EXPERIMENT 1: CONTROL ANALYSES

In order to demonstrate that the observed serial effects are truly dependent on trial history and not due to statistical artifacts in the experimental design, we performed

three control analyses on Experiment 1 data: future-related serial dependence, serial dependence with shuffled responses and serial dependence within random subsets of the data.

3.2.1. Serial dependence in relation to future (n+1) trials

Regarding the first approach, we assessed whether there was any association between the StD presented in trial n+1 and the current response zREn. Naturally, future

presentations should not be able to affect current perceptual judgments, so any supposed 'effect' found for StDn+1 should signal the presence of statistical artifacts in our

data. We analysed StDn+1-related 'serial dependence'by two methods: (i) by running a

Bayesian repeated-measures (RM) ANOVA on the influence of StDn+1 level (as within-

subject factor) on zREn (equivalent to the analyses conducted in the main text for StDn- 1) and (ii) by running a Bayesian linear mixed-effects model (LMM) for zREn (dependent

variable), with StDn+1 as independent variable (random-effects grouped by participant's

ID). Additionally, we repeated both analyses in relation to StDn+2.

Figure 4a and 4b present the average zREn as a function of StDn+1 and StDn+2,

respectively, for all trials pooled, as well as for foveal and peripheral trials separately. The flat plots strongly suggest that there is no effect of StDn+1/ StDn+2 on zREn, as

expected. In a Bayesian RM ANOVA for zREn with StDn+1 as within-subject factor, the best

model according to the analysis was the null (containing ID only: it is the reference model with BF10 null=1.000). The Bayes factor for the model containing StDn+1 (and ID)

was BF10 StDn+1=0.042; therefore, there was strong evidence against any explanatory

effect of StDn+1 compared to the null model (BFnull/StDn+1=1/0.042=23.810). Likewise, for

the RM ANOVA with StDn+2 as within-subject factor, evidence in favour of the null was

very strong: BFnull/StDn+2=1/0.031=32.258. In other words, neither StDn+1 nor StDn+2 had

any effect on zREn. Figure 4c plots zREn as a function of StDn-8, StDn-1 and StDn+1, showing

StDn-8 and StDn-1 indicate a negative (repulsive) and a positive (attractive) effect,

respectively, whereas the flat line for StDn+1 indicates no effect.

Figure 4d presents the B coefficients and 95% credible intervals for the effect of StDn+1

and StDn+2 on zREn, according to two Bayesian LMMs. They have virtually zero values:

B=2*10-5 _{(-0.0009, 0.0010) for StDn+1 and B=-0.0001 (-0.0011, 0.0009) for StDn+2, again}

confirming the absence of 'serial dependence' by trials n+1 and n+2.

3.2.2. Serial dependence in relation to shuffled datasets

As another 'sanity check' for confirming that the observed effects are truly dependent on trial history, we generated shuffled datasets from each participant's data, by permuting the responses provided by each participant within each StD level. In other words, each simulated dataset had exactly the same trial sequence as the real one, and the responses given for each StD level were identical as the participant's responses, but they were randomly assigned to any trial for the same participant and StD level. This means that everything was kept the same as in the experimental data, except for the relation between each response zREn and its trial history. Therefore, if we failed to find

serial dependence in these simulated datasets, we would need to conclude that the observations in our data cannot arise from anything else than the specific trial history leading to each zREn.

We generated 10 shuffled datasets (each encompassing the shuffled data from all 30 participants in Experiment 1) and ran 10 Bayesian LMMs for each of them, assessing the influence of StDn-t (t=1...10) on the shuffled zREn. The average and standard errors of the

obtained coefficients are presented in figure 4d together with the coefficients found for the real data. It can be observed that there is no serial dependence for the shuffled datasets.

3.2.3. Serial dependence in random data subsets

Finally, we examined serial dependencies in the real, past trial history, but within a random subsample of trials instead of the entire dataset. By this control analysis we aimed to rule out any artifact due to the trial sequence being a closed set with a fixed number of pseudorandomized, equally frequent StD presentations. Nevertheless, such an artifact would have appeared equally for future and past trials, an issue that has already been discarded; this third control serves as a mere confirmation. Note that, while for future trials and shuffled datasets we would expect true serial dependencies to disappear, in the current control the ‘sanity check’ requires the positive and negative effects to persist in the subset of trials.

We obtained twenty subsets of the Experiment 1 dataset by randomly selecting half of the trials of each experimental block (30/60). Note that this subsampling pertains to the ‘current trials’, for which the effect of their true trial history was analysed; in other words, the trial history of the selected items was not altered by the subsampling. Figure 4e summarizes the result of analysing serial dependences, up to trial n-10, in these twenty subsets. Each data point of the plot presents the average of the LMM B coefficient estimates of 20 Bayesian linear-mixed effects models for the influence of StDn-t (t=1 … 10) on the current normalized response error, zREn, with each of the 20

models ran on a different subset. The error bars represent the standard error. The ten data points correspond to the ten points of trial history (n-t, with t=1 … 10) at which serial dependencies are analysed for all 20 subsets. Figure 4e show that both the positive and negative after-effects persist in similar magnitude and timescale than for the entire dataset.

Figure 4. Experiment 1: Control analyses. 4a-4b. Normalized relative error in current response (zREn) as a function of the StD presented in the following trial (StDn+1) (4a) and in trial n+2 (4b), plotted by eccentricity: all trials pooled, foveal and peripheral trials separately. In all cases, the approximately flat plots indicate that future trials StD have no effect on current response (as expected). The error bars represent the between-subject standard error. 4c.

Normalized relative error in current response (zREn) as a function of the StD presented in trial n-8, n-1 and n+1. The slopes of past trial plots indicate a negative and positive bias in relation to n-8 and n-1, respectively, whereas the flat slope for n+1 once more indicates lack of effect. The error bars represent the between-subject standard error. 4d.

Serial dependence in real and shuffled data. The purple plot presents the fixed-effects coefficient estimates and 95% credible intervals in 12 Bayesian linear mixed-effects models (LMM) with zREn as dependent variable and the StD

presented in trials n+1, n+2, and n-1 ... n-10 as independent variable, respectively for each model. The red plot represents shuffled data: simulated datasets where everything has been kept the same as in the real experiment except for the specific association between each zREn and its corresponding trial history, which has been shuffled.

We generated 20 shuffled datasets and ran the 12 LMMs on each of them. Thus, each datapoint represents the average of the fixed-effects coefficients obtained for the 20 datasets at each trial position, and the error bars indicate the standard error for the results obtained in the 20 datasets. The plots show that, unlike in the real data, removing the true association between each response and its trial history eliminates the observed serial dependence in relation to past trials, whereas there is no serial dependence in relation to future trials, neither in the real nor in the shuffled data. 4e. Serial dependence in twenty random subsamples of the real data. The plot summarizes the result of 20 iterations consisting on randomly subsampling half of the trials of each participant’s data and running 10 Bayesian LMMs for the influence of StDn-t (t=1 … 10) on zREn, applied to the selected subset. Each datapoint represents the

average and standard error of 20 LMM B coefficient estimates for the effect of StDn-t on zREn, observed at different

points in trial history (n-t, t=1 … 10), separately for the 20 random subsets. The pattern of serial dependencies is almost identical to the one observed for the entire dataset, further confirming that serial dependencies do not arise only as a result of statistics being performed on a closed set of StD presentations.