Serial dependence

The primary outcome, estimated the way the preregistration fixes it — and what it looks like when the same data are cut by uncertainty, by response history, and by observer.

The primary model (M3adj_full) fits a derivative-of-Gaussian tuning curve to the lag-1 stimulus-history predictor while simultaneously fitting response history, attraction to cardinal orientations, regression toward the running mean at two timescales, and pull toward the dial’s random start angle. Its amplitude parameter is, by construction, the peak bias in degrees.

In the synthetic reference cohort the group amplitude is +2.56° (95% bootstrap CI +2.09 to +2.99, p < 0.001, 30 observers), peaking at a 22.5° difference. The response-history amplitude is +1.38° (+1.25 to +1.51); the observers were generated with 2.5° and 1.2° respectively.

Fig. 1 The serial-dependence curve

SIMULATED DATA

Binned mean error (± SEM across observers) against previous − current orientation, with the group DoG fit. Positive error means the report was pulled toward the previous stimulus.Source: results/serial_dependence.parquet · run serial_dependence-280346c-1

Table view (193 rows)

stratum	Δ orientation (°)	mean error (°)	SEM (°)	observers
all	−82.5	+0.284	0.264	30
all	−67.5	−1.099	0.320	30
all	−52.5	−1.417	0.227	30
all	−37.5	−2.584	0.341	30
all	−22.5	−3.015	0.441	30
all	−7.5	−1.849	0.257	30
all	+7.5	+0.968	0.319	30
all	+22.5	+2.947	0.352	30
all	+37.5	+2.919	0.363	30
all	+52.5	+1.023	0.280	30
all	+67.5	+0.039	0.216	30
all	+82.5	−0.119	0.210	30
fitted_group_dog	−90.0	−0.006	—	30
fitted_group_dog	−89.0	−0.007	—	30
fitted_group_dog	−88.0	−0.008	—	30
fitted_group_dog	−87.0	−0.009	—	30
fitted_group_dog	−86.0	−0.011	—	30
fitted_group_dog	−85.0	−0.012	—	30
fitted_group_dog	−84.0	−0.015	—	30
fitted_group_dog	−83.0	−0.017	—	30
fitted_group_dog	−82.0	−0.020	—	30
fitted_group_dog	−81.0	−0.023	—	30
fitted_group_dog	−80.0	−0.027	—	30
fitted_group_dog	−79.0	−0.031	—	30
fitted_group_dog	−78.0	−0.035	—	30
fitted_group_dog	−77.0	−0.041	—	30
fitted_group_dog	−76.0	−0.047	—	30
fitted_group_dog	−75.0	−0.054	—	30
fitted_group_dog	−74.0	−0.061	—	30
fitted_group_dog	−73.0	−0.070	—	30
fitted_group_dog	−72.0	−0.080	—	30
fitted_group_dog	−71.0	−0.091	—	30
fitted_group_dog	−70.0	−0.103	—	30
fitted_group_dog	−69.0	−0.116	—	30
fitted_group_dog	−68.0	−0.131	—	30
fitted_group_dog	−67.0	−0.148	—	30
fitted_group_dog	−66.0	−0.166	—	30
fitted_group_dog	−65.0	−0.186	—	30
fitted_group_dog	−64.0	−0.208	—	30
fitted_group_dog	−63.0	−0.232	—	30
fitted_group_dog	−62.0	−0.259	—	30
fitted_group_dog	−61.0	−0.288	—	30
fitted_group_dog	−60.0	−0.319	—	30
fitted_group_dog	−59.0	−0.353	—	30
fitted_group_dog	−58.0	−0.389	—	30
fitted_group_dog	−57.0	−0.429	—	30
fitted_group_dog	−56.0	−0.471	—	30
fitted_group_dog	−55.0	−0.516	—	30
fitted_group_dog	−54.0	−0.565	—	30
fitted_group_dog	−53.0	−0.616	—	30
fitted_group_dog	−52.0	−0.671	—	30
fitted_group_dog	−51.0	−0.729	—	30
fitted_group_dog	−50.0	−0.789	—	30
fitted_group_dog	−49.0	−0.853	—	30
fitted_group_dog	−48.0	−0.920	—	30
fitted_group_dog	−47.0	−0.990	—	30
fitted_group_dog	−46.0	−1.062	—	30
fitted_group_dog	−45.0	−1.137	—	30
fitted_group_dog	−44.0	−1.214	—	30
fitted_group_dog	−43.0	−1.293	—	30
fitted_group_dog	−42.0	−1.374	—	30
fitted_group_dog	−41.0	−1.456	—	30
fitted_group_dog	−40.0	−1.539	—	30
fitted_group_dog	−39.0	−1.623	—	30
fitted_group_dog	−38.0	−1.706	—	30
fitted_group_dog	−37.0	−1.790	—	30
fitted_group_dog	−36.0	−1.872	—	30
fitted_group_dog	−35.0	−1.952	—	30
fitted_group_dog	−34.0	−2.030	—	30
fitted_group_dog	−33.0	−2.106	—	30
fitted_group_dog	−32.0	−2.178	—	30
fitted_group_dog	−31.0	−2.245	—	30
fitted_group_dog	−30.0	−2.308	—	30
fitted_group_dog	−29.0	−2.366	—	30
fitted_group_dog	−28.0	−2.417	—	30
fitted_group_dog	−27.0	−2.461	—	30
fitted_group_dog	−26.0	−2.497	—	30
fitted_group_dog	−25.0	−2.525	—	30
fitted_group_dog	−24.0	−2.545	—	30
fitted_group_dog	−23.0	−2.555	—	30
fitted_group_dog	−22.0	−2.555	—	30
fitted_group_dog	−21.0	−2.545	—	30
fitted_group_dog	−20.0	−2.524	—	30
fitted_group_dog	−19.0	−2.493	—	30
fitted_group_dog	−18.0	−2.449	—	30
fitted_group_dog	−17.0	−2.395	—	30
fitted_group_dog	−16.0	−2.329	—	30
fitted_group_dog	−15.0	−2.251	—	30
fitted_group_dog	−14.0	−2.162	—	30
fitted_group_dog	−13.0	−2.062	—	30
fitted_group_dog	−12.0	−1.951	—	30
fitted_group_dog	−11.0	−1.830	—	30
fitted_group_dog	−10.0	−1.699	—	30
fitted_group_dog	−9.0	−1.558	—	30
fitted_group_dog	−8.0	−1.408	—	30
fitted_group_dog	−7.0	−1.251	—	30
fitted_group_dog	−6.0	−1.086	—	30
fitted_group_dog	−5.0	−0.915	—	30
fitted_group_dog	−4.0	−0.738	—	30
fitted_group_dog	−3.0	−0.558	—	30
fitted_group_dog	−2.0	−0.374	—	30
fitted_group_dog	−1.0	−0.187	—	30
fitted_group_dog	+0.0	+0.000	—	30
fitted_group_dog	+1.0	+0.187	—	30
fitted_group_dog	+2.0	+0.374	—	30
fitted_group_dog	+3.0	+0.558	—	30
fitted_group_dog	+4.0	+0.738	—	30
fitted_group_dog	+5.0	+0.915	—	30
fitted_group_dog	+6.0	+1.086	—	30
fitted_group_dog	+7.0	+1.251	—	30
fitted_group_dog	+8.0	+1.408	—	30
fitted_group_dog	+9.0	+1.558	—	30
fitted_group_dog	+10.0	+1.699	—	30
fitted_group_dog	+11.0	+1.830	—	30
fitted_group_dog	+12.0	+1.951	—	30
fitted_group_dog	+13.0	+2.062	—	30
fitted_group_dog	+14.0	+2.162	—	30
fitted_group_dog	+15.0	+2.251	—	30
fitted_group_dog	+16.0	+2.329	—	30
fitted_group_dog	+17.0	+2.395	—	30
fitted_group_dog	+18.0	+2.449	—	30
fitted_group_dog	+19.0	+2.493	—	30
fitted_group_dog	+20.0	+2.524	—	30
fitted_group_dog	+21.0	+2.545	—	30
fitted_group_dog	+22.0	+2.555	—	30
fitted_group_dog	+23.0	+2.555	—	30
fitted_group_dog	+24.0	+2.545	—	30
fitted_group_dog	+25.0	+2.525	—	30
fitted_group_dog	+26.0	+2.497	—	30
fitted_group_dog	+27.0	+2.461	—	30
fitted_group_dog	+28.0	+2.417	—	30
fitted_group_dog	+29.0	+2.366	—	30
fitted_group_dog	+30.0	+2.308	—	30
fitted_group_dog	+31.0	+2.245	—	30
fitted_group_dog	+32.0	+2.178	—	30
fitted_group_dog	+33.0	+2.106	—	30
fitted_group_dog	+34.0	+2.030	—	30
fitted_group_dog	+35.0	+1.952	—	30
fitted_group_dog	+36.0	+1.872	—	30
fitted_group_dog	+37.0	+1.790	—	30
fitted_group_dog	+38.0	+1.706	—	30
fitted_group_dog	+39.0	+1.623	—	30
fitted_group_dog	+40.0	+1.539	—	30
fitted_group_dog	+41.0	+1.456	—	30
fitted_group_dog	+42.0	+1.374	—	30
fitted_group_dog	+43.0	+1.293	—	30
fitted_group_dog	+44.0	+1.214	—	30
fitted_group_dog	+45.0	+1.137	—	30
fitted_group_dog	+46.0	+1.062	—	30
fitted_group_dog	+47.0	+0.990	—	30
fitted_group_dog	+48.0	+0.920	—	30
fitted_group_dog	+49.0	+0.853	—	30
fitted_group_dog	+50.0	+0.789	—	30
fitted_group_dog	+51.0	+0.729	—	30
fitted_group_dog	+52.0	+0.671	—	30
fitted_group_dog	+53.0	+0.616	—	30
fitted_group_dog	+54.0	+0.565	—	30
fitted_group_dog	+55.0	+0.516	—	30
fitted_group_dog	+56.0	+0.471	—	30
fitted_group_dog	+57.0	+0.429	—	30
fitted_group_dog	+58.0	+0.389	—	30
fitted_group_dog	+59.0	+0.353	—	30
fitted_group_dog	+60.0	+0.319	—	30
fitted_group_dog	+61.0	+0.288	—	30
fitted_group_dog	+62.0	+0.259	—	30
fitted_group_dog	+63.0	+0.232	—	30
fitted_group_dog	+64.0	+0.208	—	30
fitted_group_dog	+65.0	+0.186	—	30
fitted_group_dog	+66.0	+0.166	—	30
fitted_group_dog	+67.0	+0.148	—	30
fitted_group_dog	+68.0	+0.131	—	30
fitted_group_dog	+69.0	+0.116	—	30
fitted_group_dog	+70.0	+0.103	—	30
fitted_group_dog	+71.0	+0.091	—	30
fitted_group_dog	+72.0	+0.080	—	30
fitted_group_dog	+73.0	+0.070	—	30
fitted_group_dog	+74.0	+0.061	—	30
fitted_group_dog	+75.0	+0.054	—	30
fitted_group_dog	+76.0	+0.047	—	30
fitted_group_dog	+77.0	+0.041	—	30
fitted_group_dog	+78.0	+0.035	—	30
fitted_group_dog	+79.0	+0.031	—	30
fitted_group_dog	+80.0	+0.027	—	30
fitted_group_dog	+81.0	+0.023	—	30
fitted_group_dog	+82.0	+0.020	—	30
fitted_group_dog	+83.0	+0.017	—	30
fitted_group_dog	+84.0	+0.015	—	30
fitted_group_dog	+85.0	+0.012	—	30
fitted_group_dog	+86.0	+0.011	—	30
fitted_group_dog	+87.0	+0.009	—	30
fitted_group_dog	+88.0	+0.008	—	30
fitted_group_dog	+89.0	+0.007	—	30
fitted_group_dog	+90.0	+0.006	—	30

Stimulus history against response history

Plotting the same errors against the previous response instead of the previous stimulus gives a second curve. Because half the trials ask for the orthogonal orientation, the two predictors differ by 90° on cue-incongruent trial pairs, and their DoG bases correlate at only +0.31 overall (+0.88 on congruent pairs, −0.23 on incongruent ones). That is what allows the model to estimate both.

Fig. 2 Error against previous stimulus and previous response

SIMULATED DATA

previous stimulus
previous response

Two views of the same trials. These are marginal curves, not model estimates: each absorbs part of the other, which is why the adjusted model is needed to separate them.Source: results/serial_dependence.parquet · run serial_dependence-280346c-1

Table view (24 rows)

stratum	Δ orientation (°)	mean error (°)	SEM (°)	observers
all	−82.5	+0.284	0.264	30
all	−67.5	−1.099	0.320	30
all	−52.5	−1.417	0.227	30
all	−37.5	−2.584	0.341	30
all	−22.5	−3.015	0.441	30
all	−7.5	−1.849	0.257	30
all	+7.5	+0.968	0.319	30
all	+22.5	+2.947	0.352	30
all	+37.5	+2.919	0.363	30
all	+52.5	+1.023	0.280	30
all	+67.5	+0.039	0.216	30
all	+82.5	−0.119	0.210	30
response_history	−82.5	+0.454	0.306	30
response_history	−67.5	+0.977	0.309	30
response_history	−52.5	−0.302	0.303	30
response_history	−37.5	−1.144	0.267	30
response_history	−22.5	−2.380	0.297	30
response_history	−7.5	−1.473	0.263	30
response_history	+7.5	+0.475	0.294	30
response_history	+22.5	+1.867	0.389	30
response_history	+37.5	+1.341	0.378	30
response_history	+52.5	+0.080	0.292	30
response_history	+67.5	−1.258	0.260	30
response_history	+82.5	−0.674	0.301	30

Every observer

A group mean hides heterogeneity. Each observer below was generated with its own amplitude (population SD 1.2°); the estimate sits beside the generating value, so the scatter between them is estimation error, not individual difference.

Fig. 3 Individual amplitudes

SIMULATED DATA

estimate
generating value

Observers sorted by estimated amplitude. Filled: estimate. Hollow: the value used to generate that observer. The band is the group 95% CI.Source: results/individual_effects.parquet · run individual_effects-280346c-5

Table view (30 rows)

observer	estimated a_stim (°)	generating a_stim (°)	odd blocks	even blocks
SIM002	−0.29	+0.45	−1.33	−0.28
SIM013	+0.31	+0.79	+0.78	−0.81
SIM028	+0.52	+1.67	+1.24	+1.12
SIM024	+1.42	+2.55	+1.40	+1.90
SIM018	+1.44	+1.33	+0.64	+2.19
SIM027	+1.47	+0.96	+1.47	+1.64
SIM014	+1.66	+0.54	+2.42	+1.51
SIM003	+1.90	+2.19	+2.22	+1.50
SIM005	+2.03	+2.19	+2.59	+1.15
SIM008	+2.15	+2.13	+2.31	+2.44
SIM006	+2.19	+0.70	+1.04	+2.86
SIM015	+2.19	+2.90	+1.88	+2.59
SIM010	+2.20	+2.61	+1.44	+3.45
SIM016	+2.47	+3.27	+2.59	+2.55
SIM019	+2.52	+2.74	+3.25	+1.46
SIM004	+2.53	+2.66	+2.16	+2.72
SIM011	+2.66	+2.78	+2.54	+2.82
SIM017	+2.68	+2.80	+3.34	+2.28
SIM021	+2.85	+3.56	+2.55	+3.30
SIM025	+2.89	+2.88	+4.08	+1.13
SIM009	+2.93	+2.97	+2.73	+2.07
SIM026	+3.33	+3.10	+3.60	+3.28
SIM020	+3.34	+3.02	+3.74	+2.82
SIM007	+3.45	+3.59	+2.83	+4.04
SIM022	+3.56	+2.83	+3.83	+3.33
SIM029	+4.07	+3.48	+4.27	+4.12
SIM012	+4.17	+5.00	+4.66	+3.40
SIM001	+4.30	+4.27	+4.29	+4.88
SIM023	+4.41	+3.54	+4.23	+4.62
SIM000	+5.36	+5.13	+6.18	+4.26