Which design can tell the accounts apart

“Can we detect an effect?” is the easy question. “Can we tell why there is one?” is the one the design has to answer.

Nine candidate designs cross three stimulus-sequence structures with three proportions of orthogonal-report trials. For each, 60 synthetic observers were generated with both a stimulus-history amplitude (2.5°) and a response-history amplitude (1.5°), and the primary model had to recover both.

The proportion of orthogonal trials dominates everything else. With none, the two amplitudes are not separately identifiable: their recovery errors correlate at −0.88 and the joint error is 1.69°. With half the trials orthogonal the correlation falls to −0.44 and the joint error to 0.71°. That is why the protocol uses 50%.

The sequence structure matters much less than expected, and it is worth saying so plainly: balanced differences (0.71°) did not beat independent uniform sampling (0.70°) on recovery error. On a full 180° circle, independent sampling already covers every orientation difference evenly. Balanced differences are kept because they make the sequence autocorrelation exactly zero by construction rather than zero in expectation — a guarantee, not a precision gain. The autocorrelated walk (0.74°) biases the response-history estimate by −0.22°.

Fig. 1 Joint recovery error by design

SIMULATED DATA

independent uniform · 50% orthogonal0.697

balanced differences · 50% orthogonal0.707

autocorrelated walk · 50% orthogonal0.74

autocorrelated walk · 25% orthogonal0.741

independent uniform · 25% orthogonal0.878

balanced differences · 25% orthogonal0.931

autocorrelated walk · 0% orthogonal1.23

independent uniform · 0% orthogonal1.54

balanced differences · 0% orthogonal1.69

joint RMSE of (a_stim, a_resp), degrees

Joint error is √(RMSE²(a_stim) + RMSE²(a_resp)), lower is better. 60 synthetic observers per design, 600 trials each.Source: results/design_optimization.parquet · run design_optimization-8964a2e-6000

Table view (9 rows)

sequence	p(orthogonal)	RMSE a_stim	RMSE a_resp	bias a_stim	bias a_resp	r(errors)	basis r	seq. autocorr
independent uniform	0.50	0.513	0.472	+0.004	−0.029	−0.43	0.33	+0.003
balanced differences	0.50	0.516	0.484	−0.007	+0.010	−0.44	0.33	−0.009
autocorrelated walk	0.50	0.495	0.550	+0.073	−0.219	−0.45	0.51	+0.674
autocorrelated walk	0.25	0.471	0.572	+0.030	−0.040	−0.49	0.63	+0.671
independent uniform	0.25	0.633	0.610	−0.037	−0.041	−0.48	0.47	−0.004
balanced differences	0.25	0.705	0.607	−0.029	−0.078	−0.61	0.47	−0.003
autocorrelated walk	0.00	0.881	0.853	−0.143	+0.088	−0.86	0.90	+0.673
independent uniform	0.00	1.047	1.129	+0.050	−0.128	−0.82	0.90	+0.003
balanced differences	0.00	1.174	1.219	+0.257	−0.362	−0.88	0.90	−0.005

Fig. 2 Are the two accounts separable?

SIMULATED DATA

independent uniform · 50% orthogonal−0.43

balanced differences · 50% orthogonal−0.439

autocorrelated walk · 50% orthogonal−0.455

autocorrelated walk · 25% orthogonal−0.494

independent uniform · 25% orthogonal−0.483

balanced differences · 25% orthogonal−0.606

autocorrelated walk · 0% orthogonal−0.856

independent uniform · 0% orthogonal−0.822

balanced differences · 0% orthogonal−0.884

corr(recovery error a_stim, recovery error a_resp) · vertical rule: H5 threshold −0.5

Correlation between the recovery errors of the two amplitudes. Near −1, any error in one is compensated by the other and the data cannot say which process is at work. Hypothesis H5 requires |r| < 0.5.Source: results/design_optimization.parquet · run design_optimization-8964a2e-6000