# “Groupitizing”: a strategy for numerosity estimation

Aug 10, 2020

### Participants

Sixteen young adults (mean age = 26, standard deviation = 3.2, range = 23–36) participated in this study (12 male, 4 female, 13 participants were master’s students in psychology, 2 were grad-students and 1 a post-doc in neuroscience). All participants had normal or corrected-to-normal vision. All completed all tasks except one, who was unavailable to perform the two sequential numerosity conditions.

### Materials and procedure

Stimuli were created with Psychophysics toolbox for Matlab and displayed on a 60 Hz—15″ screen monitor (MacBook Pro) placed at viewing distance of 57 cm. Subjects were tested in a quite, dimly light room. The experimental procedures were approved by the local ethic committee (Comitato Etico Pediatrico Regionale—Azienda Ospedaliero-Universitaria Meyer—Firenze FI). The research was performed in accordance with Declaration of Helsinki and informed consent was obtained from all participants prior to each experiment.

Each trial started with a central fixation point that remained on screen for the entire experiment. After 500 ms a stimulus was displayed, followed by a blank screen. Participants estimated verbally the numerosity of the squares-array or square-sequence (in separate sessions with order pseudorandomized between subjects Fig. 1C, D).

The experimenter hit the spacebar when the participant responded (used to calculate reaction times), then entered the response on the numeric keypad, which initiated the following trial. Response time was measured from the stimulus offset to the beginning of vocalization. Participants were asked to respond as soon as possible, but also to concentrate on accuracy. Each condition was tested in separate blocks, and participants were never explicitly informed about the grouping cues.

Numerosity levels ranged from 4 to 16 (grain of 1, resulting in 13 numerosity levels). In the structured conditions, each numerosity was organized into clusters (between 2 and 4), each containing a variable number items (between 2 and 6), resulting in the following configurations: 2, 2−2, 2, 1−3, 3−2, 2, 2−3, 3, 1−3, 3, 2−2, 2, 2, 2−4, 4−4, 3, 2−4, 4, 1−3, 3, 3−3, 3, 3, 1−4, 4, 2, 1−3, 3, 3, 2−3, 3, 3, 3−4, 4, 4−4, 4, 3, 3−4, 4, 4, 3−5, 5, 3−4, 4, 4, 4−5, 5, 6. As numerosities 4 and 16 were not analyzed (see data analyses), each grouped pattern comprised a minimum of 2 and a maximum of 4 clusters. All clusters except one (13 = 5, 5, 3) contained from 1 to 4 elements. On each trial, a given numerosity and configuration pattern were randomly selected. Each participant completed about 150 trials for each of the six conditions (around 14,000 trials in total).

### Stimuli

#### Spatial arrays

Stimuli were arrays of squares (0.4° × 0.4°) displayed for 500 ms on each trial. Squares could not overlap and were constrained to fall within a 12°X12° virtual square area. In the conditions where spatial structure was manipulated, the individual items were white squares within black borders (so luminance was not a cue to number). In the unstructured conditions, the position of each square was randomly selected from 169 possible positions (within the stimulus area), being the centers of equally spread sectors within the 12°X12° area (each grid 1°X1°). For the spatially grouped condition, stimuli were arranged in 4 possible groups of 12 possible positions (see Fig. 1A). Each group (spanning over a max area of 4°X2°) was located in one quadrant and centered at 5° from the central fixation point. Each group was first randomly assigned to one quadrant (between 1 and 4), then the individual items positions was randomly selected between one of the 12 in the selected quadrant. Within each quadrant, the maximum center-to-center distance between each element was 4° and the minimum was 1°.

In the conditions where groups were defined by color, individual items could be red, green, blue or yellow (RGB: 255 0 0; 0 255 0; 0 0 255; 255 255 0). Color was assigned from left to right, so that similar colors appeared in vertical rows. For example, in the 3, 3, 2 condition depicted in Fig. 1B squares were first randomly located, then the first three squares (from the left border) were colored red, the next three yellow and the remaining two blue (colors randomly chosen for each group). In the unstructured color condition, positions were assigned with the same logic, but with colors assigned at random.

#### Temporal sequences

Stimuli were streams of 3° × 3° squares each presented at screen center for 70 ms, for a total trial duration of 3 s (Fig. 1D). The end of each trial was signaled by color change of the central fixation point, from white to green. Sequences were spaced pseudo-randomly: on every trial, a given number of impulses (chosen at random) were evenly spread within the 3-s sequence duration; then the timing of each impulse was randomly jittered by either ± 0, ± 20 or ± 40 ms to create a pseudorandom sequence of impulses with a minimum ISI between consecutive flashes of 50 ms. In the random condition all stimuli were black, while in the grouped condition they were grouped by color: each flash within a group could be red, green, blue or yellow (color coordinates as before), with group color randomly assigned. For example, in the 3, 3, 2 condition depicted in Fig. 1B, the first three flashes were colored red, the following were yellow and the remaining two blue.

### Data analysis

Since participants were explicitly informed about the numerical range (4–16), we eliminated the two extreme numerosities from the analyses. We controlled for response outliers by eliminating trials with RTs longer than 3 standard deviations from the average response time, calculated separately for each numerosity level and participant.

For each participant, we calculated for each numerosity the average perceived numerosity, the standard deviation of the responses and the median response time. Precision was measured by normalizing the standard deviation by the physical numerosity yielding the Coefficient of variation (CV), a dimensionless index of precision that allows comparison and averaging of performance across numerosities.

$$CV= frac{{}_{i}}{{N}_{i}}$$

(1)

where ({N}_{i}) is the analyzed numerosity and ({}_{i}) the standard deviation of responses to numerosity i. Improvement (I) by grouping was measured by a normalized index yielding the proportion improvement:

$$I= frac{{CV}_{R}-{CV}_{G}}{{CV}_{R}}$$

(2)

where ({CV}_{R}) and ({CV}_{G}) are the coefficient of variation for the random and grouped conditions.

Data were analyzed by repeated measures ANOVAs, and effect sizes were reported as η2, using JASP and Matlab.

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##### 3 thoughts on ““Groupitizing”: a strategy for numerosity estimation”
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