How far back history reaches

The pull of trial t−1 is the headline. The question here is whether t−2, t−3 and beyond also pull, and whether the pull decays smoothly.

Two estimators answer it. The first refits the primary model separately with the lag-k stimulus difference in place of lag 1, for k = 1…5, which makes no assumption about shape. The second (M4) fits lags 1–3 jointly with amplitudes constrained to decay geometrically, a·ρ^k−1. The synthetic cohort was generated with ρ = 0.4; M4 recovers ρ = 0.40 (95% CI 0.31 to 0.50).

Lags beyond 1 are secondary outcomes. Lags 1–3 are confirmatory in the preregistration; lags 4 and 5 are exploratory, and at five tests the reader should apply their own multiplicity correction to the per-lag p-values in the table.

Fig. 1 Tuning amplitude by lag

SIMULATED DATA

independent fit
M4 decay model (hollow)

Filled points: independent per-lag fits. Hollow points: amplitudes implied by the M4 geometric-decay fit. Both estimate stimulus history, so both use the stimulus-history colour; shape distinguishes the estimator. Intervals are observer-level bootstrap 95% CIs.Source: results/lag_effects.parquet · run lag_effects-280346c-2

Table view (8 rows)

estimator	lag	amplitude (°)	95% CI	p
independent_lagwise	1	+2.488	+2.02 to +2.95	< 0.001
independent_lagwise	2	+1.040	+0.75 to +1.32	< 0.001
independent_lagwise	3	+0.408	+0.10 to +0.73	0.006
independent_lagwise	4	−0.088	−0.35 to +0.22	0.534
independent_lagwise	5	−0.400	−0.62 to −0.13	< 0.001
M4_geometric_decay	1	+2.623	+2.21 to +3.06	< 0.001
M4_geometric_decay	2	+0.963	+0.73 to +1.22	< 0.001
M4_geometric_decay	3	+0.477	+0.34 to +0.64	< 0.001