LBA Model and DDM

The LBA model describes a two-choice search decision with two accumulators, corresponding to deciding a left/right target (internally), and to initiating a left/right button press response (externally). Each accumulator

begins at a starting amount of evidence (i.e., ‘starting point’, A) and

accumulates evidence at a speed which is described by a decision rate (i.e.,

‘drift rate’, v). The first accumulator to reach a common response threshold (B) determines the overt response. The decision time is then assessed by simple algebra, (𝐵 − 𝐴)/𝑣. An RT reflects the decision time plus a residual time (i.e., the ‘non-decision time’, ter). The latter non-decision time, broadly speaking, accounts for the early perceptual and late motoric processes (Brown &

Heathcote, 2008) and is presumed to be independent of the decision-making process. Accuracy is determined by the first accumulator corresponding to one of the answers for a trial. For instance, if an accumulator for the left target reaches its response threshold first in a search display containing a right target, the measured RT reflects an error response.

The measured RTs were cut at 0.2s and 2s and then fit with two specific versions of decision-making models, the DDM (Ratcliff & Tuerlinckx, 2002) and the LBA models (Heathcote & Love, 2012). Both models were assessed via the MLE method (Myung, 2003) and used the same notation, t^er and v, to refer to the parameters of mean non-decision time and of mean drift rate. As is standard practice, the diffusion model uniformly distributed variability in non-decision time with width st. The LBA model accumulates sensory evidence deterministically, with a drift rate that varies from trial-to-trial according to a normal distribution with mean v and standard deviation sv. The diffusion model specifies these same two parameters (v & sv) with a normal distribution that gives rise to a random sample of drift rate for each trial. Within each trial, evidence accumulates on average at the speed given by the drift rate sample, and a moment-to-moment random

variability. This variability sets a standard deviation s, I conventionally fix at 0.1, which ensures model identifiability (this can also be fixed at 1, see Voss & Voss, 2008).

As the drift-diffusion and LBA models differ in their accumulator structure, they describe the parameters with different symbols. The DDM represents the distance between the (positive & negative) thresholds, with parameter a. The starting point for accumulation, sometimes denoted z, was estimated by its relative position between the thresholds, denoted Z = z/a. Both models assume uniformly distributed variability in the accumulation starting point. Thus, the implementation of a starting point in the DDM varies around z, with the variability, SZ. In the case of the shorter of the distances from z to the threshold, SZ = min(z, a-z)/a. The starting points of the LBA are uniformly distributed between zero and an upper bound denoted A. The distance from A to the threshold is denoted as B (see Figure 1 in Donkin, Brown, & Heathcote, 2009 for a graphic comparison for the evidence accumulation structure between the two models).

Table 5-1. The top-level LBA and the drift-diffusion models. See text in the next paragraph for the meanings of each mathematical symbol.

LBA model Drift-diffusion model

The drift-diffusion model was fit to all variants of the factor combinations, from the most to the least complex (where all parameters were equal across all conditions), resulting in 64 models to be analysed for each participant’s data. The LBA model, likewise, resulted in 128 models per participant. It did not consider variants in which the M factor (matching vs. mismatching accumulator) was dropped for the v parameter (without this, the model is forced to predict chance accuracy). Model variants were fitted starting from the simplest, with the best fits of simpler models providing starting points for fitting more complex models (Donkin et al., 2011).

For all three experiments, I began with a complex, top-level model to fit the data as specified in Table 5-1. The decision threshold (B in the LBA and a in the

drift-diffusion models) varies with the cue factor (Q). The LBA model, because of modelling two separate accumulators, fits also the latent response (lR) factor, which corresponds to the two accumulators. lR acts like a two-level experimental factor. The drift rate (v) varies with all possible factors: stimulus (target) location (S), cue (Q), display size (N), and a scaling factor reflecting the matching/mismatching of an accumulator with correct answer (M). That is, when the left-sided target accumulator (lR-left) reaches the threshold first and a trial contains a right-sided target, the M factor will indicate a mismatched accumulator (hence its drift rate) is chosen; likewise, for the case of matched accumulator.

The between-trial standard deviation of the drift rate (sv) varies only with the scaling factor (M), presuming constant variation across the two accumulators (Brown & Heathcote, 2005). That is, the accumulators’ sv does not depend on other experimental factors, such Q, N or S. This is achieved in the DDM with an intercept. The starting point of an accumulator (A), residual time (ter) and the contamination factor (pc), presumed to be invariant with the experimental factors (Q, N, & S) and the accumulator matches (M), were modelled as intercepts. Due to the different accumulator structure in the DDM, its starting point (a) was modelled with the cue factor. Note that for both models the ensemble of the parameters forms one probability/cumulative density function, so the equations were fitted as one function to the data).

For Experiment 2 and 3 I added an ISI factor to a similar complex model.

The ISI factor (I) was added on the decision threshold and the residual time, but not the drift rate. This decision was based on an assumption that the long ISI increases the likelihood of response preparation and memory consolidation of

the target template, but not the quality of the match between a template and a search display (associated with the decision rate). A similar rationale was also applied to the DDM.

The complex models were subjected to a model selection process. The minus log-likelihoods of all possible models (different combinations of

experimental factors) were calculated, using the MLE. The most probable model was then acquired via the MLE, based on it accounting for the highest variation with minimal factors, with selection based on the lowest Akikie Information Criterion (AIC) and Bayesian Informative Criterion (BIC) values aggregated over participants. This method of model selection provides a good trade-off between goodness-and-fit and model complexity, as measured by number of parameters. The trade-off is measured differently by the two criteria (see Burnham & Anderson, 2004), with AIC tending to select more complex models than BIC.

5.2.3 HBM

In addition to applying the decision-making models, the study used a Weibull probability function to describe RT distributions. Mathematically and theoretically, the approach, compared to using the Gaussian probability function, provides a more liberal and realistic perspective to describe RT distributions. It also exploits the descriptive nature of a probability function, permitting an intuitive impression about how RT distributions changes associated with experimental manipulations. See previous Chapter 2 and Chapter 4 for more details.

5.3 Result

The result section presents the traditional analyses for the mean correct RTs and accuracy rate, which were followed by the analyses of model fitting and RT distributions.