The aim of this paper is to provide researchers with a conceptual overview of both the theoretical principles and empirical applications related to decision-making biases. More specifically, we will focus on the modelling approaches researchers use to study attention and expectation. We start by describing statistical models and then move to cognitive models of decision-making (Townsend and Ashby, 1983). Here we will describe their advantages over statistical models. We review studies that used both Signal Detection Theory (SDT; Green and Swets, 1966) and the Diffusion Decision Model (DDM; Ratcliff, 1978), as they stand out as well-established cognitive models (a thorough description of SDT and DDM is present in Section 2.3). We also compare these two cognitive models and argue that the latter offers greater insight into the data due to the analysis of response times (RT) distributions.

With respect to the latter models, response biases are particularly suitable for cognitive modelling because they provide valuable information about the underlying cognitive processes that lead to the observed behavior. This is because response biases can help identify the specific cognitive mechanisms involved in decision-making. For example, a response bias favoring one choice option over another may indicate that the participant is relying more heavily on certain perceptual cues or using a specific strategy. This information can then be incorporated into cognitive models to better understand how these cognitive mechanisms operate. Moreover, it is possible to jointly model both behavioral and neural activity measures (Turner et al., 2017). This simultaneous modelling offers the possibility of gaining a neural understanding of these cognitive processes. Response biases can also be used to test and compare different cognitive models. By comparing the fit of different models to the observed data, researchers can identify the model that best describes the putative underlying cognitive mechanisms. Lastly, response biases can be used to study individual differences in decision-making. By examining how response biases vary across different individuals or populations, researchers can identify factors that influence decision-making and tailor interventions accordingly.

Overall, response biases are a valuable tool for cognitive modelling as they provide rich information about the underlying cognitive processes that drive behavior, allow for the comparison of different cognitive models, and facilitate the study of individual differences in decision-making.

Researchers studying response biases are usually interested in testing a specific theory of decision-making. Nonetheless, theories of decision-making vary in their form. On the one hand, formal theories are characterized by mathematical tools and formal concepts (van Rooij and Blokpoel, 2020). On the other hand, verbal theories are verbally expressible intuitions (van Rooij and Blokpoel, 2020).

Verbal and formal theories are the two sides of the theoretical modelling “coin.” Intuitions about psychological phenomena need to be formally transcribed to be quantitatively tested. Similarly, the formalization process of a verbal theory can highlight practical limitations that were not previously conceptualized. The resolution of these practical limitations results in a theory that quantitatively predicts data. In this way, the process of creating a theory is characterized by the cyclical alternation between verbal and formal theories.

Nonetheless, researchers do not test formal theories solely. Contrarily, it is common to encounter studies that aim at testing a verbal theory. Testing verbal and formal theories has different implications that affect the whole empirical process, from hypotheses, to predictions, testing, and inferences (see Table 1).

Verbal theories can be, and usually are, tested and validated via statistical models. Statistical models do not make assumptions about the underlying cognitive processes that result in decision-making, but rather focus on describing the patterns observed in the data. The description and decomposition of variance in the data can in turn inform whether the data are in line with the theory and drive the further formalization of the verbal theory under investigation.

A substantial set of studies that probed decision-making biases applied statistical models to analyze their data (Hsu et al., 2018; Huang et al., 2018; Jigo et al., 2018; Garcia-Lazaro et al., 2019; Kveraga et al., 2019; Smout et al., 2019, 2020; Aitken et al., 2020; Moerel et al., 2022). An example is the event-related potential (ERP) study proposed by Doherty et al. (2005) which jointly investigated the effect of temporal and spatial expectation on decision-making. The task consists of a red ball moving in steps from the left to the right side of the screen. The ball is then occluded by a grey vertical bar for two steps. When it reappears on the other side of the occluded area, participants are asked to respond in case a black dot was present inside the ball. A 2-by-2 design produced a total of four conditions, namely temporal and spatial expectation (ST), spatial expectation only (S), temporal expectation only (T) and neither of the two (N). The four conditions were set up using the movement of the ball. The ball moved either with or without a constant spatial trajectory to manipulate spatial expectation and, with or without a constant step duration to manipulate temporal expectation. The authors analyzed the behavioral data using a repeated-measures two-way ANOVA (and post hoc paired-samples t tests), which examines the influence of two categorical independent factors (temporal and spatial expectations) on a continuous dependent variable (response time). Provided that the assumptions of the parametric estimation of p-values corresponding to the expectation manipulations have been met [which can be challenging, given the non-Gaussian nature of RT distributions and repeated measures paradigms (Girden, 1992)], these models can give valuable insights in the presence and size of biases caused by experimental manipulations. The authors reported faster response times in the S and T conditions compared to the N condition, but the interaction between the two factors was not significant. The reduction in mean response time in the ST condition was larger than the effects in the S and T conditions alone, indicating a so-called additive effect.

In sum, the authors provided evidence using a statistical model for an enhancing effect of both spatial and temporal expectation on response speed, plus an additive effect when both expectations were induced. Nonetheless, the verbal theory used for hypothesis generation cannot quantify the relative influences of these biases in the ST condition nor identify which cognitive processes (e.g., evidence accumulation, criterion or motor response time) have induced such changes. Thus, one may argue that the interpretation of these results remains an approximation while, next to statistical models, mathematical cognitive models may overcome this limitation. By specifying formal assumption about the decision process and subsequently testing which model best describes the data, mathematical cognitive models offer the possibility of decomposing behavioral data into a more structured set of cognitive processes. For the very same reason, mathematical cognitive models cannot account for any underlying cognitive factor that is not specified in the model a priori.

Mathematical cognitive models permit the decomposition of the observed performance into isolated contributions of relevant cognitive processes (Heathcote et al., 2015; Forstmann et al., 2016). Nonetheless, the relationship between statistical and cognitive models is not competitive; instead, it can be understood as a hierarchy where statistical models offer preliminary inference and cognitive models delve deeper into cognitive processes by mathematically formalizing their assumptions. The fit of the model gives insights in how well the cognitive process model can capture the empirical data. Thus, cognitive models yield falsifiable, transparent, and reproducible descriptions of the cognitive processes inferred by the behavioral data (Frischkorn and Schubert, 2018).

A diverse range of attentional tasks, from Posner Cueing to Multiple Object Tracking, is essential for comprehensively probing the intricate nature of attention. Each paradigm targets specific attentional aspects, such as spatial, feature-based, or object-based attention, enabling a more holistic understanding. As attention is a multifaceted phenomenon, employing various tasks ensures a more accurate and nuanced portrayal of its mechanisms and limitations. This broad perspective is crucial for the development of cognitive models of attention, whose primary goal is to encompass the different attentional mechanisms under a common theoretical framework.

An example of this comes from the developed of a variant of the DDM called attentional drift-diffusion model (aDDM; Krajbich et al., 2010). The aDDM is based on the idea that attention mediates the formation of visual short-term memory traces, a concept previously formalized by Smith and Ratcliff (2009). The attentional drift-diffusion model keeps the main properties of the DDM but makes further assumptions about the role of eye movements. Specifically, the model assumes that as participants allocate gaze time onto one option relative to the other, they are more likely to choose the gazed option. Such assumption is formalized as a drift rate bias towards the attended choice (see Figure 4). The aDDM predicts that participants will tend to choose the last seen option, unless the choice’s value is significantly worse than the alternative option; a prediction that has received empirically support (Smith and Krajbich, 2018) across different decision-making domains (e.g., food- and money-related, risky, and social choices), consolidating the role of eye movements in attention orienting. Crucially, promising prospects derive from the recent expansion of the aDDM, in which the probability of fixation on an alternative is expressed as a logistic function of its accumulated value (Gluth et al., 2020). Such models highlight how the attentional mechanisms not only increase evidence accumulation, but also account for the value of the available options (Smith and Krajbich, 2019).

Two more recent studies investigated how spatial attention influences perceptual decision-making with the classic DDM, while recording electroencephalography (EEG) data. The paradigm, shared between the two studies, consisted in a house/face discrimination task which featured a spatial attention cue before the stimulation (see Figure 5). The first study (Ghaderi-Kangavari et al., 2023) consisted in a model comparison test, measured with deviance information criterion (DIC) and R². The authors reported that the winning model allowed non-decision time to vary between the spatial attention conditions. This finding diverges from the idea that attention affects drift rates by increasing signal-to-noise ratio during evidence accumulation (Smith et al., 2004; Krajbich, 2019; Klatt et al., 2020) but remains in line with DDM predictions (Smith et al., 2004). Nonetheless, this result may not reflect the same attentional processes investigated in previous studies as the uncertainty factor induced by the coherence manipulation influences attentional effects (Smith and Ratcliff, 2009). It is also worth noting that the authors report a significant correlation between the amplitude of contralateral N2 component and non-decision time, corroborating previous studies (Klatt et al., 2020). These findings were extended by the second study (Ghaderi-Kangavari et al., 2022) investigating how spatial attention affects different sub-components of non-decision time. Visual encoding time was found to be influenced by spatial attention but not exclusively. In fact, the authors speculate that motor execution time might be affected as well. Considering that older adults were found to have significantly higher non-decision times compared to younger adults (Klatt et al., 2020), both hypotheses remain plausible.

Several EAM-based studies that probed prior probability manipulations converged on the idea that they mainly influence the starting point parameter (Ratcliff and McKoon, 2008; Simen et al., 2009; Forstmann et al., 2010; Mulder et al., 2012; White and Poldrack, 2014; Garton et al., 2019). More recently, researchers combined the DDM with fMRI data to reveal how prior probability affects neural dynamics during a face/house discrimination task (Dunovan and Wheeler, 2018). The authors reported that the model with the best fit allowed both starting point and drift rate to vary between prior probability conditions. At the neural level, the drift rate bias mapped onto pre- and post-stimuli BOLD activity in the Inferior Temporal Cortex (ITC), indicating that quality of evidence is reflected by physiological activity in this category-selective region. This result corroborates previous findings that found prior probability manipulations to affect the drift rate parameter (Hanks et al., 2011; Kelly et al., 2021). Another recent study (Garton et al., 2019) investigated age-related differences in prior probability bias. The results indicate that flexibility of bias control, which refers to the tendency to be biased, is minimally impaired with age.

Taken together, studies investigating the role of attention and prior probability via cognitive modelling reveal a complex picture. The discrepancies within studies on attention or expectation might be due to the heterogeneity of paradigms used, deployed strategies by participants or unknown latent factors. Nonetheless, the different impact of age on attention and expectation biases supports the view in which these psychological phenomena arise from different mechanisms.

Attentional manipulations, which have been often linked to the drift rate parameter, and thus to the speed of evidence accumulation, have also been shown to influence non-decision time. Such findings suggest that attention has a broad impact on cognition, which is not measurable by a single DDM parameter. This is in line with the generally shared view of attention as an all-round cognitive phenomenon, not attributable to a single brain region but rather to a network of regions. Components of the frontoparietal network have repeatedly been associated with spatial attention (Nobre and Kastner, 2014), including the posterior parietal cortex, intraparietal sulcus, dorsal premotor/posterior prefrontal cortex and anterior cingulate cortex. On the more temporal side, the results showing a correlation between the contralateral N2 component and non-decision time pave the way for future studies investigating the influence of attention on the motor component of non-decision time, e.g., via electromyography-based models of decision-making (Servant et al., 2021). The overall picture suggests that the selective influence of spatial attention manipulations on DDM parameters is challenged.

Several studies investigating prior probability manipulations have been shown to influence drift rate (Hanks et al., 2011; Dunovan and Wheeler, 2018; Kelly et al., 2021), in addition to the established influence on starting point (Ratcliff and McKoon, 2008; Simen et al., 2009; Forstmann et al., 2010; Mulder et al., 2012; White and Poldrack, 2014; Garton et al., 2019). The diffuse influence of prior probability manipulations on both drift rate and starting point suggests that a multistage process might be in place (Dunovan and Wheeler, 2018). In line with this hypothesis, drift rate between prior probability conditions correlates with the ITC BOLD activity (Dunovan and Wheeler, 2018) while activation in the frontoparietal network is associated with shifts in the starting point parameter (Mulder et al., 2012).