Introduction

Although the limits of cognitive capacity affect general performance, they are probably most apparent under conditions of distraction – that is, when multiple demands are simultaneously placed upon a person’s attention. Distraction and its impact on behavioural performance is a widely studied phenomenon that has been utilised to inform theoretical understandings of cognition and attention (e.g., Lavie, 1995, 2005, 2010; Näätänen, 1990; Näätänen et al., 2011); functional aberrations in a range of psychiatric conditions (Cortiñas et al., 2008; Gumenyuk et al., 2005); fatigue (Lim & Dinges, 2010); multi-tasking (Sanbonmatsu et al., 2013); and neuroanatomical function (Rinne et al., 2007; Sabri et al., 2006).

Empirically, distraction manifests in data in three ways: prolonged response times (RTs) and increased error and response-omission rates. Although prolonged RTs and increased error rates have been the subject of extensive investigation (e.g., Liete & Ratcliff, 2010; Luce, 1986; Ratcliff, 1978), increased response omissions (also known as ‘misses’ or ‘non-responses’) have not. In many cases response-omission rates are not reported, and sometimes they are even pooled with commission error responses and generically classed as errors (e.g., Mager et al., 2005; Miller et al., 2009).

We know of no reason, apart from potential rarity, that response omissions do not have the same status as a measure of distraction as accuracy and RT. On a practical level, step omissions, that is leaving out a necessary step in a task sequence but potentially still responding, are regarded as the most common human error, with a proliferation of research aimed at developing strategies to avoid such omissions (for reviews, see Reason, 2002, Reason, 1998). On a theoretical level, ignoring response omissions represents a commitment to their arising from ‘contaminant’ process, which is entirely independent from the task at hand, and so can be safely ignored when interest focuses on the process generating task-related responses.

There are often practical reasons that researchers must stipulate response windows outside of which responses are not recorded, resulting in the creation of response omissions by design related to slow but otherwise valid responses to a task. This can sometimes be justified if such ‘outlying’ (i.e., unusually fast or slow) responses are classed as being in part or whole generated by a contaminant process. However, to the degree that this is not true, there is the risk of potentially misleading impacts on summary statistics for RTs and error rates particularly when RT distributions have been cut off (Ulrich & Miller, 1994). Although there are ways that these impacts can be avoided (see, Heathcote, 1996; Kendall & Stuart, 1967; Ulrich & Miller, 1994), response omissions that occur as a consequence of design can be hard to detect, particularly when they occur at relatively low levels, as they are not always apparent in visual inspections of the RT distribution (Ulrich & Miller, 1994).

In the current paper we explore the degree to which response omissions arise from a contaminant process versus from the same task mechanisms producing responses, with the latter being described as ‘task-related’ omissions. Note that contaminant omissions are defined by a process of elimination, that is, as what is not accounted for by task-related omissions, so the processes that give rise to them are unspecified except that they are not the processes giving rise to task responses. Consequently, a defining feature of contaminant omissions is that they do not differ as a function of task-related manipulations, whereas (definitionally) task-related omissions may do so.

We use evidence-accumulation models (EAMs) to provide an explicit account of the process by which responses are usually produced. EAMs have long offered a comprehensive solution for analysis of RT and accuracy data. EAMs describe the process of choice, assuming evidence favouring different choice options accumulates over time until a threshold is reached, and an associated response is triggered. EAMs provide a comprehensive account of responses, including the probabilities of correct and error responses and the shapes of the corresponding RT distributions but, in most cases, omissions have been ignored. An additional benefit of EAMs is that they also allow quantification of latent psychological processes involved in decision making and response execution (e.g., Brown & Heathcote, 2008; Donkin & Brown, 2018; Leite & Ratcliff, 2010; Luce, 1986). Exploring how different experimental manipulations modulate parameter estimates, or how parameter estimates differ between different populations is ultimately what enables insights into latent psychological processes and informs broader theoretical understandings.

By augmenting an EAM to provide an account of omissions data, we leverage their explanatory ability to elucidate the latent psychological processes that underpin the occurrence of omissions. This approach also has the potential advantage of using any extra information available in the omission rate – information that has previously been considered noise. We use this modelling framework to examine two data sets – one from an experiment reported by Castro et al. (2019) where participants made a detection response and a new experiment requiring choice responses. In both experiments, factors causing distraction from the task at hand were manipulated. Our model is based on Brown and Heathcote’s (2008) Linear Ballistic Accumulator (LBA), and provides a unified account of the probability of omissions and responses, including the distribution of RTs for both detection and choice responses. The model also allows us to differentiate between two types of task-related omissions, design omissions occurring due to slow RT cut-offs imposed in the experiments we examined, and intrinsic omission caused by a failure of the task process to produce a response.

We aim to establish a methodology to determine the prevalence of the three types (contaminant, design, and intrinsic) of omissions. In the next section, we provide a brief overview of the relatively small body of research that has acknowledged the occurrence of omissions. We then propose our cognitive model of omissions and apply it to data. Table 1 provides an overview of our framework for understanding response omissions.

Table 1 A taxonomy of omissions. The levels of response-task omissions are both mediated by effects on the accumulation rate parameters of the LBA response process, whereas non-task omissions are mediated by a parameter unrelated to the response process
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Omissions in past distraction research

Past research using omissions as a dependent variable to measure attention and/or distraction has used both detection and choice tasks. Examples of detection tasks include the psychomotor-vigilance task (e.g., Dinges & Powell, 1985; Lim & Dinges, 2008), the continuous-performance task (e.g., Rosvold et al., 1956), and the detection-response task (DRT, e.g., Castro et al., 2019; Howard et al., 2020). Examples of choice tasks include the distraction paradigm (e.g., Schröger & Wolff, 1998), the n-back task (Evans et al., 2018), and dual-task perceptual choice (Howard et al., 2020). Our focus here is on the DRT and the distraction paradigm; we consider the other paradigms in the General discussion.

The DRT is an International Standards Organization (ISO, 2016) method of quantifying cognitive workload in driving (e.g., Strayer et al., 2015). In the DRT participants are required to respond to a target stimulus occurring randomly every 3–5 s while concurrently performing a primary task or tasks (e.g., driving and talking on a cell phone). The target can be presented in the visual or tactile modalities. Increased load in the primary task is found to increase the number of omissions (Castro et al., 2016; Castro et al., 2019; Cooper et al., 2016).

The distraction paradigm (e.g., Schröger & Wolff, 1998) is used to investigate how selective attention systems filters extraneous changes in stimulus features. In classic versions a task-irrelevant auditory odd-ball sequence accompanies a primary choice task (e.g., numerical parity; Andrés et al., 2006). Participants are instructed to focus exclusively on the primary task. The occurrence of task-irrelevant oddballs has been found to increase omission rates in the primary task (e.g., Domínguez-Borràs et al., 2009; Gumenyuk et al., 2005; Parmentier et al., 2010; Schröger, 1996; cf. Escera et al., 2002; Muller-Gass et al., 2006).

For both detection and choice tasks, there is evidence of changes in omission rates with response-task manipulations. Howard et al. (2020) studied both a DRT with a shorter 2.5-s response window and a modified version that required a choice rather than detection response, again with a 2.5-s response window. They manipulated the difficulty of a primary object-motion tracking task, but only reported omission results for the detection task, which increased in the highest load condition. Castro et al. (2019) also examined a choice modification of the DRT and reported substantially reduced omissions relative to the standard DRT.

Increased omission rates are interpreted, at least in part, as being indicative of inattention – an inability to sustain attention and inhibit distraction. However, even in the areas of research that occasionally feature omissions as a dependent variable, very little effort has gone into their further exploration. Any additional information omissions might contain about decision making processes, particularly when distracted, has been underutilised. Although omissions are often collapsed with commission errors (e.g., Mager et al., 2005; Miller et al., 2009), there is evidence that, at least in some instances, patterns of omission and commission errors can differ (Meule, 2016; Meule et al., 2012), suggesting that it is better to consider omissions separately and that there might be extra psychological insights to be gained by understanding their cause or causes.

Castro et al. (2019) incorporated an account of omissions into an evidence-accumulation model of the standard and choice versions of the DRT using a diffusion process with a single positive barrier (Heathcote, 2004), but they did not differentiate among different possible causes of omissions. Evans et al. (2018) and Howard et al. (2020) took account of design omissions associated with slow RT cut-offs in the data they examined by applying the approach for fitting censoredFootnote 1 distributions proposed by Kendall and Stuart (1967; also Heathcote, 1996, and Ulrich & Miller, 1994) to modify the likelihood equations of a version of Brown and Heathcote’s (2008) LBA model proposed by Heathcote and Love (2012) that allows only positive accumulation rates. Ratcliff and Van Dongen (Ratcliff & Van Dongen, 2011; see also Ratcliff & Strayer, 2014) fit the single-barrier diffusion model to psychomotor-vigilance task data and accounted for ‘lapses’ (responses slower than 0.5 s) through both censoring and intrinsic omissions. Intrinsic omissions are caused by negative evidence-accumulation rates due to Gaussian trial-to-trial variability, which can result in a failure to ever reach the positive barrier.

In the next section we combine these approaches to provide a more comprehensive account of omissions. We use Brown and Heathcote’s (2008) original LBA, which allows negative accumulation rates due to Gaussian trial-to-trial variability, to account for intrinsic omissions. We use Kendall and Stuart’s (1967) approach to account for design omissions through censoring. Finally, we use a version of the approach of Castro et al. (2019) to model contaminant omissions (see also Matzke et al., 2019).

A comprehensive cognitive model of omissions

In the LBA parameters corresponding to different psychological processes include: the rate evidence is accumulated at, which can be broken down to the average rate over accumulators for each response (quantifying the urgency with which responses are made) and the difference between the rates for the accumulators that match and mismatch the stimulus (quantifying the quality of the evidence on which the decision is based); the threshold (b) amount of evidence required to trigger a response, which can be broken down into the average over accumulators (which quantifies response caution) and the difference between accumulator threshold (which quantifies response bias); and the non-decision time (ter) required for processes such as stimulus encoding and response execution. For detection tasks a single threshold and accumulation process is assumed so only the rate, threshold, and non-decision parameters are relevant. Before we outline how these parameters are expressed and their underlying mathematical properties, it is useful to provide a review of the three types of omissions we have defined. The first two – intrinsic omissions (due to between-trial variation in accumulation rates making it impossible to ever reach the evidence threshold) and design omissions (due to a priori response windows that cause slow responses not to be recorded) – are determined by the same process that produces responses; and a third, contaminant omissions, allows for a cause unrelated to the response process.

Often, one or more of the aforementioned parameters is assumed to randomly vary between trials, with extra parameters corresponding to the level of variation. For the LBA the starting points (or equivalently thresholds) of evidence accumulation have a uniform distribution (between 0 and A), rates have a normal distribution (with mean v and standard deviation sv), and both assumed independent over accumulators. As the rate distribution is unbounded the sampled rate for a trial can be negative with probability \(\Phi \left(\frac{v}{s_v}\right)\) , where Φ(x) is the integral from −∞ to zero of a normal distribution with mean x and a unit standard deviation. In the case of the single-accumulator model for detection tasks (e.g., the DRT) this means that the probability of an intrinsic omission is \({p}_I=\Phi \left(\frac{v}{s_v}\right)\) . In the case of the two-accumulator model for choice tasks (e.g., the distraction paradigm), the probability of an intrinsic omission is given by the probability that both matching and mismatching accumulator have negative rates at the same time, \({p}_I=\Phi \left(\frac{v_T}{s_{vT}}\right)\times \kern0.5em \Phi \left(\frac{v_F}{s_{vF}}\right)\) where the subscript T indicates the matching accumulator and F the mismatching accumulator. Brown and Heathcote (2008) noted that in their fits to data in which omissions were rare or unreported, pIwas negligible, and so they did not take account of this possibility in the likelihood equations they used to obtain fits.

The probability of design omissions is determined by the response window and the LBA accumulator probability density function, f(t | A, b, v, sv), and cumulative distribution function, F(t | A, b, v, sv) (see Brown & Heathcote, 2008, for the corresponding equations), where t is the decision time (i.e., RT – ter). In the cases we address here, the response window excluded only slow RTs greater than an upper cut-off, U. For the detection case the probability of a design omission is pD = 1 − F(U). For the choice case we first require the probability densities for correct, fc(t) = f (t | AT, bT, vT, svT)(1 − F(t | AF, bF, vF, svF)) and error, fE(t) = f(t | AF, bF, vF, svF) (1 − F (t | AT, bT, vT, svT)), responses. The probability of a design omission is then the sum of the probabilities that correct and error responses exceed \(U,{p}_D={\int}_U^{\infty }{f}_E(x)\ dx+{\int}_U^{\infty }{f}_c(x)\ dx\), where each probability is obtained by numerical integration.

Both the probability of design and intrinsic omissions, and hence the overall probability of a task-related omission, pT, have different value for each cell of the experimental design that differ on one or more accumulator parameters (i.e., A, b, v, or sv), so for design cell i, pTi = pIi + (1 − pIi) pDi. Note that no extra parameters need to be estimated to produce predictions about task-related omissions. The ability of the model to discriminate task-related from contaminant omissions comes from the former being constrained by the observed responses (i.e., having to fit response probabilities and RT distributions). The contaminant component adds extra freedom to explain omissions that cannot easily be accommodated by the response process. We make the assumption that only the evidence accumulation process is affected by the factors that constitute the experimental design, and hence that the probability of omission due to contamination, pC, applies to all design cells. Hence, the overall probability of omission for design cell i is pOi = pC + (1 − pC)pTi . Therefore, differences in omission rates between design cells are entirely accounted for by the evidence accumulation process, whereas some or all of the overall level of omissions is accounted for by contamination at the cost of requiring one extra parameter estimate.

This set of assumptions helps to make it possible to identify the parameters of the model. The supplemental materials (available at https://osf.io/hb4dw/) provide detailed methods and results for a series of parameter-recovery simulations. We evaluated the recovery of the three types of omission parameters on two dimensions: accuracy and the calibration of uncertainty. The detection response task performed well on both dimensions. Accuracy was evident in posterior mean estimates of omission parameters from fits to simulated data falling close to the data-generating values for all parameters. Calibration of uncertainty was evident in the proportion of times that the 95% credible intervals encompass the data-generating omission parameter value being close to the nominal value. Estimation in the distraction paradigm was similarly accurate, but the uncertainty of the omission parameter estimates was slightly underestimated (i.e., the percentage of data-generating values falling in 95% credible intervals was slightly less than nominal, indicating that their width was underestimated). Given the relatively minor variations from ideal behavior these results indicate that the omission parameter estimates we report can be interpreted with confidence.

In the following two sections we assess the evidence for each of the model’s three types of omissions, first in the simpler case of Castro et al.’s (2019) DRT data and then in more complex case of our new distraction-paradigm data. In the General discussion we consider alternative assumptions about how omissions are generated.

Omissions in the detection response task

Castro et al. (2019) manipulated cognitive load by having 20 participants count backwards by 3s in one condition and not in another. These conditions were crossed with four other conditions, three of which required participants to perform versions of the DRT – detecting either a lower or a higher intensity light in two and discriminating the lower versus higher intensity lights in a third choice-DRT condition. No extra task was performed in the fourth condition. Omissions were most frequent in the detection conditions, at an average of around 5%, so we analyse only those conditions, which form a 2 (stimulus: low vs. high intensity) × 2 (load: none vs. 3s) within-subjects design.

Omissions were more common with the secondary-task load (6%) than without (4.3%), χ2(1) = 19.8, p < .001. RT was also much slower with the load, by 0.16s, χ2(1) = 1080, p < .001, and slower for low- than high-intensity stimuli, by 0.02s, χ2(1) = 37.9, p < .001. These results suggest that RT and omission rates are related (i.e., both RT and omissions increase with load) but not entirely redundant (stimulus intensity affects RT but not omissions). Note that throughout this paper tests on manifest measures were carried out using linear mixed models (Bates et al., 2015), assuming Gaussian error for the log of RT and a binomial model with a probit link function for binary measures, with Type III Wald test statistics in both cases (Fox & Weisberg, 2018).

We first fit seven different versions of the LBA-Omission (LBAO) model that differed in the parameters that were affected by the load factor (i.e., separate parameter values were estimated for each level of the load factor), constituting all possible combinations in the power set of one or more of the three mean rate, rate standard deviation, and non-decision time parameter types (23−1 = 7, omitting the model with no effect of any of the three parameters). We then report analyses based on two additional follow-up models that enable us to follow up on results obtained with the power-set models.Footnote 2

All models assumed a common A parameter across all conditions, and we assumed different mean rate and rate standard deviation parameters for high and low stimuli. In order for the standard LBA model to be identifiable either a rate mean or standard deviation or threshold parameter must be fixed, with the choice of which being arbitrary as a simple re-parametrization makes the effect of fixing one exactly equivalent to fixing another (see Donkin et al., 2009), and a common choice being a rate standard deviation parameter. However, in the LBAO model both rate parameters can affect intrinsic omissions, whereas the threshold does not, and so the equivalence does not hold. Hence, we choose to fix the threshold for identifiability. As low and high stimulus conditions had almost entirely separate parameters, we fixed the threshold at 1 in both. We fit the models using the software and Bayesian methods described in Heathcote et al. (2019) with relatively non-informative priors (see Online Supplementary Materials (OSM) for details, and files at https://osf.io/hb4dw/ for all of the data and R code for all of the model fits described in this paper).

Detection response task modeling results

Table 2 reports model-selection results based on the Deviance Information Criterion (DIC; Spiegelhalter et al., 2002), where smaller values indicate a better trade-off between goodness-of-fit and a penalty for additional model complexity. However, DIC alone does not provide an absolute indication of the best model, and should be considered only as a guide, especially when models differ by a relatively small margin (see Spiegelhalter et al., 2014, for further discussion). Hence, we also consider other evidence, particularly posterior predictions (i.e., the distribution of data outcomes predicted by the fitted model), to assess whether a model adequately captures key features related to our focus here, omission probability (see Morey et al., 2012, for a discussion of the importance of such model checks). Open availability of our data and modelling code enables readers to reach independent conclusions based on the types of evidence and assumptions they favour.

Table 2 DIC model selection results (better models have smaller values) for seven models constituting all possible combinations of load effects on Linear Ballistic Accumulator (LBA) rate means (v) and standard deviations (sv) and non-decision time (ter) parameters with the DIC for the best model, min(DIC) = - 1515, subtracted
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The full model is selected with the largest load effect being on mean rates and the least on non-decision time (i.e., the increase in DIC caused by dropping a single parameter from the full model is least for non-decision time). This pattern is consistent with Castro et al.'s (2019) DIC model-selection results, except that they fixed the moment-to-moment variability parameter of the single-barrier diffusion model (which does not include a trial-to-trial variability parameter) to make it identifiable, and found that load affected the threshold parameter.

Figure 1 plots average omissions and mean RT along with fits of the selected model. In the Bayesian framework, uncertainty (indicated by 95% credible intervals) is indicated for the model rather than the data. The model provides a good account of the positively skewed RT distributions and the increase in RT with load as well as the percentage of omissions, although there is slight underestimation for low load stimuli.

Fig. 1
figure 1

Top row: Response time (RT) distributions (lower, middle and upper lines and points are 10th, 50th and 90th percentiles, respectively). Bottom left panel: Omission percentages data (open symbols), with model fits of the selected model in Table 2 as posterior predictive medians (solid points) and 95% credible intervals. Bottom right panel: Individual participant omission percentages against the posterior median of the contaminant omission parameters estimated by the model. Load conditions: none and 3s (i.e., count backwards by 3s)

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Figure 1 also plots individual omissions, showing that three participants had particularly high omission rates, and that these were almost entirely accounted for by the model’s contaminant omission parameter. However, the figure makes it clear that for most other participants the task-related omissions are necessary. This is also clear from the fact that omissions vary as a function of the experimental manipulations, which cannot be explained by the contaminant parameter as it is assumed fixed across these conditions.

Figure 2 shows how the two types of task-related omission were estimated to vary over experimental conditions by the selected model. In every case, omissions appear to increase with load, consistent with the effect of load on the observed omission rates. To provide a test we calculated task-related omission rates for every posterior parameter sample and computed medians and Bayesian 95% credible intervals on the posterior predictive distributions of differences over manipulations. Credible intervals (provided in square brackets below) were estimated by the range between the 2.5th and 97.5th percentiles of the resulting distribution. There was a main effect of load with both stimuli for intrinsic (low: 0.74% [0.42, 1.04]; high: 0.71% [0.48, 0.99]) and design (low: 1.18% [0.9, 1.41]; high: 1.24% [1.02, 1.24]) omissions. No main effects or interactions with the stimulus factor were supported except a slightly higher rate of design omissions for low than high stimuli with no load (0.21%, [0.04, 0.42]). Intrinsic omissions were less common than design omissions both without load (low: 0.26% [0.16, 0.34]; high: 0.18% [0.08, 0.25]) and more so with load (low: 0.69% [0.53, 0.84]; high: 0.7% [0.56, 0.83]).

Fig. 2
figure 2

Median and 95% credible intervals for the posterior predictive distribution of task-related omissions computed for the model selected in Table 2. The left column displays intrinsic omissions and the right column design omissions with the upper row containing results for low stimulus and the lower row for high stimuli

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We also examined posterior difference distributions to test effects on model parameters. As expected from the model selection results, and consistent with the idea that a secondary task drains cognitive capacity as reflected in accumulation rates (Castro et al., 2019), increased load decreased mean rates from 5.96 to 4.06 (a difference of 1.89 [1.78, 2.02]). Also consistent with their being allowed to vary with stimulus in the models and the standard interpretation of rates reflecting stimulus strength, mean rates were greater for high than low intensity stimuli (0.26 [0.18, 0.34]). Although there was no support for a main effect of stimulus on rate standard deviations, there was a bigger load effect for low than high stimuli (0.18 [0.05, 0.32]). The main effect of load was caused by a decrease in rate standard deviations with load from 2.1 to 1.58 (0.52 [0.43, 0.61]), but the decrease was largely proportional to the mean, so the ratio of mean to standard deviation decreased only slightly with load (0.13 [-0.02, 0.27]).

In light of the results just reported we fit two extra models. Because of the relatively weak intrinsic omission rates, we fit a model in which they are absent as the rate distribution is truncated to be positive (Heathcote & Love, 2012). Consistent with the observed small effects, this model, which was parameterized as for the winning model in Table 2, was preferred by the DIC by a small margin (32). However, as shown in the OSM (Fig. S1), this model had an exaggerated under-prediction of omissions for low-intensity stimuli. Given this, we conclude that the inclusion of intrinsic omissions is warranted but that its effect is smaller than that for design omissions.

Second, because three participants had much higher omission rates (15%, 26% and 32%) whose overall level was largely explained by the contaminant parameter (see Fig. 1), we fit a model without the contaminant parameter (but otherwise the same as the model selected in Table 2) to see if it was necessary in the remaining participants. The support for the model with the contaminant parameter remained, with the model selected in Table 2 winning on DIC by 55, but the DIC difference was much smaller than for the fit including all participants (369). Further, without contaminant omissions the misfit to observed omission rates was very marked (see OSM Fig. S1) suggesting that it was impossible to increase the level of task-based omissions sufficiently to compensate for contaminant omissions. Hence, we conclude that contaminant omissions are necessary, at least for a subset of participants.

Detection response task discussion

The LBAO model was able to provide a quite accurate representation of all aspects of performance in the DRT, ranging from RT distribution to omission rates. The model even accurately captured participants with high omission rates who might ordinarily be excluded from analyses due to being deemed non-compliant with experimental instructions. Given that participants had to simultaneously perform the tracking task and counting backwards in one condition, higher omission rates do not seem unreasonable, and may simply represent part of the normal continuum of methods for coping with an attention demanding task. The LBAO model was able to provide a good fit to data from these participants (see OSM Fig. S2 for individual plots), including the RT data for the responses they made. This provides evidence that the LBAO was able to successfully separate out the effects of the contaminant process.

The LBAO model suggested that omissions related to the primary task were increased by the distraction caused by a secondary-task load (counting backward by 3s). This was true of both intrinsic omissions, occurring because the stimulus did not cause any positive evidence accumulation towards the response threshold, and of design omissions, where the positive accumulation was so weak that it did not result in threshold crossing in under the 3-s time limit dictated by the ISO standard. This result is striking because even the slow 90th percentile of RT was little more than 1 s on average (see Fig. 1), indicating that the LBAO produced distributions with a long thin slow tail that contains non-negligible probability mass at quite long RTs.

The incidence of intrinsic omissions was much less, and indeed there was some evidence that they were not a necessary part of a parsimonious model, although their exclusion led to some noticeable misfit in omission rates for low-intensity stimuli. Note that we did not allow contaminant omissions to vary with load in order to determine if task-related omissions alone could account for its effect. Given the good fit we obtained this seems to be the case here.

In the next section we investigate whether the LBAO model can also provide a good description of choice behaviour in the distraction paradigm and how the relative levels of all three types of omissions differ in this case.

Omissions in the distraction paradigm

In the distraction paradigm, as well as examining the effect of distraction, we also explored differences in omission behaviour due to aging, comparing the performance of a young and healthy older groups. Participants made a choice response classifying tones as long (0.25 s) or short (0.1 s). On a minority of ‘deviant’ trials the task irrelevant tone pitch could either be higher or lower than on the majority of trials, causing an oddball distraction effect. As reviewed in the opening section of this paper, the occurrence of a deviant can increase omission rates, and is known to prolong RT and decrease accuracy rates (Schröger & Wolff, 1998) in the primary task.Footnote 3

In light of evidence that distraction also affects performance on the trial after a deviant occurs, we conducted a 2 × 2 × 5 mixed model analysis with the between subject factor of age (young vs. old), and the within-subject factors of stimulus (long vs. short) and deviance (S = standard after standard, HD = high deviant after standard, LD = low deviant after standard, SAHD = standard after high deviant, and SALD = standard after low deviant). Before reviewing our EAM results, below we provide an overview of the methodology used and results from traditional analysis techniques.

Method

Participants

Participants were 34 younger (18−38 years, M = 23 years, SD = 4.85 years; 22 females) volunteer community members and undergraduate students from the University of Newcastle, and 23 older (59−74 years, M = 67 years, SD = 4.02 years; 12 females) volunteer community members.

Apparatus

Assessments of hearing thresholds were conducted using an Earscan 3 ES3S pure tone audiometer in line with the American Speech-Language-Hearing Association Guidelines for Manual Pure-Tone Threshold Audiometry (2005). Experiments were delivered using Presentation© version X by a standard PC running Windows XP, on a 27-in. LED monitor (60 Hz). Sound stimuli were delivered binaurally via Sennheiser HD 280 professional headphones.

Stimuli

Sounds were equiprobable short (0.1s) and long (0.25s) pure tones with .005s rise/fall times. Tone frequency (i.e., pitch) was manipulated to produce an auditory oddball sequence. Regular tones had a frequency of 700 Hz (p = .75), while rare tones were either of a lower (613 Hz; p = .125) or higher (1,560 Hz; p = .125) frequency. All frequencies were presented equiprobably across short and long tones. All tones were delivered at an intensity of 75 dB SPL.

Task

Participants completed an auditory duration discrimination task. Specifically, they were asked to indicate whether presented tones were short or long in duration by pressing a left or right button on a custom built two-button response terminal. Response mapping was counterbalanced between participants with the hand-to-button-press requirements (i.e., left index finger to left button and right index finger to right button) explained and demonstrated to participants prior to task onset. Participants completed 800 duration discrimination trials, split equally across four blocks. Blocks were separated by a 1-min rest break. Tones had an inter-stimulus interval of 1.6 s. Participants were instructed to respond as quickly and accurately to tones as possible.

Procedure

Participants attended two testing sessions, each approximately 2 h in duration. In session one, participants completed assessment of suitability for inclusion based on mental health history, as well as several other psychometric measures for use in another study (Todd et al., in preparation). In session two, participants completed the duration discrimination task. Prior to the beginning of the task participants were fitted with a 64-channel ActiveTwo Biosemi EEG system to record continuous electroencephalogram (EEG) data during completion of the duration discrimination task, also for use in another study (Todd et al., in preparation). Finally, participants were fitted with headphones and asked to stay as still as possible while they followed instructions presented on the screen and completed the duration discrimination task.

Results

The first three trials of every block were removed from analysis as is standard with most distraction paradigm tasks so that the pattern is established that makes deviant tones distracting. Responses faster than 0.2 s were then removed from analysis, which reflected 0.78% of responses by old and 0.075% of responses by young participants. The programmed response window closed at 1.45 s; however, due to computer timing jitter, eight responses slightly greater than 1.45-s duration were identified and removed from analysis. Of these, five were for old (0.03%) and three for young (0.01%) participants. See Figs. 3 and 4 for plots of the results for which inferential results are reported next.

Fig. 3
figure 3

Fit of the most flexible intrinsic and contaminant model in Table 3. The three lines in the left column are the 10th, 50th and 90th percentiles of response time (RT) distributions. Data are open circles joined by lines. Fits are shown with 95% credible intervals. HighD = High Deviant, LowD = Low Deviant, SAHD = Standard After High Deviant, SALD = Standard After Low Deviant

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Fig. 4
figure 4

Parameters and fit of the most flexible intrinsic and contaminant model in Table 3. The left column shows individual participant omission percentages against the posterior median of the contaminant omission parameters estimated by the model. The middle and right columns show differences between the standard and deviant conditions omission percentages for the data (solid symbols) and the model (open symbols with 95% credible intervals). HighD = High Deviant, LowD = Low Deviant, SAHD = Standard After High Deviant, SALD = Standard After Low Deviant

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Mean RT was faster for short than long stimuli (0.616 s vs. 0.633 s), χ2(1) = 96, p < .001, and was clearly shorter for standard trials than other trials (S: 0.601 s, HD: 0.645 s, LD: 0.651 s, SAHD: 0.656 s, SALD: 0.635 s), χ2(4) = 925, p < .001. Although the main effect of age just failed conventional significance, the interaction of age and stimulus was significant; older participants responded 0.044 s slower to short stimuli and 0.035 s slower to long stimuli, χ2(1) = 13.4, p < .001.

Accuracy was greater for short than long stimuli (90% vs. 88%), χ2(1) = 41, p < .001, highest in the standard condition, and slightly lower in the other deviance conditions except in the standard-after-high-deviant condition, where it was markedly lower (S = 91%, HD = 78%, LD = 90%, SAHD = 89%, SALD = 90%), χ2(4) = 937, p < .001. The latter effect was driven by an interaction between tone length and frequency whereby long-duration high deviants were perceived as short-duration tones (accuracy 61%), χ2(3) = 899, p < .001. Older participants were less accurate on average than younger participants (82% vs. 93%), χ2(1) = 19.7, p < .001. The misperception effect was also stronger, with accuracy for long-duration high deviants less than chance for older participants (45%) but better for younger participants (72%), χ2(1) = 49.6, p < .001.

Omissions were more common for older than younger participants (6.4% vs. 2.2%), χ2(1) = 7.3, p = .007, and for short than long stimuli (4.2% vs. 3.6%), χ2(1) = 15.7, p < .001. They were most common following the misperceived high-deviant in the standard-after-high-deviant condition, χ2(4) = 49.7, p < .001, particularly for young participants, χ2(4) = 23.8, p < .001. Detailed patterns are shown along with model fits below.

Distraction paradigm modeling results

We fit young and old participants separately, and as with the DRT we fixed threshold parameters in order to make the LBAO model identifiable. We reasoned that thresholds could mediate a response bias (i.e., long and short response accumulators could have different thresholds) and that thresholds might differ on trials after deviants relative to other trials (as participants could easily detect the occurrence of a deviant and then had sufficient time before the next trial to adjust their threshold). As the most complex model we considered had different rate parameters for all stimuli, we fixed the threshold at 1 for the short accumulator for all conditions and estimated separate long accumulator thresholds for trials after a deviant and after a standard trial to allow for differential response bias.

In light of the strong differences among the ten stimulus and deviance conditions we allowed separate mean rate parameters for each, resulting in a total of 20 estimated parameters, ten for matching accumulators and ten for mismatching accumulators. Given their critical role in determining both design and intrinsic omissions, we explored three models with progressively more complex rate standard deviations, either only differing between matching and mismatching accumulators (two estimated parameters), also differing with the deviance factor (ten estimated parameters), or also differing with deviance and stimulus factors (20 estimated parameters). We fully crossed these three cases with three types of omission model: the full omissions model allowing for contaminant and task-related (design and intrinsic) omissions and models removing one mechanism, a second with only contaminant and design mechanisms to test the necessity of our new proposal of an intrinsic mechanism, and a third removing the contaminant mechanism to test the efficacy of the task-related mechanisms in isolation. All nine resulting models we assumed the same value of non-decision time and start-point noise for each condition.

Table 3 shows that for every combination of contaminant mechanism, the most complex model of rate standard deviations was preferred. Older participants showed the same pattern of results as for the DRT, with support for design and contaminant omissions but not for intrinsic omissions, whereas younger participants displayed support for design and intrinsic but not contaminant omissions.

Table 3 DIC minus minimum DIC separately for young (minimum DIC = -19156) and old (minimum DIC = 2744) participants for the nine models crossing the three combinations of omission mechanisms with the three parameterizations of rate standard deviations (M = match vs. mismatch, D = deviance and S = stimulus factors). N = number of estimated parameters
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Figure 3 shows the fit of the full omission model. Although its account of responses (i.e., accuracy and RT distributions) is good, it underestimates omissions by a constant amount across all conditions, particularly for older participants and short stimuli. We speculated that the perceptual interaction between tone length and frequency might be the cause, and so fit a model in which the probability of contaminant omissions, which modulates the overall level of omissions across the different deviance conditions, differed for long and short stimuli. This model was preferred by DIC, with the same type of models winning by an extra 192 and 128 units for old and young participants, respectively. Although the fit to omissions was better misfit remained (see OSM Fig. S3). Given the fact that allowing contaminant omissions to differ as a function of experimental manipulations blurs the boundary between them and decision-process related omissions, we chose to focus further analysis on the model with fits depicted in Fig. 3.

The left column of Fig. 4 plots the individual participant omission percentages against the model’s contaminant estimates. In contrast to the DRT data, although one of the higher omission-rate cases was predominantly explained by contamination, two were not. The remaining columns of Fig. 4 focus on the model’s ability to fit the pattern of omission differences across deviant conditions by subtracting out the standard condition. As the contaminant process adds a constant to all conditions it does not play a role in this explanation. The figure shows that the model’s task-related omission components are generally quite good at accommodating the pattern of differences, with the exception of the underestimation for old participants with long stimuli in the high deviant condition and for young participants for short stimuli in the standard after high deviant condition.

Figure 5 plots a breakdown of task-related omission estimates for correct and error responses. Intrinsic omissions are notably rare for older participants, and although generally low for younger participants there are noticeable elevations in some deviant conditions. This is consistent with the DIC support for the inclusion of intrinsic omissions in Table 3. However, the large credible intervals indicate that the elevation in intrinsic omissions applies to only some participants. Design omissions are elevated for older participants, consistent with their generally slower responses.

Fig. 5
figure 5

Task-related omissions with 95% credible intervals for the most flexible intrinsic and contaminant model in Table 3 for correct and error responses in each condition. HighD = High Deviant, LowD = Low Deviant, SAHD = Standard After High Deviant, SALD = Standard After Low Deviant

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Detailed parameter estimates are found in the OSM (see Figs. S4, S5, and S6) and we summarize the most relevant aspects here. On average over accumulators mean rates were higher for short than long stimuli but did not vary greatly over deviance conditions. The difference between mean rates for the matching and mismatching accumulators also did not vary much over stimulus and deviance conditions. The only exception to this trend is for a sharp decrease in the high deviant condition for long stimuli, particularly for older participants. This reflects the marked tendency for old participants to classify long high deviant stimuli as short. Rate standard deviations averaged over accumulators did not differ much between conditions. As is usually found with the LBA, the mismatch accumulator had a greater standard deviation than the match accumulator, with this difference being greater for short than long stimuli. Again, for the difficult high-deviant condition things differed, with no difference for short stimuli and a reversal for long stimuli. As also shown in the OSM, participants were generally biased towards short responses. Consistent with all models allowing thresholds for each accumulator to differ on trials following a standard and following a deviant, the bias for young participants was greater after a deviant, whereas for old participants it was greater after a standard.

General discussion

We proposed a variant of Brown and Heathcote’s (2008) LBA evidence-accumulation model in which participants fail to make a response due to factors related to the decision process or to a contamination process. We defined three types of omission: the first two – intrinsic omissions (due to between-trial variation in accumulation rates making it impossible to ever reach the evidence threshold) and design omissions (due to a priori response windows that cause slow responses not to be recorded) – are determined by the same process that produces responses; the third – contaminant omissions – allows for a cause unrelated to the response process. Because of the latter stipulation, contaminant omissions were assumed to be unrelated to the factors affecting task-related omissions, and so were fixed to be constant across experimental condition that we examined.

We fit this LBA Omission (LBAO) model to both detection data from a DRT task (Castro et al., 2019) and to new choice response data from a distraction paradigm task (Schröger & Wolff, 1998). Both paradigms manipulated experimental factors that were likely to cause omissions through causing distraction. In Castro et al. this was the load from other tasks that had to be performed in parallel with the detection responses. In the distraction paradigm this was relatively rare (and so attention attracting but response-task-irrelevant) variations in an otherwise repetitive auditory stream. In both data sets RT for responses longer than a slow cut off (3s in the DRT and 1.45 s in the distraction paradigm) was not recorded. Also in both paradigms, a few participants had unusually high omission rates, including cases in which almost half of the response were omitted. Commonly, the latter participants are classified as non-compliant with experimental instructions and excluded from analysis. However, in the present study we included these participants in our analysis in order to see if the LBAO model was able to accommodate such a wide variation of omission rates.

We found that the LBAO model displayed varying degrees of success. It provided a quite accurate account of the detection data, both in terms of the distribution of RTs and the probability of omissions, with both being elevated under conditions of increased load. Although the unusually large omission rates displayed by three participants were predominantly accounted for by the contaminant process, variation in omissions rates for the remaining 17 participants was related to task-related omissions. Design omissions played a more prominent role than intrinsic omissions; however, both were necessary to provide an accurate account. Both were also elevated under load, consistent with their common cause lying in the rate of evidence accumulation. The effect of design omissions is perhaps surprising – the associated slow cut off was more than twice as long as the 90th percentile of observed RTs (see Fig. 1) and histograms of individual participants’ RT distributions did not reveal any obvious signs of slow responses being cut off. This result suggests that it can be hard to detect design omissions by a simple visual inspection of the RT distribution (Ulrich & Miller, 1994), particularly when they occur at relatively low levels.

The LBAO model was less successful with the choice data from the distraction paradigm. It did provide a good account of choice accuracy and RT distributions in all conditions. This is the first demonstration that EAMs in general, and an LBA-type model in particular, can be successful and informative with data from the distraction paradigm. The LBAO model also captured the high error rates and slowing apparent when long stimuli were accompanied by a high deviant. This was apparently due to a tendency for higher frequency deviants with a long duration to be perceived as shorter than they really were. Interactions between tone length and frequency, and a greater susceptibility to this effect with increasing age has been observed in a prior distraction paradigm study (Mager et al., 2005). This type of effect has previously gone unexplained, and so our results point to new avenues for further applications of EAM analysis within distraction paradigm data.

The LBAO model, did, however, have a tendency to underestimate overall omission rates by a small amount that was approximately the same amount for all levels of the deviance factor. This global tendency was most marked for older participants with short choice stimuli. In contrast, the model’s account of differences in omission rates among the different levels of the deviance factor was quite good. The latter account, which relies only on omissions related to the evidence-accumulation process, suggests that intrinsic omissions were essentially absent for older participants, and played a predominant role only for some younger participants that were susceptible to misperception of long high deviant stimuli. Design omissions played a more generally prominent role, being present in all conditions. These were most pronounced in older participants, perhaps less surprisingly, given that the slow cut off was less than twice the 90th RT percentile (see Fig. 3) and older participants had comparatively slower RTs.

Omissions in other paradigms

Omissions are a common occurrence in many experimental paradigms. For example, the psychomotor-vigilance task is typically used to study fatigue due to sleep loss. Participants are required to monitor a blank computer screen and respond to a target that occurs at randomly spaced intervals. Omissions in the psychomotor-vigilance task, also referred to as ‘sleep attacks’ or ‘lapses’, are typically the dependent variable most impacted by fatigue (Lim & Dinges, 2010).

The N-back task is used to study sustained attention and working memory (Kirchner, 1958). In N-back task, omissions are found to be more common than commission error responses (Meule, 2017), increase with increasing task difficulty (Meule, 2017), and have a strong genetic basis (Evans et al., 2018).

In the continuous-performance task (e.g., Beale et al., 1987) participants are required to respond to a rare target that appears in a continuous stream of non-targets that are presented at a fixed rate. Increased omission rates in the continuous-performance task occur with longer task durations, low signal-to-noise ratio between targets and non-targets, and in participants with clinical diagnoses such as attention deficit hyperactivity disorder, traumatic brain injury, and a history of cerebrovascular accidents (for reviews, see Ballard, 2001; Riccio et al., 2002).

In each of these paradigms, the LBAO has the potential to provide novel insights into the nature of these omissions. That said, we do not think that the scope of the LBAO is limited to these paradigms. Rather we suggest that wherever omission rates are elevated, researchers should consider making use of a model such as the LBAO that can account for omissions to leverage any extra information they might provide. Importantly, using the LBAO can make it possible to avoid excluding participants and trials, potentially avoiding biases caused by such exclusions.

Conclusions and future directions

Overall, our results suggest that omission rates can provide information relevant to the psychological processes related to distraction. Further, we have demonstrated that it is possible to integrate the information provided by omissions with the information provided by accuracy and RT through an EAM approach. It is important to note, however, that not all of the potential causes of omission my vary among experimental settings. Of the two types of task-related omission in our experiments, support for intrinsic omissions was weaker, suggesting that further work is required to extend investigation into the factors that determine their occurrence with detection responses (see Ratcliff & Strayer, 2014; Ratcliff & Van Dongen, 2011) versus choice responses. Our results clearly support consideration of design omissions through determining the proportion of responses that fall outside of response windows (see also Evans et al., 2018; Howard et al., 2020). An interesting avenue to further understand design omissions is to examine the effect of manipulating the response window. Importantly, the results of our parameter-recovery studies indicate that these two types of task-related omission can be separately identified by the LBAO model.

The role played by contaminant omissions is less clear, in large part because they are defined as what is not explained by task-specific omissions, with no specific underlying process model. However, our work suggests that they are certainly necessary to explain the occurrence of high omission rates in some participants, at least given our assumptions about task-related omission processes. Our parameter-recovery studies also show that the LBAO model is able to differentiate contaminant from task-related omissions, likely because the latter are also constrained by RTs for responses.

Further empirical and modelling work might investigate our assumption that contaminant processes occur entirely independently of the decision process and experimental manipulations of distraction. For example, a manipulation like load might plausibly have a broad effect, including on processes leading to contaminant omissions. The accuracy of our account suggests that extra freedom was not needed for Castro et al.’s (2019) data, but that may not hold in other cases. Finding ways to differentiate contaminant from task-related omissions on a trial-by-trial basis, perhaps using extra constraint from physiological measures, is another interesting avenue for future research.

On the theoretical front, an intriguing possibility is that the passage of time itself plays some role in contaminant processes. Recently, Hawkins and Heathcote (2021) proposed the addition of a ‘timing’ accumulator (Simen et al., 2016) to an evidence-accumulation race architecture, so that responses can be based on the passage of time as well as on the accumulation of evidence. In their proposal, this enables a time-out criterion with an associated guessed response. A possible extension of Hawkins and Heathcote’s proposition to omissions is that a time-out might also sometimes result in a failure to respond.

Finally, the LBAO is a novel addition to the body of research that seeks to further constrain EAMs (Evans et al., 2018; Evans et al., 2020; Howard et al., 2020; Ratcliff & Rouder, 1998) by including omissions data. We conclude that omissions provide a theoretically sensible constraint, particularly in paradigms that examine distraction. Future studies might refine the methods we have developed here, and even potentially combine them with other approaches to providing extra constraint, such as taking account of the effects of the passage of time.