Today’s post will analyze two orthogonal dimensions in signal detection theory. To ensure that I don’t move us along before providing some key takeaways, let’s clarify the two metrics first:

“Response bias” is quantified by ß(likelihood ration) and C(report criterion), reflecting decision strategy;

“perceptual sensitivity” measured by d’(d prime), representing true detection ability.

1 What does signal detection theory assume?

In our previous article, we mentioned that any process involving stimulus reception and response encompasses both a sensory process and a decision-making process.

• Sensory process: Depends on the nature of the external stimulus;

• Decision-making process: Influenced by the Subjective factor.

2 What are the four outcomes of signal detection theory?

The Signal Detection Theory(SDT) assumes that the sensory and decision-making processes are independent. Consequently, the corresponding “response bias” and “Perceptual sensitivity”(d’) are also independent.

As we have discussed multiple times before, there are four possible types of response:

The probabilities of these four response types are related as follows:

P(H)+P(M) = 1

P(FA)+P(CR) = 1

Where:

3 What is response bias in signal detection theory?

There are two ways to calculate response bias in SDT: the likelihood ratio (ß) and the report criterion(C).

3.1 What is the likelihood ratio?

Each time the subject receives a stimulus, a psychological perceptual magnitude x is generated, and x always falls within the overlapping region of the noise distribution (N) and the signal-plus-noise distribution (SN). The blue curve represents the pure noise distribution (N), while the red curve represents the signal-plus-noise distribution (SN). Now, consider a randomly presented stimulus that elicits a perceptual value x:

• If x is weak and lies to the left of X₁ (far from the SN distribution), the subject will judge it as noise;

• If x is strong and lies to the right of X₃ (far from the N distribution), the subject will judge it as a signal;

• Depending on the likelihood ratio, if x falls between X₁ and X₃, the subject may judge it as either noise or a signal.

For an x located between X₁ and X₃, a vertical line is drawn from the x-axis, intersecting both the SN and N distributions at two points:

• The height of the intersection with the SN distribution is denoted as O(SN);

• The intersection height with the N distribution is denoted as O(N).

The ratio O(SN)/O(N) is the likelihood ratio, denoted as lₓ.

To better visualize the impact of different **likelihood ratios (lₓ)**, we analyze three key points:

1. Point A (between original X₂ and X₃)

   1. ß = 4.38

   2. Interpretation: A ß significantly >1 indicates a strict criterion, where the subject avoids false alarms at the cost of more misses.

      
      

2. Point B (between original X₁ and X₂)

   1. ß = 1.00

   2. Interpretation: ß = 1 reflects a neutral criterion, balancing hit rates and false alarm rates.

   
   

3. Point C (exactly at original X₂)

   1. ß = 0.28

   2. Interpretation: A ß <1 suggests a lenient criterion, prioritizing hits but accepting higher false alarms.

General Rule:

• ß > 1 → Strict criterion (conservative strategy);

• ß ≈ 1 → Neutral criterion;

• ß < 1 → Lenient criterion (liberal strategy).

3.2 The payoff matrix

Under constant signal strength (d') and sensitivity levels, the optimal decision criterion ß_opt for maximizing expected payoff is determined by:

ß_opt = [P(N)/P(S)] × [(V(CR) + C(FA))/(V(H) + C(M))]

Key dynamic relationships:

1. When signal probability P(S) increases:

   1. Rational decision-makers should relax the criteria (decrease ß)

   2. Behavioral manifestation: Rightward shift of decision threshold

   3. Example: ß_opt may drop from 1.5 to 0.6 when P(S) increases from 20% to 60%

   
   

2. When P(S) decreases:

   1. Stricter criteria (increase ß) should be adopted

   2. Behavioral manifestation: Leftward threshold shift

   3. Example: ß_opt may rise from 0.6 to 2.2 when P(S) decreases from 60% to 10%

   
   

3. Payoff adjustments:

   1. Increasing hit reward V(H) reduces ß_opt

   2. Increasing the cost of FA, C(FA), elevates ß_opt.

   3. Security system case: Setting high C(M) can drive ß_opt to approximately 0.3.

   
   

4. Boundary condition:

   1. When V(CR)+C(FA) = V(H)+C(M), ß_opt = P(N)/P(S)

   2. Medical diagnosis example: ß≈0.1 ensures minimal miss rate when disease risk >> test cost

   
   

   

To maintain clarity and conciseness, this article focuses on the first key metric of response bias - the likelihood ratio and its relationship with the payoff matrix. Through the concrete examples provided above, we hope readers have better understood how to calculate the optimal decision criterion ß_opt for maximizing expected benefits.

In our next article, we will further explore:

1. The nature and application of the report criterion

2. Perceptual sensitivity in signal detection theory

3. The calculation method and interpretation of the core metric d' (d prime)

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