# Faster ≠ Smarter: Children with Higher Levels of Ability Take Longer to Give Incorrect Answers, Especially When the Task Matches Their Ability

^{1}

^{2}

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^{†}

## Abstract

**:**

## 1. Introduction

We hear school men very authoritatively saying that the fast students make the best grades and the slow ones the poorest. Statements of this kind are usually based on the assumption that if a student knows the subject in which he is being tested it should follow that he requires but a short time to make his answer. Needless to say, this assumption merits confirmation(Longstaff and Porter 1928, p. 638; as cited in Gernsbacher et al. 2020).

#### 1.1. The Uncertain Role of Intelligence

#### 1.2. Refuting the Stereotype

#### 1.3. The F > C Phenomenon

_{ij}= μ + γ FC

_{ij}+ ε

_{ij},

_{ij}is the response time of person j on item i, μ is an intercept (average time across each item and person), FC

_{ij}is a binary variable that indicates whether the answer to item i of person j was false or correct, γ is the unstandardised regression coefficient that could be interpreted as a mean difference in response time between the false and correct answers, and ε

_{ij}is normally distributed residuals of the model.

#### 1.4. The Distance–Difficulty Hypothesis

_{j}) and the item’s difficulty (b

_{i}). In other words, people should take more time to solve tasks closer to their ability level. Conversely, people should spend less time on tasks that are substantially easy or difficult for them. Since the formal representation of the hypothesis includes an absolute value, the model implies that the predictive time differences should be the same and symmetrical. Thissen’s model is formally represented by Equation (2), which assumes a person who answers a set of items in a test within a certain time:

_{ij}= μ + τ

_{j}+ β

_{i}− γ|θ

_{j}− b

_{i}| + ε

_{ij},

_{ij}is a logarithmic transformation of the response time of person j spent on item i (the transformation is used to achieve normally distributed errors, ε

_{ij}, since the response times are assumed to be log-normally distributed); μ is the intercept, which could be interpreted as the mean time spent on all items among the whole sample; τ

_{j}is a parameter for the general speediness of person j (how much the person spent on the items on average); β

_{i}is the time required to answer item i by the person of average ability; and γ is the magnitude of the linear relationship between the ability (θ

_{j}) and difficulty (b

_{i}) absolute distance and the response time (expected to be negative by definition).

#### 1.5. The Proposed Model

## 2. Materials and Methods

#### 2.1. Participants

#### 2.2. Measures

#### Triton and the Hungry Ocean

#### 2.3. Data Management

#### 2.4. Analysis plan

#### 2.4.1. Preliminary IRT Models

_{j}) and each item’s difficulty parameter (b

_{i}). We initially estimated the dichotomous Rasch model (Bond and Fox 2013) in R version 4.2.2 (R Core Team 2021) using package mirt (version 1.37.1; Chalmers 2012). This model estimates the probability of solving an item as a function of the participant’s ability. The dichotomous Rasch model is defined by Equation (3) as:

_{ij}is the probability of the correct answer of person j to item i, θ

_{j}is the ability of person j, and b

_{i}is the difficulty parameter of item i.

_{ij}is the probability of the correct answer of person j to item i, θ

_{j}is the ability of person j, b

_{i}is the difficulty parameter of item i, and c

_{i}is the guessing parameter of item i.

_{i}) and the ability parameter (θ

_{j}) of every participant under this model and used these in the main analyses.

#### 2.4.2. Main Analyses

_{ij}). In the next step (Model A2), we added the predictors of item difficulty (b

_{i}) and person’s ability (θ

_{j}) as control variables. Finally (Model A3), we added the interaction term of the answer correctness and the person’s ability to investigate whether the effect of the F > C phenomenon increases or decreases with higher/lower levels of participant’s ability. Model A3 is represented by Equation (5):

_{ij}= μ + τ

_{j}+ β

_{i}+ γ

_{1}FC

_{ij}+ γ

_{2}b

_{i}+ γ

_{3}θ

_{j}+ γ

_{13}FC

_{ij}θ

_{j}+ ε

_{ij},

_{ij}is a logarithmic transformation of the response time of person j on item I; μ is the fixed intercept (response time of average-ability person spent on average-difficulty item); τ

_{j}is the random intercept for each person (general speediness of each person); β

_{i}is the random intercept for each item (average time required to answer each item); γ

_{1}, γ

_{2}, γ

_{3}, and γ

_{13}are the fixed effects of corresponding predictors; and ε

_{ij}is normally distributed residuals. The previous models, A1 and A2, could be derived from this equation by setting select regression coefficients to zero (see Appendix A).

_{j}− b

_{i}|) in the first model (Model B1). We extended the second model (Model B2) by a binary variable representing the F > C phenomenon (FC

_{ij}) to assess the incremental validity of both concepts against each other. The model series ended with the last model (Model B3), where we added the interaction term of the distance and answer correctness (FC

_{ij}). By including this term, we examined whether the distance difficulty effect followed a different pattern with correct and incorrect responses. Model B3 is represented by Equation (6) as follows:

_{ij}= μ + τ

_{j}+ β

_{i}+ γ

_{4}|θ

_{j}− b

_{i}| + γ

_{1}FC

_{ij}+ γ

_{14}FC

_{ij}|θ

_{j}− b

_{i}| + ε

_{ij},

_{ij}is a logarithmic transformation of the response time of person j on the item i; μ is the fixed intercept; τ

_{j}is the random intercept for each person; β

_{i}is the random intercept for each item; γ

_{1}, γ

_{4}, and γ

_{14}are the fixed effects of corresponding predictors; and ε

_{ij}is a normally distributed residual2. The previous models, B1 and B2, could be derived from this equation by setting select regression coefficients to zero (see Appendix A).

## 3. Results

#### 3.1. Ability Estimates

_{j}) and item difficulty (b

_{i}) parameters for the main analyses. The Rasch model did not fit the item data well (M

_{2}(405) = 1255.32, p < .001, RMSEA = 0.065, SRMSR = 0.069, TLI = 0.867, AIC = 14,200.33, and BIC = 14,327.60). The empirical reliability of the sum score was rather high (r = 0.847).

_{2}(405) = 888.39, p < .001, RMSEA = 0.049, SRMSR = 0.075, TLI = 0.924, AIC = 14,000.78, and BIC = 14,128.04). Moreover, the empirical reliability of this model, r = 0.871, was slightly higher than that of the previous Rasch model. Item descriptive statistics with fixed guessing parameters and estimated difficulty parameters are listed in Table A1 (in Appendix B). The parameters from this model were used in multilevel regression models.

#### 3.2. Null Model

_{i}) = 0.20, 95% CI [0.12, 0.33]) explained more variance of the response time than the individual differences of children in that characteristic (var(τ

_{j}) = 0.07, 95% CI [0.06, 0.08]).

#### 3.3. Models Assessing the F > C Phenomenon

_{1}= −0.04, 95% CI [−0.06, −0.01]). However, the effect size was relatively small, and the transformed parameter indicated that the expected average difference in response times between wrongly and correctly answered items was 0.87 s.

_{1}= −0.05, 95% CI [−0.08, −0.03]). In addition, we found that response time was significantly higher in children with higher ability (γ

_{2}= 0.06, 95% CI [0.05, 0.07]). On the other hand, response time did not have a significant relationship with item difficulty (γ

_{3}= 0.05, 95% CI [0.00, 0.12]).

_{13}= −0.12, 95% CI [−0.13, −0.11]). Adding the interaction also slightly suppressed the F > C phenomenon effect (γ

_{1}= −0.07, 95% CI [−0.09, −0.04]), as well as the relationship of children’s ability with response time (γ

_{3}= 0.12, 95% CI [0.11, 0.14]). All effects combined, the relationship between children’s ability and response time was negligible when the item was answered correctly. However, response time increased with higher children’s ability in case of false answers. Figure 2 illustrates these patterns.

**Table 1.**Parameters of the models assessing the F > C phenomenon (interacting with a person’s ability).

Model 0 | Model A1 | Model A2 | Model A3 | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

95% CI | 95% CI | 95% CI | 95% CI | ||||||||||||||

coef. | est. | LL | UL | est. | LL | UL | est. | LL | UL | est. | LL | UL | |||||

Fixed effects | |||||||||||||||||

intercept | μ | 3.16 | *** | 3.00 | 3.33 | 3.18 | *** | 3.02 | 3.35 | 3.15 | *** | 2.99 | 3.31 | 3.19 | *** | 3.03 | 3.36 |

correct answer (FC) | γ_{1} | −0.04 | ** | −0.06 | −0.01 | −0.05 | *** | −0.08 | −0.03 | −0.07 | *** | −0.09 | −0.04 | ||||

item difficulty | γ_{2} | 0.05 | 0.00 | 0.12 | 0.05 | 0.00 | 0.11 | ||||||||||

person ability | γ_{3} | 0.06 | *** | 0.05 | 0.07 | 0.12 | *** | 0.11 | 0.14 | ||||||||

FC × ability | γ_{13} | −0.12 | *** | −0.13 | −0.11 | ||||||||||||

Random effects | |||||||||||||||||

person intercept variance | var(τ_{j}) | 0.07 | *** | 0.06 | 0.08 | 0.07 | *** | 0.06 | 0.08 | 0.06 | *** | 0.05 | 0.07 | 0.06 | *** | 0.05 | 0.07 |

item intercept variance | var(β_{i}) | 0.20 | *** | 0.12 | 0.33 | 0.19 | *** | 0.11 | 0.32 | 0.17 | *** | 0.09 | 0.28 | 0.17 | *** | 0.10 | 0.29 |

residual variance | var(ε_{ij}) | 0.37 | *** | 0.36 | 0.38 | 0.37 | *** | 0.36 | 0.38 | 0.37 | *** | 0.36 | 0.38 | 0.36 | *** | 0.35 | 0.37 |

Goodness of fit | |||||||||||||||||

conditional R^{2} | 0.417 | 0.415 | 0.422 | 0.439 | |||||||||||||

marginal R^{2} | 0.000 | 0.001 | 0.055 | 0.070 | |||||||||||||

log-likelihood | −14,193 | −14,189 | −14,154 | −14,008 | |||||||||||||

AIC | 28,395 | 28,389 | 28,323 | 28,031 | |||||||||||||

BIC | 28,425 | 28,427 | 28,376 | 28,092 | |||||||||||||

Δχ^{2} (df) | 8.09 (1) | ** | 70.08 (2) | *** | 293.23 (1) | *** |

_{23}= 0.04, 95% CI [0.03, 0.04]). This means the relationship between the ability and response time was stronger for more difficult items. Adding this information also significantly improved the model fit in comparison with the previous Model A3.

_{1}= −0.02, 95% CI [−0.04, 0.01]). The strength of the interaction between answer correctness and children’s ability also noticeably decreased (γ

_{13}= −0.03, 95% CI [−0.04, −0.01]). Figure 3 aids the interpretation of the additional term. It further expands the interpretation of Model A3, indicating that the effect of the ability on time required to answer incorrectly answered items applies only to moderately and highly difficult items.

**Table 2.**Parameters of the exploratory Model A4 (including the interaction of item difficulty and person ability).

95% CI | |||||
---|---|---|---|---|---|

coef. | est. | LL | UL | ||

Fixed effects | |||||

intercept | μ | 3.14 | *** | 2.98 | 3.30 |

correct answer (FC) | γ_{1} | −0.02 | −0.04 | 0.01 | |

item difficulty | γ_{2} | 0.06 | 0.00 | 0.12 | |

person ability | γ_{3} | 0.05 | *** | 0.03 | 0.06 |

FC × ability | γ_{13} | −0.03 | *** | −0.04 | −0.01 |

difficulty × ability | γ_{23} | 0.04 | *** | 0.03 | 0.04 |

Random effects | |||||

person intercept variance | var(τ_{j}) | 0.06 | *** | 0.05 | 0.07 |

item intercept variance | var(β_{i}) | 0.17 | *** | 0.10 | 0.28 |

residual variance | var(ε_{ij}) | 0.34 | *** | 0.33 | 0.35 |

Goodness of fit | |||||

conditional R^{2} | 0.466 | ||||

marginal R^{2} | 0.097 | ||||

log-likelihood | −13,606 | ||||

AIC | 27,231 | ||||

BIC | 27,300 | ||||

Δχ^{2} (df) | 802.28 (1) | *** |

#### 3.4. Models Assessing the Distance–Difficulty Hypothesis

_{13}= −0.13, 95% CI [−0.14, −0.12]). This means that the response time decreased 1.14 times with each logit unit of the absolute ability–difficulty distance, which is a moderately strong effect. The fixed intercept showed the average response time of 33.14 s for zero distance (the item difficulty equivalent to the person’s ability), where the time was at its maximum. The estimated response time decreases to 29.03 when the ability–difficulty distance is one logit unit, to 25.45 when the distance is two logit units, and so on.

_{1}= −0.02, 95% CI [−0.04, 0.01]); the ability–difficulty distance effect remained unchanged (γ

_{4}= −0.13, 95% CI [−0.14, −0.12]).

_{14}= 0.05, 95% CI [0.04, 0.07]). Including the interaction also suppressed the F > C phenomenon effect, which became significant (γ

_{1}= −0.10, 95% CI [−0.13, −0.07]); small suppression was also visible in the ability–difficulty distance effect (γ

_{4}= −0.16, 95% CI [−0.17, −0.15]). Interpretation-wise, the interaction effect means that, for false answers, the negative relationship between ability–difficulty distance and response time is stronger than for correct answers. A graphical description of these effects is in Figure 4.

^{2}(1) = 2.14, p = 0.144).

## 4. Discussion

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

_{ij}= μ + τ

_{j}+ β

_{i}+ ε

_{ij},

_{ij}= μ + τ

_{j}+ β

_{i}+ γ

_{1}FC

_{ij}+ ε

_{ij},

_{ij}= μ + τ

_{j}+ β

_{i}+ γ

_{1}FC

_{ij}+ γ

_{2}b

_{i}+ γ

_{3}θ

_{j}+ ε

_{ij},

_{ij}= μ + τ

_{j}+ β

_{i}+ γ

_{1}FC

_{ij}+ γ

_{2}b

_{i}+ γ

_{3}θ

_{j}+ γ

_{13}FC

_{ij}θ

_{j}+ ε

_{ij},

_{ij}= μ + τ

_{j}+ β

_{i}+ γ

_{1}FC

_{ij}+ γ

_{2}b

_{i}+ γ

_{3}θ

_{j}+ γ

_{13}FC

_{ij}θ

_{j}+ γ

_{23}b

_{i}θ

_{j}+ ε

_{ij},

_{ij}= μ + τ

_{j}+ β

_{i}+ γ

_{4}|θ

_{j}− b

_{i}| + ε

_{ij},

_{ij}= μ + τ

_{j}+ β

_{i}+ γ

_{4}|θ

_{j}− b

_{i}| + γ

_{1}FC

_{ij}+ ε

_{ij},

_{ij}= μ + τ

_{j}+ β

_{i}+ γ

_{4}|θ

_{j}− b

_{i}| + γ

_{1}FC

_{ij}+ γ

_{14}FC

_{ij}|θ

_{j}− b

_{i}| + ε

_{ij},

## Appendix B

Response Correctness | Response Time (in Seconds) | |||||||
---|---|---|---|---|---|---|---|---|

Item | Sample | Guessing | Difficulty | M | SD | M | SD | Mdn |

item 1 | 514 | 0.200 | −1.28 | 0.75 | 0.43 | 21.19 | 24.57 | 15.00 |

item 2 | 514 | 0.200 | −1.91 | 0.81 | 0.39 | 19.52 | 17.80 | 14.50 |

item 3 | 514 | 0.067 | −3.35 | 0.91 | 0.28 | 31.75 | 24.68 | 24.00 |

item 4 | 514 | 0.067 | −3.12 | 0.90 | 0.30 | 20.80 | 21.49 | 16.50 |

item 5 | 514 | 0.050 | −1.26 | 0.71 | 0.46 | 47.71 | 38.18 | 35.50 |

item 6 | 514 | 0.067 | −2.32 | 0.83 | 0.37 | 20.07 | 15.59 | 15.50 |

item 7 | 514 | 0.200 | −1.97 | 0.84 | 0.37 | 15.96 | 25.88 | 10.00 |

item 8 | 514 | 0.100 | −2.54 | 0.86 | 0.35 | 24.81 | 21.55 | 18.00 |

item 9 | 514 | 0.200 | 0.17 | 0.59 | 0.49 | 13.52 | 10.57 | 10.00 |

item 10 | 514 | 0.050 | −1.62 | 0.76 | 0.43 | 34.38 | 19.46 | 28.50 |

item 11 | 514 | 0.200 | −0.54 | 0.68 | 0.47 | 22.89 | 18.04 | 17.50 |

item 12 | 514 | 0.200 | −1.55 | 0.79 | 0.41 | 10.87 | 8.86 | 8.50 |

item 13 | 514 | 0.200 | 2.91 | 0.25 | 0.44 | 15.92 | 16.14 | 11.00 |

item 14 | 514 | 0.050 | −0.17 | 0.56 | 0.50 | 51.22 | 30.05 | 44.20 |

item 15 | 514 | 0.200 | 0.99 | 0.49 | 0.50 | 27.23 | 19.43 | 21.00 |

item 16 | 514 | 0.017 | 0.31 | 0.46 | 0.50 | 50.40 | 33.98 | 42.50 |

item 17 | 514 | 0.200 | 1.40 | 0.44 | 0.50 | 31.63 | 24.05 | 26.00 |

item 18 | 514 | 0.200 | 1.29 | 0.42 | 0.49 | 25.93 | 20.69 | 20.00 |

item 19 | 514 | 0.050 | 0.35 | 0.48 | 0.50 | 42.44 | 28.52 | 34.25 |

item 20 | 514 | 0.200 | 1.74 | 0.38 | 0.49 | 42.19 | 32.50 | 33.75 |

item 21 | 514 | 0.200 | 2.19 | 0.32 | 0.47 | 29.14 | 23.76 | 22.50 |

item 22 | 514 | 0.067 | 3.15 | 0.19 | 0.39 | 61.93 | 53.74 | 45.75 |

item 23 | 514 | 0.200 | 1.44 | 0.40 | 0.49 | 25.45 | 19.60 | 19.50 |

item 24 | 510 | 0.100 | 3.60 | 0.16 | 0.37 | 43.21 | 36.16 | 31.75 |

item 25 | 507 | 0.050 | 3.09 | 0.15 | 0.36 | 60.81 | 52.38 | 46.00 |

item 26 | 504 | 0.200 | 4.88 | 0.15 | 0.36 | 39.15 | 42.53 | 26.50 |

item 27 | 502 | 0.200 | 3.71 | 0.28 | 0.45 | 31.27 | 28.12 | 22.50 |

item 28 | 500 | 0.050 | 5.23 | 0.06 | 0.24 | 44.33 | 37.55 | 33.50 |

item 29 | 496 | 0.017 | 5.80 | 0.04 | 0.19 | 45.70 | 40.60 | 35.00 |

## Notes

1 | In this study, we use the terms ‘item’ and ‘task’ semi-interchangeably. The word ‘item’ refers to a clearly demarcated part of the test whose psychometric difficulty can be empirically extracted. The word ‘task’ refers to the content of the item. In the case of Triton, children solve the same ‘task’ (balance both sides of the equation) many times, though are administered ‘items’ of varying difficulty. |

2 | Please note that the individual effects of item difficulty (b _{i}) and a person’s ability (θ_{j}) are not included in the model, as they are already used to form the distance–difficulty difference term. |

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**Figure 2.**Predicted response time values according to Model A3 for correct (green) and incorrect (red) responses depending on children’s ability. With correctly answered items (green line), there is no substantial relationship between a person’s ability and the time needed to solve an item. On the other hand, with incorrectly answered items (red line), the time required to answer an item increases with the ability level. Children with greater ability, therefore, take longer to answer an item incorrectly.

**Figure 3.**Predicted response time values according to the exploratory model A3 for correct (green line) and incorrect (red line) responses depending on children’s ability. The graph is divided into three panels based on different item difficulty levels. The observed difference in slopes of correct and incorrect responses is visibly weaker in Model A4. This further expands Model A3, as the proposition that the time required to answer incorrectly answered items (red line) increases with ability level is applicable only for moderately (e.g., difficulty = 0) and highly (e.g., difficulty = 2) difficult items.

**Figure 4.**Predicted response time values according to Model B3 as a function of ability–difficulty distance separated by whether the answer was correct (green) or incorrect (red). Regardless of the response correctness, items whose difficulty matches the participant’s ability level take the longest time to solve (with incorrect answers taking the longest). The relationship changes for very difficult and easy items—with these, correctly answered items take longer than the incorrect ones.

**Table 3.**Parameters of the models assessing distance–difficulty hypothesis interacting with the F > C phenomenon.

Model 0 | Model B1 | Model B2 | Model B3 | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

95% CI | 95% CI | 95% CI | 95% CI | ||||||||||||||

coef. | est. | LL | UL | est. | LL | UL | est. | LL | UL | est. | LL | UL | |||||

Fixed effects | |||||||||||||||||

intercept | μ | 3.16 | *** | 3.00 | 3.33 | 3.50 | *** | 3.31 | 3.69 | 3.51 | *** | 3.32 | 3.70 | 3.56 | *** | 3.37 | 3.75 |

correct answer (FC) | γ_{1} | −0.02 | −0.04 | 0.01 | −0.10 | *** | −0.13 | −0.07 | |||||||||

ability–difficulty distance | γ_{4} | −0.13 | *** | −0.14 | −0.12 | −0.13 | *** | −0.14 | −0.12 | −0.16 | *** | −0.17 | −0.15 | ||||

distance × FC | γ_{14} | 0.05 | *** | 0.04 | 0.07 | ||||||||||||

Random effects | |||||||||||||||||

person intercept variance | var(τ_{j}) | 0.07 | *** | 0.06 | 0.08 | 0.07 | *** | 0.06 | 0.08 | 0.07 | *** | 0.06 | 0.08 | 0.07 | *** | 0.06 | 0.08 |

item intercept variance | var(β_{i}) | 0.20 | *** | 0.12 | 0.33 | 0.25 | *** | 0.15 | 0.42 | 0.25 | *** | 0.15 | 0.42 | 0.26 | *** | 0.16 | 0.45 |

residual variance | var(ε_{ij}) | 0.37 | *** | 0.36 | 0.38 | 0.34 | *** | 0.33 | 0.35 | 0.34 | *** | 0.33 | 0.35 | 0.34 | *** | 0.33 | 0.34 |

Goodness of fit | |||||||||||||||||

conditional R^{2} | 0.417 | 0.527 | 0.525 | 0.542 | |||||||||||||

marginal R^{2} | 0.000 | 0.082 | 0.081 | 0.092 | |||||||||||||

log-likelihood | −14,193 | −13,563 | −13,562 | −13,537 | |||||||||||||

AIC | 28,395 | 27,136 | 27,136 | 27,088 | |||||||||||||

BIC | 28,425 | 27,174 | 27,181 | 27,142 | |||||||||||||

Δχ^{2} (df) | 1260.82 (1) | *** | 2.14 (1) | 49.49 (1) | *** |

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**MDPI and ACS Style**

Tancoš, M.; Chvojka, E.; Jabůrek, M.; Portešová, Š.
Faster ≠ Smarter: Children with Higher Levels of Ability Take Longer to Give Incorrect Answers, Especially When the Task Matches Their Ability. *J. Intell.* **2023**, *11*, 63.
https://doi.org/10.3390/jintelligence11040063

**AMA Style**

Tancoš M, Chvojka E, Jabůrek M, Portešová Š.
Faster ≠ Smarter: Children with Higher Levels of Ability Take Longer to Give Incorrect Answers, Especially When the Task Matches Their Ability. *Journal of Intelligence*. 2023; 11(4):63.
https://doi.org/10.3390/jintelligence11040063

**Chicago/Turabian Style**

Tancoš, Martin, Edita Chvojka, Michal Jabůrek, and Šárka Portešová.
2023. "Faster ≠ Smarter: Children with Higher Levels of Ability Take Longer to Give Incorrect Answers, Especially When the Task Matches Their Ability" *Journal of Intelligence* 11, no. 4: 63.
https://doi.org/10.3390/jintelligence11040063