Contribution
LIT-834 stated the QQ equality and tested it on six data sets, four of them picked because they showed order effects. This paper tests it on all 66 Pew surveys of a decade that varied the order of two questions, plus six other studies. The equality holds across the set while order effects are present, and the paper presents the result as a regularity in survey data in its own right, whatever model explains it.
Key insight
Read the context-effect table, not the marginals. Order effects are usually reported as shifts in “yes” rates. The QQ equality is a constraint on the joint table: within each diagonal the two shifts cancel. That constraint can be checked on data whatever model one holds, and an unconstrained model would not produce it.
Assumptions
- Two binary questions asked back to back to two random halves of a sample, with nothing inserted before or between them.
- A common initial state across the two halves; individual differences are allowed as a mixed state (density matrix), and the equality still holds on average.
- Complete two-way tables are needed. Most published order-effect studies report only marginals, which limits the studies available.
Key results
- Table 1 (three Gallup polls, Moore 2002): order effects significant in all three (χ²(3) = 10.14, 73.04, 67.19). q = −0.003 (χ²(1) = 0.01, p = 0.91) for Clinton–Gore, −0.02 (χ²(1) = 0.56, p = 0.46) for white–black, and 0.1514 (χ²(1) = 28.57, p < 0.001) for Rose–Jackson, where background information was read out before each question. The text gives that last value as “−0.15”; the table, and the 2013 paper, give +0.1514.
- Algebraic facts about q (from the main text; proofs in the SI): the main-diagonal and off-diagonal q are equal and opposite; |q| cannot exceed the size of the larger diagonal’s order effect; the tables meeting the equality are a triangular plane inside the three-dimensional pyramid of possible context-effect tables.
- Fig. 1, left: for 72 studies, one diagonal cell against the other falls along the a priori line y = −x, r = −0.82 (−0.73 without two extremes). Fig. 1, right: for the 17 studies with an order effect above 0.10, q/(order effect) falls towards zero as the effect grows.
- The unbiased test: across the 66 Pew surveys, the χ² distribution test for order effects gives p = 0.0004 and the test for q gives p = 0.4625. Including the four selected national surveys does not change the conclusion (SI, not read).
- Robustness of the model (SI, not read): if the post-answer state moves only part of the angular distance towards the projection, q is no longer exactly zero but stays “very close to zero” in every case the authors tried.
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | In survey question-order experiments, the probability of agreeing answers is order-invariant while order effects are present | strong for this population of surveys: 66 unselected Pew surveys, p = 0.4625 for q against p = 0.0004 for order effects | Results, Fig. 1 |
| C2 | The equality fails when information is inserted between questions | weak: one case | Table 1, Rose–Jackson |
| C3 | No existing cognitive theory imposes this symmetry | weak: an argument from absence; the authors concede a classical model can be built to satisfy it | Results, last paragraph |
| C4 | The result reveals a “quantum nature” of human judgments | not supported: the evidence supports the equality, which a constrained classical model could also meet, and which LIT-264 shows is a noncontextual pattern | title, Discussion |
| C5 | The equality is robust to partial updating | not checkable here | SI, not read |
Concepts
- context effect: the difference between the two orders in the proportion of a response pair, cell by cell of the 2 × 2 table.
- q value: the sum of the two context effects on a diagonal. The QQ equality is E(q) = 0.
- order effect (size): the sum of the absolute context effects on the diagonal with the larger sum.
- law of reciprocity: |⟨T|V⟩|² = |⟨V|T⟩|² for T = P_A S, V = P_B S; the paper presents it as the principle the equality tests.
Connections
The model and the derivation are LIT-834’s. LIT-264 obtained these data from the authors, re-derived the equality from the identity P Q P + (I − P)(I − Q)(I − P) = I − (P + Q) + (P Q + Q P), and showed it implies the Contextuality-by-Default noncontextuality criterion for these rank-2 cyclic systems. The Discussion leans on beim Graben and Atmanspacher’s account of incompatible observables arising from coarse measurement of classical systems. The paper cites LIT-316 for proofs.
Bearing on the record
- Primary source of THEORY-166: survey order effects satisfy the QQ equality, and the regularity does not discriminate quantum probability from every classical model, nor does it show contextuality.
- THEORY-013. The QQ data are the “73 poll question-order pairs” of LIT-264 §3. This reading adds that the data’s own authors present the regularity as quantum, while the equality is exactly what makes the system noncontextual. The two papers agree on the data and differ on what it shows.
- On the title. “Reveal quantum nature” is stronger than anything in the body. The Discussion says that quantum probability “may provide a better description” even if the brain is classical, and the Results invite alternative accounts. The record should cite this paper for the equality and not for the title.
- No instruction for machine-learning practice; nothing for the anthology.
Limitations
- The SI, which holds the proofs, the per-study table and the χ² tests, was not read here.
- The 66 Pew surveys are unselected for outcome, but they are all one organisation’s question pairs and all in one country.
- The equality is a null hypothesis that is “accepted” by failing to reject. Many Pew order effects are small, and the paper itself notes that q is uninformative when the order effect is small; Fig. 1, right, addresses this for 17 studies only.
- The sample sizes are given as 651–3,006 participants per national study. LIT-264 gives the Pew N as 125–927. The two figures may count different things (whole samples, one order, or only yes/no respondents); neither paper says, and the discrepancy is unresolved here.
Open questions
- A classical model fitted to the same 72 tables, constrained to satisfy the equality, and compared with the quantum model on the order effects themselves. The paper names this need and leaves it to others.
- Pre-registered question pairs with information inserted between the questions, to test C2 beyond one poll.