The ex-Gaussian distribution of reaction times in adolescents with attention-deficit/hyperactivity disorder

Highlights

• The positive skew in reaction time (RT) distribution is most sensitive to differentiate individuals with ADHD from controls.

• A smaller mu (negatively correlated with commission errors), and larger tau (positively correlated with omission errors) were found in ADHD.

• As the ISI/Block increased, the magnitude of greater tau in ADHD than controls increased.

• The findings suggest tau would be related to the attention lapses due to effort regulation problems.

• The findings suggest that mu would be related to the impulsive response style.

Abstract

We investigated the three parameters (mu, sigma, tau) of ex-Gaussian distribution of RT derived from the Conners’ continuous performance test (CCPT) and examined the moderating effects of the energetic factors (the inter-stimulus intervals (ISIs) and Blocks) among these three parameters, especially tau, an index describing the positive skew of RT distribution. We assessed 195 adolescents with DSM-IV ADHD, and 90 typically developing (TD) adolescents, aged 10–16. Participants and their parents received psychiatric interviews to confirm the diagnosis of ADHD and other psychiatric disorders. Participants also received intelligence (WISC-III) and CCPT assessments. We found that participants with ADHD had a smaller mu, and larger tau. As the ISI/Block increased, the magnitude of group difference in tau increased. Among the three ex-Gaussian parameters, tau was positively associated with omission errors, and mu was negatively associated with commission errors. The moderating effects of ISIs and Blocks on tau parameters suggested that the ex-Gaussian parameters could offer more information about the attention state in vigilance task, especially in ADHD.

Introduction

Increased intra-individual variability in reaction time (RT) is one of highly consistent abnormalities in ADHD (Castellanos et al., 2005, Kuntsi et al., 2009). Most studies that assumed that RT was distributed in a Gaussian distribution found that RT was typically slower and more variable (represented by larger means and standard deviations) in individuals with ADHD than in controls (Andreou et al., 2007, Castellanos et al., 2005, Epstein et al., 2010). However, recent studies on ex-Gaussian distribution of RT showed that children with ADHD did not respond slower (Geurts et al., 2008, Leth-Steensen et al., 2000), but faster (Hervey et al., 2006) than controls when the contribution of exponential distribution was factored out; nevertheless, children with ADHD still had greater RT variability (Buzy et al., 2009, Vaurio et al., 2009), and a larger amount of abnormally slow RT (Buzy et al., 2009, Epstein et al., 2011b, Leth-Steensen et al., 2000). Some studies suggested that the infrequent slow RT might not be a cautious respose style, but related to lapses in attention to the perfomance instead (Epstein et al., 2011b, Hervey et al., 2006, Leth-Steensen et al., 2000). The psychopathological explanation for the right tail in RT distribution expressed by these ex-Gaussian parameters remains unclear.

The ex-Gaussian distribution is an estimated distribution combining a normal distribution and an exponential distribution (Heathcote, Popiel, & Mewhort, 1991; Luce, 1986). Three parameters are derived from the ex-Gaussian distribution: mu (μ) and sigma (σ), the mean and the standard deviation of the normal distribution, respectively; and tau (τ), a value describing the mean and the standard deviation of the exponential distribution. Analyses of ex-Gaussian distributions have been used to describe the distribution of RT in a variety of neuropsychological tasks (Epstein et al., 2011b), including choice reaction time (Geurts et al., 2008, Leth-Steensen et al., 2000), vigilance tasks (Hervey et al., 2006), Go/No Go (Vaurio et al., 2009), and working memory (Buzy et al., 2009), etc. The most consistent finding is that τ, a more sensitive measure of RT variability, is higher in individuals with ADHD (Buzy et al., 2009, Epstein et al., 2011b, Leth-Steensen et al., 2000); however, the findings of μ and σ, the parameters of a normal distribution have not been consistent. Some studies using working memory (Buzy et al., 2009) and Go/No Go tasks (Vaurio et al., 2009) showed smaller μ and bigger σ in individuals with ADHD than in controls, but other studies using choice RT tasks did not demonstrate a group difference in either parameter (Geurts et al., 2008, Leth-Steensen et al., 2000).

In light of these findings, infrequent slow RT represents important inattentive phenomena in RT responses in ADHD. Leth-Steensen et al. (2000) proposed that the τ effect could represent the dysfunctional regulation of adequate effort allocated to meet task demands, so the fluctuating response would easily appear to be an individual one in the non-optimal energetic state proposed by the cognitive-energetic model. In contrast, Schmiedek, Oberauer, Wilhelm, Suss, & Wittmann (2007) believed that τ was strongly related to working memory capacity and predicted control of attention in the structural equation model; therefore, as the cognitive load of the task increased, the value of τ might disproportionately increase in individuals with ADHD. Hervey et al.’s study (2006) used the Conners’ continuous performance test (CCPT) as the experimental task, and showed that as the inter-stimulus interval (ISI) lengthened, the value of τ increased disproportionately in ADHD. Using different versions of difficulty of the Go/No Go tasks (Vaurio et al., 2009) and working memory tasks (Buzy et al., 2009), other studies did not find that τ had increased significantly with increased task complexity (cognitive load) in ADHD. These results might support the hypothesis that τ is related more to a deficit in effort allocation in non-optimal states for individuals with ADHD, rather than to working memory capacity. The aim of this study was to examine this issue further, using not only event rates (ISI), but also using time-on-task (Block).

From the perspective of the cognitive-energetic model, both ISI and Block would be contextual factors that altered the psychophysiological state by both the level of arousal (defined as a time-locked, phasic, physiological response to input stimuli) and the level of activation (defined as a tonic, voluntary readiness for motor action) (Andreou et al., 2007, Kuntsi et al., 2009, Sergeant, 2005). In order to respond consistently as the state was changed by contextual variables, the effort system needed to mobilize extra energy to compensate for the discrepancy between the actual state and the required state (Sergeant & van der Meere, 1990; Van der Meere, 2002). Studies have shown that, for individuals with ADHD, performance was easily impacted by contextual variables, such as event rate (Andreou et al., 2007, Kuntsi et al., 2009) and time-on-task (Borger et al., 1999), because they had less extra energy to compensate for the non-optimal state. In the present study, all the values of RT in each trial of all the ISI (1, 2, 4 s) and all the block levels (1–2, 3–4, 5–6 Blocks) were used to compute the parameters of an ex-Gaussian distribution. Based on existent research, we hypothesized that individuals with ADHD would have more periods of slower RT than controls, especially if ISI lengthened or the Block number increased. This would be demonstrated by significant interactions between groups (ADHD versus typically developing, TD) and ISI/Blocks in the τ parameter.

Some studies have suggested that periodic slow RT might not be a cautious response style; however, few studies have examined the association between the ex-Gaussian parameters and response errors (errors of commission and omission). Epstein et al. (2010) found that children with ADHD exhibited a greater degree of RT slowing than did controls before and after errors of omission on the CCPT, suggesting that periodic long RT would be associated with the number of omission errors. Other studies suggested that the faster μ in ADHD children was related to an impulsive response style (Hervey et al., 2006), and that μ would have a negative association with the number of commission errors, but not omission errors on the CCPT.

Although the positive skew in RT distribution has been recognized as the most sensitive index to differentiate ADHD from controls by using the τ parameter of ex-Gaussian distribution (Buzy et al., 2009, Epstein et al., 2011b, Leth-Steensen et al., 2000, Vaurio et al., 2009), further investigation is warranted to clarify its psychological rather than mathematical meaning. From the perspective of the cognitive-energetic model, we hypothesized that τ would be directly moderated by energetic levels manipulated by the ISIs and Block effects. We also hypothesized that τ would be positively associated with omission errors, because both indexes implicate similar phenomena in attention lapses (Epstein et al., 2010), and μ would be negatively associated with commission errors, because it might be related to an impulsive response style (Hervey et al., 2006).