Drift diffusion model
The drift diffusion model is a model of decision making in 2-alternative forced choice tasks, in which participants must choose between 2 options. We will use it to simulate choice data in a lexical decision-making task, in which participants must decide whether a stimulus is a real word or not.
In the drift diffusion model, a hypothetical quantity called X changes over time. If it reaches some threshold a, one of the choices is made (for this project, the participant decides the stimulus is a real word); if it drops below 0, the other choice is made (the participant decides the stimulus is not a real word). X changes over time according to the following equation:
Xt+∆t = Xt + A∆t + cW ∆t (1)
where ∆t is the time step (the time between successive updates to X), Xt+∆t is the value of X at the next time step, Xt is the value of X at the current time step, A determines the average rate of change of X, W is white noise (i.e., the value of W is sampled from a normal/Gaussian distribution with mean 0 and variance 1 each time X is updated), and c controls the magnitude of the noise. X starts from some value z, 0 , at the first time step and X is repeatedly updated for as long as X is not less than 0 and not greater than a.
Various alterations in these parameters, and in their statistical distributions, can re-create accuracy and reaction time effects found in human participants.
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In the lexical decision task, there are typically speed–accuracy trade-offs both within participants (quicker decisions are more likely to be erroneous) and be- tween participants (quicker participants make more errors overall).
Use the drift diffusion model to simulate a lexical decision task in which 40 participants complete 300 trials each, 150 with words and 150 with non-words. ∆t will always be 0.1 and c will always be 1.2. Each participant will have a different individual value of a, which will be drawn at random from a uniform distribution between 0.5 and 1.5. At the beginning of each trial z and A are initialized at random: z is drawn at random from a uniform distribution between 0 and a; A is drawn from a normal distribution with mean μ and standard deviation 0.1, where μ = 0.8 if the stimulus is a real word and μ = −0.8 if it is not.
Each trial proceeds as follows: t is initialized to 0 and X is initialized to z. For as long as X is greater than 0 and less than a, t is incremented by ∆t and X is updated according to equation 1. If the stimulus is a word, X exceeding z is a true positive and X dropping below 0 is a false negative. If the stimulus is not a word, X exceeding z is a false positive and X dropping below 0 is a true negative.
To demonstrate the within-participant speed–accuracy trade-off, compute each participant’s mean reaction time for correct responses (true positives and true negatives) and their mean reaction time for incorrect responses (false pos- itives and false negatives). Then compute a difference score by subtracting the mean reaction time for correct responses from the mean reaction time for in- correct responses. Thus each participant will have their own difference score. Do a t-test on these difference scores to demonstrate that they are on average greater than 0.
To demonstrate the between-participants speed–accuracy trade-off, compute each participant’s mean reaction time across all their 300 trials and their overall accuracy (where overall accuracy is computed as the number of true positives plus the number of true negatives divided by the total number of trials). Com- pute the correlation between mean reaction time across all trials and overall accuracy.
Outline questions
How will you make sure there are 150 words and 150 non-words for each participant? Note that you don’t have to have real stimuli anywhere in your code; you could just have a variable called stim type that would take on a value of either "word" or "nonword".
How will you make sure a given participant has the same value of a for each trial and different values of z and A for each trial?
How will you compute a participant’s mean reaction time?
How will you figure out if a participant’s response on a given trial is a true positive, true negative, false positive, or false negative?
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5. How will you collect participants’ difference scores, mean overall reaction times, and overall accuracies for later statistical analysis?