Worked Example 1 — Finding the driver, and losing two of the three you thought you'd found
The case. A hospital pharmacist, six months after a near-miss dispensing error that was caught by someone else. No one was hurt. She checks and rechecks, and the checking has begun to cost her more time than the work. When she brings the memory to mind she reports a still-to-moving visual image, arm's length, straight ahead, in colour, with the sound of the other pharmacist's voice.
Step 1 — Establish the noise floor before you establish anything else
Four readings of S with the memory untouched, taken across two sessions, with a genuine break state and an unrelated conversation between each:
62, 68, 65, 71
The mean:
S₀ = (62 + 68 + 65 + 71) / 4 = 266 / 4 = 66.5
The deviations from the mean, in order: −4.5, +1.5, −1.5, +4.5. Their squares: 20.25, 2.25, 2.25, 20.25, summing to 45. The sample variance uses n−1 = 3 in the denominator, because one degree of freedom was spent estimating the mean:
s² = 45 / 3 = 15
s = √15 = 3.873
Call this ε, the within-session measurement noise. It is not her instability. It is the instrument's.
Step 2 — Convert the noise into a threshold
Every probe below is a difference between two readings: one baseline, one after a change. Variances of independent quantities add, so
Var(Δ) = Var(S₀) + Var(Sᵢ) = ε² + ε² = 2ε² = 30
SE(Δ) = √30 = 5.477
A two-tailed 95% interval is 1.96 standard errors wide on each side:
MDE₉₅ = 1.96 × 5.477 = 10.735 ≈ 10.7
Any Δ smaller than about 11 points is indistinguishable from the instrument breathing. Write that number on the page before you run a single probe, because after you run them you will want it to be smaller.
Step 3 — The one-at-a-time sweep
Each probe: change exactly one submodality, read S, restore the original, break state, re-read the baseline to confirm it came back. Nine probes, one of which is a sham — a change with no plausible mechanism, inserted precisely so that you can see what a null looks like in this particular client on this particular day.
| # | submodality | change made | S | Δ | D = Δ/5.477 |
|---|---|---|---|---|---|
| 1 | brightness | halve | 63.0 | 3.5 | 0.64 |
| 2 | size | shrink to ¼ of the field | 61.5 | 5.0 | 0.91 |
| 3 | colour | full colour → greyscale | 57.5 | 9.0 | 1.64 |
| 4 | distance | arm's length → across the room | 42.5 | 24.0 | 4.38 |
| 5 | location | straight ahead → low left | 47.5 | 19.0 | 3.47 |
| 6 | motion | film → freeze-frame | 53.5 | 13.0 | 2.37 |
| 7 | volume | halve the voice | 60.5 | 6.0 | 1.10 |
| 8 | timbre | deepen the voice | 64.5 | 2.0 | 0.37 |
| 9 | sham | widen the image's frame by 2 mm | 65.5 | 1.0 | 0.18 |
Against the 10.7 threshold, three probes clear: distance (24.0), location (19.0), motion (13.0). Colour, at 9.0, is the one everybody wants to keep and the one you must let go.
Step 4 — Pay the multiplicity tax
You ran nine tests. At α = 0.05 each, the expected number of false positives when nothing whatever is happening is
E[false positives] = 9 × 0.05 = 0.45
Just under one probe in every two sweeps will clear the bar by luck. The Bonferroni correction divides the error budget across the family of tests:
α' = 0.05 / 9 = 0.00556
The two-tailed critical value at α' = 0.00556 puts 0.00278 in each tail, which is z ≈ 2.77. So the corrected threshold is
MDE_corrected = 2.77 × 5.477 = 15.17
Distance (24.0) and location (19.0) survive. Motion does not. Thirteen points, which felt like a great deal in the room, is not a finding; it is a candidate for a second sweep. Bonferroni is conservative — it assumes the worst about the dependence among your tests, and your tests are almost certainly correlated — so the honest verdict on motion is unconfirmed, not refuted. Note it and move.
Step 5 — Test whether two drivers are two drivers
Combine the two survivors: image across the room and low left.
Observed: S = 42.0, Δ(d,l) = 24.5
If the two acted independently on the state, their reductions would add:
Predicted (additive): Δd + Δl = 24.0 + 19.0 = 43.0
Interaction: I = 24.5 − 43.0 = −18.5
Strongly sub-additive. The redundancy coefficient normalises this against the smaller of the two effects:
R = −I / min(Δd, Δl) = 18.5 / 19.0 = 0.974
Ninety-seven per cent of the smaller effect was already contained in the larger one.
The floor check, which is not optional. Sub-additivity is exactly what a floor effect looks like from the outside: if the scale bottoms out, two large effects cannot add because there is nowhere left to fall. So you need the client's effective floor. This client reads a neutral memory of the same age at S = 8. The combined probe landed at 42.0 — thirty-four points clear of her floor. The sub-additivity is real, not an artifact of the ruler running out.
Step 6 — The isolation probe, and the collapse of one of the drivers
Two changes that overlap this completely are usually not two changes. The natural hypothesis: when she moved the image low left, she also moved it away. Almost everyone does, unprompted, because human imagery treats the periphery as farther. Test it. Move the image low left while explicitly holding apparent size and apparent distance fixed — same visual angle, same felt proximity, different position.
Observed: S = 60.0, Δ = 6.5 (D = 1.19)
Location, isolated, is not a driver. It never was. It was distance wearing a costume, and the sweep could not tell the difference because the sweep never asked her to hold anything constant.
What this case actually shows
Nine probes produced one driver. Not three, and certainly not "she's a visual, so brightness and colour." The mechanism, so far as it is understood: imagery is not a slide projector but a re-enactment, and there is good evidence that visual imagery recruits retinotopically organised early visual cortex — Kosslyn's imaging work through the 1990s and 2000s is the standard reference. If imagining is partly re-running the perceptual machinery, then the parameters that matter will be the ones that machinery treats as consequential. Egocentric distance is the single most consequential parameter a threat-detecting animal computes. Frame width is not. That the sham dial returned 1.0 and distance returned 24.0 is not a coincidence, and it is also not proof — it is one client, one memory, one day.
The failure mode, named in the same breath. The sweep is a within-session instrument, and its cheapness is entirely a consequence of that. Every number above is a difference between two readings taken twenty minutes apart, which means her day cancels, her general disposition toward this memory cancels, and only instrument noise remains. The moment you carry any of these numbers out of the room and call it a change in her life, that cancellation stops, and the arithmetic becomes brutal. Which is Worked Example 2.