Welcome back to the AI Bayeslab Statistics series.
After covering the content yesterday, discussing the likelihood ratio and one of its main criteria, the "payoff matrix," we will continue our exploration today. In today's post, we will delve deeper into:
The nature and application of the report criterion
Perceptual sensitivity in signal detection theory
The calculation method and interpretation of the core metric d' (d prime)
Examples of signal detection theory in real life
What is the report criterion?
In Signal Detection Theory, the Report Criterion (C) is another key metric for measuring response bias. Unlike the likelihood ratio, which is defined by the ratio of two vertical likelihood values, the report criterion (C) is a judgment threshold point on the horizontal axis.
As previously mentioned, each signal reception generates a random observation within the overlapping region of the noise distribution (Distribution N) and the signal-plus-noise distribution (SN). This observation is then compared against the receiver’s subjectively defined report criterion (C):
If the observation falls to the right of C, it is judged as a signal;
It is judged as noise if it falls to the left of C.
In other words, the C value represents the decision threshold, reflecting the minimum physical stimulus intensity required to elicit a response from the subject.
Example: Suppose you ask a child to lie to their teacher about dreaming of a dinosaur last night. Since the child has been taught that "lying = bad," they have an internal motivation threshold to lie (i.e., their C value). Different children have different C values:
Some may lie for just one piece of candy;
Others (like me) would refuse even for 10 pieces.
Here, the number of candies represents each child’s C value. Individuals vary in stimulus sensitivity and subjective response tendencies, leading to different C values. For instance:
If Tom’s C value is 6 candies, offering 5 will inevitably fail to persuade him.
The formula for the report criterion in Signal Detection Theory
Formula 1: Classic Report Criterion ©
Where:
Z(H): Z-score of Hit Rate (probability of correct detection when signal is present)
Z(FA): Z-score of False Alarm Rate (probability of false detection when only noise is present)
Interpretation:
Quantifies the respondent’s decision threshold
Negative values indicate liberal bias (tendency to respond “yes”)
Positive values indicate conservative bias (tendency to respond “no”)
Zero represents neutral bias
Typical Application:
Binary detection tasks (signal present/absent)
Basic sensitivity analysis
Formula 2: Stimulus-Contingent Criterion ©
Parameters:
S_max: Maximum stimulus intensity
S_min: Minimum stimulus intensity
d′: Sensitivity index (d-prime)
Z_CR: Z-score of Correct Rejection rate for minimum stimulus
Interpretation:
Map the decision criterion onto the physical stimulus dimension
Accounts for the stimulus intensity range
Provides a criterion in stimulus units
Application Context:
Experiments with graded stimulus intensities
Studies requiring a physical scale reference
What is d’(d prime)?
d’ (d-prime) is the fundamental sensitivity measure in Signal Detection Theory (SDT) that quantifies how well an observer can distinguish between signal and noise. Mathematically, it represents the standardized distance between:
The noise distribution (N) — when only background noise exists
The signal noise distribution (SN) — when a target signal is present
Formula for the perceptual sensitivity index (d’)
Mathematical Definition:
Where:
μSN: Mean of signal+noise distribution
μN: Mean of noise distribution
σN: Standard deviation of noise distribution
Key Properties:
Normal Distribution Assumption: When both distributions are normal with equal variance:
Interpretation:
Higher d’ = Better discriminability
d’ = 0: No detection ability (chance performance)
Typical range: 0–4 (varies by task)
Calculation from Response Data:
Where Φ−1 is the inverse normal CDF (z-score)
Practical Example for d’
If a subject has:
Hit Rate = 0.80 → Z(H) ≈ 0.84
False Alarm Rate = 0.20 → Z(F) ≈ -0.84
Then:
d′=0.84−(−0.84)=1.68
Important Notes:
d’ is independent of response bias (unlike C)
Requires the equal variance assumption for simple calculation
For unequal variances, use generalized d’:
How can I determine the best d’ value?
Similar to response bias, reaction sensitivity also has an optimal level. Refer to the diagram below: the red shaded area beneath the red curve on the right indicates the probability of hits, while the blue shaded area under the blue curve signifies false alarms. We will now keep the β value at 1 and adjust the d’ value to observe how both shaded areas change. It is evident that as the β value remains constant, a larger d’ value brings the S distribution closer to the SN distribution. In contrast, with a smaller d’, the hit rate declines while the false alarm rate rises.
Signal Detection Theory identifies two types of indicators that differentiate between discrimination power and response bias in decision-making. Discrimination power, denoted as d’, reflects an individual’s genuine sensitivity to the information presented. Provided that the intensity of the information stimulus stays constant, d’ remains a stable measure.
β and C denote an individual’s inclination to respond when confronted with information; this response can be modified by altering the reward conditions in the payoff matrix or by adjusting the ratio of noise to signal occurrence (prior probability).
In the following article, we will present a clear example from company management, such as a challenge related to bonus distribution, to illustrate these two types of indicators in a manner more relevant to real-life scenarios.
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