Notes on Thresholds, Weber's/Fechner's/Stevens' Laws, and Signal Detection Theory

Method of Constant Stimuli (Absolute Threshold)

• Definition: A psychophysical method that measures thresholds by presenting fixed, preselected levels of stimulus intensity in random order across many trials.
• Procedure details from lecture:
  • Select several intensity levels (brightness, loudness, etc.).
  • Present each level many times to the subject in random order.
  • Ask the subject whether they perceived the stimulus (e.g., did you hear it, did you see it).
  • Pure psychophysical experiments can require thousands of trials (e.g., listening to headphones for 1000 trials in a course).
  • Later innovations in the late 1980s reduced testing time, but the core approach remains: many trials to estimate detection probability at each level.
• Expected results:
  • If a level is always detected (e.g., 6 out of 6) or never detected (e.g., 1 out of 6), the data would form a step function across levels.
  • In practice, the data form an S-shaped (sigmoid) psychometric function instead of a sharp step.
  • If you plot detection probability vs. intensity, you obtain an S-curve that differs across sensory modalities and even across individuals.
• Absolute threshold concept in practice:
  • A common graphical definition is the 50% detection point: the intensity at which the subject detects the stimulus on 50% of trials.
  • Example discussion: Treating a test where 5 and 6 are tested may yield a threshold around 5.4, illustrating that the true threshold lies between tested levels.
  • The y-intercept of the psychophysical regression line (often denoted as b in y = mx + b or a in p = ax + b) corresponds to the absolute threshold in the given setup.
• Lab vs real life:
  • In lab, thresholds are concrete detections, but real-world thresholds vary with context (lighting, background noise, etc.).
• Variability and context:
  • There is variability within a person (intra-individual) and across people (inter-individual).
  • Example demonstrations of real-world thresholds:
  • Seeing a candle at 30 miles.
  • Hearing a ticking watch at about 20 feet.
  • Hearing the wing of a fly landing on your back from ~3 inches above.
• Summary takeaway:
  • Absolute threshold marks the starting point of perception (detection) along a psychophysical axis; it is not a fixed, universal constant due to variability.

Absolute Threshold and Threshold Estimation in Psychophysics

• Absolute threshold is the starting point of perception, i.e., the intercept on the psychophysical axis (y-axis) in the basic psychophysical equation.
• The common regression analogy: y = mx + b (or p = ax + b, etc.). Here, the intercept (b or a) represents the absolute threshold—where detection begins.
• After obtaining the absolute threshold, the next question is the slope of the relationship between stimulus magnitude and perception, which brings in the difference threshold (Just Noticeable Difference, JND).
• The JND (just noticeable difference) is the amount by which the stimulus must change for the observer to notice a difference:
  • $ext{JND} = \frac{ riangle I}{I} = k$ (Weber's law constant)
• The goal is to understand how much more (or less) of a stimulus is needed to say there's more (or less) of it.

Variability of Thresholds

• Thresholds are not fixed; there is:
  • Within-subject variability: the same person may have slightly different thresholds across trials or days.
  • Between-subject variability: different people have different thresholds.
• Real-life examples to illustrate variability:
  • Candle visibility in different contexts (30 miles away).
  • Ticking a watch at different distances.
  • Fly wing on a back at different positions
• Implication: The psychophysical system is context-dependent, and no single universal absolute threshold exists.

Psychophysical Laws: From Constant Stimulus to Perceived Intensity

• The problem: thresholds are not strictly linear, and the slope (how perception changes with stimulus) is not constant.
• Weber's Law (Weber's fraction/constant):
  • Core idea: The just noticeable difference ΔI scales with the baseline intensity I.
  • Formula: $k = \frac{ riangle I}{I}$ where k is Weber's constant.
  • Interpretation: A constant proportion of the original stimulus is required to notice a change.
  • Example constants (approximate across modalities):
  • Salt taste: $k \,\approx\, 0.083 = 8.3\%$
  • Brightness: $k \,\approx\, 0.079 = 7.9\%$
  • Loudness: $k \,\approx\, 0.048 = 4.8\%$
  • Line length (visual): $k \,\approx\, 0.029 = 2.9\%$
  • Heaviness: $k \,\approx\, 0.020 = 2\%$
  • Electric shock: $k \,\approx\, 0.013 = 1.3\%$
  • Caveats:
  • The constants vary by sensory modality and context; Weber's law is not perfectly universal but historically provided a close-to-constant relation for many conditions.
• Fechner's Law (classical law following Weber):
  • Idea: Sensation increases as the logarithm of the stimulus intensity.
  • Formula: $S = k \log I$
  • Interpretation: Equal ratios of increments in physical intensity produce equal increments in sensation when viewed on a logarithmic scale.
  • Relationship to Weber: Fechner’s law uses Weber's constant as the proportionality factor and applies a logarithmic transformation to I.
  • Historical note: Fechner's law reigned for about a century with strong influence; it was later refined by Stevens and others.
• Stevens' Power Law (modern alternative to Weber/Fechner):
  • Idea: Perceived intensity follows a power function of the physical stimulus.
  • Formula: $S = k I^b$
  • Characteristics:
  • The exponent b determines the rate of growth of perception with stimulus.
  • If b < 1, perception grows more slowly than the stimulus (compressive relationship; e.g., brightness in some ranges).
  • If b > 1, perception grows faster than the stimulus (superlinear relationship; e.g., some psychological magnitude estimations such as electric shock in some ranges).
  • The historical shift from Fechner's logarithmic mapping to Stevens' power law reflects a more nuanced view of cross-modal magnitude estimation.
• Summary of the historical progression:
  • Weber proposed a constant ratio for JND relative to baseline intensity.
  • Fechner formalized this into a logarithmic relation between stimulus and sensation.
  • Stevens showed that the exponent can vary by modality and range, leading to a power-law description of perceived magnitude.
  • The century-long view shifted from a single constant to a family of laws depending on the sensory modality and range.

Just Noticeable Difference (JND) and Difference Thresholds

• JND is the smallest detectable difference in stimulus intensity that a person can reliably notice.
• Weber's law expresses JND as a constant fraction of the baseline intensity: $\frac{ riangle I}{I} = k$
• The JND is not a single fixed number; it scales with the baseline stimulus and can vary across modalities and contexts.
• Relationship to slopes of psychophysical functions:
  • The difference threshold defines the slope of the psychophysical function; a larger JND implies a shallower slope, a smaller JND implies a steeper slope.

Example: The Cave Candlelight Demonstration (Weber-like intuition)

• In a dark cave, the perception of brightness is highly context-dependent.
• A single candle light may be barely noticeable against deep cave darkness; ten or more candles may be required before brightness is clearly detectable against the ambient dark.
• This illustrates that the threshold for detecting brightness is not fixed, but depends on ambient conditions and available cues (background illumination).

Signal Detection Theory (SDT)

• Core idea: Distinguishing signal from noise, separating sensory information from decision criteria and response biases.
• Key components:
  • Signal vs Noise: The physical stimulus (signal) must be detected against background noise (random neural activity, environmental noise).
  • A decision criterion (beta) is used to decide whether to respond yes (signal present) or no (signal absent).
• Mental vs physical components:
  • Physically, a stimulus may or may not be present (signal vs noise).
  • Mentally, a person applies a criterion to decide whether the observed sensory evidence is strong enough to report a detection.
• SDT decision framework (2x2 table):
  • True state: Signal present or Signal absent (noise only).
  • Observer's decision: Report signal present (Yes) or report signal absent (No).
  • Four outcomes:
  • Hit: Signal present and reported as present.
  • Miss: Signal present but reported as absent.
  • False Alarm: Signal absent but reported as present.
  • Correct Rejection: Signal absent and reported as absent.
• The beta criterion (β) controls the decision threshold:
  • A higher threshold makes hits less likely but reduces false alarms (more conservative).
  • A lower threshold makes hits more likely but increases false alarms (more liberal).
• Type I vs Type II errors (in SDT terminology):
  • Type I error: False alarm (reporting a signal when there is none).
  • Type II error: Miss (failing to report a signal that is present).
  • In common statistics language: Type I = false alarm; Type II = miss.
• Visual example: A grid illustrating signal vs noise distributions and a criterion line (beta) that separates detections from non-detections. Adjusting beta shifts the balance between hits and false alarms and between misses and correct rejections.
• Practical implications:
  • SDT separates sensory capability from decision strategy, allowing assessment of sensitivity independent of response bias.
  • The same observer can show different hit/false alarm rates depending on their criterion (e.g., in high-stakes environments, people may adopt stricter criteria).

Real-World Examples and Applications of SDT and Threshold Concepts

• Camouflage and signal reduction:
  • Military camouflage aims to reduce the signal relative to environmental noise, making detection harder.
  • The goal is to minimize the observer’s ability to distinguish the signal from noise, effectively shifting the signal distribution closer to the noise distribution.
• Olfaction (smell):
  • In a forest or greenhouse with many odors, detecting a specific scent is challenging due to competing olfactory signals.
• Taste and flavor: wine tasting example
  • A two-step process: initial impression and a deeper, subsequent evaluation after the first sip.
  • The perceptual system can distinguish changes in taste intensity or quality with practice and context.
• Pain, touch, and other modalities:
  • The same SDT framework applies across modalities: selecting how to respond to potential sensory events depends on both sensory evidence and decision criteria.
• Practical laboratory and classroom notes:
  • Students should be able to draw and interpret the SDT 2x2 table and define hits, misses, false alarms, and correct rejections.
  • Recognize that the same physical stimulus can be perceived differently depending on context, criterion, and prior expectations.

Key Takeaways and Connections

• Absolute threshold marks the rough starting point for sensation; it is informed by the intercept of the psychophysical relationship but is not a fixed, universal value due to variability.
• The slope of the relationship between stimulus intensity and perception is characterized by the JND and is modality-dependent.
• Weber's Law provides a historically influential constant ratio for JND relative to baseline intensity, but it is not perfect and context-dependent.
• Fechner's Law maps stimulus to sensation via a logarithmic function, introducing the idea that sensory increments are perceived more evenly on a log scale.
• Stevens' Power Law generalized magnitude estimation by allowing the exponent to vary by modality, capturing both compressive and expansive psychophysical relationships.
• SDT separates the sensory process (signal/noise) from decision processes (criterion/bias), enabling clearer interpretation of detection performance via hits, misses, false alarms, and correct rejections.
• Real-world examples illustrate how thresholds and detection are context-dependent and subject to variability across individuals and environments.

Quick Reference: Core Equations and Terms

• Just Noticeable Difference (JND) / Difference Threshold:
  • $riangle I = k I$ or $k = \frac{\triangle I}{I}$
• Weber's Law:
  • $k = \frac{\triangle I}{I}$
  • Typical constants: salt 8.3%, brightness 7.9%, loudness 4.8%, line length 2.9%, heaviness 2%, electric shock 1.3%
• Fechner's Law:
  • $S = k \log I$
• Stevens' Power Law:
  • $S = k I^b$
• SDT and decision theory:
  • Four outcomes: Hit, Miss, False Alarm, Correct Rejection
  • Criterion (beta) and its effect on sensitivity and bias
  • Type I error: False Alarm; Type II error: Miss
• Absolute threshold (conceptual):
  • The detection threshold at 50% across trials, corresponding to the y-intercept in a psychophysical regression