Measuring temporal bias in sequential numerosity comparison

While several methods have been proposed to assess the influence of continuous visual cues in parallel numerosity estimation, the impact of temporal magnitudes on sequential numerosity judgments has been largely ignored. To overcome this issue, we extend a recently proposed framework that makes it possible to separate the contribution of numerical and non-numerical information in numerosity comparison by introducing a novel stimulus space designed for sequential tasks. Our method systematically varies the temporal magnitudes embedded into event sequences through the orthogonal manipulation of numerosity and two latent factors, which we designate as “duration” and “temporal spacing”. This allows us to measure the contribution of finer-grained temporal features on numerosity judgments in several sensory modalities. We validate the proposed method on two different experiments in both visual and auditory modalities: results show that adult participants discriminated sequences primarily by relying on numerosity, with similar acuity in the visual and auditory modality. However, participants were similarly influenced by non-numerical cues, such as the total duration of the stimuli, suggesting that temporal cues can significantly bias numerical processing. Our findings highlight the need to carefully consider the continuous properties of numerical stimuli in a sequential mode of presentation as well, with particular relevance in multimodal and cross-modal investigations. We provide the complete code for creating sequential stimuli and analyzing participants’ responses. Supplementaryinformation The online version contains supplementary material available at 10.3758/s13428-024-02436-x.

the number of events (n) in the sequence and its Total event duration (TED) and Total stimulus duration (TSD), specified in frames.Sequences are created as a structure with two fields correspon ding to a vector of individual event durations and a vector of inter-event intervals.

Figure S1
. Secondary function to generate one sequence.The function seq_stim_creator takes as input parameters the number of events (n), the total event duration (ted), the total stimulus duration (tsd), and the intended variability in individual event duration (event_method) and individual interval duration (interval_method) to return a structure array that defines the sequence in terms of durations in frames.
Both regular and irregular sequences can be created modifying the corresponding parameters referring to events or intervals: 'Fixed' for homogeneous individual durations and 'Sum' for heterogeneous individual durations, obtained through an iterative process.The output sequence is defined by two vectors: • ied_vec, containing n individual event durations in frames.
Through the main script, users can save the stimuli in a spreadsheet containing, for each stimulus, the timestamps of events and intervals and the stimulus features (see Figure S2).Moreover, the script allows the visualization of the intended and real features of the stimulus set.

Figure S2. Example of output spreadsheet.
Users can save a table containing complete stimulus information such as the timestamps of events and intervals (in frames), number of events and continuous features (in frames).
The manipulation of duration in frames allows the generation of sequential numerical stimuli in different sensory modalities from the same timestamps.However, this can make the sequences dependent on the system used, especially for visual stimuli.The example provided and the default values used in all the scripts assume the correspondence: 1 frame = 0.01667 s (for a screen refresh rate of 60 Hz).Values of Duration and Temporal Spacing in the main script, as well as minimum IED or Interval values used in the secondary function should therefore be changed according to the system, screen refresh rate and the intended use of the output sequences.
An example on how to generate auditory stimuli (as .wavfiles) from the output timestamps (assuming that 1 frame = 0.01667 s) is provided in an additional script.In the script, users can easily modify event tone, sound sampling frequency, and amplitude, as well as visually inspect the generated audio signal.
Alternatively, the stimulus set spreadsheet can be imported in the preferred software or tool for running experiments, to create stimuli on the fly from timestamps.

Power analysis
The sample size and the number of trials were selected based on a power analysis conducted using Monte Carlo simulations.To estimate the sample size required to detect a non -numerical bias within a group and to detect a difference in non-numerical bias between two groups, we extracted 1000 samples, for several sample sizes and trial numbers, from a population with mean numerical acuity and variability based on a pilot study with 5 participants performing a sequential numerosity comparison task in visual modality on 120 trials, for which we estimated a mean w of 0.35 (SD = 0.05).For each sample, we simulated individual trialby-trial responses in the current comparison task from a psychophysical model of numerosity discrimination for half participants (G1) and from a psychophysical model of total stimulus duration for th e other half (G2).
We then estimated the individual parameters of the described GLM with binomial distribution and probit link function with the log of Numerosity, Duration, and Temporal Spacing as regressors.
We can consider the power of detecting a non-numerical bias from a given sample size and a certain number of randomly selected stimuli, as the proportion of samples where we could individuate a significant Temporal Spacing coefficient in G2.Based on this procedure we estimated a minimum sample size of 20 participants and 90 trials to achieve a power above 0.90 to detect a significant non-numerical bias.The power of detecting a difference in strategy between the two groups was instead defined as the proportion of samples where we could individuate a significant difference between groups in the numerosity coefficient and the Temporal Spacing coefficient.We estimated a minimum sample size of 20 participants per group and 90 trials to achieve a power above 0.90 to detect a group difference in non-numerical bias.
To estimate the sample size required to detect a difference in numerical acuity between two groups, we extracted 1000 samples, for several sample sizes and trial numbers, from two populations characterized by different numerical acuity of w = 0.35 (SD = 0.05) and w = 0.40 (SD = 0.05), with a standardized difference between groups below the effect size reported in previous studies that found a significant difference in numerical acuity in response to visual or auditory sequences (Tokita et al., 2013).Following the same procedure as the previous simulation, the power of detecting a difference in numerical acuity between the two groups was defined as the proportion of samples, for a given sample size and a certain number of randomly selected stimuli, where we could individuate a significant difference between groups in the numerosity coefficient.We estimated a minimum sample size of 40 participants per group and 120 trials to achieve a power above 0.7 to detect a group difference in acuity with Cohen's d = 1.