# Signal Detection Theory

## Separating sensitivity from decision bias

<p class="big"><span class="signal">Is the signal there?</span><br>How sure do we need to be before we say “yes”?</p>

<p class="small">An accessible primer for biologists · 2026</p>

Notes:
Signal detection theory, or SDT, is a framework for decisions made with uncertain evidence. The central idea is simple: performance depends both on the quality of the evidence and on where the decision-maker draws a line.

---

## Where we are going

<div class="overview">
  <a class="card" href="#/2">
    <span class="overview-number">1</span>
    <strong>Why SDT?</strong>
    <span>Decisions under uncertainty—from cancer screening to hearing in noise</span>
  </a>
  <a class="card" href="#/7">
    <span class="overview-number">2</span>
    <strong>The core model</strong>
    <span>Outcomes, evidence distributions, sensitivity (d′), and criterion (c)</span>
  </a>
  <a class="card" href="#/15">
    <span class="overview-number">3</span>
    <strong>From theory to experiment</strong>
    <span>ROC curves and an auditory tone-in-noise task</span>
  </a>
  <a class="card" href="#/18">
    <span class="overview-number">4</span>
    <strong>Put it into practice</strong>
    <span>A worked example, practical cautions, and interpretation</span>
  </a>
</div>

<p class="small center">Select a section to jump ahead, continue with →, or use ↓ when an optional deep dive appears.</p>

Notes:
We will begin with why signal detection theory is useful, build the core model, connect that model to ROC curves and auditory experiments, and finish by calculating and interpreting the key parameters ourselves.

---

## The recurring problem

<div class="flow">
  <div>🧬<br><strong>Biology varies</strong><br><span class="small">Signals are rarely identical</span></div>
  <div class="arrow">+</div>
  <div>🌫️<br><strong>Measurements vary</strong><br><span class="small">Noise enters every instrument and nervous system</span></div>
  <div class="arrow">→</div>
  <div>⚖️<br><strong>We must decide</strong><br><span class="small">Often with only “yes” or “no”</span></div>
</div>

> SDT asks what the responses reveal about both the evidence and the decision rule, **assuming an optimal observer**.

Notes:
Signal and noise are not necessarily literal sounds. “Signal” is the condition we want to detect; “noise” is everything that can be mistaken for it. The optimal observer provides the normative benchmark; real observers can—and often do—depart from it.

---

## Why people use SDT

### Cancer detection

A faint feature in an image may indicate disease—or normal variation.

Missing disease and triggering an unnecessary biopsy have different consequences.

</div>
<div class="card">

### Airport screening

A rare threat must be found among many harmless bags.

Low prevalence can shift the willingness to report a target.

</div>
</div>

<p class="small">The same logic applies to laboratory assays, quality control, and clinical tests.</p>

---

## A neuroscience workhorse

SDT has helped researchers study:

- sensory detection and discrimination — Green & Swets (1966)
- recognition memory — Macmillan & Creelman (1991)
- value-based decisions — Ackermann & Landy (2015)
- information carried by neural spikes — Britten et al. (1992)
- confidence and metacognition — Maniscalco & Lau (2012)

</div>
<div class="callout center">
<p class="big">One framework</p>
<p>behavior ↔ neural responses</p>
</div>
</div>

<p class="small">Landy describes SDT as the primary performance model for sensory experiments in audition, vision, and other modalities.</p>

<div class="deep-dive-cue">Optional deep dive ↓</div>

-v-

<div class="section-tag section-1">1 · Why SDT?</div>
<div class="supplement-tag">Optional deep dive</div>

## From spikes to decisions

- Build a neural ROC by sweeping a criterion across spike-count distributions.
- Plot AUROC against stimulus strength to obtain a **neurometric function**.
- On zero-signal trials, group activity by the animal’s eventual choice; AUROC is then called **choice probability**.

<p class="callout"><strong>One scale:</strong> compare a neuron’s discriminability with the animal’s behavioral performance.</p>

</div>
<div class="center">
<img src="images/landy_fig11.png" alt="Spike-count histograms, their ROC curve, a neurometric function, and spike counts grouped by behavioral choice">
</div>
</div>

<p class="tiny">Landy, Fig. 11. Choice probability is an association with choice; by itself it does not prove that one neuron caused the decision.</p>

Notes:
This provides a bridge between neural and behavioral measurements. A neuron’s ROC can quantify how well its responses distinguish two stimuli. Choice probability asks whether trial-to-trial neural variability covaries with the animal’s choice when the stimulus itself is uninformative.

---

## Why we use it for hearing in noise

On each trial, a listener hears:

- <span class="signal"><strong>signal + noise</strong></span>: a tone embedded in noise, or
- <span class="noise"><strong>noise alone</strong></span>

Then reports: **“tone present”** or **“tone absent.”**

</div>
<div class="callout">
Two listeners can have the same percent correct but use very different decision strategies.
</div>
</div>

<div class="deep-dive-cue">Optional deep dive ↓</div>

Notes:
Percent correct blends sensory ability and response preference. SDT helps us distinguish a listener who truly hears the tone better from one who simply says “present” more often.

-v-

<div class="section-tag section-1">1 · Why SDT?</div>
<div class="supplement-tag">Optional deep dive</div>

## Why SDT instead of a threshold model?

### Discrete threshold model

Evidence enters either a “detect” or “no-detect” state. Varying response bias predicts straight ROC segments.

### Signal detection theory

Evidence varies continuously and overlaps across conditions. Varying the criterion predicts a smooth ROC curve and separates **d′** from **c**.

<p class="callout">Confidence-rating ROCs let the data test these competing descriptions.</p>

</div>
<div class="center">
<img src="images/landy_fig8.png" alt="Piecewise linear ROC predicted by a discrete threshold model">
</div>
</div>

<p class="tiny">Landy, Fig. 8. Curved empirical ROCs often favor continuous-evidence models, but the appropriate model should be checked rather than assumed.</p>

Notes:
Threshold theories treat detection as an all-or-none internal state. SDT instead treats evidence as graded. Multiple confidence criteria reveal the ROC shape and allow these predictions to be compared.

---

## A common behavioral framework

On each sensory trial:

1. present **noise** or **signal + noise**
2. record the animal’s choice
3. classify the outcome as a hit, miss, false alarm, or correct rejection
4. measure dopaminergic activity during evidence and choice formation

<p class="callout"><strong>Question:</strong> Does higher dopamine accompany more accurate perception—perhaps by changing sensitivity, criterion, or confidence?</p>

</div>
<div class="center">
<img src="images/CBE_auditory.png" alt="Behavioral timeline comparing a self-timed motor task with a noisy auditory perceptual task containing hit, miss, false-alarm, and correct-rejection outcomes">
</div>
</div>

Notes:
The broader experiment asks whether dopaminergic activity is related to sensory perception, rather than movement alone. Trial-by-trial SDT outcomes let us ask whether higher dopamine predicts response accuracy and potentially confidence in detecting the stimulus. Separating d-prime and criterion is important because a change in licking alone need not mean better perception.

---

<div class="section-tag section-2">2 · The core model</div>

## Four possible outcomes

<table class="matrix">
  <tr><th></th><th>Say “present”</th><th>Say “absent”</th></tr>
  <tr><th>Signal present</th><td class="good"><strong>Hit</strong><br><span class="small">detected the tone</span></td><td class="bad"><strong>Miss</strong><br><span class="small">overlooked the tone</span></td></tr>
  <tr><th>Signal absent</th><td class="bad"><strong>False alarm</strong><br><span class="small">“heard” a tone that was not there</span></td><td class="good"><strong>Correct rejection</strong><br><span class="small">correctly said absent</span></td></tr>
</table>

<p class="center"><strong>“Correct” is not enough:</strong> we need to know which kind of correct—and which kind of error.</p>

---

<div class="section-tag section-2">2 · The core model</div>

## Start with two rates

### Hit rate

<div class="equation">hits ÷ signal-present trials</div>

“When the tone was present, how often did the listener say present?”

</div>
<div>

### False-alarm rate

<div class="equation">false alarms ÷ signal-absent trials</div>

“When the tone was absent, how often did the listener say present?”

</div>
</div>

$$
H = \frac{n_{\mathrm{Hit}}}{n_{\mathrm{Signal}}}
\qquad
F = \frac{n_{\mathrm{FA}}}{n_{\mathrm{Noise}}}
$$

<p class="callout center">A useful detector has a high hit rate <em>and</em> a low false-alarm rate.</p>

<div class="deep-dive-cue">Optional deep dive ↓</div>

-v-

<div class="section-tag section-2">2 · The core model</div>
<div class="supplement-tag">Optional deep dive</div>

## SDT ↔ machine-learning metrics

<table class="matrix compact">
  <tr><th>Signal detection theory</th><th>Machine learning</th></tr>
  <tr><td>Four outcomes</td><td>Confusion matrix</td></tr>
  <tr><td>Hit rate</td><td>True-positive rate · recall · sensitivity</td></tr>
  <tr><td>False-alarm rate</td><td>False-positive rate · 1 − specificity</td></tr>
  <tr><td>Criterion c</td><td>Classification threshold</td></tr>
  <tr><td>ROC / AUROC</td><td>Threshold sweep / ranking performance</td></tr>
</table>

$$
\mathrm{AUROC} = \Phi\!\left(\frac{d'}{\sqrt{2}}\right)
\qquad \text{(equal-variance Gaussian model)}
$$

<p class="callout center"><strong>d′ and AUROC are related, not interchangeable:</strong> d′ is model-based; empirical AUROC can be nonparametric.</p>

Notes:
The vocabulary changes across fields, but the outcome table and ROC are the same objects. Under the equal-variance Gaussian SDT model, d-prime maps monotonically to AUROC through the standard normal cumulative distribution. Outside that model, there is no universal conversion.

---

<div class="section-tag section-2">2 · The core model</div>

## Evidence varies from trial to trial

- The same stimulus does not produce exactly the same internal response every time.
- Noise and signal evidence therefore form overlapping distributions.
- The standard model assumes **equal variance** for signal and noise: the distributions differ only in their means.

$$
X_N \sim \mathcal{N}(\mu_N,\sigma^2),
\qquad
X_S \sim \mathcal{N}(\mu_S,\sigma^2)
$$

</div>
<div class="center">
<img src="images/landy_fig1a.png" alt="Overlapping Gaussian noise and signal evidence distributions with different means and the same standard deviation">
</div>
</div>

<p class="tiny">Landy, Fig. 1A.</p>

Notes:
Panel A is the measurement model. Repeated noise trials produce the distribution N and repeated signal-plus-noise trials produce S. Standard equal-variance SDT gives both distributions the same sigma and separates only their means.

---

<div class="section-tag section-2">2 · The core model</div>

## The optimal criterion lies midway

When signal and noise have:

- equal prior probability
- equal costs and rewards
- equal variance

the optimal decision boundary is where the two likelihoods are equal—midway between their means.

$$
x_{\mathrm{opt}} = \frac{\mu_N+\mu_S}{2}
\qquad \Longleftrightarrow \qquad c=0
$$

</div>
<div class="center">
<img src="images/landy_fig1c.png" alt="Equal-variance signal and noise distributions crossing at the optimal midpoint criterion, with no responses to the left and yes responses to the right">
</div>
</div>

<p class="tiny">Landy, Fig. 1C. Unequal priors or payoffs move the optimal criterion away from the midpoint.</p>

Notes:
Panel C shows the maximum-likelihood observer for equal priors and payoffs. Measurements below the crossing point produce “no”; measurements above it produce “yes.” In the centered criterion convention used in this deck, that neutral midpoint is c equals zero.

---

<div class="section-tag section-2">2 · The core model</div>

## The criterion turns evidence into a choice

Evidence **above** the criterion → say “present”

Evidence **below** the criterion → say “absent”

Moving the criterion changes both hits and false alarms.

</div>
<div class="center">
<img src="images/landy_fig2.png" alt="Decision outcomes for neutral, liberal, and conservative criteria">
</div>
</div>

<p class="tiny">Landy, Fig. 2.</p>

Notes:
Panel A partitions signal-present trials into misses to the left of the criterion and hits to the right. Panel B partitions signal-absent trials into correct rejections to the left and false alarms to the right. Panel C moves the criterion left, producing a liberal strategy with more hits and more false alarms. Panel D moves it right, producing a conservative strategy with fewer false alarms but more misses.

---

<div class="section-tag section-2">2 · The core model</div>

## Sensitivity: d′ (“d-prime”)

**d′ measures separation** between noise and signal-plus-noise evidence.

- d′ = 0: no discriminability
- larger d′: less overlap, better discriminability
- d′ is designed to be independent of the response criterion

$$
d' = \frac{\mu_S-\mu_N}{\sigma}
   = z(H)-z(F)
$$

</div>
<div class="center">
<img src="images/landy_fig3.png" alt="Two standardized distributions separated by d-prime">
</div>
</div>

<p class="tiny">Landy, Fig. 3.</p>

---

<div class="section-tag section-2">2 · The core model</div>

## Bias: criterion c

### Liberal criterion · c < 0

Say “present” readily

↑ hits · ↑ false alarms

Useful when misses are especially costly

</div>
<div class="card">

### Conservative criterion · c > 0

Require stronger evidence

↓ false alarms · ↓ hits

Useful when false alarms are especially costly

</div>
</div>

$$
c = -\frac{1}{2}\left[z(H)+z(F)\right]
$$

<p class="callout center"><strong>c = 0</strong> is unbiased only under the model’s equal-prior, equal-cost assumptions.</p>

<div class="deep-dive-cue">Optional deep dive ↓</div>

-v-

<div class="section-tag section-2">2 · The core model</div>
<div class="supplement-tag">Optional deep dive</div>

## Put priors and payoffs into the criterion

Keep the same yes/no task, but vary across blocks:

- signal prevalence: P(S) versus P(N)
- rewards for hits and correct rejections
- costs of misses and false alarms

These define an optimal likelihood-ratio threshold:

$$
\beta_{\mathrm{opt}}
= \frac{P(N)}{P(S)}
  \frac{V_{\mathrm{CR}}-V_{\mathrm{FA}}}
       {V_{\mathrm{Hit}}-V_{\mathrm{Miss}}}
$$

</div>
<div class="callout center">
<p class="big">Optimal criterion</p>
<p>Accounts for both probability and value</p>
<p class="small">for equal-variance Gaussian SDT and this deck’s criterion convention</p>
</div>
</div>

$$
c_{\mathrm{opt}} = \frac{\ln\!\left(\beta_{\mathrm{opt}}\right)}{d'}
$$

<p class="callout center">Changing prevalence or value should shift <strong>c</strong> without changing sensory sensitivity <strong>d′</strong>.</p>

Notes:
No new response paradigm is required. Manipulating priors or payoffs tests whether the observer moves the criterion in the direction predicted by statistical decision theory. Actual shifts are often smaller than optimal.

---

<div class="section-tag section-2">2 · The core model</div>

## Sensitivity and bias answer different questions

<table class="matrix">
  <tr><th>Quantity</th><th>Question</th><th>Changes when…</th></tr>
  <tr><td><strong>d′</strong></td><td>How separable are signal and noise?</td><td>the stimulus or sensory system changes</td></tr>
  <tr><td><strong>c</strong></td><td>How much evidence triggers “present”?</td><td>priors, instructions, payoffs, or strategy change</td></tr>
</table>

> SDT’s key contribution is estimating these separately from the same choices.

<div class="deep-dive-cue">Optional deep dive ↓</div>

-v-

<div class="section-tag section-2">2 · The core model</div>
<div class="supplement-tag">Optional deep dive</div>

## Confidence and meta-d′

- **First-order decision:** “Was the signal present?”
- **Second-order decision:** “How confident am I?”
- Confidence criteria divide evidence into low- and high-confidence responses.
- **meta-d′** asks what d′ would reproduce the observed confidence ROC.
- **M-ratio** $M = \mathrm{meta}\text{-}d' / d'$ summarizes metacognitive efficiency relative to task sensitivity.

</div>
<div class="center">
<img src="images/landy_fig12.png" alt="First-order and confidence criteria on evidence distributions with a second-order ROC curve">
</div>
</div>

<p class="tiny">Landy, Fig. 12. Meta-d′ is a model-based estimate and should be reported with its assumptions and uncertainty.</p>

Notes:
Distance from the first-order criterion can support a confidence judgment. Meta-d-prime expresses metacognitive sensitivity on the same scale as first-order d-prime. Dividing them gives the M-ratio, which is intended to separate metacognitive efficiency from raw task performance.

---

<div class="section-tag section-3">3 · From theory to experiment</div>

## ROC curves show the trade-off

A **receiver operating characteristic (ROC)** plots:

- false-alarm rate on the x-axis
- hit rate on the y-axis

Changing the criterion moves performance **along** a curve.

Changing sensitivity moves performance **between** curves.

</div>
<div class="center">
<img src="images/landy_fig4.png" alt="ROC curves for d-prime values from zero to three">
</div>
</div>

<p class="tiny">Landy, Fig. 4.</p>

---

<div class="section-tag section-3">3 · From theory to experiment</div>

## Read an ROC at a glance

### The diagonal

Chance-level discrimination: gaining hits requires an equal increase in false alarms.

### The upper-left corner

Perfect performance: all signals detected, no false alarms.

</div>
<div class="callout center">
<p class="big">Area under the curve</p>
<p>0.5 = chance</p>
<p>1.0 = perfect</p>
<p class="small">A criterion-free summary of ranking performance</p>
</div>
</div>

<div class="deep-dive-cue">Optional deep dive ↓</div>

-v-

<div class="section-tag section-3">3 · From theory to experiment</div>
<div class="supplement-tag">Optional deep dive</div>

## When the data are not Gaussian

### Keep the decision logic

The optimal rule still compares likelihoods, even when evidence is skewed, discrete, or better described by a Poisson distribution.

$$
\Lambda(x) = \frac{p(x\mid S)}{p(x\mid N)}
\quad \underset{\text{say “noise”}}{\overset{\text{say “signal”}}{\gtrless}} \quad \beta
$$

### Use an empirical ROC

Sweep a criterion across observed values and compute AUROC without fitting Gaussian curves.

<p class="callout"><strong>AUROC = P(a random signal value ranks above a random noise value)</strong>, with ties handled explicitly.</p>

</div>
<div class="center">
<img src="images/landy_fig10.png" alt="Discrete Poisson evidence distributions for noise and signal with a decision criterion">
</div>
</div>

<p class="tiny">Landy, Fig. 10. Nonparametric does not mean assumption-free: representative sampling, independence, and uncertainty still matter.</p>

Notes:
Gaussian evidence is convenient, not mandatory. For discrete spike or photon counts, likelihood ratios can be computed from an appropriate count model. An empirical ROC avoids a distributional form, but AUROC alone does not estimate response bias or explain the data-generating mechanism.

---

<div class="section-tag section-3">3 · From theory to experiment</div>

## Auditory experiment: one trial

<div class="flow">
  <div><strong>1</strong><br>Play noise alone or tone + noise</div>
  <div class="arrow">→</div>
  <div><strong>2</strong><br>Listener accumulates uncertain evidence</div>
  <div class="arrow">→</div>
  <div><strong>3</strong><br>“Tone present” or “tone absent”</div>
</div>

Manipulate sensory evidence:

- tone intensity
- background noise
- attention or adaptation

</div>
<div>

Manipulate decision context:

- signal prevalence
- reward or penalty
- response instructions

</div>
</div>

<div class="deep-dive-cue">Optional deep dive ↓</div>

-v-

<div class="section-tag section-3">3 · From theory to experiment</div>
<div class="supplement-tag">Optional deep dive</div>

## Alternative paradigm: 2AFC

Present two intervals on every trial—one with noise and one with signal + noise—then ask **which interval** contained the signal.

- removes a simple tendency to say “present”
- can still show a first/second-interval preference
- under independent, equal-variance evidence: $d'_{\mathrm{2AFC}} = \sqrt{2}\,d'_{\mathrm{YN}}$
- predicted $P(\mathrm{correct\ in\ 2AFC}) = \mathrm{AUROC}_{\mathrm{YN}}$

</div>
<div class="center">
<img src="images/landy_fig6.png" alt="Two-dimensional evidence space for a two-alternative forced-choice task">
</div>
</div>

<p class="tiny">Landy, Fig. 6. The √2 relation depends on the stated model and d′ definitions; real observers may also differ in sensitivity across intervals.</p>

Notes:
2AFC compares two noisy measurements directly. It avoids a yes-versus-no response criterion, but is not literally bias-free because observers may prefer one interval. Under the standard model, the geometry links forced-choice discriminability and accuracy to yes-no d-prime and AUROC.

---

<div class="section-tag section-4">4 · Put it into practice</div>

## Worked example: collect responses

Eighty trials: 40 contain a tone and 40 contain noise alone.

<table class="matrix">
  <tr><th></th><th>Say “present”</th><th>Say “absent”</th><th>Total</th></tr>
  <tr><th>Tone present</th><td class="good">32 hits</td><td class="bad">8 misses</td><td>40</td></tr>
  <tr><th>Tone absent</th><td class="bad">8 false alarms</td><td class="good">32 correct rejections</td><td>40</td></tr>
</table>

$$
H = \frac{32}{40} = 0.80
\qquad
F = \frac{8}{40} = 0.20
$$

---

<div class="section-tag section-4">4 · Put it into practice</div>

## Worked example: transform the rates

Convert each proportion to its corresponding standard-normal **z score**:

$$
z(0.80)=+0.842
\qquad
z(0.20)=-0.842
$$

Then estimate sensitivity:

$$
d' = z(H)-z(F) = 0.842-(-0.842)=\mathbf{1.68}
$$

And criterion:

$$
c = -\frac{1}{2}\left[z(H)+z(F)\right]
  = -\frac{1}{2}[0.842-0.842]
  = \mathbf{0}
$$

---

<div class="section-tag section-4">4 · Put it into practice</div>

## Worked example: interpret, don’t just calculate

<div class="columns">
<div class="callout center">
<p class="big">d′ = 1.68</p>
<p>The listener has meaningful—but imperfect—sensitivity to the tone.</p>
</div>
<div class="callout center">
<p class="big">c = 0</p>
<p>The listener has no estimated yes/no bias under the equal-variance model.</p>
</div>
</div>

> Report the counts and rates with d′ and c. The parameters are summaries, not substitutes for the data.

---

<div class="section-tag section-4">4 · Put it into practice</div>

## Practical details that matter

- Compute hit and false-alarm rates using their **own denominators**.
- A rate of exactly 0 or 1 gives an infinite z score; use a documented correction, such as adding 0.5 to the relevant counts.
- Standard d′ assumes continuous, Gaussian evidence distributions with **equal variance**.
- Small samples produce unstable estimates; report uncertainty when possible.
- For confidence ratings, fit an ROC rather than relying on one operating point.

<div class="deep-dive-cue">Optional deep dive ↓</div>

Notes:
A common correction replaces a rate x/n with (x + 0.5)/(n + 1). The choice of correction should be stated in methods.

-v-

<div class="section-tag section-4">4 · Put it into practice</div>
<div class="supplement-tag">Optional deep dive</div>

## Biological caveats: variance and conservatism

### Unequal variance

Signal evidence may be more variable than noise. The ROC becomes asymmetric and the zROC slope differs from 1.

### Criterion conservatism

When prevalence or payoffs change, people often move c in the predicted direction—but **less far** than the optimal model predicts.

Possible causes include model mismatch, distorted probability weighting, learning, and a criterion that drifts over time.

</div>
<div class="center">
<img src="images/landy_fig9.png" alt="Unequal-variance evidence distributions, their ROC, and a zROC with slope below one">
</div>
</div>

<p class="tiny">Landy, Fig. 9. Diagnose assumptions with multiple operating points; do not treat one hit/false-alarm pair as a full model check.</p>

Notes:
Equal-variance Gaussian SDT is a baseline. Confidence ratings or criterion manipulations provide multiple ROC points that can reveal unequal variance. Separately, observed criterion shifts are commonly conservative relative to normative predictions, and several mechanisms could produce that pattern.

---

<div class="section-tag section-4">4 · Put it into practice</div>

## What SDT can—and cannot—tell us

### It can separate

- discriminability from response bias
- a sensory change from a strategic shift
- operating point from overall ROC performance

</div>
<div>

### It cannot identify by itself

- the biological source of noise
- whether the Gaussian model is correct
- the “right” criterion without costs and goals

</div>
</div>

---

<div class="section-tag section-4">4 · Put it into practice</div>

## Take-home message

<p class="big center">Every uncertain yes/no decision combines<br><span class="signal">evidence quality</span> and a <span class="noise">decision rule</span>.</p>

<div class="flow">
  <div><strong>Count</strong><br>hits and false alarms</div>
  <div class="arrow">→</div>
  <div><strong>Estimate</strong><br>d′ and c</div>
  <div class="arrow">→</div>
  <div><strong>Interpret</strong><br>sensitivity and bias separately</div>
</div>

> For auditory detection in noise, SDT tells us whether perception improved—or the listener simply changed when they said “yes.”

---

<div class="section-tag section-4">4 · Put it into practice</div>

## Sources and further reading

<div class="sources-grid">
<div>
<ul>
<li>Abdi, H. (2009). <em>Signal Detection Theory</em>.</li>
<li>Landy, M. S. <em>Signal Detection Theory</em>. Provided chapter and figures.</li>
<li>Green, D. M., & Swets, J. A. (1966). <em>Signal Detection Theory and Psychophysics</em>.</li>
<li>Macmillan, N. A., & Creelman, C. D. (1991). <em>Detection Theory: A User’s Guide</em>.</li>
</ul>
</div>
<div>
<ul>
<li>Ackermann, J. F., & Landy, M. S. (2015). <em>Attention, Perception, & Psychophysics, 77</em>, 638–658.</li>
<li>Britten, K. H., et al. (1992). <em>Journal of Neuroscience, 12</em>, 4745–4765.</li>
<li>Maniscalco, B., & Lau, H. (2012). <em>Consciousness and Cognition, 21</em>, 422–430.</li>
<li>Wolfe, J. M., et al. (2007). <em>JEP: General, 136</em>, 623–638.</li>
</ul>
</div>
</div>

<p class="small">The first two sources and all original figures are included in <code>text_and_figs/</code>.</p>