# Modeling Mental Speed: Decomposing Response Time Distributions in Elementary Cognitive Tasks and Correlations with Working Memory Capacity and Fluid Intelligence

^{*}

## Abstract

**:**

## 1. Introduction

_{RT}) is slightly more correlated with gf than the mean (M

_{RT}) [2]. Variability in task performance is of interest as it may indicate impaired stability of the cognitive system. It was shown that SD

_{RT}is lowest in young adults and is higher in both younger children and older adults [14]. Thereby, the developmental trajectory of SD

_{RT}as an inverse marker of cognitive functioning resembles that of cognitive ability. A recent meta-analysis on the basis of 27 independent samples [15] confirms a moderate relation between SD

_{RT}and intelligence, but does not support the previously held notion [2] that variability is a better predictor of intelligence than mean RT. Generally, M

_{RT}and SD

_{RT}are highly collinear across participants (i.e., r ≈ 0.9; [16]). Additionally, both scores are affected by extreme values, which can be expected to contribute to their correlation. Therefore, the common practice in some RT measures to scale means by the individual’s variability may function as a pragmatic remedy [17]. Theoretically, more satisfactory modeling approaches are discussed below. Since M

_{RT}, SD

_{RT}, and gf are highly related, it was suggested that, from a psychometric perspective, a common factor could account for their relation [18].

_{RT}) than with the median (Mdn

_{RT}) RT [18,25], where the former but not the latter is known to be biased by extreme values.

_{er}) subsumes processes before and after the actual decision phase (encoding of stimuli and execution of the motor response). In turn, the actual decision process is characterized by a continuous sampling of information. A decision process, originating from starting point z, fluctuates over time as a function of systematic stimulus information and random noise (see gray sample path). When it hits either the lower or the upper response boundary (at 0 or a, respectively), the according response is elicited. The mean slope of the decision process across trials denotes the drift rate (ν).

_{RT}over M

_{RT}and the latter over Mdn

_{RT}, and the linear correlation of M

_{RT}and SD

_{RT}. All these effects were argued to be driven by a single latent relation between individual differences in drift rate and individual differences in general intelligence.

_{er}). These can result from the by-chance occurrence of a few extra slow error response times that would bias all parameters jointly in the same direction [34]. Additionally, estimating drift rate (ν) and response caution (a) can be a challenge in case of only few errors values, as different combinations of both parameters (both jointly increasing) could account for the observed distribution of correct RT values.

_{er}) is positively related with μ but not with the other ex-Gaussian parameters.

_{RT}). In contrast, non-decision time (T

_{er}) was most highly correlated with the faster quantiles of the RT distribution. Analogous correlations were obtained for μ and τ of the ex-Gaussian distribution, respectively. Unfortunately, the authors did not report these correlations with accuracy. Nevertheless, these findings are definitely informative with respect to the correspondence of parameters with observable scores. However, since identical RT distributions were entered, this does not distinguish between state/contamination and the stable/generalizable portions of parameters as (indicators of) trait-like dispositions (cf. [42]) that can be predicted to be considerably more moderate. Another challenge is that the correlations of parameters with observed RT scores were shown to depend on the variance in the other parameters, as demonstrated in simulation studies [35]. Therefore, observed correlations in experimental datasets can be expected to vary as a function of the variance in the other parameters. The resulting range of relationship estimates is therefore an empirical question.

#### Goals and Hypotheses of the Present Study

_{RT}) were expected to be somewhat more highly correlated with ability than with their robust counterparts (e.g., Mdn

_{RT}or M

_{log(RT)}) [18,25]. However, in line with the current meta-analysis, we did not predict RT variability to be consistently more highly correlated with ability than with mean RT [15]. Error scores were less frequently employed in the literature, supposedly because of their reduced variability and, consequently, their reduced reliability and validity. We still included them in this study and investigated their correlations with model parameters and cognitive ability to uncover possible speed-accuracy trade-offs.

## 2. Experimental Section

#### 2.1. Sample

#### 2.2. Speed Tasks

#### 2.3. WMC and gf Measures

#### 2.4. Scoring and Modeling of Response Times

_{RT}) is frequently used, but it can be biased towards higher values by a few extreme RT values. Therefore, the mean of individually log-transformed RTs (M

_{log(RT)}) or the Median of the RT distribution (Mdn

_{RT}) are frequently computed when a more robust estimate of the location of the distribution is desired. Distributions of mean response times across participants can be positively skewed.

_{RT}) and the interquartile range (IQR

_{RT}) of the response times as two conventional scores. The first can be biased by just a few outliers, whereas the second may be a more robust estimate of the spread of the RT distribution.

_{er}). Dependencies are less of a problem using EZ, since parameters are not estimated, but their respective equivalents are directly computed from moments of the RT distribution and accuracy. Additionally, a number of simulation studies converge in showing that EZ estimates are robust for the purpose of modeling individual differences [54,55], even when only a moderate number of trials are available [56]. Since the diffusion model is only suited for binary choice tasks, it could only be fitted to the data from the Search and Comparison tasks.

#### 2.5. Data Preparation and Descriptive Analyses

_{RT}score is in part driven by extreme values from the right tail of the RT distribution. The same holds for the M

_{RT}, which appears to be biased towards higher values compared with the μ parameter.

_{er}) was higher in the Comparison tasks, possibly indicating longer encoding time of the more complex stimuli.

^{2}(6) = 10.03, p = 0.12.)

## 3. Results

#### 3.1. Correlations of Mental Speed with WMC and Fluid Intelligence (H1)

^{2}(1) = 7.17, p < 0.01).

#### 3.2. Differential Validity of RT Scores (H2)

^{2}(1) = 2.11, p = 0.15). Correlations of the RT variability scores were somewhat smaller than the location scores. Error scores were not consistently correlated with WMC and gf.

#### 3.3. Validity of Parameters Sensitive to the Right Tail of the Distribution (H3)

#### 3.4. Moderation of the WPR by Task Complexity (H4)

#### 3.5. Testing Model Parameters as Trait-Like Dispositions (H5)

_{er}). Cross-task correlations tended to be more moderate for the ex-Gaussian parameter estimates and were less specific for the respective parameters. The highest correlations were observed for the τ parameter, while the σ parameter was only moderately correlated across task classes.

_{er}) was positively correlated with μ, but there was no substantial correlation with the other parameters.

_{RT}exceeded the average correlation obtained for M

_{RT}. Response caution (a) was positively correlated with all location scores in a comparable magnitude. Non-decision (T

_{er}) time also showed a moderate positive relation with the location scores, while its relation was SD

_{RT}was somewhat weaker. All diffusion model parameters were negatively correlated with the error rate in the other tasks, in particular drift rate and response caution.

_{RT}. The σ parameter was moderately related with all RT location scores, but weaker than μ. Actually, the τ parameter showed the highest correlations with the RT scores compared with the other ex-Gaussian parameters. Its correlations were stronger with the slow quantiles compared with the fast quantiles of the RT distribution in the other tasks. Please note that the average correlations with error rate were virtually zero.

## 4. Discussion

_{RT}), which is potentially biased by extreme RT values, is more highly correlated with cognitive ability than those location scores that are less affected by extreme RT values (Mdn

_{RT}, M

_{log(RT)}) [18,25]. However, there was virtually no difference in their correlations. Apparently, a few outlier values did not distort the rank order of participants. This finding suggests that all of the conventional mean RT scores can be used as predictors of cognitive ability and none of them is clearly superior over the others. In addition, scores of RT variability (SD

_{RT}) were not found to be more strongly related with ability than with mean RT (M

_{RT}), confirming findings from a recent meta-analysis [15]. As expected, the error scores were less reliable than the RT scores. Not surprisingly, their correlations obtained with WMC and gf were low. The fit of the error score model was still good. Consequently, the probit transformation did not considerably improve the fit of the model, but slightly decreased validity of the scores. This result suggests that the correlations in error variables are largely driven by participants with more extreme error proportions.

_{RT}. However, comparably moderate retest correlations were also observed using other estimation procedures [42].

_{RT}) than with mean response times (M

_{RT}) [35], even when correlations were estimated across non-overlapping data sets. Still then, SD

_{RT}was not more highly related with ability than M

_{RT}, in line with the recent meta-analysis on RT and intelligence [15]. This finding suggests that drift rate is not the only “determinant” of a high task score. It is of note for the assessment of mental speed that the correlations of the response caution (a) parameter with virtually all location scores suggests the setting of speed-accuracy trade-off contributes as well to the observed RT scores in elementary tasks. The virtual null correlations of the ex-Gaussian parameters with error rate might be due to the constraint that ex-Gaussian parameters are estimated from the distribution of correct response times only. This observation suggests that errors contain independent (possibly incremental) information to RT scores. However, the null relations could also be the result of an artifact from the joint effects of processing efficiency (which would result in a positive relation of RT and error rate) and of the speed–accuracy trade-off (which would result in a negative relation of RT and error rate). The diffusion model is better suited to dissociate both components.

## 5. Limitations

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

Measurement Models | Model Fit | |||||||
---|---|---|---|---|---|---|---|---|

# | Model | Saturation | Loadings | χ^{2} | df | RMSEA | CFI | |

Cognitive Ability | ||||||||

0 | WMC | 0.74 | 0.69 (0.55–0.78) | 12 | 8 | 0.048 | 0.991 | |

gf | 0.83 | 0.77 (0.75–0.79) | ||||||

Speed | ||||||||

1 | +1/RT | 0.74 | 0.64 (0.46–0.95) | 132 | 81 | 0.056 | 0.975 | |

Response Time | ||||||||

2 | +M_{RT} | 0.74 | 0.64 (0.46–0.93) | 130 | 81 | 0.055 | 0.975 | |

3 | +M_{log(RT)} | 0.73 | 0.64 (0.45–0.95) | 129 | 81 | 0.054 | 0.976 | |

4 | +Mdn_{RT} | 0.70 | 0.61 (0.42–0.92) | 122 | 81 | 0.050 | 0.977 | |

Quartiles | ||||||||

5 | +Q1 | 0.64 | 0.56 (0.35–0.92) | 140 | 81 | 0.060 | 0.965 | |

6 | +Q2 | 0.69 | 0.61 (0.43–0.94) | 125 | 81 | 0.052 | 0.976 | |

7 | +Q3 | 0.71 | 0.62 (0.43–0.92) | 122 | 81 | 0.050 | 0.978 | |

8 | +Q4 | 0.76 | 0.64 (0.47–0.88) | 152 | 81 | 0.066 | 0.960 | |

Variability | ||||||||

9 | +SD_{RT} | 0.67 | 0.53 (0.34–0.82) | 154 | 81 | 0.067 | 0.948 | |

10 | +IQR_{RT} | 0.54 | 0.44 (0.29–0.74) | 155 | 81 | 0.068 | 0.926 | |

Errors | ||||||||

11 | +Error Rate | 0.66 | 0.54 (0.34–0.69) | 126 | 81 | 0.052 | 0.953 | |

12 | +Probit (Error) | 0.64 | 0.54 (0.46–0.66) | 112 | 81 | 0.044 | 0.963 | |

Ex-Gaussian Model | ||||||||

13a | +Search/CRT | μ | 0.83 | 0.79 (0.71–0.89) | 133 | 71 | 0.066 | 0.959 |

σ | 0.50 | 0.48 (0.23–0.71) | ||||||

τ | 0.55 | 0.58 (0.53–0.66) | ||||||

13b | +Comparison | μ | 0.81 | 0.76 (0.54–0.94) | 84 | 71 | 0.030 | 0.991 |

σ | 0.55 | 0.54 (0.40–0.64) | ||||||

τ | 0.84 | 0.79 (0.69–0.92) | ||||||

13c | +Substitution | μ | 0.83 | 0.79 (0.70–0.86) | 136 | 71 | 0.068 | 0.958 |

σ | 0.61 | 0.55 (0.35–0.74) | ||||||

τ | 0.63 | 0.61 (0.58–0.64) | ||||||

Diffusion Model | ||||||||

14a | +Search/CRT | a | 0.80 | 0.74 (0.72–0.77) | 125 | 71 | 0.062 | 0.959 |

ν | 0.81 | 0.75 (0.70–0.82) | ||||||

T_{er} | 0.86 | 0.83 (0.76–0.93) | ||||||

14b | +Comparison | a | 0.87 | 0.83 (0.78–0.92) | 99 | 71 | 0.044 | 0.976 |

ν | 0.79 | 0.75 (0.65–0.82) | ||||||

T_{er} | 0.70 | 0.66 (0.41–0.92) |

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**Figure 4.**Parameters of the diffusion model fit to Comparison tasks as predictors of WMC and gf. Significant parameters are displayed in black font.

**Figure 5.**Cross-task correlations of latent variables for the diffusion model and the ex-Gaussian parameters.

**Figure 6.**Correlations of latent variables for the diffusion model, the ex-Gaussian parameters, and with RT and accuracy scores across different tasks.

Tasks | Excluded | Descriptive Statistics | Ex-Gaussian Model | Diffusion Model | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Post-Err | Extreme | M_{RT} | SD_{RT} | Accuracy | μ | σ | τ | a | ν | T_{er} | |

Search/CRT | |||||||||||

Numbers | 0.04 (0.08) | 0.01 (0.02) | 377 (46) | 78 (20) | 0.97 (0.03) | 307 (46) | 32 (18) | 70 (27) | 0.09 (0.01) | 0.39 (0.07) | 263 (35) |

Letters | 0.04 (0.05) | 0.02 (0.04) | 386 (43) | 79 (21) | 0.96 (0.04) | 317 (45) | 37 (18) | 69 (30) | 0.09 (0.01) | 0.38 (0.07) | 272 (34) |

Symbols | 0.05 (0.04) | 0.01 (0.02) | 469 (49) | 90 (19) | 0.95 (0.04) | 397 (44) | 51 (20) | 72 (26) | 0.09 (0.01) | 0.34 (0.06) | 343 (34) |

Comparison | |||||||||||

Numbers | 0.04 (0.04) | 0.00 (0.01) | 871 (145) | 218 (81) | 0.95 (0.04) | 676 (102) | 83 (47) | 196 (84) | 0.15 (0.03) | 0.23 (0.05) | 561 (81) |

Letters | 0.05 (0.04) | 0.01 (0.01) | 1009 (196) | 258 (101) | 0.95 (0.04) | 773 (129) | 93 (44) | 236 (105) | 0.16 (0.04) | 0.20 (0.04) | 643 (102) |

Symbols | 0.07 (0.04) | 0.02 (0.02) | 1230 (208) | 331 (110) | 0.93 (0.04) | 924 (139) | 151 (62) | 306 (136) | 0.17 (0.04) | 0.17 (0.03) | 769 (101) |

Substitution | |||||||||||

Num→Sym | 0.02 (0.03) | 0.01 (0.02) | 1473 (254) | 447 (111) | 0.98 (0.02) | 1127 (251) | 253 (110) | 347 (169) | — | — | — |

Sym→Let | 0.03 (0.03) | 0.00 (0.01) | 1252 (210) | 403 (113) | 0.97 (0.03) | 940 (221) | 222 (102) | 312 (161) | — | — | — |

Let→Num | 0.03 (0.03) | 0.01(0.01) | 1345 (275) | 420 (125) | 0.97 (0.03) | 1006 (254) | 225 (107) | 339 (171) | — | — | — |

_{RT}= mean response time, SD

_{RT}= within-person standard deviation in RT. The ex-Gaussian model decomposes RT distributions into the parameters μ and σ (M and SD of the normal distribution) and τ (as the parameter of an exponential component). The diffusion model yields parameters a (boundary separation), ν (drift rate), and T

_{er}(non-decision time), for the binary-choice Search/CRT and Comparison tasks.

# | Score/Parameter | WMC | gf | ||||
---|---|---|---|---|---|---|---|

Speed | |||||||

1 | 1/RT | 0.62 *** | 0.43 *** | ||||

Response Time | |||||||

2 | M_{RT} | −0.69 *** | −0.46 *** | ||||

3 | M_{log(RT)} | −0.65 *** | −0.45 *** | ||||

4 | Mdn_{RT} | −0.67 *** | −0.45 *** | ||||

RT Quartiles | |||||||

5 | Q1 | −0.59 *** | −0.41 *** | ||||

6 | Q2 | −0.65 *** | −0.44 *** | ||||

7 | Q3 | −0.69 *** | −0.46 *** | ||||

8 | Q4 | −0.68 *** | −0.44 *** | ||||

RT Variability | |||||||

9 | SD_{RT} | −0.50 *** | −0.30 *** | ||||

10 | IQR_{RT} | −0.50 *** | −0.31 *** | ||||

Errors | |||||||

11 | Error Rate | −0.18 * | −0.09 | ||||

12 | Probit (Error) | −0.06 | −0.02 | ||||

Ex-Gaussian Model | Search | Comp | Subst | Search | Comp | Subst | |

13 | μ | −0.10 | −0.09 | −0.55 *** | −0.11 | −0.02 | −0.39 *** |

σ | −0.16 | −0.02 | 0.01 | −0.12 | 0.23 | 0.03 | |

τ | −0.32 ** | −0.27 | −0.43 *** | −0.23 * | −0.33 * | −0.24 ** | |

Diffusion Model | Search | Comp | Subst | Search | Comp | Subst | |

14 | a | −0.20 | −0.01 | — | −0.21 | −0.02 | — |

ν | 0.35 *** | 0.41 *** | — | 0.15 | 0.29 ** | — | |

T_{er} | −0.19 | −0.12 | — | −0.10 | 0.11 | — |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Schmitz, F.; Wilhelm, O. Modeling Mental Speed: Decomposing Response Time Distributions in Elementary Cognitive Tasks and Correlations with Working Memory Capacity and Fluid Intelligence. *J. Intell.* **2016**, *4*, 13.
https://doi.org/10.3390/jintelligence4040013

**AMA Style**

Schmitz F, Wilhelm O. Modeling Mental Speed: Decomposing Response Time Distributions in Elementary Cognitive Tasks and Correlations with Working Memory Capacity and Fluid Intelligence. *Journal of Intelligence*. 2016; 4(4):13.
https://doi.org/10.3390/jintelligence4040013

**Chicago/Turabian Style**

Schmitz, Florian, and Oliver Wilhelm. 2016. "Modeling Mental Speed: Decomposing Response Time Distributions in Elementary Cognitive Tasks and Correlations with Working Memory Capacity and Fluid Intelligence" *Journal of Intelligence* 4, no. 4: 13.
https://doi.org/10.3390/jintelligence4040013