A participant reports more faint signals after receiving new instructions. That could reflect better information, a greater willingness to answer “yes,” or several changes together. The hit rate alone cannot separate those explanations.
This lesson uses an original, deliberately simplified evidence model to make the distinction visible. Signal detection theory appears in the AAMC's sensory-processing content outline; the examples here are independent teaching material, not official questions. AAMC content outline
Keep the stimulus and the response in separate columns
In a yes/no detection task, two facts define each outcome: whether a signal was present and whether the participant reported it.
| Actual trial | Response: signal present | Response: signal absent |
|---|---|---|
| Signal present | Hit | Miss |
| Signal absent | False alarm | Correct rejection |
A false alarm is not a weak hit. It is a “present” response on a trial without the signal. A miss is not a correct rejection. Both involve saying “absent,” but the actual trial differs. These four outcomes are part of the standard detection framework. David Heeger's NYU teaching notes
Write the denominators before calculating:
- Hit rate: hits divided by all signal-present trials.
- False-alarm rate: false alarms divided by all signal-absent trials.
- Overall accuracy: hits plus correct rejections, divided by all trials.
A useful terminology warning: in a diagnostic table, “sensitivity” often means the hit rate. In signal detection theory, sensory sensitivity or discriminability concerns how distinguishable the evidence distributions are. An increased hit rate need not mean increased discriminability.
Fictional worked case: freeze the evidence first
Imagine a teaching simulation of detecting a brief tone in background noise. Each trial produces one evidence score: low, middle or high. Higher scores favour a “signal present” response. The simulation supplies the following complete records:
| Evidence score | Signal-present trials | Signal-absent trials |
|---|---|---|
| Low | 20 | 60 |
| Middle | 30 | 30 |
| High | 50 | 10 |
| Total | 100 | 100 |
These are invented counts, not measurements from a participant. The scores are model evidence, not direct recordings of a person's private experience. Their distributions overlap: even a high score sometimes occurs without a signal.
We will apply two deterministic response rules to these same records. No stimulus, score or trial label changes. That construction isolates the effect of the decision rule without claiming a human experiment has held every internal process fixed.
Rule S: answer “present” only for high scores.
The 50 high-scoring signal trials become hits; the other 50 become misses. Among absent trials, 10 become false alarms and 90 become correct rejections.
Rule L: answer “present” for middle or high scores.
Now there are 30 + 50 = 80 hits and 20 misses. There are also 30 + 10 = 40 false alarms and 60 correct rejections.
| Rule | Hits | Misses | False alarms | Correct rejections | Hit rate | False-alarm rate |
|---|---|---|---|---|---|---|
| S: high only | 50 | 50 | 10 | 90 | 50% | 10% |
| L: middle or high | 80 | 20 | 40 | 60 | 80% | 40% |
The decision criterion became more permissive. Thirty previously missed signal trials became hits, while thirty previously rejected absent trials became false alarms. The evidence itself did not become more discriminable: we have not changed any record.
Two summaries that can conceal the change
Both rules have 140 correct responses:
S: (50 + 90)/200 = 70%.
L: (80 + 60)/200 = 70%.
Equal accuracy does not imply equal behaviour. One rule produces 60 “present” responses; the other produces 120. Their error patterns are different even though their overall accuracy agrees.
There is another denominator trap. Under S, 50/60, approximately 83.3%, of “present” responses came from signal-present trials. That is not the 50% hit rate: it conditions on the response rather than the actual trial. Under L, the corresponding proportion is 80/120, approximately 66.7%.
Those response-conditioned proportions also depend on how often signals occur. If a new simulation contains different proportions of present and absent trials, they need not carry over even if the conditional hit and false-alarm rates remain the same. Do not silently treat a laboratory's equal trial allocation as a universal signal frequency.
Neither rule is automatically preferable. A choice would require an objective, including the consequences of misses and false alarms. The labels “liberal” and “conservative” describe response policies; they do not describe a person's moral character or political beliefs.
What the model establishes—and what a real comparison would not
Changing a criterion on fixed evidence moves the operating point: the pair of false-alarm and hit rates. This is the logic behind comparing points on a receiver-operating-characteristic plot. The original decision-theory treatment distinguishes the evidence available from the criterion used to classify it. Swets, Tanner and Birdsall, 1961
Our conclusion follows especially strongly because the two rules reclassify identical records. If two separate human sessions instead produced these percentages, that design fact would be missing. Instructions might change attention, fatigue might change evidence quality, and sampling variation might change the observed counts.
The pattern “more hits and more false alarms” is therefore compatible with a more permissive criterion, but does not by itself prove that criterion was the only change. Ask what was manipulated, what remained comparable and how much uncertainty surrounds the rates.
Likewise, a higher hit rate accompanied by a lower false-alarm rate is better performance at those observed operating points. It is not, without an appropriate comparison and model, a complete explanation of which sensory or decision process changed.
Use d′ only with its interpretation attached
In the equal-variance Gaussian model, d′ describes the separation between the signal-present and signal-absent means in units of their common standard deviation. That interpretation has distributional assumptions. The original paper explicitly discusses both the equal-variance model and reasons to consider alternatives. Swets and colleagues
Our three-category table does not stipulate Gaussian evidence with equal variance. It supports exact counts and criterion comparisons, but does not establish that latent model. Applying a normal-quantile calculation mechanically would not validate the resulting sensory interpretation. And a passage supplying only a hit rate has not supplied the false-alarm information needed for the usual yes/no d′ estimate.
The appropriate response to missing assumptions is to state the narrower conclusion the data support, not invent a discriminability value.
Transfer exercise: check a changed denominator
A new fictional simulation has 80 signal-present and 120 signal-absent trials. Applying a strict rule to fixed evidence produces 48 hits and 12 false alarms. Lowering the threshold adds 16 hits and 24 false alarms on those same records.
Which conclusion follows?
A. The hit rate rises from 24% to 32%, proving better discrimination.
B. The hit rate rises from 60% to 80%, while the false-alarm rate rises from 10% to 30%; this reclassification changes the criterion.
C. The false-alarm rate rises to 18%, so fewer absent trials are misclassified.
D. The new rule must be preferable because it finds more signals.
Feedback: B. Initially, 48/80 = 60% and 12/120 = 10%. After reclassification, 64/80 = 80% and 36/120 = 30%. The underlying evidence records are unchanged by stipulation.
A divides hits by all 200 trials and then overstates the mechanism. C similarly divides false alarms by all trials and reverses the direction. D supplies no objective for balancing errors.
As a further check, accuracy falls from (48 + 108)/200 = 78% to (64 + 84)/200 = 74%. More hits can coexist with lower overall accuracy, and accuracy alone still does not determine which error trade-off matters.
Make one precise reading note
For your next detection passage, record both conditional rates, the criterion or instruction manipulation, and the evidence needed to distinguish a policy change from changed information. That small habit makes the conclusion auditable without turning every passage into a calculation of d′.