# Developmental Trajectory of Anticipation: Insights from Sequential Comparative Judgments

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods—Experiment 1

#### 2.1. Dataset and Participants

#### 2.2. Tasks

#### 2.3. Trial Sequence

#### 2.4. Analyses

^{2}, with the continuity correction) with response profile, difficulty level and age as independent variables, we did not find changes in the relative proportion of participants showing a fast or slow response profile with the difficulty level when controlling for age (X

^{2}= 0.18, p = 0.67). Consequently, the difficulty level factor was removed from all subsequent analyses.

_{p}

^{2}) for each ANOVA. η

_{p}

^{2}= 0.01 indicates a small effect, η

_{p}

^{2}= 0.06 indicates a medium effect, and η

_{p}

^{2}= 0.14 indicates a large effect [32].

^{2}with response profile, task and age as independent variables. It allowed us to determine whether the proportion of participants changed as a function of response profile, age and task. Chi-square tests were also applied when the CMH χ

^{2}test showed significant results. Additional post hoc comparisons were conducted when needed. Cramer’s V was calculated to estimate the effect sizes for each chi-square test. A commonly used interpretation is to qualify effect sizes as small for V > 0.06, medium for V > 0.17, and large for V > 0.29 (with our degree of freedom being equal to 3) [33].

^{2}test and the associated chi-square and post hoc tests, allowing us to assess whether independent factors (response profile, age and task) influence the probability of belonging to one or the other Gaussian sub-distribution.

_{p}

^{2}) for each ANOVA.

## 3. Results—Experiment 1

^{2}tests. We also check if the two responder’s profiles are associated with different levels of response accuracy (percentage of correct responses). Finally, the mixture model is fitted to the data to consolidate the results and notably the developmental course of the response strategy.

#### 3.1. Group-Level Distributions

#### 3.2. Individual Distributions

^{2}tests to evaluate the effects of age, task and response profile on the proportion of participants. We first investigated if there was an association between response profile and task that was dependent upon age. We found that the relative proportion of participants showing a fast or slow response profile significantly changed with the task when controlling for the age (CMH: M

^{2}= 21.29, p < 0.001). Specifically, follow-up chi-square tests revealed a significant response profile × task interaction in 8- to 9-year-old children (X

^{2}= 18.37, p < 0.001, V = 0.12), in 10- to 11-year-old children (X

^{2}= 8.71, p = 0.03, V = 0.08) and in adults (X

^{2}= 8.2, p = 0.04, V = 0.08), but not in 12- to 15-year-old children (X

^{2}= 3.54, p = 0.31, V = 0.05). According to Cramer’s V, the effect sizes were small for this interaction [33]. Post hoc tests showed a higher proportion of fast responders in multiplication than in the other tasks among 8- to 9- and 10- to 11-year-old children (8–9 years: p < 0.001; 10–11 years: p = 0.05). In adults, post hoc tests pointed to a non-significant trend toward a higher proportion of fast responders in the multiplication and subtraction tasks than in the numerosity and rhyme tasks.

^{2}= 257.18, p < 0.001). Specifically, follow-up chi-square tests revealed a significant response profile × age interaction in all four tasks (numerosity: X

^{2}= 61.9, p < 0.001, V = 0.22; rhyme: X

^{2}= 61.75, p < 0.001, V = 0.23; multiplication: X

^{2}= 48.67, p < 0.001, V = 0.20; subtraction: X

^{2}= 93.37, p < 0.001, V = 0.28). For all tasks, Cramer’s V estimated this interaction as a medium effect higher than 0.20, indicating that age relatively strongly influences the proportion of fast and slow responders. All post hoc tests indicated that the two youngest groups of children (8- to 9-year-olds and 10- to 11-year-olds) had fewer fast responders than adults (all p’s < 0.05 except numerosity for 10–11-year-olds, where p = 0.06), whereas the 12- to 15-year-old children did not differ from adults (all p’s = 1). Indeed, there was a similar proportion of fast and slow responders among 12- to 15-year-old children, the intermediate profile between the profile of the two youngest child groups and the profile of adults.

#### 3.3. No Main Effect of Response Profile on Accuracy

_{p}

^{2}= 0.14), revealing an expected improvement in performance with age. A main effect of task was also present (F (3519) = 134.74, p < 0.001, η

_{p}

^{2}= 0.41), such that accuracy was higher for numerosity (mean = 96.43%, E-C = 5.12) and rhyme (mean = 95.47%, E-C = 6.55) than for multiplication (mean = 84.48%, E-C = 13.28) and subtraction (mean = 86.35%, E-C = 14.59). An age x task interaction (F (9519) = 43.11, p < 0.001, η

_{p}

^{2}= 0.43) further indicated that the improvement with age was particularly potent for multiplication and subtraction (two skills that are arguably acquired later in children than numerosity comparison and rhyme comparison).

_{p}

^{2}= 0.02) and almost significant when considered a between-subject factor (F (1161) = 3.57, p = 0.06, η

_{p}

^{2}= 0.02), being associated with a response profile x age interaction (F (3519) = 4.39, p = 0.005, η

_{p}

^{2}= 0.02). Slow responders were more accurate than the fast responders, but this effect seemed to be reduced with age (Figure 5). Post hoc analyses revealed that this difference mostly occurred in 10–11-year-old children, but reached significance only when FDR correction was used (Bonferroni: p = 0.09, FDR: p = 0.02).

_{p}

^{2}= 0.01). No age × response profile interaction was found for any task (all p’s > 0.05). Overall, these results indicate a speed–accuracy trade-off where being a fast responder was associated with lower accuracy. However, this effect (which was particularly found in numerosity) seemed to mainly occur in children. This suggests that adults were able to adopt an optimized strategy with a performance that was both fast and accurate.

#### 3.4. Mixture Model

^{2}tests with age, task and response profile as factors. As with the previous CMH χ

^{2}test, the results show a significant change in lambda values according to response profile and task when controlling for age (CMH: M

^{2}= 14.41, p = 0.002). Follow-up chi-square tests revealed a significant response profile × task interaction in all four age groups (8–9 years: X

^{2}= 11.08, p = 0.01, V = 0.09; 10–11 years: X

^{2}= 13.3, p = 0.004, V = 0.10; 12–15 years: X

^{2}= 11.78, p = 0.008, V = 0.10; adults: X

^{2}= 13.63, p = 0.003, V = 0.11). Effect sizes were categorized as a small effect for this interaction. For 8- to 9-year-old children, post hoc tests suggested a marginal, not significant, difference between the (higher) probability participants would respond quickly in numerosity compared to in the other tasks (p = 0.08). For both 10- to 11- and 12- to 15-year-old children, post hoc tests indicated a significantly lower probability of being a fast responder for subtraction than for the three other tasks (10–11 years: p = 0.005; 12–15 years: p = 0.02). Finally, for adults, post hoc tests showed a significantly lower probability of being a fast responder for rhyme (~50%) than for the other tasks (p = 0.006).

^{2}tests were used with response profile, age and task as factors, we found a significant change in lambda values according to response profile and age when controlling for the task (CMH: M

^{2}= 503.73, p < 0.001). Follow-up chi-square tests revealed a significant response profile × age interaction in all four tasks (numerosity: X

^{2}= 92.69, p < 0.001, V = 0.28; rhyme: X

^{2}= 80.78, p < 0.001, V = 0.26; multiplication: X

^{2}= 149.67, p < 0.001, V = 0.35; subtraction: X

^{2}= 215.68, p < 0.001, V = 0.42), with medium or large effect sizes according to Cramer’s V. For numerosity and multiplication, post hoc tests indicated a difference between adults and both 8- to 9-year-old and 10- to 11-year-old children (numerosity: p < 0.001 for 8–9 years and p = 0.02 for 10–11 years; multiplication: p < 0.001 for both 8–9 and 10–11 years) but not between adults and 12- to 15-year-old children (numerosity: p = 1; multiplication: p = 0.55). For rhyme, post hoc tests showed a difference between 8- to 9-year-old children and all other age classes (p < 0.001), as well as a difference between adults and other age classes (p < 0.001). Finally, for subtraction, all age classes differed from each other (all p’s < 0.01).

#### 3.5. Diffusion Model

_{p}

^{2}= 0.33; as a between-subject factor: F (1158) = 430.98, p < 0.001, η

_{p}

^{2}= 0.73), associated with a significant triple interaction of age × task × response profile (F (9515) = 3.16, p = 0.001, η

_{p}

^{2}= 0.05). Post hoc analyses indicated that t0 was shorter for fast than slow responders for all tasks and all ages (all p’s < 0.05), except for adults in the subtraction task, where the same tendency did not reach significance (which, however, could be explained by the weak proportion of slow adults in this task). We also found, for the “a” parameter, significant interactions between age and response profile (F (3515) = 2.82, p = 0.04, η

_{p}

^{2}= 0.02) and between task and response profile (F (3515) = 3.58, p = 0.01, η

_{p}

^{2}= 0.02). Follow-up analyses revealed that the difference between fast and slow responders reached significance only in adults (p = 0.05) and only for multiplication (p = 0.04) and subtraction (p < 0.001). The “a” parameter was higher for fast adults (1.28) than slow adults (1.02), whereas it was lower for fast responders than slow responders in multiplication (fast: 1.46, slow: 1.62) and subtraction (fast: 1.35, slow: 1.70). No main effect nor interaction was found for the “v” parameter.

## 4. Discussion—Experiment 1

#### 4.1. Two Response Profiles

#### 4.2. Developmental Differences

## 5. Materials and Methods—Experiment 2

#### 5.1. Pilot Study

#### 5.2. Participants

#### 5.3. Procedure

#### 5.4. Analyses

_{p}

^{2}).

## 6. Results—Experiment 2

#### 6.1. Descriptive Statistics

#### 6.2. Analysis of RT Distributions

^{2}= 10.14, p = 0.02, V = 0.12), with the effect size categorized as small, and post hoc analyses disclosed significant results only for the (less represented) fast–slow subgroup, revealing that the proportion of fast responders during Part 1 becoming slow responders during Part 2 was higher for the “wait and see” rule (4.2%) than the “anticipate” rule (0.4%). Therefore, fast participants became slow responders only if specifically instructed to do so. Interestingly, a chi-square test with rule and response profiles as factors conducted in participants who changed their response profile between the two parts (fast–slow and slow-fast) was significant (X

^{2}= 7.8, p = 0.005, V = 0.10), with an effect size categorized as small, showing that the number of participants for each response profile was different according to the received rule. Thus, initially fast responders who changed their profile to slow were more likely to have received the “wait and see” rule than the “anticipate” rule. Conversely, initially slow responders who changed their profile to fast were more likely to have received the “anticipate” rule than the “wait and see” rule.

^{2}= 0.61, p = 0.43, V = 0.03). During Part 2, the effect of rule on the proportion of fast and slow responders did not reach significance (X

^{2}= 2.95, p = 0.09, V = 0.06). Finally, we asked participants to rate their success in following the rule during the second part between 0 (not at all) and 10 (totally). We found a similar mean score across subgroups, whatever the received rule (“anticipate”: fast–fast = 6.13, fast–slow = 6, slow–fast = 6.92, slow–slow = 5.26; “wait and see”: fast–fast = 6.52, fast–slow = 6.75, slow–fast = 6.14, slow–slow = 6.23). This suggests that the subjective rating scores were low in all subgroups and did not predict the profile participants adopted following the rule.

#### 6.3. Analysis of Percentage of Correct Responses

_{p}

^{2}= 0.07), indicating that participants became less accurate after receiving the “anticipate” rule (Part 1: 86.47%, Part 2: 79.61%) compared to the “wait and see” rule (Part 1: 87.18%, Part 2: 87.25%). We also found a main effect of profile pattern (F (3228) = 2.70, p = 0.05, η

_{p}

^{2}= 0.03). Pairwise comparisons indicated a significant difference between the slow responders who stayed slow during Part 2 and participants who were fast during Part 2 (fast–fast: p = 0.009 and slow–fast: p = 0.02) (Figure 10). No difference was found between fast–slow and slow–slow subgroups. However, we did not find a significant rule × profile pattern interaction. These results suggest that participants were globally less accurate after receiving the “anticipate” rule and that participants who were or became fast during Part 2 were less accurate than consistently slow responders.

## 7. Discussion—Experiment 2

## 8. General Discussion

#### 8.1. Strategy Preference Is Influenced by Age and Environment

#### 8.2. Underlying Cognitive Processes and Neural Substrates

#### 8.3. Methodological Consideration

## 9. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## AppendixA

Adults | Easy | Hard | Medium |

Numerosity judgment | RT = 733.22 +/− 11.62 Acc = 96.70 +/− 0.47 | 816.9 +/− 11.24 95.91 +/− 0.63 | 761.08 +/− 11.64 93.47 +/− 1.00 |

Phonological judgment | 995.66 +/− 14.18 95.95 +/− 0.68 | 1064.36 +/− 16.45 88.22 +/− 1.11 | -- |

Multiplication | 706.83 +/− 10.84 97.22 +/− 0.49 | 780,18 +/− 13.54 92.15 +/− 1.27 | -- |

Subtraction | 681.26 +/− 9.45 97.48 +/− 0.55 | 722.21 +/− 10.89 96.44 +/− 0.74 | -- |

12–15-year-old children | |||

Numerosity judgment | RT = 943.81 +/− 15.80 Acc = 90.04 +/− 1.61 | 1003.82 +− 14.70 86.96 +/− 1.72 | 958.38 +/− 14.89 88.84 +/− 1.61 |

Phonological judgment | 1145.07 +/− 13.2 88.84 +/− 1.08 | 1245.20 +/− 18.41 89.68 +/− 3.48 | -- |

Multiplication | 1017.59 +/− 13.49 91.86 +/− 1.73 | 1257.86 +/− 17.93 75.13 +/− 2.21 | -- |

Subtraction | 1089.29 +/− 14.29 87.37 +/− 1.74 | 1195.34 +/− 16.61 84.76 +/− 2.23 | -- |

10–11-year-old children | |||

Numerosity judgment | RT = 1001.65 +/− 12.48 Acc = 90.97 +/− 1.31 | 1065.34 +/− 12.60 86.42 +/− 1.70 | 1015.03 +/− 12.08 89.82 +/− 1.54 |

Phonological judgment | 1156.12 +/− 11.47 86.41 +/− 1.74 | 1267.50 +/− 16.96 68.40 +/− 2.44 | -- |

Multiplication | 1088.05 +/− 12.53 86.57 +/− 1.88 | 1241.26 +/− 15.20 71.46 +/− 2.25 | -- |

Subtraction | 1225.90 +/− 14.39 83.43 +/− 1.81 | 1339.28 +/− 16.30 76.06 +/− 2.41 | -- |

8–9-year-old children | |||

Numerosity judgment | RT = 1025.89 +/− 17.06 Acc = 89.74 +/− 1.44 | 1133.07 +/− 18.30 86.24 +/− 1.87 | 1086.52 +/− 17.75 90.18 +/− 1.52 |

Phonological judgment | 1312.75 +/− 16.79 83.12 +/− 2.94 | 1363.59 +/− 23.29 66.74 +/− 3.30 | -- |

Multiplication | 1269.16 +/− 19.80 88.65 +/− 2.39 | 1364.01 +/− 25.60 64.32 +/− 3.10 | -- |

Subtraction | 1314.08 +/− 19.68 81.04 +/− 3.13 | 1469.65 +/− 21.10 76.31 +/− 3.49 | -- |

## Appendix B

**Figure A1.**Individual RT distribution plots for Experiment 2 combining both parts of the task (Part 1/before rule, Part 2/after rule). The 3 different colors identify participants who remained fast (blue) or slow (orange) during the entire experiment and those who changed their response profile (green).

## Appendix C

**Figure A2.**Effect of rule on the group-level and individual RT distributions of incorrect responses in Experiment 2. Panel (

**A**): group-level distributions obtained before (purple) and after (green) the rule are superimposed. Panel (

**B**): individual level distributions (fast and slow responders are color-coded) before (

**left**) and after (right) the rule. Panel (

**C**): group-level distributions as in (

**A**) but plotted separately for the 2 rule types: “wait and see” (

**left**) and “anticipate” (

**right**).

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**Figure 1.**The trials sequence in the four tasks: numerosity judgment (

**A**), phonological judgment (

**B**), multiplication (

**C**) and subtraction (

**D**).

**Figure 2.**RT distributions at the group level for each task (panels) and each age subgroup (colored distributions).

**Figure 3.**RT distributions at the individual level for each task (rows) and each age subgroup (columns). Participants were classified as fast (blue distributions) or slow (orange distributions) responders.

**Figure 4.**Proportion of participants in each peak (fast or slow responder) according to the tasks (panels) and age subgroups. Blue = fast responder (peak 1), orange = slow responder (peak 2).

**Figure 5.**Accuracy, expressed in log-transformed percentages of correct responses, of participants for each peak (fast or slow responder) according to the age subgroup. A high value indicates a lower level of performance. Slow responders were more accurate than fast responders, particularly in 10–11-year-old children, suggesting a speed–accuracy trade-off which decreases with age. Blue = fast responder (peak 1), orange = slow responder (peak 2).

**#**: p = 0.09 with Bonferroni correction and p = 0.02 with FDR correction.

**Figure 6.**Mixture model fitted to the group-level RT distributions for each task and age. Blue curve corresponds to fast responders, orange curve corresponds to slow responders, as estimated by the model. Distributions on the back correspond to the actual group distributions as plotted in Figure 2.

**Figure 7.**Mixture model lambda values (i.e., probability in each age group and task that a given RT taken randomly from the actual distribution belongs to the fast or the slow responder fitted Gaussian distribution) plotted separately for age groups and tasks. Blue = fast responder (Lambda 1), orange = slow responder (Lambda 2).

**Figure 8.**Superimposed RT distributions between the first (before rule, purple distribution) and the second blocks of trials (after rule, green distribution) for all participants, whatever the received rule (

**A**). Individual RT distributions before (

**left**) and after (

**right**) the rule. (

**B**) Participants were classified as fast (blue distributions) or slow (orange distributions) responders. Note the 16% increase (55 to 71%) in the proportion on fast responders in the second part of Experiment 2.

**Figure 9.**Superimposed RT distributions before (purple distribution) and after (green distribution) subjects were asked to either “anticipate” (

**left**) or “wait and see” (

**right**) for all subjects together (

**A**) or separately for each response profile during the first part of the experiment (fast responders on the left, slow responders on the right) (

**B**).

**Figure 10.**Accuracy after receiving the rule (Part 2), expressed in log-transformed percentages of correct responses, according to the “anticipate” or “wait and see” rule and the profile pattern between Part 1 and Part 2 (fast–fast, fast–slow, slow–fast, slow–slow). Participants were less accurate after receiving the “anticipate ” rule, and participants who were or became fast during Part 2 were less accurate than consistently slow–slow responders (*: p < 0.05, **: p < 0.01).

**Table 1.**Number (N) and proportion (%) of participants according to their response profile (slow or fast responder) in Part 1 and Part 2 and the rule they received (“wait and see” or “anticipate”). “Fast-Fast” and “Slow-Slow” are participants who did not change their response profile during the entire experiment, whereas “Fast-Slow” and “Slow-Fast” are participants who did change their response profile.

Profile in Part 1—Profile in Part 2 | ||||||
---|---|---|---|---|---|---|

Fast—Slow | Slow—Fast | Fast—Fast | Slow—Slow | Total | ||

Rule | “Wait and see” | N = 10 4.2% | 19 8.1% | 58 24.6% | 31 13.1% | 118 |

“Anticipate” | 1 0.4% | 30 12.7% | 60 25.4% | 27 11.5% | 118 | |

Total | 11 | 49 | 118 | 58 | 236 |

**Table 2.**Number (N) of participants for Part 1 (top) and Part 2 (bottom) separately according to their response profile (slow or fast responder) and the received rule (“wait and see” or “anticipate”).

Part 1 Only (before Rule) | Response Profile | |||
---|---|---|---|---|

Fast Responders | Slow Responders | Total | ||

Rule | “Wait and see” | N = 68 | 50 | 118 |

“Anticipate” | 61 | 57 | 118 | |

Total | 129 | 107 | 236 | |

Part 2 only(after rule) | Response profile | |||

Fast responders | Slow responders | Total | ||

Rule | “Wait and see” | N = 77 | 41 | 118 |

“Anticipate” | 90 | 28 | 118 | |

Total | 167 | 69 | 236 |

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## Share and Cite

**MDPI and ACS Style**

Tricoche, L.; Meunier, M.; Hassen, S.; Prado, J.; Pélisson, D.
Developmental Trajectory of Anticipation: Insights from Sequential Comparative Judgments. *Behav. Sci.* **2023**, *13*, 646.
https://doi.org/10.3390/bs13080646

**AMA Style**

Tricoche L, Meunier M, Hassen S, Prado J, Pélisson D.
Developmental Trajectory of Anticipation: Insights from Sequential Comparative Judgments. *Behavioral Sciences*. 2023; 13(8):646.
https://doi.org/10.3390/bs13080646

**Chicago/Turabian Style**

Tricoche, Leslie, Martine Meunier, Sirine Hassen, Jérôme Prado, and Denis Pélisson.
2023. "Developmental Trajectory of Anticipation: Insights from Sequential Comparative Judgments" *Behavioral Sciences* 13, no. 8: 646.
https://doi.org/10.3390/bs13080646