U-Shape by Quadratic Equation: A Best-fit for Modelling He- modynamic Changes in the DCCS Task

General Linear Modelling (GLM) has been widely employed to estimate the hemodynamic changes evoked by cognitive processing, which are more likely to be nonlinear than linear. First, this study re-analyzed the fNIRS data (N = 38, Mage = 5.0 years, SD = 0.69 years, 17 girls) collected in the Mixed-Order Design Dimensional Change Card Sort (DCCS) task. The results indicated that the quadratic equation was better than GLM to model HbO changes in this task. Second, analysis of a new set of data indicated that the Habit-DisHabit design of DCCS was more effective in identifying the neural correlates of cognitive shifting than the Mixed-Order Design. Third, this study found that the Non-users were more attentive and engaged than the Heavy-users, with a slower but more steady increase of brain activation in BA8 and BA9.


Introduction
The functional near-infrared spectroscopy (fNIRS) technology is a portable and comfortable way to measure the hemodynamic changes in targeted brain areas [1][2][3]. It can generate time-sensitive data that could be analyzed using general linear modeling (GLM) to estimate the oxygenated hemoglobin (HbO) changes between the task and baseline. For instance, Li et al. examined the effect of heavy tablet use on preschoolers' executive function during the Dimensional Change Card Sort (DCCS) task using fNIRS [4]. They conducted t-tests and GLM to compare the hemodynamic changes in the Non-user and the Heavyuser groups and found a significant difference between the two groups in activating the prefrontal cortex (BA 9). The 'Non-user' activation pattern was 'normal and healthy', whereas the 'Heavy-user' pattern was 'not normal thus needs further exploration' [4]. However, they presented no further statistical evidence to demonstrate how 'abnormal' the 'Heavy-user' pattern was. In addition, the hemodynamic changes over time in brain areas are a nonlinear rather than linear relationship; thus, the GLM analysis conducted by Li et al. [4] might not be appropriate or accurate. Therefore, this study aims to re-analyze the same data with a quadratic or cubic equation to identify the best-fit way to model the hemodynamic responses in the DCCS task. In addition, a set of data with a new design DCCS task was also collected and analyzed with a quadratic equation.

Modeling Hemodynamic Changes with GLM
The fNIRS technology allows us to monitor brain activation by measuring hemodynamic changes such as the concentration of oxy-hemoglobin (HbO) and deoxy-hemoglobin (HbR) in targeted brain areas. The HbO and HbR data are dynamic and changing over time; thus, careful and advanced statistical analyses are needed to examine this type of time-sensitive data [1][2][3]. However, no systematic and standardized approaches were established in the first decade of this millennium; thus, fNIRS scientists had the liberty to choose the statistical methods they believed to be appropriate and adequate. Later, Schroeter et al. [5] proposed employing the general linear modeling (GLM) as the standard statistical approach for fNIRS data. Accordingly, Pouliot et al. [6] adopted GLM as the standard hemodynamic response function to examine the fNIRS data for spikes and seizures. They concluded that GLM could be used to analyze the fNIRS data for posterior epileptic activity. In 2014, Tak and Ye [7] systematically reviewed the commonly used fNIRS statistics such as principal component analysis, independent component analysis, false discovery rate, in addition to the inference statistics such as the standard t-test, F-test, analysis of variance, and statistical parameter mapping framework. Eventually, they proposed adopting GLM mixed-effect model with restricted maximum likelihood variance estimation [7].
Since then, employing GLM to estimate the hemodynamic changes has become the standard inference statistics in fNIRS studies. GLM empowers scientists to estimate the subject, channel, and task-specific evoked hemodynamic responses and robustly separate the evoked brain activity from systemic physiological interference using independent measures of nuisance regressors [1][2][3]. In addition, GLM can significantly enhance the contrast to noise ratio of the brain signal, improve feature separability, and ultimately lead to better classification accuracy. Recently, von Lühmann et al. [8] found that GLM could provide better single-trial estimates of brain activity as well as a new feature type, such as the weight of the individual and channel-specific hemodynamic response function regressor. Therefore, GLM has become the well-established and well-accepted method for fNIRS data analysis.

Modeling Hemodynamic Changes in the DCCS Task
The DCCS task asks children to sort a set of two-dimensional (i.e., color and shape) test cards according to two target cards that match the former in one dimension but not the other. Normally, a set of white paper cards (3.5 x 7.0 cm) is used as the stimuli, which have two dimensions: shape and color. Children are asked to sort the test cards according to one dimension that matches the target card (e.g., red/blue, boat/rabbit). And the rule for matching is changed according to the experimenter's instruction. Initially, Morriguchi and his colleagues [9] conducted fNIRS studies on the DCCS task. They conducted t-tests and correlation analyses to compare HbO changes between the task and baseline conditions. Even recently, Moriguchi & Lertladaluck [10] and Xie et al. [11] conducted the same t-tests and correlation analysis of the HbO changes for the same DCCS task. But the results were contradictory: Moriguchi & Lertladaluck [10] found no significant relationship between prefrontal activations and English proficiency, whereas Xie et al. [11] found a significant correlation. This inconsistency indicated that either the data analysis or the DCCS task paradigm employed by the two teams might not be rigorous or sensitive enough to identify the specific neural correlates responsible for the cognitive shifting of the DCCS task.
Therefore, Li et al. [12] developed the "habituation-dishabituation paradigm of DCCS task" ( "Habit-Dishabit Design" hereafter) and proposed a more direct and critical indicator -the "V shape of General Linear Modelling" to identify the cognitive shifting. This paradigm has improved the arrangement of testing items to maximize the chances of habituation and dishabituation in the participating children. In the pre-switch period (20'), the children were asked to sort six or more cards using the same rule thus tended to be habituated before. Then, they were asked to use the other rule to sort another six cards in the pos-switch period (20'). And the three sessions followed the same sorting rule of the second period of the previous session: Session 1: color (6 cards)  shape (6 cards); Session 2: shape (6 cards)  color (6 cards); and Session 3: color (6 cards)  shape (6 cards). The children tended to be habituated when they knew the second round of sorting cards should follow the same rule. In another word, this paradigm has helped to trigger the occurrence of habituation and dishabituation. Accordingly, Li et al. (2020) proposed a pair of GLMs to estimate HbO changes (∆HbO) for the pre-and post-switch periods, using the same regression formula: YΔHbO = aX time + b + ε. If apre-switch is negative (-a), whereas apost-switch is positive (+a), and both models are significant, a perfect V shape could be verified. The corresponding channel was identified as the neural correlate of 'cognitive shifting' [12]. Using this new paradigm, they found V shape in BA 6, BA8, BA 9, BA 10, BA 40, and BA 44, which should be regarded as the neural correlations of cognitive shifting during the DCCS task.

The Context of This Study
Recently, Li et al. [4] adopted the Mixed-order design DCCS and GLM by Moriguchi & Lertladaluck [10] and Xie et al. [11] to examine the impact of tablet use on preschoolers' executive function. They found that the Non-users outperformed the Heavy-users with a significantly higher correct rate in the DCCS task. And the two groups differed significantly in the activation of BA9 (ch 16), indicating that the Non-user pattern was 'normal and healthy' [4]. In contrast, the Heavy-user pattern was 'not normal and needs further exploration'. However, the hemodynamic changes in each channel should be a kind of nonlinear relationship [1][2][3]; thus, polynomial regression via quadratic function might be more appropriate for estimating the HbO changes in targeted brain areas. Accordingly, in this study, we proposed a quadratic function for better modeling the hemodynamic changes: . If a > 0, the curve is a typical U shape; if a = 0, the quadratic function does not exist, indicating a linear relationship that GLM could model; if a < 0, the curve is a kind of reversed U shape. We hypothesized that this U-shape curve by quadratic function might be more statistically appropriate for modeling hemodynamic changes in the DCCS task than the V-shape by GLM [12]. In addition, we also hypothesized that the "Habit-DisHabit Design" might be more appropriate for identifying the neural correlates of cognitive shifting. Accordingly, this study dedicated to addressing the following questions: 1. GLM, quadratic, or cubic equations, which is the best fit for estimating the hemodynamic changes in the DCCS task? 2. Mixed-Order Design versus Habit-DisHabit design, which is more effective in identifying the neural correlates of cognitive shifting in the DCCS task?
3. What are the significant differences in brain activation between the heavy-users and non-users evidenced by fNIRS?

Participants
Li et al. [4] reported that 38 children (ages 4 to 6.3 years, Mage = 5.0 years, SD = 0.69 years, 17 girls, 21 boys) completed this experiment. The parents of 38 participating children completed the survey to identify the heavy-users or non-users. Eight children never used tablets; thus were included in the 'Non-user' group (2 girls, 6 boys). About 16 (12 girls, 4 boys) were classified into the 'Heavy-user' group as their daily screen time was more than the mean level, and their use was neither controlled nor limited [4].

The Mixed-Order Design DCCS Task
This task included two target cards and 24 test cards, each different in shape and color. One pair of target trays was used for the three consecutive test sessions, and each session consisted of a rest (20 s) phase and mix (25 s) phase. During the rest phase, children were asked to be still, doing nothing. As shown in Figure 1, during the mix phase, the children were asked to sort the cards according to the instructed rule (color or shape). In each phase, the children were given the rule before each trial. In each block, the rule order was fixed and mixed: shape, shape, color, shape, shape, color, shape, shape (totally 8 cards per block). This fixed-order was applied to all the participants, resulting in more color-toshape switches in total [4].

The Habit-DisHabit Design DCCS Task
This task is different from the Mixed-order design. As shown in Figure 2, the children were asked to sort eight to twelve cards using the same rule within the pre-switch period (20s). Then, they were asked to use the other rule to sort another eight to twelve cards within the pos-switch period (20s). And the three sessions followed the same sorting rule of the second period of the previous session: Session 1: color (20s)  shape (20s); Session 2: shape (20s)  color (20s); and Session 3: color (20s)  shape (20s). This design was very inductive to cognitive habituation and dishabituation [12].
In Li et al. [4], the mean of z-scores (HbO & HbR) was calculated for each DCCS task block separately for each participant. Then, the mean of z-scores (HbO & HbR) was calculated by averaging across the three task blocks for each participant. Finally, the mean of z-scores (HbO & HbR) across all channels were compared using t-tests between 'Non-user' and 'Heavy-user' groups using SPSS. Li et al. [4] conducted GLM analysis predicting zscores (HbO & HbR) in channel 16 in R ( ∆ = + + ). This study explored two sets of polynomial regression to better fit the nonlinear relationship between hemodynamic changes and time. The first set was quadratic equation using R [ ∆ = + + + ( ) ]. If a > 0, the curve is a typical U shape; if a = 0, the quadratic function does not exist, indicating a GLM linear relationship; if a < 0, the curve is a kind of reversed U shape. The second set was cubic equation analysis using R [ ∆ = + + + + ( )]. If a = 0, the cubic function does not exist, indicating a quadratic equation or GLM (if b = 0) should be considered.

Re-Analysis of Li et al. [4] Data with Quadratic and Cubit Equations
First, as shown in Table 1, the quadratic regression results for the Non-user group indicated that: (1) a U-shaped curve (a > 0) was observed in 9 channels (ch 1, 2, 5, 6,7,8, 9, 11, & 14), the quadratic predictor (time) could significantly explained 7.5% to 94.3% of the variance (HbO), R 2 s > .075, Fs > 5.94, ps < .001; (2) the quadratic function does not exist in channels 3 and 4 (a = 0), R 2 s >. 66, Fs = 145.65, ps < .001; and (3)  Second, as shown in Table 2, the quadratic regression results for the Heavy-user group indicated that: (1) a U-shaped curve (a > 0) was observed in channels 2-5, 8, 12, Table 2). Note: YHbO change = ax 2 +bx+c. Third, two sets of cubic equation analyses were conducted for the Non-user and Heavy-user groups, separately, with X 3 as the cubic term in addition to X 2 as the quadratic term and X as the linear term. However, the results for the Non-user group indicated that only two channels (ch 11 & 13) could apply to the cubic regression (a ≠ 0). In contrast, all the other channels should be modeled with a quadratic equation (a = 0). Similarly, the results for the Heavy-user group indicated that no channels could be estimated by the cubic regression, as a = 0. Therefore, we adopted a quadratic equation to analyze the new data collected in this study.
Last, we calculated the local maximum of HbO and HbR changes and the estimated time for the Heavy-user and Non-user groups separately and compared the betweengroup differences in the estimated time using t-tests. As shown in Table 3, the results indicated a significant between-group difference in the estimated time for channels 10 and 11. The Non-user group was significantly slower than the Heavy-user group to achieve the local maximum of HbO changes for channels 10 [Mdifference = 10.47 (second), t = -2.825, p < .05] and 11 [Mdifference = 9.66 (second), t = -3.337, p < .05].

Quadratic Regression Modelling New DCCS Data
First, we compared the behavioral performance in the Habit-DisHabit design DCCS task with t-test and found no significant differences between the 'Non-user' (Mcorrect rate = .989, SD = .017) and 'Heavy-user' (Mcorrect rate = .979, SD = .045) groups, t = -.587, p > .562. Then, a set of independent-samples t-tests with FDR correction was conducted to determine whether there were significant differences in HbO increases in the 17 channels between the 'Non-user' and 'Heavy-user' groups. However, no significant differences were found in 17 channels between the Non-user and the Heavy-user, ts < 0.44, ps > .05. These results jointly indicated that the two groups had the same behavioral performance and total HbO changes during the DCCS task.

Quadratic: The Best-fit Modelling
Li et al. [4] conducted GLM analysis to model the HbO changes in channel 16 and found significant between-group differences in a set of t-tests. The GLM results could only demonstrate a significant HbO increase for the Non-user group and a significant decrease for the Heavy-user group. However, HbO changes evoked by cognitive shifting are not a linear function of time; instead, there is a nonlinear relationship [1][2][3]. Therefore, first, a set of quadratic equation analyses was conducted to re-analyze Li et al. [4] data. The Nonuser group results indicated that 15 channels could be perfectly modeled, whereas only two channels (ch 3 & 4) were suitable for GLM. The Heavy-user group results indicated that 13 channels could be perfectly modeled by the quadratic equation, whereas only four channels (ch 1, 6, 7, & 13) should be modeled by GLM. Second, a set of cubic equation analyses was conducted to re-analyze Li et al. [4] data. However, the results indicated that the cubic equation could model only channels 11 and 13 for the Non-user. In addition, all the other channels for both groups were not suitable for this cubic equation. This finding implies that the cubic equation might not be appropriate for modeling the HbO changes in the DCCS mixed-order design task. Thus, these results jointly indicated that the quadratic equation might be the best-fit modeling for the HbO changes in the DCCS tasks. This study first re-analyzed Li et al. [4] data with quadratic equation and identified U-shape in 9 channels for the Non-user group. Then, analysis of the new Habit-DisHabit design data found a U-shape in 14 channels. Similarly, within the Heavy-user group, the re-analysis identified U-shape in 8 channels, whereas the new design data demonstrated it in 9 channels. The within-group increases in the U-shape indicated that the Habit-Dis-Habit design could help identify more neural correlates of the cognitive shifting in the DCCS task. In addition, within the Non-user group, the Mixed-Order Design data indicated that six channels had a reversed U-shape, indicating that the HbO changes increased over time for the corresponding channels. In contrast, the Habit-DisHabit design data indicated only one channel (ch 10) had a reversed U-shape. Therefore, this accumulative increase cannot reflect the rise and fall of HbO over time corresponding to the cognitive shifting. For the Heavy-user group, the Mixed-Order Design data indicated that five channels had a reversed U-shape, indicating that the HbO changes increased over time. In contrast, the Habit-DisHabit design data indicated no channel had a reversed U-shape, indicating a tendency of decreasing HbO in all the channels. This tendency might reflect the unique brain activation pattern for the Heavy-user, which will be discussed in the next section.
In particular, Li et al. [4] employed linear GLM analyses to model the HbO changes in BA 9 (ch 16) and found a significant increase for the Non-user group. In contrast, a significant decrease was found for the Heavy-user group, which demonstrated a significant increase after the 12th second. Thus, they concluded that BA 9 was significantly activated only in the Non-user group during the DCCS task [4]. However, we believe that this contrast might indicate a U-shape (fall and rise) of HbO changes in channel 16. Re-analyzing Li et al. [4] data and analyzing the U-shape data with quadratic equation have jointly indicated a U-shape in channel 16 for the Heavy-user group (See Table 2 ). In contrast, reanalysis found a reversed U-shape in channel 16 for the Non-user group (See Table 1), whereas the new analysis of Habit-DisHabit design found a U-shape in this channel. Therefore, this study has proved that the Habit-DisHabit design DCCS by the quadratic equation could effectively identify the neural correlates of cognitive shifting.

Two Brain Activation Patterns: Heavy-user Vs. Non-User
First, Li et al. [4] found that the Non-users group outperformed the Heavy-users with a significantly higher correct rate in the Mixed-Order Design DCCS task. This study found no significant between-group differences in the behavioral performance using the Habit-DisHabit design DCCS tasks. The two designs differed in the arrangement of changing rules: Habit-DisHabit design followed a simple pattern of changing the sorting rules thus was more likely to cause habituation, whereas the Mixed-Order Design had no pattern of changing rules; thus, the children had to be highly attentive. In this study, both groups completed the Habit-DisHabit design with almost the same success rate, indicating that they shared the same level of cognitive shifting. But the Heavy-user group had a significantly lower rate in completing the Mixed-Order Design, indicating that they might not be so attentive to the frequently changing rules thus made more mistakes in sorting the cards. Therefore, the first conclusion is that the Non-users tend to be more attentive than the Heavy-users.
Second, the analyses of HbO changes found that a quadratic equation could perfectly model 15 channels, and only two channels were found fit for GLM linear analysis in both Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 19 July 2021 doi:10.20944/preprints202107.0417.v1 mixed-and Habit-DisHabit designs. Thus, the Heavy-user group had 13 or 9 channels perfectly modeled by a quadratic equation in the Mixed-order and Habit-DisHabit designs, respectively. Meanwhile, four or eight channels should be modeled by GLM for the Mixed-order and Habit-DisHabit designs, respectively. This contrast indicated that the Non-user group had more channels engaged in the cognitive shifting in both Mixed-and Habit-DisHabit designs. Therefore, the second conclusion is that the Non-users tend to be more engaged in the DCCS task than the Heavy-users. Third, this study found that the Non-user group was significantly later to achieve the local maximum HbO changes for BA8 (ch 10) and BA9 (ch 11) than the Heavy-user group. To the contrary, the Heavy-users achieved the local maximum of HbO changes very early, followed by a significant decrease in the following period. This finding implies that the Non-users tend to respond to the DCCS tasks slower but more steadily than the Heavyusers, whose HbO achieved the maximum very early but dropped very quickly. Therefore, the third conclusion is that the Non-users tend to be more slowly and steadily increasing their hemodynamic responses than the Heavy-users. As it is widely accepted that HbO increase reflects the enhancement of brain activation thus is the most sensitive indicator in NIRS measurements [1][2][3]. These findings suggest that the Non-users are more attentive and engaged than the Heavy-users, with a slower but steady increase of brain activation in BA8 and BA9.

Conclusions, Limitations, and Implications
First, this study found that the quadratic equation might be the best-fit statistics for modeling the HbO changes in the DCCS tasks, by re-analyzing Li et al. [4] data and analyzing new data. Second, this study has proved that the Habit-DisHabit design DCCS in conjunction with the quadratic equation effectively identified the neural correlates of cognitive shifting. Third, this study found that the Non-users were more attentive and engaged than the Heavy-users, with a slower but steady increase of brain activation in BA8 and BA9. However, these results must be interpreted with caution, as the sample size was very small. The unexpected COVID-19 lockdown in China has stopped the experiment in late January 2020. Thus, only 38 cases were included in this study. A post hoc power analysis on G*Power 3.1 (https://www.psychologie.hhu.de) with effect size as d = 0.50 and an alpha of 0.05 was conducted. And the result indicated a power of 0.32, which might have compromised the significance of the statistical comparisons in this study.
Nevertheless, the findings have some implications for future study and practical improvement. First, the quadratic equation should be considered a standard nonlinear model to estimate hemodynamic changes in the DCCS tasks. Second, the Habit-DisHabit design DCCS could be widely used and further developed to better identify the neural correlates of cognitive shifting. Third, the finding that Non-users were more attentive and engaged than Heavy-users implies that we should consider limiting and regulating young children's tablet use.  Informed Consent Statement: Informed consent was obtained from all the participating parents involved in the study.
Data Availability Statement: All the data for this study will be available upon request.