How does cognitive fatigue affect Young, Middle-aged, and Older people? A distribution analysis of Time-on-Task effect by fitting the ex-Gaussian parameters to the response time distributions.

Introduction.
Young people classically experiment cognitive fatigue after long-lasting cognitive tasks (Time-on-Task effect), as evidenced by changes in response time (RT) distribution (Wang et al., 2014). As RT distributions are often positively skewed, the ex-Gaussian distribution (the convolution of the exponential and the Gaussian distributions) fits RT data relatively well. The ex-Gaussian distribution comprises three parameters: mu & sigma are mean & SD of the Gaussian component while tau describes the exponential component (the increase in extreme RT). In Young people, mu and sigma are not fatigue sensitive while tau is well known to suffer from Time-on-Task. Aim. To replicate the results previously found in Young people (Wang et al., 2014) and to test this Time-on-Task effect in Middle-aged and Older people by fitting the three ex-Gaussian parameters to RT distributions.
Method.
Procedure. Twenty-one Young (8 males; MAge=22.43), 17 Middle-aged (7 males; MAge= 50.47), and 17 Older (9males; MAge=65.06) were recruited. Based on Wang et al. (2014) and Wang et al. (2016), we administered a modified version of a computerized Stroop task (Stroop, 1935) during 160 min. In our task, different words (“BLUE”, “RED”, “YELLOW”, “GREEN”) or the symbol “XXXX” appeared one at a time printed in one of the following colors: blue, red, yellow, or green. The task was composed of facilitatory (FA) items (“BLUE” written in blue), interfering (I) items (“BLUE” written in red), and neutral (NE) items (“XXXX” symbol printed in color), these latter ones systematically appearing one item out of two. Since RT data are rarely normally distributed but are often positively skewed, classical parameters like mean and SD are not good candidates to properly describe the data (Heathcote et al., 1991). Therefore, in a very similar way to Wang et al. (2014) in a Young group, we decided to fit the ex-Gaussian distribution to our RT data in order to fully capture the Time-on-Task effect on RT distribution in our Young, Middle-aged, and Older groups. Among the available techniques, the SIMPLEX algorithm is well used for fitting the ex-Gaussian distribution (Lacouture & Cousineau, 2008). However, in certain cases, the parameter search with this method may fail to converge. To obtain a very efficient algorithm, we decided to base our search procedure both on the algorithm of Nelder & Mead (1965) which is gradient-free but also on the Greedy algorithm technique allowing to be flexible in the starting values it takes.
Statistical analyses. The duration of the task was divided into four blocks of 40min each and we fitted the ex-Gaussian parameters (mu, sigma, tau) to the RT distributions in those four blocks, separately for each item types (FA, I, NE). Analyses were performed with the GLIMMIX procedure of SAS on mu and tau but not on sigma because of homoscedasticity violation. Each of the 6 remaining measures (FA_mu, I_mu, NE_mu, FA_tau, I_tau, NE_tau) was then analyzed with the GLIMMIX repeated measure and multiple comparisons post-hoc of Tukey. Each parameter was explained by Group, Block (the repeated variable), Group X Block interaction, Sex, Education, and Depression status.
Results
Mu parameter. All three FA_mu, I_mu, and NE_mu measures showed a significant effect of Group (all p < .01). Tukey post-hoc further showed significant differences between the Young group and both the Middle-aged and the Older groups, with the Young group having a smaller value of mu than the two other groups for the three measures. Moreover, the Group X Block interaction showed a tendency (p = .077) of significance on the I_mu measure. Tukey post-hoc on this interaction showed that the Older group had a greater mu in Block 4 as compared to Block 1 and Block 2 while the Young and the Middle-aged groups did not show any difference. Finally, none of the demographical variables was significant on mu parameter.
Tau parameter.
FA_tau showed a significant effect of Block (p = .002). Tukey post-hoc showed an almost significant difference between the 1st and the 4th Block (p = 0.05) and a significant difference between the 2nd and the 4th Block (p<.001). Exploratory Tukey post-hoc on the Group X Block interaction showed that Middle-aged had a significant difference between the 2nd and the 4th Block (p = .002) but also between the 3rd and the 4th Block (p = .015) while the Young and the Older groups did not show any difference. None of the demographical variables was significant on FA_tau.
I_tau showed a tendency of significance on the Group X Block interaction (p = .074). Tukey post-hoc showed that Middle-aged had a significant difference between the 1st and the 4th (p = .041) but also between the 2nd and the 4th Block (p = .03). I_tau also showed a significant effect of Sex (p = .048), with the male status being associated with greater values of I_tau as compared to the woman status.
NE_tau showed a significant effect of Block (p<.0001). Tukey post-hoc showed significant increase in tau between the 1st and the 3rd Block, the 1st and the 4th Block, but also between the 2nd and the 4th Block (all p < .001). Exploratory Tukey post-hoc on the Group X Block interaction showed that Young people had an increase in tau between the 1st and the 3rd Block (p = .007) but also between the 1st and the 4th Block (p = .001). Moreover, Middle-aged people had an increase in tau between the 1st and the 4th Block (p = .008) but also between the 2nd and the 3rd (p = .015) and between the 2nd and the 4th Block (p<.001). Finally, the Older group had a significant increase between the 1st and the 4th Block (p = .02). None of the demographical variables was significant on NE_tau.
Discussion
We demonstrated for the first time that mu parameter undergoes a Group effect. Indeed, all three measures (FA_mu, I_mu, and NE_mu) showed a significant effect of Group, with Young people showing less slowdown than Middle-aged and Older. It is consistent with literature showing the classic age effect on the slowdown of reaction time (Salthouse, 2000). Furthermore, mu did not show any Block effect, which is in agreement with the “ex-Gaussian” literature (Wang et al., 2014). As compared to the classic mean, mu is free from extreme values given that they are comprised in the tau parameter, what can explain the absence of Time-on-Task effect on the mu parameter. As proposed by Hohle (1965), mu would represent the peripheral/motor processes while tau would represent the decisional component of RT data. Therefore, if mu tends to represent procedural processes, our study is also in great agreement with that of Borragán et al. (2016) showing an improvement in procedural learning – in our case, an absence of impairment in procedural learning - in situation of cognitive fatigue. Only the Older group showed difference between the two first and the last Block on I_mu, probably because they are more prone to suffer from interference and decrements in inhibition (Hasher & Zacks, 1988) as compared to the two other groups, leading them to show fatigue installation specifically on the Interfering items.
By contrast, FA_tau and NE_tau suffered well from Time-on-Task effect, which is also in agreement with the ex-Gaussian literature showing increases in tau with the time spent on a task (Wang et al., 2014). Exploratory Tukey post-hoc analyses showed that the Middle-aged group had significant increases in tau with Time-on-Task on the three tau measures (FA_tau, I_tau, and NE_tau) while the two other groups only got significant difference with Time-on-Task for NE_tau. We made the hypothesis that the three groups showed significant differences in the NE_tau measure because NE items were the most predictable. Indeed, as they were appearing one out of two in our task, it possible that their greater predictability have led participants to a lack of motivation and then a lack of top-down attention necessary to counteract task-disengagement for NE items, making the motivation drop and boredom good candidates to explain cognitive fatigue effects (Boksem & Tops, 2008). Moreover, we did not observe a classic age affect in the sense that Middle-aged seemed to suffer more than Older from Time-on-Task. We claim it is possible that Middle-aged people be the more fatigued group. Indeed, it is a population fully busy, immersed in active life but also having generally great responsibilities, leading them to be more susceptible to cognitive fatigue and to show fatigue signs on the three measures as compared to the other groups.
In conclusion, the combined use of mu and tau parameters to characterize RT distributions of Young, Middle-aged, and Older has allowed to more accurately understanding fatigue effect in those populations. First, we demonstrated that the three groups were not impaired in their motor/peripheral processes with Time-on-task except for the Older on the Interfering items, what is not surprising given their baseline difficulties in inhibition (Hasher & Zacks, 1988). Secondly, the three groups were disabled in their decisional processes as indexed by increases in their extreme RT, Middle-aged being the most cognitively fatigued.