01-09-2015 | Original Article | Uitgave 5/2015 Open Access

# Empirical validation of the diffusion model for recognition memory and a comparison of parameter-estimation methods

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- Psychological Research > Uitgave 5/2015

## Introduction

## The diffusion model

_{ v }(Ratcliff et al., 2004).

_{ z }(Ratcliff, 1978).

_{0}. It contains the non-decisional proportion of the reaction time (Ratcliff, 1978). The total reaction time RT equals RT

_{decision}+ t

_{0}. Like drift rate v and starting point z, this parameter differs between trials. t

_{0}is uniformly distributed with range s

_{ t }(Ratcliff et al. 2004). Ratcliff (2013) showed that, in most cases, these standard assumptions about the distributions of drift rate, starting point, and response-time constant lead to the same predictions as different distributional assumptions.

## Data analysis and parameter estimation with the diffusion model

_{v}and s

_{z}, and a tendency to yield smaller differences between conditions, especially for the drift rate, with a small number of trials.

_{0}via three equations. As an advantage, these equations do not require any parameter fitting and can be used even if the error rate is very small. To achieve this, the model makes some simplifications. That is, (1) it assumes there is no between-trial variability, and thus, s

_{ v }, s

_{ z }and s

_{ t }are set to zero. (2) The starting point is assumed to be unbiased, and thus, z/a is set to 0.5.

_{0}are provided by the EZ model (Wagenmakers et al., 2007). Realistic values are chosen as starting points for the other parameters.

_{0}) by the desired diffusion constant. We converted the fast-dm results to s = 0.1 to make the results more comparable.

## The validity of the model

_{0}. Their participants had to decide whether a dot stimulus was dominated by orange or by blue dots. There were four conditions, namely one standard condition and three other conditions that each targeted one specific model parameter. Task difficulty was increased to decrease the drift rate (difficult condition). An instruction to be very accurate was aimed at increasing the threshold parameter a exclusively (accuracy condition). Finally, by allowing participants to press the response keys with one finger only, the authors strove to increase the response-time constant t

_{0}(handicap condition). They found the predicted pattern. That is, higher task difficulty decreased drift rate, accuracy instructions led to a higher threshold parameter, and the handicap condition led to an increased response-time constant t

_{0}. However, the authors also found unexpected results. In the accuracy condition, the t

_{0}parameter was higher than in the standard condition. In the handicap condition, the drift rate for blue dominated stimuli v

_{blue}and the starting point z/a differed significantly from those in the standard condition. The increased t

_{0}parameter was easily explained because if participants have more time to respond they execute their responses more slowly. Differences in drift rate and starting point in the handicap condition, however, could not be explained that easily. However, all individual models revealed good model fit as assessed via the goodness-of-fit statistic T (see Voss et al., 2004, for a detailed description). In a second experiment, Voss et al. (2004) manipulated the starting point by promoting one response over the other. They found that the starting point was biased towards the promoted response. Overall, the models described the empirical data well. The authors concluded that the parameters of the diffusion model represent the process components of the perceptual task well. The study supported the convergent and partly the discriminant validity of the diffusion-model parameters in the perceptual domain.

_{0}, and overestimated drift rate and boundary separation. It covered group differences well. Fast-dm showed the largest bias and showed smaller group differences than there were in the simulated data sets. Van Ravenzwaaij and Oberauer concluded that all three methods show reasonable accuracy when they have sufficient data points. DMAT required a large number of data points, whereas EZ and fast-dm needed only 80 data points to produce reasonable estimates. EZ and DMAT proved better at detecting group differences. Thus, it is not easy to decide which toolbox to use. EZ seems to be very accurate but cannot detect differences in the bias parameter. DMAT is better than fast-dm at detecting group differences but needs more trials to yield reasonable estimates.

## Experiment 1

### Methods

#### Participants

#### Design

#### Materials

#### Procedure

### Results

#### Performance measures

#### Parameter estimation and model fit

_{old}, the mean drift rate for new items v

_{new}, the mean response-time constant t

_{0}, the range of the bias parameter s

_{ z }, the range of the response-time constant s

_{ t, }and the standard deviation of the drift rates s

_{ v }. Like Voss et al. (2004), we present z/a instead of z because z/a is easier to interpret. A bias parameter of z/a = 0.5 represents an unbiased starting point. Values greater than 0.5 indicate a bias towards the old response; values lower that 0.5 indicate a bias towards the new response.

#### Parameter analyses with fast-dm

#### Parameter analyses with DMAT

### Discussion

## Experiment 2

### Methods

#### Participants

#### Design

#### Materials

#### Procedure

### Results

#### Performance measures

#### Parameter estimation and model fit

#### Parameter analyses with fast-dm

_{0}, M (speed) = 0.54, SD (speed) = 0.08, M (accuracy) = 0.67, SD (accuracy) = 0.12, t(58) = −5.10, p < 0.01, d = 1.19. The effect size (measured as Cohen’s d) was about twice as large for the threshold parameter a as for drift rate v

_{new}and response-time constant t

_{0}. However, all effect sizes represent large effects according to Cohen (1988). No other difference was significant (all p > 0.05). Averaged mean parameter estimates are shown in Fig. 4.

#### Parameter analyses with DMAT

_{0}, M (speed) = 0.55, SD (speed) = 0.09, M (accuracy) = 0.72, SD (accuracy) = 0.18, t(43.91) = −4.33, p < 0.01, d = 1.15. No other difference was significant (all p > 0.05). Averaged mean parameter estimates are shown in Fig. 5.

#### Parameter analyses with EZ

_{0}. Averaged mean parameter estimates are shown in Fig. 6.

### Discussion

_{0}in the accuracy condition. Because of the time pressure in the speed condition, participants carried out the non-decisional components of the task (such as motor response) faster than participants in the accuracy condition. This result was also obtained by Voss et al. (2004) and is easy to explain in psychological terms without questioning the discriminant validity of the parameters. However, with the EZ-diffusion model analysis this parameter did not show significant differences.

_{ ν }of the drift rates for old and new items in a recognition-memory test. Heathcote and Love (2012) found that a speed–accuracy manipulation affected rate variability in the linear ballistic accumulator model which can be seen as a simplified diffusion model.

## Experiment 3

### Methods

#### Participants

#### Materials

#### Design and procedure

### Results

#### Performance measures

_{ p }

^{2}= 0.25.

#### Parameter estimation and model fit

_{ 0 }and the ranges of bias parameter s

_{ z }and response-time constant s

_{ t }as well as the standard deviation of the drift rates s

_{ v }. The KS test used in fast-dm showed a good fit (p > 0.05) for all models. The Chi-square test used in DMAT showed a good fit for all calculated models. However, for some participants, DMAT failed to estimate parameter values in some conditions due to too few error responses. Thus, we included only participants with given parameter estimates in all conditions, resulting in only 11 participants.

^{1}

#### Parameter analyses with fast-dm

_{ p }

^{2}= 0.46. As Helmert contrasts revealed, the difference between old versus new items was significant, F(1,27) = 19.99, p < 0.01. The difference between once- and twice-presented items was significant as well, F(1,27) = 30.52, p < 0.01.

_{ p }

^{2}= 0.36, dfs Greenhouse–Geisser corrected. The more often the items were presented (not, once, twice) the higher was the bias-parameter z/a. The response-time constant t

_{0}also showed significant differences, M (not presented) = 0.63, SD (not presented) = 0.07, M (presented once) = 0.60, SD (presented once) = 0.06, M (presented twice) = 0.59, SD (presented twice) = 0.07, F(2,54) = 6.18, p < 0.01, η

_{ p }

^{2}= 0.19. As expected, the threshold parameter a did not differ significantly between the conditions.

_{0}). This was recommended by Ratcliff (1978) and is possible for within-subject designs only. We expected that the model in which v was free to vary would show the best model fit. The models that allowed the threshold parameter a or the response-time constant t

_{0}to vary, did not fit the data. We found satisfactory model fit only for models that allowed either the drift rate v or the starting point z to vary. The model that allowed for drift rate variation had a model fit that was more than four times better than the model that allowed for variation of the starting point z. Averaged mean parameter estimates are shown in Fig. 7.

#### Parameter analyses with DMAT

_{ p }

^{2}= 0.31, dfs Greenhouse–Geisser corrected. As Helmert contrasts revealed, the difference between old versus new items was not significant, F(1,10) = 0.07, p = 0.79. However, as predicted, the difference between once- and twice-presented items was significant, F(1,10) = 12.30, p < 0.01.

_{ p }

^{2}= 0.41 due to the difference between non-presented items and presented items, F(1,10) = 8.53, p = 0.02. There was no difference between items presented once and items presented twice, F(1,10) = 0.59, p = 0.47.

_{ p }

^{2}= 0.38, dfs Greenhouse–Geisser corrected. Again, this difference resulted due to the difference between non-presented items and presented items, F(1,10) = 6.04, p = 0.03. There was no difference between items presented once and items presented twice, F(1,10) = 0.59, p = 0.81. As expected, the response-time constant t

_{0}did not differ significantly between the conditions.

_{0}). The model that allowed for drift-rate variation had––with four exceptions––the best (and acceptable) model fit. Averaged mean parameter estimates are shown in Fig. 8.

#### Parameter analyses with EZ

_{ p }

^{2}= 0.57 (dfs Greenhouse–Geisser corrected). Helmert contrasts revealed that new and old items were significantly different, F(1,25) = 24.54, p < 0.01. Additionally, items presented once and items presented twice differed significantly, F(1,25) = 73.54, p < 0.01. Threshold parameter a and response-time constant t

_{0}did not show significant differences. Averaged mean parameter estimates are shown in Fig. 9.

### Discussion

_{0}and the bias parameter z/a. These results were contrary to predictions, but are in line with a finding by Criss (2010) who found a correlation between bias parameter z and drift-rate parameter v. The difference in the response-time constant t

_{0}can be explained easily. Differences in t

_{0}between the conditions are probably due to enhanced encoding. Items that have been presented before are more readily accessible for encoding.

_{0}, the absolute differences were quite small (see Fig. 9). The effect of the frequency manipulation on v was considerably larger than those on the other two parameters.

_{ 0 }, most likely due to the small absolute difference in this parameter between conditions. A difference in the bias parameter z/a cannot be detected by the EZ-diffusion model, of course. The restricted model versions showed best model fit for the model that allowed the predicted parameter v to vary between the conditions.

## General discussion

_{0}in Experiment 2 and the effect of the number of presentations on t

_{0}in Experiment 3. However, other effects were harder to reconcile psychologically, namely the effect of the number of presentations (Experiment 3) on the bias parameter z/a and on the threshold parameter a. Whereas the threshold effect may be explained within the diffusion model by invoking additional assumptions (see above), the bias effect is clearly unexplainable within the model.

_{0}that were psychologically plausible. DMAT showed discriminant validity for Experiment 1 and Experiment 2, but failed to do so for Experiment 3. Parameters t

_{0}, z, and v all represent cognitive processes that affect decision times. However, when they must be recovered from a noisy response-time distribution, their respective influences presumably cannot be clearly separated. This does not necessarily undermine the validity of the diffusion model in certain applications, though, given specific conditions discussed below. To put our findings into perspective, the reader is reminded that in every case, the effect sizes were considerably larger for the target parameters than for the side effects. Hence, the lion’s share of variation in the data could always be attributed to the correct parameter, and it may, thus, be warranted to conclude that the model has some discriminant validity, although it is rather weak. Voss et al. (2004) concluded that their findings supported the validity of the diffusion model. However, their results were more straightforward in that their manipulations only affected the hypothesized parameters (except for the response handicap condition which we did not test).