Extending SDT: Sequential Sampling

The flexibility of SDT is both a strength and a weakness. It specifies the relationship between variables but not the absolute values themselves; so as long as the variables scale with each other it is mathematically unconstrained (Gold & Shadlen, 2007). One way to further constrain these models is to account for reaction times as well as decision probabilities. Models that capture both reaction times and choices are called sequential sampling models (SSM), because they are extending SDT to integrate several pieces of evidence before discriminating between H0 and H1. SSM first asks, “Is there enough information to make a successful discrimination?”. If so the appropriate choice is selected; if not another piece of evidence is accumulated and integrated. The easiest way to visualise a sequential sampling model is to imagine this gradual accumulation of evidence as the movements of a particle (see Figure 2.4.). The starting point of the particle represents our belief before we receive any evidence, then for each unit of evidence the particle moves towards or away from a threshold where the threshold represents a choice. There are

49

many classes of sequential sampling models (as discussed in the introduction), but I will focus on drift diffusion models (DDM). The distinguishing feature of drift diffusion models is that they measure the relative evidence between two options, so that the evidence accumulation for different options is anti-correlated (the two thresholds are opposite to each other in decision- space, see Figure 2.4.). This can be contrasted with race models, which model evidence accumulation separately for each option and allow for some degree of independence between these accumulators (Forstmann et al., 2016).

Figure 2.4. Schematic of a Drift Diffusion Process

A drift diffusion process model human decision making as a particle moving between two bounds. 4 key parameters are estimated from the data, bias (z), boundary separation (a) drift rate (v) and non-decision-time (t).

There are 4 primary parameters in a DDM, the drift rate (v) determines the average amount of evidence per unit of time and can be considered a sensitivity parameter (like d’ in basic SDT). The boundary separation (a) captures the speed accuracy trade–off; the further away the decision boundaries are the greater the chance that the choice will reflect the drift rate (be accurate) but it will take longer for the particle to hit a boundary, meaning slower responses. As mentioned above, the starting point (z) of the particle captures prior beliefs or preference favouring one option over the other, with the result that choices in the direction of the starting point are more likely, and will happen more quickly when they happen. The boundary separation and starting point together correspond to c in SDT because they jointly capture the response criterion.

50

Finally, to accurately capture human decision data SDT requires a fourth parameter, non- decision time (t) reflecting processing independent from the decision process itself (such as planning the motor response). In other words response times are a function of the time it takes the evidence to accumulate to a boundary and the non-decision time. Besides the primary free parameters described above there are three additional free parameters: inter-trial variability in drift rates, starting points and non-decision times. These additional parameters are necessary to capture the differences in response time distributions for correct responses and error responses. It might seem as if DDMs can capture response distributions and response probabilities simply because they have so many free parameters, so any results derived from this approach are psychologically uninformative because the parameters are not sufficiently constrained by the data. However, there are several pieces of evidence to the contrary. First, Ratcliff (2002) used simulations to show that the DDM failed to account for plausible but fake data. Second, the model can be constrained by theoretical considerations (e.g. only allowing task difficulty to influence drift rate, and changing accuracy incentives to influence the speed accuracy trade-off). Third, experiments that vary a specific psychological construct show that most of the resulting variance is being captured by the appropriate parameter (so if difficulty varies between trials, that is captured by the drift rate parameter but not by the threshold parameter; Voss, Rothermund, & Voss, 2004)

The DDM framework has several benefits for behavioural research. Just like SDT, DDM captures both accuracy and bias in an intuitive way, and it also captures the speed accuracy trade- off. This allows researchers to explore questions relating to psychological mechanisms. For example, it is well-known that older people have slower response times for binary decision making tasks than younger people, but is this difference caused by slower evidence integration, slower motor responses, or because they are more careful to avoid errors? DDM analyses show that the differences in response time are best captured by differences in non-decision-times and boundary separation, suggesting that older people have slower motor responses and are more careful, but evidence integration is unaffected by aging (Ratcliff, Thapar, & McKoon, 2010). For a list of research that has benefited from a DDM approach see Forstmann, Ratcliff and

Wagenmakers (2016).

The role of experimental manipulations on the DDM parameters can be examined by applying linear regression models to the parameters. For example, if we have two difficulty conditions in a 2AFC experiment and we want to test whether these affect the evidence accumulations we can fit two models. First a null model that ignores trial difficulty when estimating drift-rate and then

51

a second model that estimates drift rate as an intercept and a coefficient multiplied by a dummy- variable, coded 0 for hard trials and 1 for easy trials. The extent to which the second model fits the data better than the first model would quantify whether the difficulty manipulation has worked, and the size of the coefficient would quantify how much the difficulty manipulation influences evidence accumulation. As with the GLM frameworks we could also draw various intercepts and slopes for different participants, and therefore capture both general trends across people and individual differences between people in the same model.