = 4).
Figure 3
Predictions for criterion placement and proportion of low-reward responses for different incorrect beliefs. Predicted results shown for low-reward stimulus A (red) and high-reward stimulus B (blue), with equal priors (P(A) = P(B) = 0.5), correct B responses being rewarded twice as much as correct A responses (VA,A = 2 and VB,B = 4), and perceptual sensitivity of d′ = 1. A) Criterion placement for different beliefs about the prior probability of stimulus A. The greater the estimated probability of A , the greater the rightward shift in the criterion. B) The proportion of low-reward responses, for different beliefs about the prior probability of A. Prediction for the use of the optimal criterion with correct beliefs shown by the red marker, and incorrect prior beliefs by the black markers. The more the observer believes A is probable, the more low-reward responses. C) Criterion placement for different beliefs about perceptual performance. Under-estimations of performance lead to leftward criterion shifts, and over-estimations of performance lead to rightward criterion shifts. D) The proportion of low-reward responses, for different beliefs about performance. Predicted proportion with correct beliefs shown by the red marker, and incorrect beliefs by the black markers. Over-estimations of performance lead to more A responses and under-estimations to less.
Figure 4
The effect of subjective value on decision making. Predicted results shown for low-reward stimulus A (red) and high-reward stimulus B (blue), with equal priors (P(A) = P(B) = 0.5), correct B responses being rewarded twice as much as correct A responses (VA,A = 2 and VB,B = 4), and perceptual sensitivity of d′ = 1. A) Example subjective utility functions. When α < 1, the subjective value of the two reward outcomes (A/B) is more similar. B) Reward landscape is affected by the subjective utility function. Dashed lines show the criterion placement expected to maximise expected gain with the subjective reward values. When α < 1, there is a dampening affect, with the perceived consequence of criterion misplacement being small (i.e., loss incurred is minimal for other values of c). C) Optimal criterion placement according to the subjective reward ratios. As α → 0, the criterion placement that maximises expected reward with these distorted reward values shifts towards the neutral position between the distributions. D) The proportion of low-reward responses when criterion is adjusted according to the subjective value of reward. When the subjective value matches objective value (red), the proportion of A responses is lower than if the subjective value is used with α < 1 (black).