Chapter 20 A simple accumulator model to account for choice response time

As mentioned in chapter 19, the most popular class of cognitive-process models that can incorporate both response times and accuracy are sequential sampling models (for a review, see Ratcliff et al. 2016). This class of model includes, among others, the drift diffusion model (Ratcliff 1978), the linear ballistic accumulator (Brown and Heathcote 2008), and the log-normal race model (Heathcote and Love 2012; Rouder et al. 2015). We discuss the log-normal race model in the current chapter. Sequential sampling or evidence-accumulation models are based on the idea that decisions are made by gathering evidence from the environment (e.g., the computer screen in many experiments) until sufficient evidence is gathered and a threshold of evidence is reached. The log-normal race model seems to be the simplest sequential sampling model that can account for the joint distribution of response times and response choice or accuracy (Heathcote and Love 2012; Rouder et al. 2015).

This model belongs to the subclass of race models, where the evidence for each response grows gradually in time in separate racing accumulators, until a threshold is reached. A response is made when one of these accumulators first reaches the threshold, and wins the race against the other accumulators. This model is sometimes referred as deterministic (or non-stochastic, and ballistic), since the noise only affects the rate of accumulation of evidence before each race starts, but once the accumulator starts accumulating evidence, the rate is fixed. This means that a given accumulator can be faster or slower in different trials (or between choices) but its rate of accumulation will be fixed during a trial (or within choices). Brown and Heathcote (2005) claim that even though it is clear that a range of factors might cause within-choice noise, the behavioral effects might sometimes be small enough to ignore (this is in contrast to models such as the drift diffusion model, where both types of noise are present).

The two main advantages of the log-normal race model in comparison with other sequential sampling models are that: (i) the log-normal race model is very simple, making it easy to extend hierarchically; (ii) it is relatively easy to avoid convergence issues; and (iii) it is straightforward to model more than two choices. This specific model is presented next for pedagogical purposes because it is relatively easy to derive its likelihood given some reasonable assumptions. However, even though the log-normal race is a “legitimate” cognitive model (see Further Readings for examples), the majority of the literature fits choice response times with the linear ballistic accumulator and/or the drift diffusion model, which provide more flexibility to the modeler.

The next section explains how the log-normal race model is implemented, using data from a lexical decision task.

20.1 Modeling a lexical decision task

In a lexical decision task, a subject is presented with a string of letters on the screen and they need to decide whether the string is a word or a non-word; see Figure 20.1. In the example developed below, a subset of 600 words and 600 non-words from 20 subjects (\(600\times2 \times 20\) data points) are used from the data of the British Lexicon project (Keuleers et al. 2012). The data are stored as the object df_blp in the package bcogsci. In this data set, the lexicality of the string (word or non-word) is indicated in the column lex. The goal is to investigate how word frequency, shown in the column freq (frequency is counted per million words using the British National Corpus), affects the lexical decision task as quantified by accuracy and response time. For more details about the data set, type ?df_blp on the R command line after loading the library bcogsci.

Two trials in a lexical decision task. For the first trial, rurble, the correct answer would be to press the key on a keyboard or a response console that is mapped to the “non-word” response, for the second trial, monkey, the correct answer would be to press the key that is mapped to the “word” response.

FIGURE 20.1: Two trials in a lexical decision task. For the first trial, rurble, the correct answer would be to press the key on a keyboard or a response console that is mapped to the “non-word” response, for the second trial, monkey, the correct answer would be to press the key that is mapped to the “word” response.

## # A tibble: 24,000 × 8
##    subj block lex      trial string    acc    rt  freq
##   <dbl> <dbl> <chr>    <dbl> <fct>   <dbl> <dbl> <dbl>
## 1     1    57 non-word 28263 paybods     1   591     0
## 2     1    53 non-word 26414 lunned      1   621     0
## 3     1    49 non-word 24333 pertrax     1   575     0
## # … with 23,997 more rows

The following code chunk adds \(0.01\) (which corresponds to a word that appears only once in the corpus) to avoid word frequencies of zero, and then log-transforms the frequencies to compress their range of values (see Brysbaert, Mandera, and Keuleers 2018 for a more in-depth treatment of word frequencies) and centers them. It also creates a new variable that sum-codes the lexicality of the each given string (either a word, \(0.5\), or a non-word, \(-0.5\)).

If one wants to study the effect of frequency on words, the “traditional” way to analyze these data would be to fit response times and choice data in two separate models on words, ignoring non-words. One model would be fit on the response times of correct responses, and a second model on the accuracy. These two models are fit below.

To fit the response times model, subset the correct responses given to strings that are words:

Fit a hierarchical model with a log-normal likelihood and log-transformed frequency as a predictor (using brms here) and relatively weak priors.

Show the estimate of the effect of log-frequency on the log-ms scale.

##           Estimate Est.Error    Q2.5   Q97.5
## b_c_lfreq  -0.0379   0.00267 -0.0431 -0.0326

To fit the accuracy model, subset the responses given to strings that are words:

Fit a hierarchical model with a Bernoulli likelihood (and logit link) using log-transformed frequency as a predictor (using brms) and relatively weak priors:

Show the estimate of the effect of log-frequency on the log-odds scale:

##           Estimate Est.Error  Q2.5 Q97.5
## b_c_lfreq    0.573    0.0258 0.523 0.627

For this specific data set, it does not matter whether response times or accuracy are chosen as the dependent variable, since both yield results with a similar interpretation: More frequent words are identified more easily, that is, with shorter reading times (this is evident from the negative sign on the estimate of the mean effect), and with higher accuracy (positive sign on the estimate). However, it might be the case that some data set shows divergent directions in response times and accuracy. For example, more frequent words might take longer to identify, leading to a slowdown in response time as frequency increases, but might still be identified more accurately.

Furthermore, two models are fit above, treating response times and accuracy as independent. In reality, there is plenty of evidence that they are related (e.g., the speed-accuracy trade-off). Even in these data, as frequency increases, correct answers are given faster, and most errors are for low-frequency words (see Figure 20.2).

The distribution of response times for words and non-words, and correct and incorrect answers.

FIGURE 20.2: The distribution of response times for words and non-words, and correct and incorrect answers.

A powerful way to convey the relationship between response times and accuracy is using quantile probability plots (Ratcliff and Tuerlinckx 2002; these are closely related to the latency probability plots of Audley and Pike 1965).

A quantile probability plot shows quantiles of the response times distribution (typically \(0.1\), \(0.3\), \(0.5\), \(0.7\), and \(0.9\)) for correct and incorrect responses on the y-axis against probabilities of correct and incorrect responses for experimental conditions on the x-axis. The plot is built by first aggregating the data.

To display a quantile probability plot, create a custom function qpf() that takes as arguments a data set grouped by an experimental condition (e.g., words vs non-words, here by lex), and the quantiles that need to be displayed (by default, \(0.1\), \(0.3\), \(0.5\), \(0.7\), \(0.9\)). The function works as follows: First, calculate the desired quantiles of the response times for incorrect and correct responses by condition (these are stored in rt_q). Second, calculate the proportion of incorrect and correct responses by condition (these are stored in p); because this information is needed for each quantile, repeat it for the number of quantiles chosen (here, five times). Last, record the quantile that each response time and response probability corresponds to (this is recorded in q), and whether it corresponds to an incorrect or a correct response (this information is stored in response).

The aggregated data look like this:

## # A tibble: 20 × 5
##    lex       rt_q      p     q response 
##    <chr>    <dbl>  <dbl> <dbl> <chr>    
##  1 non-word  433. 0.0521   0.1 incorrect
##  2 non-word  521. 0.0521   0.3 incorrect
##  3 non-word  613  0.0521   0.5 incorrect
##  4 non-word  779. 0.0521   0.7 incorrect
##  5 non-word 1110  0.0521   0.9 incorrect
##  6 non-word  448  0.948    0.1 correct  
##  7 non-word  513  0.948    0.3 correct  
##  8 non-word  575  0.948    0.5 correct  
##  9 non-word  666  0.948    0.7 correct  
## 10 non-word  905  0.948    0.9 correct  
## # … with 10 more rows

Plot the data by joining the points that belong to the same quantiles with lines. Given that incorrect responses in most tasks occur in less than 50% of the trials and correct responses occur in a complementary distribution (i.e., in more than 50% of the trials), incorrect responses usually appear in the left half of the plot, and correct ones in the right half. The code that appears below produces Figure 20.3.

Quantile probability plots showing \(0.1\), \(0.3\), \(0.5\), \(0.7\), and \(0.9\)-th response time quantiles plotted against proportion of incorrect responses (left) and proportion of correct responses (right) for strings that are words and non-words.

FIGURE 20.3: Quantile probability plots showing \(0.1\), \(0.3\), \(0.5\), \(0.7\), and \(0.9\)-th response time quantiles plotted against proportion of incorrect responses (left) and proportion of correct responses (right) for strings that are words and non-words.

The vertical spread among the lines shows the shape of the response time distribution. The lower quantile lines correspond to the left part of the response time distribution, and the higher quantiles to the right part of the distribution. Since the response time distribution is long tailed and right skewed, the higher quantiles are more spread apart than the lower quantiles.

A quantile probability plot can also be used to corroborate the observation that high-frequency words are easier to recognize. To do that, subset the data to only words, and group the strings according to their “frequency group” (that is, according to the quantile of frequency that the strings belong to). Whereas we previously aggregated over all the observations, ignoring subjects, we can also aggregate by subjects first, and then average the results. This will prevent some idiosyncratic subjects from dominating in the plot. (We can also plot individual quantile probability plots by subject). Apart from the fact that the aggregation is by subjects, the code below follows the same steps as before, and the result is shown in Figure 20.4. The plot shows that for more frequent words, accuracy improves and responses are faster.

Quantile probability plot showing \(0.1\), \(0.3\), \(0.5\), \(0.7\), and \(0.9\) response times quantiles plotted against proportion of incorrect responses (left) and proportion of correct responses (right) for words of different frequency. Word frequency is grouped according to quantiles: The first group is words with frequencies smaller than the \(0.2\)-th quantile, the second group is words with frequencies smaller than the \(0.4\)-th quantile and larger than the \(0.2\)-th quantile, and so forth.

FIGURE 20.4: Quantile probability plot showing \(0.1\), \(0.3\), \(0.5\), \(0.7\), and \(0.9\) response times quantiles plotted against proportion of incorrect responses (left) and proportion of correct responses (right) for words of different frequency. Word frequency is grouped according to quantiles: The first group is words with frequencies smaller than the \(0.2\)-th quantile, the second group is words with frequencies smaller than the \(0.4\)-th quantile and larger than the \(0.2\)-th quantile, and so forth.

So far, several ways were shown to describe the data by representing them graphically. Next, we turn to modeling the data.

20.1.1 Modeling the lexical decision task with the log-normal race model

The log-normal race model is used here to examine the effect of word frequency in both response times and choice (word vs. non-word) in the lexical decision task presented earlier. In this example, the log-normal race model is limited to fitting two choices; as mentioned earlier, this model can in principle fit more than two choices. When modeling a task with two choices, there are two ways to account for the data: either fit the response times and the accuracy (i.e., accuracy coding: correct vs. incorrect), or fit the response times and actual responses (i.e., stimulus coding: in this case word vs. non-word). In this example, we will use the stimulus-coding approach.

The following code chunk adds a new column that incorporates the actual choice made (as word vs non-word in choice and as 1 vs 2 in nchoice):

To start modeling the data, think about the behavior of one synthetic subject. This subject simultaneously accumulates evidence for the response, “word” in one accumulator, and for “non-word” in another independent accumulator. Unlike other sequential sampling models, an increase in evidence for one choice doesn’t necessarily reduce the evidence for the other choices. Rouder et al. (2015) points out that it might seem odd to assume that we accumulate evidence for a non-word in the same manner as we accumulate evidence for a word, since non-words may be conceptualized as the absence of a word. However, they stress that this approach is closely related to novelty detection, where the salience of never-before experienced stimuli seems to indicate that novelty is psychologically represented as more than the absence of familiarity. Nevertheless, notions of words and non-word evidence accumulation are indeed controversial (see Dufau, Grainger, and Ziegler 2012). The alternative approach of fitting accuracy rather than stimuli discussed before doesn’t really circumvent the problem. This is because when the correct answer is word, we assume that the “correct” accumulator accumulates evidence for word, and the incorrect one for non-word, and the other way around when the correct answer is non-word.

20.1.2 A generative model for a race between accumulators

To build a generative model of the task based on the log-normal race model, start by spelling out the assumptions. In a race of accumulators model, the assumption is that the time \(T\) taken for each accumulator of evidence to reach the threshold at distance \(D\) is simply defined by

\[\begin{equation} T = D/V \end{equation}\]

where the denominator \(V\) is the rate (velocity, sometimes also called drift rate) of evidence accumulation.

The log-normal race model assumes that the rate in each trial is sampled from a log-normal distribution:

\[\begin{equation} V \sim \mathit{LogNormal}(\mu_v, \sigma_v) \end{equation}\]

A schematic illustration of the log-normal race model for the lexical-decision task with a word stimulus. A larger rate of accumulation (V) leads to a larger angle. Here, the choice of word is selected.

FIGURE 20.5: A schematic illustration of the log-normal race model for the lexical-decision task with a word stimulus. A larger rate of accumulation (V) leads to a larger angle. Here, the choice of word is selected.

A log-normal distribution is partly justified by the work by Ulrich and Miller (1993) (also see Box 4.3), and as discussed later, it is very convenient mathematically.

For simplicity, assume that the distance \(D\) to the threshold is kept constant. This might not be a good assumption if the experiment is designed so that subjects change their threshold depending on speed or accuracy incentives (that was not the case in this experiment), or if the subject gets fatigued as the experiment progresses, or if there is reason to believe that there might be random fluctuations in this threshold. Later in this chapter, we will discuss what happens if this assumption is relaxed.

Assume that, for trial \(n\), the location \(\mu_{w}\) of the distribution of rates of accumulation of evidence for a string \(w\) is a function of the lexicality of the string presented (only a word will increase this rate of accumulation and not a non-word), frequency (i.e., high-frequency words might be easier to identify, leading to a faster rate of accumulation than with low-frequency words), and bias (i.e., a subject might have a tendency to answer that a string is a word rather than non-word or vice-versa, regardless of the stimuli). This assumption can be modeled with a linear regression over \(\mu_{w}\), with parameters that represent the bias, \(\alpha_{w}\), the effect of lexicality, \(\beta_{lex_{w}}\), and the effect of log-frequency \(\beta_{\mathit{lfreq}_{w}}\).

\[\begin{equation} \mu_{w,n} = \alpha_w + \mathit{lex}_n \cdot \beta_{lex_{w}} + \mathit{\mathit{lfreq}}_n \cdot \beta_{\mathit{lfreq}_{w}} \end{equation}\]

The location for the rate of accumulation of evidence for the non-word accumulator is defined similarly:

\[\begin{equation} \mu_{nw,n} = \alpha_{nw} + \mathit{lex}_n \cdot \beta_{lex_{nw}} + \mathit{lfreq}_n \cdot \beta_{\mathit{lfreq}_{nw}} \end{equation}\]

Thus the rates are generated as follows:

\[\begin{equation} \begin{aligned} V_{w,n} &\sim \mathit{LogNormal}(\mu_{w,n}, \sigma)\\ V_{nw,n} &\sim \mathit{LogNormal}(\mu_{nw,n}, \sigma) \end{aligned} \end{equation}\]

The accumulators reach the threshold in times:

\[\begin{equation} \begin{aligned} T_{w,n} &= D/V_{w,n}\\ T_{nw,n} &= D/ V_{nw,n} \end{aligned} \end{equation}\]

The choice for trial \(n\) corresponds to the accumulator with the shortest time for that trial,

\[\begin{equation} \mathit{choice}_n = \begin{cases} \mathit{word}, & \text{ if } T_{w,n} < T_{nw,n} \\ \mathit{non}\text{-}{word}, & \text{ otherwise } \end{cases} \end{equation}\]

and the decision for trial \(n\) is made in time

\[\begin{equation} T_n = min(T_{w,n},T_{nw,n}) \end{equation}\]

We also need to take into account that not all the time spent in the task involves making the decision: Time is spent fixating the gaze on the screen, pressing a button, etc. We’ll add a shift to the distribution, representing the minimum amount of time that a subject needs for all the peripheral processes that happened before and after the decision (also see Rouder 2005). We represent this with \(T_{nd}\); “nd” stands for non-decision. Although some variation in the non-decision time is highly likely, we use a constant as an approximation that will be reasonable if its variation is small relative to the variation associated with the decision time (Heathcote and Love 2012).

\[\begin{equation} rt_n = T_{nd} + T_n \end{equation}\]

The following chunk of code generates synthetic data for one subject, by setting true values to the parameters and translating the previous equations to R. The true values are relatively arbitrary and were decided by trial and error until a relatively realistic distribution of response times was obtained. Considering that this is only one subject (unlike what was shown in previous figures), Figure 20.6 looks relatively fine. (One can also inspect the quantile probability plots of individual subjects in the real data set and compare it to the synthetic data).

First, set a seed to always generate the same pseudo-random values, take a subset of the data set to keep the same structure of the data frame for our simulated subject, and set true values:

Second, generate the locations of both accumulators, mu_w and mu_nw, for every trial. This means that both variables are vectors of length, N, the number of trials:

Third, generate values for the rates of accumulation, V_w and V_nw, for every trial. Use those rates to calculate T_w and T_nw, how long it will take for each accumulator to reach its threshold for every trial:

Fourth, calculate the time it takes to reach to a decision in every trial, T_winner as the by-trial minimum between T_w and T_nw. Similarly, store the winner accumulator for each trial in accumulator_winner:

Finally, add this information to the data frame that now indicates choice, time, and accuracy for each trial:

The distribution of response times for words and non-words, and correct and incorrect answers for the synthetic data of one subject.

FIGURE 20.6: The distribution of response times for words and non-words, and correct and incorrect answers for the synthetic data of one subject.

20.1.3 Fitting the log-normal race model

A first issue that we face when we attempt to fit the log-normal race model, is that we need to fit its likelihood to a ratio of the random variables \(D\) and \(V\); that is, we need a ratio or quotient distribution function. Although for arbitrary distributions this requires solving (sometimes extremely complex) integrals (see, for example, Nelson 1981), there are two situations that are compatible with our assumptions and are mathematically simple:

  1. If we assume that \(D\) is a constant \(k\), then \(T = k/V\), and

\[\begin{equation} \log(T) = \log(k/V) = \log(k) - \log(V) \end{equation}\]

Since \(V\) is log-normally distributed, \(\log(V) \sim \mathit{Normal}(\mu_v, \sigma_v)\), and \(\log(k)\) is a constant:

\[\begin{equation} \begin{aligned} \log(T) &\sim \mathit{Normal}(\log(k) - \mu_v, \sigma_v)\\ T &\sim \mathit{LogNormal}(\log(k) - \mu_v, \sigma_v) \end{aligned} \tag{20.1} \end{equation}\]

  1. A log-normally distributed time is not uniquely predicted by assuming that distance is a constant. It also follows if distance is also a log-normally distributed variable: If we assume that \(D \sim \mathit{LogNormal}(\mu_d, \sigma_d)\) then \(T\) is the ratio of two random variables \(D/V\), and

\[\begin{equation} \log(T) = \log(D/V) = \log(D) - \log(V) \end{equation}\]

We have a difference of independent, normally distributed random variables. It follows from random variable theory that:

\[\begin{equation} \begin{aligned} \log(T) &\sim \mathit{Normal}(\mu_d - \mu_v, \sqrt{\sigma_d^2+\sigma_v^2})\\ T &\sim \mathit{LogNormal}(\mu_d - \mu_v, \sqrt{\sigma_d^2+\sigma_v^2}) \end{aligned} \tag{20.2} \end{equation}\]

From Equations (20.1) and (20.2), it should be clear that the threshold and accumulation rate cannot be disentangled: a manipulation that affects the rate or the decision threshold will affect the location of the distribution in the same way (also see Rouder et al. 2015). Another important observation is that \(T\) won’t have a log-normal distribution when \(D\) has any other distributional form.

Following Rouder et al. (2015), we assume that the noise parameter is the same for each accumulator, since this means that contrasts between finishing time distributions are captured completely by contrasts of the locations of the log-normal distributions. We discuss at the end of the chapter why one would need to relax this assumption (also see Exercise 20.2).

In each trial \(n\), with an accumulator for words, indicated with the subscript \(w\), and one for non-words, indicated with \(nw\), we can model the time it takes for each accumulator to get to the threshold \(D\) in the following way. For the word accumulator,

\[\begin{equation} \begin{aligned} \mu'_{w,n} &= \mu_{d_w} - \mu_{w,n}\\ \mu'_{w,n} &= \mu_{d_{w}} - (\alpha_w + \mathit{lex}_n \cdot \beta_{lex_{w}} + \mathit{lfreq}_n \cdot \beta_{\mathit{lfreq}_{w}})\\ \mu'_{w,n} &= (\mu_{d_w} - \alpha_w) - \mathit{lex}_n \cdot \beta_{lex_{w}} - \mathit{lfreq}_n \cdot \beta_{\mathit{lfreq}_{w}}\\ \mu'_{w,n} &= \alpha'_w - \mathit{lex}_n \cdot \beta_{lex_{w}} - \mathit{lfreq}_n \cdot \beta_{\mathit{lfreq}_{w}}\\ T_{w,n} &\sim \mathit{LogNormal}(\mu'_{w,n}, \sigma) \\ \end{aligned} \tag{20.3} \end{equation}\]

The parameter \(\alpha'_w\) absorbs the location of the threshold distribution minus the intercept of the rate distribution, and represents a bias. As \(\alpha'_w\) gets smaller, the accumulator will be more likely to reach the threshold first all things being equal, biasing the responses to word.

Similarly, for the non-word accumulator,

\[\begin{equation} \begin{aligned} \mu'_{nw,n} &= \alpha'_{nw} - \mathit{lex}_n \cdot \beta_{lex_{nw}} - \mathit{lfreq}_n \cdot \beta_{\mathit{lfreq}_{nw}}\\ T_{nw,n} &\sim \mathit{LogNormal}( \mu'_{nw,n}, \sigma) \end{aligned} \end{equation}\]

The only observed time is the one associated with the winner accumulator, the response selected \(s\), which corresponds to the faster accumulator:

\[\begin{equation} T_{\mathit{accum}=s,n} \sim \mathit{LogNormal}(\mu_{\mathit{accum}=s,n}, \sigma) \end{equation}\]

If we only fit the observed finishing times of the accumulators, we’re always ignoring that in a given trial the accumulator that lost was slower than the accumulator for which we have the latency; this means that we underestimate the time it takes to reach the threshold and we overestimate the rate of accumulation of both accumulators. We can treat this as a problem of censored data, where for each trial we don’t know the slower observations.

Since the potential decision time for the accumulator that wasn’t selected is definitely longer than the one of the winner accumulator, we obtain the likelihood for each unobserved time by integrating out all the possible decision times that the accumulator could have, that is, from the time it took for the winner accumulator to reach the threshold to infinitely large decision times:

\[\begin{equation} P(T_{\mathit{accum} \neq s,n}) = \int_{T_{\mathit{accum}=s,n}}^{\infty} \mathit{LogNormal}(T|\mu_{a=s,n}, \sigma) \, dT \end{equation}\]

This integral is the complement of the CDF of the log-normal distribution evaluated at \(T_{\mathit{accum} = s,n}\).

\[\begin{equation} P(T_{\mathit{accum} \neq s,n}) = 1 - \mathit{LogNormal}\_CDF(T_{\mathit{accum}=s,n}| \mu_{\mathit{accum} =s,n}, \sigma) \end{equation}\]

So far we have been fitting the decision time \(T\), but our dependent variable is response times, \(rt\), the sum of the decision time \(T\) and the non-decision time \(T_{nd}\). This requires a change of variables in our model, \(T_{n} = rt_{n} - T_{nd}\), since \(rt\) but not \(T\) is available as data. Here, the Jacobian is 1, since adjusting the likelihood is equal to multiplying the likelihood by one (or adding zero to the log likelihood). This is because \(|d\frac{\mathit{rt}_n - T_{0}}{d\mathit{rt}_n}| = 1\); for more details, see section 12.1. So, although one could write in the Jacobian adjustment in the Stan code, it will always evaluate to zero.

To sum up, our model can be stated as follows:

\[\begin{equation} \begin{aligned} T_n &= rt_n - T_{nd}\\ \mu'_{w,n} &= \alpha'_w - \mathit{lex}_n \cdot \beta_{lex_{w}} - \mathit{lfreq}_n \cdot \beta_{\mathit{lfreq}_{w}}\\ \mu'_{nw,n} &= \alpha'_{nw} - \mathit{lex}_n \cdot \beta_{lex_{nw}} - \mathit{lfreq}_n \cdot \beta_{\mathit{lfreq}_{nw}}\\ T_n &\sim \begin{cases} \mathit{LogNormal}(\mu'_{w,n}, \sigma) \text{ if } \mathit{choice}= \text{word}\\ \mathit{LogNormal}(\mu'_{nw,n}, \sigma) \text{ otherwise } \end{cases}\\ T_{censored,n} &= rt_{censored,n} - T_{nd}\\ \end{aligned} \end{equation}\]

Rather than trying to estimate all the censored observations, we integrate them out:51

\[\begin{equation} P(T_{censored}) = \begin{cases} 1 - \mathit{LogNormal}\_CDF(T_{n}| \mu'_{nw,n}, \sigma) \text{, if } \mathit{choice}= \text{word}\\ 1 - \mathit{LogNormal}\_CDF(T_{n}| \mu'_{w,n}, \sigma) \text{, otherwise }\\ \end{cases} \end{equation}\]

We need priors for all the parameters. An added complication here is the prior for the non-decision time, \(T_{nd}\): we need to make sure that it’s strictly positive and also that it’s smaller than the shortest observed response time. This is because the decision time for each observation, \(T_n\) should also be strictly positive:

\[\begin{equation} \begin{aligned} T_n = rt_n - T_{nd} &>0 \\ rt_n &> T_{nd}\\ min(\mathbf{rt}) &> T_{nd} \end{aligned} \end{equation}\]

We thus truncate the prior of \(T_{nd}\) so that the values lie between zero and \(min(\mathbf{rt})\), the minimum value of the vector of response times. Given the time it takes to fixate the gaze on the screen and a minimal motor response time, centering the prior in 150 ms seems reasonable. The rest of the priors are in log-scale. One should use prior predictive checks to verify that the order of magnitude of all the priors is appropriate. We skip this step here and present the priors below:

\[\begin{equation} \begin{aligned} T_{nd} &\sim \mathit{Normal}(150, 100) \text{ with } 0 < T_{nd} < min(rt_n)\\ \boldsymbol{\alpha} &\sim \mathit{Normal}(6, 1) \\ \boldsymbol{\beta} &\sim \mathit{Normal}(0, .5) \\ \sigma &\sim \mathit{Normal}_+(.5, .2)\\ \end{aligned} \end{equation}\]

where \(\boldsymbol{\alpha}\) is a vector \(\langle \alpha'_{n}, \alpha'_{nw} \ \rangle\), and \(\boldsymbol{\beta}\) is a vector of all the \(\beta\) used in the likelihoods.

To translate the model into Stan, we need a normal distribution truncated so that the values lie between zero and \(min(\mathbf{rt})\) for the prior of T_nd. This means dividing the original distribution by the difference of the CDFs evaluated at these two points; see Box 4.1. In log-space, this is a difference between the log-transformed original distribution and the logarithm of the difference of the CDFs. The function log_diff_exp is a more stable version of this last operation. What log_diff_exp does is to take the log of the difference of the exponent of two functions. In this case the functions are two log-CDFs.

target += normal_lpdf(T_nd | 150, 100)
          - log_diff_exp(normal_lcdf(min(rt) | 150, 100),
                         normal_lcdf(0 | 150, 100));

We implement the likelihood of each joint observation of response time and choice with an if-else clause that calls the likelihood of the accumulator that corresponds to the choice selected in the trial n, and the complement CDF for the accumulator that was not selected:

    if(nchoice[n] == 1)
      target += lognormal_lpdf(T[n] | alpha[1] -
                               c_lex[n] * beta[1] -
                               c_lfreq[n] * beta[2] , sigma)  +
        lognormal_lccdf(T[n] | alpha[2] -
                        c_lex[n] * beta[3] -
                        c_lfreq[n] * beta[4], sigma);
    else
       target += lognormal_lpdf(T[n] | alpha[2] -
                                c_lex[n] * beta[3] -
                                c_lfreq[n] * beta[4], sigma) +
        lognormal_lccdf(T[n] | alpha[1] -
                        c_lex[n] * beta[1] -
                        c_lfreq[n] * beta[2], sigma);
  }

The complete Stan code for this model is shown below as lnrace.stan:

data {
  int<lower = 1> N;
  vector[N] c_lfreq;
  vector[N] c_lex;
  vector[N] rt;
  array[N] int nchoice;
}
parameters {
  array[2] real alpha;
  array[4] real beta;
  real<lower = 0> sigma;
  real<lower = 0, upper = min(rt)> T_nd;
}
model {
  vector[N] T = rt - T_nd;
  target += normal_lpdf(alpha | 6, 1);
  target += normal_lpdf(beta | 0, .5);
  target += normal_lpdf(sigma | .5, .2)
    - normal_lccdf(0 | .5, .2);
  target += normal_lpdf(T_nd | 150, 100)
    - log_diff_exp(normal_lcdf(min(rt) | 150, 100),
                   normal_lcdf(0 | 150, 100));
  for(n in 1:N){
    if(nchoice[n] == 1)
      target += lognormal_lpdf(T[n] | alpha[1] -
                               c_lex[n] * beta[1] -
                               c_lfreq[n] * beta[2], sigma)  +
        lognormal_lccdf(T[n] | alpha[2] -
                        c_lex[n] * beta[3] -
                        c_lfreq[n] * beta[4], sigma);
    else
       target += lognormal_lpdf(T[n] | alpha[2] -
                                c_lex[n] * beta[3] -
                                c_lfreq[n] * beta[4], sigma) +
        lognormal_lccdf(T[n] | alpha[1] -
                        c_lex[n] * beta[1] -
                        c_lfreq[n] * beta[2], sigma);
  }
}

Store the data in a list and fit the model. Some warnings might appear during the warm-up, but these warnings can be ignored since they no longer appear afterwards, and all the convergence checks look fine (omitted here):

Print the parameters values:

##            mean   2.5%  97.5% n_eff Rhat
## alpha[1]   6.67   6.60   6.73  4580    1
## alpha[2]   6.53   6.46   6.59  4211    1
## beta[1]    0.36   0.16   0.55  3372    1
## beta[2]    0.21   0.18   0.25  3210    1
## beta[3]   -0.46  -0.69  -0.22  3207    1
## beta[4]   -0.07  -0.13  -0.02  3125    1
## T_nd     143.41 131.36 152.30  3372    1
## sigma      0.78   0.74   0.82  3673    1

As in previous chapters, mcmc_recover_hist() can be used to compare the posterior distributions of the relevant parameters of the model with their true values (Figure 20.7). First, however, we need to reparameterize the true values, since \(D\) cannot be known, and we don’t fit \(V\), but rather \(D/V\), with \(V\) log-normally distributed. Then, obtain an estimate of \(\alpha'\), rather than \(\alpha\), such that \(\alpha' = log(mu_{d}) - \alpha\).

Posterior distributions of the main parameters of the log-normal race model fit_blp1_sim together with their true values.

FIGURE 20.7: Posterior distributions of the main parameters of the log-normal race model fit_blp1_sim together with their true values.

Before moving on to a more complex version of this model, it’s worth spending some time making the code more modular. Encapsulate the likelihood of the log-normal race model by writing it as a function. The function has four arguments, the decision time T, the choice nchoice (this will only work with two choices, 1 and 2), an array of locations mu (which again we implicitly assume that has two elements), and a common scale sigma.

functions {
  real lognormal_race2_lpdf(real T, int nchoice, 
  real[] mu, real sigma){
    real lpdf;
    if(nchoice == 1)
        lpdf = lognormal_lpdf(T | mu[1] , sigma)  +
          lognormal_lccdf(T | mu[2], sigma);
      else
        lpdf = lognormal_lpdf(T | mu[2], sigma) +
          lognormal_lccdf(T | mu[1], sigma);
    return lpdf;
  }
}

Next, for each iteration n of the original for loop, generate an auxiliary variable T which contains the decision time for the current trial, and mu as an array of size two that contains all the parameters that affect the location at each trial. This will allow us to use our new function as follows in the model block:52

  real log_lik[N];
  for(n in 1:N){
    real T = rt[n] - T_nd;
    real mu[2] = {alpha[1]  -
                    c_lex[n] * beta[1]  -
                    c_lfreq[n] * beta[2],
                    alpha[2]  -
                    c_lex[n] * beta[3] -
                    c_lfreq[n] * beta[4]};
    log_lik[n] = lognormal_race2_lpdf(T | nchoice[n], mu, sigma);
  }

The variable log_lik contains the log-likelihood for each trial. We must not forget to add the total log-likelihood to the target variable. This is done simply by target += sum(log_lik).

The complete Stan code for this model can be found in the bcogsci package as lnrace_mod.stan, it is left for the reader to verify that the results are the same as from the non-modular model lnrace.stan fit earlier.

20.1.4 A hierarchical implementation of the log-normal race model

A simple hierarchical version of the previous model assumes that that all the parameters \(\alpha\) and \(\beta\) have by-subject adjustments:

\[\begin{equation} \begin{aligned} \mu'_{w,n} &= \alpha'_w + u_{subj[n], 1} - \mathit{lex}_n \cdot (\beta_{lex_{w}} + u_{subj[n], 2}) - \mathit{lfreq}_n \cdot (\beta_{\mathit{lfreq}_{w}} + u_{subj[n], 3}) \\ \mu'_{nw,n} &= \alpha'_{nw} + u_{subj[n], 4} - \mathit{lex}_n \cdot (\beta_{lex_{nw}} + u_{subj[n], 5}) - \mathit{lfreq}_n \cdot (\beta_{\mathit{lfreq}_{nw}}+ u_{subj[n], 6}) \\ \end{aligned} \end{equation}\]

Similarly to the hierarchical implementation of the fast-guess model in section 19.1.5, assume that \(\boldsymbol{u}\) is a matrix with as many rows as subjects and six columns. Also assume that \(u\) follows a multivariate normal distribution centered at zero. For lack of more information, we assume the same (weakly informative) prior distribution for the six variance components \(\tau_{u_{1,...,6}}\) with a somewhat smaller effect than we assumed for the prior of \(\sigma\). As with previous hierarchical models, we assign a regularizing LKJ prior for the correlations between the adjustments:53

\[\begin{equation} \begin{aligned} \boldsymbol{u} &\sim\mathcal{N}(0, \Sigma_u)\\ \tau_{u_{1..6}} & \sim \mathit{ \mathit{Normal}}_+(.1, .1)\\ \rho_u &\sim \mathit{LKJcorr}(2) \end{aligned} \end{equation}\]

Before we fit the model to the real data, we’ll verify that it works with simulated data. To create synthetic data of several subjects, we repeat the same generative process we used before and we add the by-subject adjustments u in the same way as in section 19.1.5. This version of the log-normal race model assumes that all the parameters \(\alpha\) and \(\beta\) have by-subject adjustments; that is, 6 adjustments. To simplify the model, we ignore the possibility of an adjustment for the non-decision time \(T_{nd}\), but see Nicenboim and Vasishth (2018) for an implementation of the log-normal race model with a hierarchical non-decision time. For simplicity, all the adjustments u are normally distributed with the same standard deviation of \(0.2\), and they have a \(0.3\) correlation between pairs of u’s; see tau_u and rho below.

First, set a seed, take a subset of the data set to keep the same structure, set true values, and auxiliary variables that indicate the number of observations, subjects, etc.

Second, generate the locations of both accumulators, mu_w and mu_nw, for every trial:

Third, generate values for the rates of accumulation and use those rates to calculate T_w and T_nw.

Fourth, calculate the time it takes to reach to a decision and the winner accumulator for each trial.

Finally, add this information to the data frame.

The Stan code for this model implements the non-centered parameterization for correlated adjustments (see section 11.1.3 for more details). The model is shown below as lnrace_h.stan:

functions {
  real lognormal_race2_lpdf(real T, int nchoice, real[] mu, real sigma){
    real lpdf;
    if(nchoice == 1)
        lpdf = lognormal_lpdf(T | mu[1] , sigma)  +
          lognormal_lccdf(T | mu[2], sigma);
      else
        lpdf = lognormal_lpdf(T | mu[2], sigma) +
          lognormal_lccdf(T | mu[1], sigma);
    return lpdf;
  }
}
data {
  int<lower = 1> N;
  int<lower = 1> N_subj;
  vector[N] c_lfreq;
  vector[N] c_lex;
  vector[N] rt;
  array[N] int nchoice;
  array[N] int subj;
}
transformed data{
  real min_rt = min(rt);
  real max_rt = max(rt);
  int N_re = 6;
}
parameters {
  array[2] real alpha;
  array[4] real beta;
  real<lower = 0> sigma;
  real<lower = 0, upper = min(rt)> T_nd;
  vector<lower = 0>[N_re] tau_u;
  matrix[N_re, N_subj] z_u;
  cholesky_factor_corr[N_re] L_u;
}
transformed parameters {
  matrix[N_subj, N_re] u;
  u = (diag_pre_multiply(tau_u, L_u) * z_u)';
  }
model {
  array[N] real log_lik;
  target += normal_lpdf(alpha | 6, 1);
  target += normal_lpdf(beta | 0, .5);
  target += normal_lpdf(sigma | .5, .2)
    - normal_lccdf(0 | .5, .2);
  target += normal_lpdf(T_nd | 150, 100)
    - log_diff_exp(normal_lcdf(min(rt) | 150, 100),
                   normal_lcdf(0 | 150, 100));
  target += normal_lpdf(tau_u | .1, .1)
    - N_re * normal_lccdf(0 | .1, .1);
  target += lkj_corr_cholesky_lpdf(L_u | 2);
  target += std_normal_lpdf(to_vector(z_u));
  for(n in 1:N){
    real T = rt[n] - T_nd;
    real mu[2] = {alpha[1] + u[subj[n], 1] -
                    c_lex[n] * (beta[1] + u[subj[n], 2]) -
                    c_lfreq[n] * (beta[2] + u[subj[n], 3]),
                    alpha[2] + u[subj[n], 4] -
                    c_lex[n] * (beta[3] + u[subj[n], 5]) -
                    c_lfreq[n] * (beta[4] + u[subj[n], 6])};
     log_lik[n] = lognormal_race2_lpdf(T | nchoice[n], mu, sigma);
  }
  target += sum(log_lik);
}
generated quantities {
  corr_matrix[N_re] rho_u = L_u * L_u';
}

Store the simulated data in a list and fit it.

The code below compares the posterior distributions of the relevant parameters of the model with their true values, and plots them in Figure 20.8. The true value for all the correlations was \(0.3\), but we need to correct the sign depending on whether there was a minus sign in front of the adjustment when we built mu_w and mu_wn or not: For example, there is no minus before u[subj, 1], but there is one before u[subj, 2], thus the true correlation between u[subj, 1] and u[subj, 2] we generated should be negative (plus times minus is minus); and there is a minus before u[subj, 3] and thus the correlation between u[subj, 2] and u[subj, 3] should positive (minus times minus is positive).

Posterior distributions of the main parameters of the log-normal race model fit_blp_h_sim together with their true values.

FIGURE 20.8: Posterior distributions of the main parameters of the log-normal race model fit_blp_h_sim together with their true values.

Figure 20.8 shows that we can recover the true values quite well, even if there is a great deal of uncertainty over the posteriors of the correlations. As mentioned in previous chapters, a more principled (and computationally demanding) approach uses simulation based calibration; this was introduced in section 12.2 (also see Talts et al. 2018; Schad, Betancourt, and Vasishth 2020).

We are now ready to fit the model to the observed data.

Create a list with the real data and fit the same Stan model:

Print the summary (omit the correlations for now).

##           mean  2.5% 97.5% n_eff Rhat
## alpha[1]  6.83  6.75  6.91   610    1
## alpha[2]  6.64  6.58  6.69  1109    1
## beta[1]   0.37  0.31  0.42  2110    1
## beta[2]   0.06  0.06  0.07  1334    1
## beta[3]  -0.14 -0.18 -0.09  2732    1
## beta[4]  -0.07 -0.07 -0.06  4032    1
## T_nd     11.31 10.00 12.45  9007    1
## sigma     0.34  0.34  0.35  7575    1
## tau_u[1]  0.17  0.13  0.23  1505    1
## tau_u[2]  0.10  0.07  0.15  3076    1
## tau_u[3]  0.02  0.01  0.02  2108    1
## tau_u[4]  0.14  0.10  0.18  1840    1
## tau_u[5]  0.08  0.05  0.12  3344    1
## tau_u[6]  0.01  0.00  0.02  1335    1

Even though the model converged, the posterior summary shows a clear problem with the model: The estimate for the non-decision time, \(T_{nd}\) is less than \(12\) milliseconds! This is just not possible; physiological research (Clark, Fan, and Hillyard 1994) shows that the eye-to-brain lag, the time it takes for the visual features on the screen until they are propagated from the retina to the brain is at least \(50\) milliseconds. Besides identifying the stimuli, the subjects also need to initiate a motor response which takes at least \(100\) milliseconds. Then how is it possible that we obtained this extremely fast non-decision time? The reason is that the parameter T_nd is constrained to be between zero and the shortest reaction time.

Verify what is the shortest reaction time in the data set and how many observations are below 150 milliseconds.

## [1] 16
## [1] 2

This shows that some responses must have been initiated even before the stimulus was presented! Next, we deal with contaminant observations (Ratcliff and Tuerlinckx 2002).

20.1.5 Dealing with contaminant responses

So far we have assumed that all the observations were coming from responses done after a decision was made. But what happens if there are anticipations or lapses of attention where the subject responds either before the stimuli is presented or after the stimulus was presented, but without attending to the stimulus? We are in a situation analogous to what we described before in chapter 19 with the fast-guess model (Ollman 1966). There, we assumed that the behavior of a subject would be the mixture of two distributions, one corresponding to a guessing mode of responses and another one to a task-engaged mode. There are two major differences with the fast guess model, however. First, we assume here that guesses can be fast (e.g., anticipations) as well as slow. Second, here guesses occur in a minority of the cases and choice and response times are mostly explained by the log-normal race model. The distribution that corresponds to these guesses is sometimes called a contaminant distribution (Ratcliff and Tuerlinckx 2002). When the contaminant response times are outside the usual range of response times (either shorter or longer), they can cause major problems in data analysis, distorting estimates. As we saw before, extremely short response times caused by anticipating the response can make it virtually impossible to estimate the non-decision time.

A recommended approach (e.g., Ratcliff and Tuerlinckx 2002) for dealing with this problem that we follow here is to assume that the responses come from a mixture between the sequential sampling model (in this case the log-normal race model) and a uniform distribution bounded at the minimum and maximum observed response time.

The new likelihood function will look as follows:

\[\begin{equation} \begin{aligned} p(rt_n, choice_n) =& \theta_c \cdot \mathit{Uniform}(rt_n | min(rt), max(rt)) \cdot \mathit{Bernoulli}(choice_n | \theta_{bias}) +\\ & (1-\theta_c) \cdot p_{lnrace}(rt_n, choice_n | \mu', \sigma) \end{aligned} \tag{20.4} \end{equation}\]

The first term of the sum represents the contaminant component that occurs with probability \(\theta_{c}\) and has a likelihood that depends on the response time, represented with the uniform PDF, and on the response given, represented with a Bernoulli PMF. When a subject is guessing, the likelihood of each choice depends on \(\theta_{bias}\).

The second term of the likelihood represents the log-normal race model that occurs with probability \(1-\theta_c\). We use \(p_{lnrace}(rt_n, choice_n)\) as a shorthand for the following function (which we have already used in the models before):

\[\begin{equation} \begin{aligned} p_{lnrace}(rt_n, choice_n)& =\\ &\begin{cases} \mathit{LogNormal}(\mu'_{w,n}, \sigma) \cdot \\ (1 - \mathit{LogNormal}\_CDF(T_{n}| \mu'_{nw,n}, \sigma)) \text{, if } \mathit{choice}= \text{word}\\ \\ \mathit{LogNormal}(\mu'_{nw,n}, \sigma) \cdot \\ (1 - \mathit{LogNormal}\_CDF(T_{n}| \mu'_{w,n}, \sigma))\text{, otherwise }\\ \end{cases} \end{aligned} \end{equation}\]

To simplify the model, we assume that contaminant responses are completely random (i.e., there is no bias to word or non-word). This assumption is encoded in the model by setting \(\theta_{bias} = 0.5\).54 This makes \(\mathit{Bernoulli}(choice_n | \theta_{bias}) = .5\).

For this model to converge, we need to assume that \(\theta_{c}\) is much smaller than \(1\). This is a sensible assumption for this particular model, since the contaminant distribution is assumed to only happen in a minority of the cases. We set the following prior to \(\theta_{c}\):

\[\begin{equation} \theta_c \sim \mathit{Beta}(0.9, 70) \end{equation}\]

By setting the first parameter of the beta distribution to a number smaller than \(1\), we get a distribution of possible probabilities with a “horn” on the left; see Figure 20.9. Our prior belief for \(\theta_{c}\) has mean \(0.007\), and its 95% CrI is \([0, 0.048]\).55

Prior distribution for \(\theta_{c}\). Most of the probability mass is close to \(0\).

FIGURE 20.9: Prior distribution for \(\theta_{c}\). Most of the probability mass is close to \(0\).

We also want to “push” the non-decision time further from zero to get more realistic values. For this reason we increase the informativity of the prior of \(T_{nd}\). A log-normal prior discourages values too close to zero, even with a similar location (on log-scale) than the truncated normal prior. We settle on the following prior:

\[\begin{equation} T_{nd} \sim \mathit{LogNormal}(log(150), .6) \end{equation}\]

Prior distribution for \(T_{nd}\). The dashed lines show the 95% credible interval.

FIGURE 20.10: Prior distribution for \(T_{nd}\). The dashed lines show the 95% credible interval.

Finally, we want to be able to account for response times that are actually faster than the non-decision time. If the observed response time, rt is smaller than the non-decision time, T_nd, we can be sure that the observation belongs to the contaminant distribution, because otherwise the decision time T should be negative. This means that in this case, the log-normal race likelihood is \(0\) (and its logarithm is negative infinity). When \(T<0\), the log-likelihood of our model is \(log(\theta \cdot \mathit{Uniform}(rt_{n} | min, max) \cdot 0.5 + (1-\theta) \cdot p_{lnrace})\) with \(p_{lnrace} =0\). This means that we only use the following code:

   target += log(theta_c) + uniform_lpdf(rt[n] | min_rt, max_rt) +
                 log(0.5);

This also means that we need to relax the constraints on T_nd; it doesn’t need to be smaller than the smallest observed response time, since some of the observations are responses from the contaminant distribution:

  real<lower = 0> T_nd;

When \(T>0\), the likelihood is a mixture of the contaminant distribution and the log-normal race model as defined in (20.4). We use log_sum_exp exactly as we did in sections 19.1.2 and 19.1.3 for the fast-guess model. We fit a mixture distribution between a contaminant distribution and the log-normal race model.

 target += log_sum_exp(
                  log(theta_c) + uniform_lpdf(rt[n] | min_rt, max_rt)
                           + log(0.5),
                  log1m(theta_c) + lognormal_race2_lpdf(T[n] | nchoice[n],
                                                        mu, sigma));

Finally, the complete Stan code for this model is shown below as lnrace_cont.stan:

functions {
  real lognormal_race2_lpdf(real T, int nchoice, real[] mu, real sigma){
    real lpdf;
    if(nchoice == 1)
        lpdf = lognormal_lpdf(T | mu[1] , sigma)  +
          lognormal_lccdf(T | mu[2], sigma);
      else
        lpdf = lognormal_lpdf(T | mu[2], sigma) +
          lognormal_lccdf(T | mu[1], sigma);
    return lpdf;
  }
}
data {
  int<lower = 1> N;
  int<lower = 1> N_subj;
  vector[N] c_lfreq;
  vector[N] c_lex;
  vector[N] rt;
  array[N] int nchoice;
  array[N] int subj;
}
transformed data{
  real min_rt = min(rt);
  real max_rt = max(rt);
  int N_re = 6;
}
parameters {
  array[2] real alpha;
  array[4] real beta;
  real<lower = 0> sigma;
  real<lower = 0> T_nd;
  real<lower = 0, upper = 1> theta_c;
  vector<lower = 0>[N_re] tau_u;
  matrix[N_re, N_subj] z_u;
  cholesky_factor_corr[N_re] L_u;
}
transformed parameters {
  matrix[N_subj, N_re] u;
  u = (diag_pre_multiply(tau_u, L_u) * z_u)';
}
model {
  array[N] real log_lik;
  target += normal_lpdf(alpha | 6, 1);
  target += normal_lpdf(beta | 0, .5);
  target += normal_lpdf(sigma | .5, .2)
    - normal_lccdf(0 | .5, .2);
  target += lognormal_lpdf(T_nd | log(150), .6);
  target += beta_lpdf(theta_c | .9, 70);
  target += normal_lpdf(tau_u | .1, .1)
    - N_re * normal_lccdf(0 | .1, .1);
  target += lkj_corr_cholesky_lpdf(L_u | 2);
  target += std_normal_lpdf(to_vector(z_u));
    for(n in 1:N){
    real T = rt[n] - T_nd;
    if(T > 0){
    real mu[2] = {alpha[1] + u[subj[n], 1] -
                    c_lex[n] * (beta[1] + u[subj[n], 2]) -
                    c_lfreq[n] * (beta[2] + u[subj[n], 3]),
                    alpha[2] + u[subj[n], 4] -
                    c_lex[n] * (beta[3] + u[subj[n], 5]) -
                    c_lfreq[n] * (beta[4] + u[subj[n], 6])};
    log_lik[n] = log_sum_exp(
                  log(theta_c) + uniform_lpdf(rt[n] | min_rt, max_rt)
                         + log(.5),
                  log1m(theta_c) + lognormal_race2_lpdf(T | nchoice[n], mu, sigma));
    } else {
      // T < 0, observed time is smaller than the non-decision time
      log_lik[n] = log(theta_c) + uniform_lpdf(rt[n] | min_rt, max_rt)
                   + log(.5);
    }
  }
  target += sum(log_lik);
}

In practice, we should verify that this model can recover the true values of its parameters by first simulating data and fitting the model to simulated data, and using simulation based calibration. We skip these steps here.

Store the real data in a list and fit the model:

This model takes more than a day to finish in a relatively powerful computer and, disappointingly, it doesn’t converge; this non-convergence is apparent from the traceplots in Figure 20.11. We’ll see later that even if the converging model finishes faster, it still takes a considerable amount of time. If one has a powerful computer (with for example multiple cores) available, it is possible to parallelize the sampling further than what we did so far. This is possible with special functions that allow for multithreading, which is discussed in the Stan’s user guide (Stan Development Team 2021, ch. 25).

The traceplots of the fit_blp_h shows that the chains are clearly not mixing well for the real data set.

FIGURE 20.11: The traceplots of the fit_blp_h shows that the chains are clearly not mixing well for the real data set.

The traceplots in Figure 20.11 shows that the chains get stuck and don’t mix well. It seems that there is not enough information to constraint the model. If we look at the parameter theta_c, which represents the mixing proportion between the contaminant and the log-normal distribution, we see that its chains are getting stuck at very unlikely values that are over \(0.25\). Although in general is not recommended to cut off values from a prior just because they’re unlikely, in this case restricting the parameter theta_c to be smaller than \(0.1\) helps solving convergence problems. To truncate the prior for \(\theta_c\) in Stan we declare the parameter to have an upper bound of \(0.1\):

real<lower = 0, upper = 0.1> theta_c;

Change its prior distribution in the model block to the following:

target += beta_lpdf(theta_c | 0.9, 70) -
  beta_lcdf(0.1 | 0.9, 70);

If we were to fit this new model, we would see that some chains of T_nd mix in values around 300 ms and some other chains (sometimes) get stuck in values very close to zero. This indicates that the model needs more information regarding the non-decision time. Another issue that slows down convergence is that this parameter T_nd is on a different scale (with a value above 100) than the rest of the parameters (with values below 10). Rather than sampling from T_nd, we sample from lT_nd in a new model, so that T_nd = exp(lTnd). We assign the following prior to lT_nd:

\[\begin{equation} lT_{nd} \sim \mathit{Normal}(log(200), 0.3) \end{equation}\]

This is mathematically equivalent to assigning the following prior to \(T_{nd}\):

\[\begin{equation} T_{nd} \sim \mathit{LogNormal}(log(200), 0.3) \end{equation}\]

The new version of the model is omitted here, but can be found as lnrace_h_contb.stan in the bcogsci package. Fit the new modified model to the same data.

This time the model takes considerably less time, but it still takes nine hours. However, the model does converge and the posterior distribution does make sense now.

Print the summary of the main parameters:

##            mean   2.5%  97.5% n_eff Rhat
## alpha[1]   6.36   6.23   6.48   565 1.01
## alpha[2]   6.02   5.91   6.14  1133 1.00
## beta[1]    0.60   0.52   0.68  2339 1.00
## beta[2]    0.13   0.11   0.14  1969 1.00
## beta[3]   -0.23  -0.30  -0.16  3036 1.00
## beta[4]   -0.12  -0.14  -0.11  3454 1.00
## T_nd     315.66 312.48 318.63  5504 1.00
## sigma      0.62   0.61   0.62  5065 1.00
## theta_c    0.01   0.00   0.01  6811 1.00
## tau_u[1]   0.29   0.23   0.37  1718 1.00
## tau_u[2]   0.15   0.10   0.21  2321 1.00
## tau_u[3]   0.02   0.02   0.04  1879 1.00
## tau_u[4]   0.26   0.20   0.33  2173 1.00
## tau_u[5]   0.12   0.07   0.19  3071 1.00
## tau_u[6]   0.02   0.00   0.04  1255 1.00

Print the summary of the correlations:

##             mean  2.5% 97.5% n_eff Rhat
## rho_u[1,2] -0.06 -0.42  0.29  2798    1
## rho_u[1,3]  0.21 -0.15  0.53  2295    1
## rho_u[1,4]  0.61  0.33  0.80  2288    1
## rho_u[1,5]  0.00 -0.38  0.37  3742    1
## rho_u[1,6] -0.07 -0.54  0.42  5335    1
## rho_u[2,3] -0.39 -0.73  0.09  1935    1
## rho_u[2,4]  0.06 -0.30  0.43  2046    1
## rho_u[2,5] -0.51 -0.84 -0.06  2565    1
## rho_u[2,6] -0.34 -0.79  0.28  3303    1
## rho_u[3,4]  0.04 -0.33  0.40  2213    1
## rho_u[3,5]  0.35 -0.12  0.73  3550    1
## rho_u[3,6]  0.00 -0.53  0.53  3869    1
## rho_u[4,5] -0.33 -0.67  0.06  3976    1
## rho_u[4,6] -0.06 -0.53  0.44  4631    1
## rho_u[5,6]  0.04 -0.52  0.62  3615    1

What can we say about the fit of the model now?

Under the assumptions that we have made, we can look at the parameters and conclude the following:

  • All other things being equal, there is an overall bias to respond non-word rather than word. We can deduce this because the parameters \(\alpha\) represent the boundary separation of each accumulator minus their rate of accumulation (see equation (20.3)). A smaller alpha indicates a closer boundary of evidence and/or a faster rate for a given accumulator. In this case alpha[2] is smaller than alpha[1]. However, the fact that tau_u[1] is relatively large suggests large individual differences.
  • The task seems to have been well-understood given that, when a word appears, the rate of accumulation of the word accumulator increases (beta[1] \(> 0\)), and the rate of the non-word accumulator decreases (beta[3] \(<0\)).
  • As expected, and replicating previous findings in the literature, words with higher frequency are easier to identify correctly as words, compared to lower frequency words (beta[2] \(>0\) and beta[4] \(<0\)).
  • The non-decision time (T_nd) is relatively long, considering that the normal reading of words in a sentence takes around 200-400 ms (Rayner 1998).
  • The proportion of contaminant responses is quite small (\(1\%\)), but without taking them into account, the non-decision time would not be possible to estimate.
  • Subjects that are faster to answer in the word trials tend to be also faster to answer in the non-word trials. We can deduce this given the high correlation between by-subject adjustments to the parameters alpha (rho_u[1,4] is much larger than \(0\)).

Our assumptions include both the likelihood we have chosen and the priors. It’s clear from the difficulties fitting the data that the model is very sensitive to the choice of priors. Since there seem to be not enough information in the data, we need to provide information through the prior distributions.

20.2 Posterior predictive check with the quantile probability plots

As in the previous chapter, in a new file, we can write the generated quantities block for posterior predictive checks. The advantage is that we can generate as many observations as needed after estimating the parameters. There is no model block in the following Stan program. The gqs() function needs a Stan model without transformed parameters. For this reason, this model includes u in the parameters block rather than in the transformed parameters. The complete Stan code for this model is shown below as lnrace_h_contb_gen.stan.

data {
  int<lower = 1> N;
  int<lower = 1> N_subj;
  vector[N] c_lfreq;
  vector[N] c_lex;
  vector[N] rt;
  array[N] int nchoice;
  array[N] int subj;
}
transformed data{
  real min_rt = min(rt);
  real max_rt = max(rt);
  int N_re = 6;
}
parameters {
  array[2] real alpha;
  array[4] real beta;
  real<lower = 0> sigma;
  real<lower = 0> T_nd;
  real<lower = 0, upper = .1> theta_c;
  vector<lower = 0>[N_re] tau_u;
  matrix[N_subj, N_re] u;
}
model {
}
generated quantities {
  array[N] real rt_pred;
  array[N] real nchoice_pred;
  for(n in 1:N){
    real T = rt[n] - T_nd;
    real mu[2] = {alpha[1] + u[subj[n], 1] -
                    c_lex[n] * (beta[1] + u[subj[n], 2]) -
                    c_lfreq[n] * (beta[2] + u[subj[n], 3]),
                    alpha[2] + u[subj[n], 4] -
                    c_lex[n] * (beta[3] + u[subj[n], 5]) -
                    c_lfreq[n] * (beta[4] + u[subj[n], 6])};
    real cont = bernoulli_rng(theta_c);
    if(cont == 1){
      rt_pred[n] = uniform_rng(min_rt, max_rt);
      nchoice_pred[n] = bernoulli_rng(0.5) + 1;
    } else {
      real accum1 = lognormal_rng(mu[1], sigma);
      real accum2 = lognormal_rng(mu[2], sigma);
      rt_pred[n] = fmin(accum1, accum2) + T_nd;
      nchoice_pred[n] = (accum1 > accum2) + 1;
    }
  }
}

Compile the model lnrace_h_contb_gen.stan and subset 200 samples of each parameter appearing in the parameters block using the previous model:

Use the function gqs() (this function draws samples of generated quantities from the Stan model) to generate responses from 200 simulated experiments:

One can examine the general distribution of response times generated by the posterior predictive model, or the effect of the experimental manipulations on response times, as we did in section 19.1.5.3.

However, it is more informative to look at the quantile probability plot of the posterior predictive distribution. In order to create this plot, first extract the predicted response times and choice, and match each one to their corresponding observation of the corresponding simulation. Before we can use qpf(), we need a data frame identical to the one with the observed data, df_blp, but one that includes response time and choice for each observation of each simulation (indicated with the column sim). We do this in the piece of code below. This piece of code yields a data frame called df_blp_pred_qpf, which has the quantile probability information of the observed data and each one of the 200 simulations.

First, extract the array of predicted reading times from gen_race_data and transform it to a long format:

## # A tibble: 4,800,000 × 3
##     sim obs_id rt_pred
##   <int>  <dbl>   <dbl>
## 1     1      1    483.
## 2     1      2    488.
## 3     1      3    623.
## # … with 4,799,997 more rows

Second, extract the array of predicted choice (1 for words and 2 for non-words) and transform it into a long format:

## # A tibble: 4,800,000 × 3
##     sim obs_id nchoice_pred
##   <int>  <dbl>        <dbl>
## 1     1      1            2
## 2     1      2            2
## 3     1      3            2
## # … with 4,799,997 more rows

Third, create a new data frame with the characteristics of the stimuli the predictions and observations. The predictions come from \(200\) simulated dataset, each simulation is indexed with sim (whereas for the empirical observations, sim is set to NA):

## # A tibble: 4,824,000 × 8
##     sim obs_id    rt nchoice  subj lex      lfreq   acc
##   <int>  <dbl> <dbl>   <dbl> <dbl> <chr>    <dbl> <dbl>
## 1     1      1  483.       2     1 non-word -4.61     1
## 2     1      2  488.       2     1 non-word -4.61     1
## 3     1      3  623.       2     1 non-word -4.61     1
## # … with 4,823,997 more rows

Finally, create a data frame with the results of the quantile probability function applied by simulation and by frequency:

Now, plot the results with the code shown below (Figure 20.12).

Quantile probability plots showing 0.1, 0.3, 0.5, 0.7, and 0.9 response times quantiles plotted against proportion of incorrect responses (left) and proportion of correct responses (right) for only words of different frequency. Word frequency are grouped according to quantiles: The first group are words with frequencies smaller than the 0.2-th quantile, the second group are words with frequencies smaller than the 0.4-th quantile and larger than the 0.2-th quantile, and so forth. The summary of the observed data is plotted in black and the summaries of the synthetic data sets are plotted in gray.

FIGURE 20.12: Quantile probability plots showing 0.1, 0.3, 0.5, 0.7, and 0.9 response times quantiles plotted against proportion of incorrect responses (left) and proportion of correct responses (right) for only words of different frequency. Word frequency are grouped according to quantiles: The first group are words with frequencies smaller than the 0.2-th quantile, the second group are words with frequencies smaller than the 0.4-th quantile and larger than the 0.2-th quantile, and so forth. The summary of the observed data is plotted in black and the summaries of the synthetic data sets are plotted in gray.

Figure 20.12 shows the quantile probability plots of the observed and simulated data. We would expect the observed quantile probability summaries (in black) to fall within those of the posterior predictive distribution (in grey).

The fit is clearly bad. One major issue is the misfit at the left side of the plot, this is because the model is unable to capture fast errors: Low probability responses must be slower, because having a slow rate and “losing” the race often is what makes their probability low. Other sequential sampling models such as the linear ballistic accumulator and the drift diffusion model can account for fast errors, under the assumption that they happen in the trials where there is a strong initial bias toward the wrong response; this bias occurs due to random variation in the starting points of the accumulators. This characterization of fast errors is not possible for the log-normal race model, because bias and rate effects combine to determine distribution location (Heathcote and Love 2012). Heathcote and Love (2012) point out that the log-normal race model can still produce fast errors if the scale of the accumulator that corresponds to the incorrect choice is larger than the scale of the accumulator of the correct choice. We leave it as an exercise for the reader to verify that a log-normal race model with a scale that depends on the stimuli improves the fit (see Exercise 20.2).

20.3 Summary

In this chapter, we learned to fit what seems to be the simplest sequential sampling model, a log-normal race model with equal scale (or variances) starting with one subject, continuing with a fully hierarchical model, and finally incorporating a contaminant distribution by using a mixture model. We saw how to evaluate model fit of response time and choice using quantile probability plots. We saw that the log-normal race model with equal scale is unable to account for fast errors and a more complex model is needed. Crucially, many of the techniques explained in this chapter (e.g., including a shift in the distribution, mixture distributions for dealing with contaminated responses, and quantile probability plots) can be used with virtually any type of model that fits response times and choice. One downside of this type of model is that they take a long time to fit.

20.4 Further reading

The log-normal race model was first introduced in Heathcote and Love (2012), and its first Bayesian implementation is described in Rouder et al. (2015). The log-normal race model is closely connected to the retrieval process from memory that is assumed in the cognitive architecture ACT-R; see, for example, Nicenboim and Vasishth (2018); Fisher, Houpt, and Gunzelmann (2022); Lissón et al. (2021). Heathcote et al. (2019) outline how to fit evidence-accumulation models in a Bayesian framework with a custom R package that relies on a Differential-Evolution sampler (DE-MCMC; Turner et al. 2013).

20.5 Exercises

Exercise 20.1 Can we recover the true values of the parameters of a model when dealing with a contaminant distribution?

In Section 20.1.5, we fit a hierarchical model that assumed a contaminant distribution (lnrace_h_cont.stan) without first verifying that we can recover the true values of its parameters if we simulate data. An important first step would be to work with a non-hierarchical version of this model.

  1. Generate data of one subject as in section 20.1.2, but assume a contaminant distribution as in section 20.1.5.
  2. Fit a non-hierarchical version of lnrace_h_cont.stan without restricting the parameter theta_c to be smaller than 0.1.
  3. Plot the posterior distributions of the model and verify that you can recover the true values of the parameters.

Exercise 20.2 Can the log-normal race model account for fast errors?

Subject 13 shows fast errors for incorrect responses. This can be seen in the left side of the quantile probability plot in Figure 20.13.

  1. Fit a log-normal race model (with equal scales for the two accumulator) that accounts for contaminant responses.
  2. Fit a variation of this model, where whether the lexicality of the string matches or not the accumulator affects its scale.
  3. Visualize the fit of each model with quantile probability plots.
  4. Use cross-validation to compare the models.

Notice that the models should be fit to only one subject and they should not have a hierarchical structure.

Quantile probability plot showing 0.1, 0.3, 0.5, 0.7, and 0.9 response times quantiles plotted against proportion of incorrect responses (left) and proportion of correct responses (right) for only words of different frequency for subject number 13.

FIGURE 20.13: Quantile probability plot showing 0.1, 0.3, 0.5, 0.7, and 0.9 response times quantiles plotted against proportion of incorrect responses (left) and proportion of correct responses (right) for only words of different frequency for subject number 13.

Exercise 20.3 Accounting for response time and choice in the lexical decision task using the log-normal race model.

In Chapter 19, we modeled the data of the global motion detection task from Dutilh et al. (2011) (df_dots) using a mixture model. Now, we’ll investigate what happens if we fit a log-normal race model to the same data. As a reminder, in this type of task, subjects see a number of random dots on the screen from which a proportion of them move in a single direction (left or right) and the rest move in random directions. The goal of the task is to estimate the overall direction of the movement. In this data set, there are two difficulty levels (diff) and two types of instructions (emphasis) that focus on accuracy or speed. (More information about the data set can be found by loading the bcogsci package and typing ?df_dots in the R console). For the sake of speed, we’ll fit only one subject from this data set.

  1. Before modeling the data, show the relationship between response times and accuracy with a quantile probability plot that shows quantiles and accuracy of easy and hard difficulty conditions.

  2. Fit a non-hierarchical log-normal race model to account for how both choice and response time are affected by task difficulty and emphasis. Assume no contaminant distribution of responses.

Note that the direction of the dots is indicated with stim, when stim and resp match, both are L, left, or both are R, right, the accuracy, acc, is 1. For modeling this task with a log-normal race model, the difficulty of the task should be coded in a way that reflects that the stimuli will be harder to detect for the relevant accumulator. One way to do it is the following:

  1. Expand the previous model including a contaminant distribution of responses.

  2. Visualize the fit of the two previous models by doing posterior predictive checks using quantile probability plots.

  3. Use cross-validation to compare the models.

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  1. One could estimate the censored times by fitting log-normal distributions that are truncated at \(T_{n}\), since this is the minimum possible time for each censored observation: \[\begin{equation} T_{censored,n} \sim \begin{cases} \mathit{LogNormal}(\mu'_{nw,n}, \sigma) \text{ with } T_{censored,n} > T_n \text{, if } \mathit{choice}= \text{word}\\ \mathit{LogNormal}(\mu'_{w,n}, \sigma) \text{ with } T_{censored,n} > T_n \text{, otherwise } \end{cases} \end{equation}\]

  2. This for-loop can also be implemented in the transformed parameter block; the advantage of doing this is that the log-likelihood of each observation can be used, for example, for cross-validation; the disadvantage is that the R object might be very large, because it will store the log-likelihood during the warm-up period as well.

  3. There are 15 correlations since there are 15 ways to choose 2 variables out of 6 for specifying the pairwise correlations, where order doesn’t matter. This is calculated with \({6 \choose 2}\) which is choose(6, 2) in R.

  4. This is, of course, just an assumption that could be verified. But we’ll see that the model is already quite complex and achieving convergence is not trivial.

  5. The average is calculated as follows: \(0.9/(0.9+70)\); this is because the mean of a beta distribution with parameters \(a,b\) is \(a/(a+b)\). The 95% quantile can be calculated in R with qbeta(c(0.025, 0.975), .9, 70).