This page includes plots of observed and simulated data from an experiment on working memory.

About the working memory task

The working memory task we used is a variant of the task designed by Jane Oakhill and her colleagues. The task is administered via computer and requires participants to recall the final digits from a series of digit strings. For example, given the following four digit strings:

3 1 7

6 4 9

5 0 1

2 3 8

the correct recall response would be 7 9 1 8. Note that the final digits must be recalled in the order that the corresponding strings appeared. Also, note that the digits appeared one at a time and participants were instructed to read each digit aloud.

The factors manipulated in this experiment (within-subjects) were

About the model

We developed a model of this task within the ACT-R theory of cognition. The model specifies that a certain, limited amount of source activation (labeled W; you can think of this as attentional capacity) influences the activation levels of the various memory elements. The activation of the various memory elements, in turn, influences, performance measures such as accuracy and latency of recall.

We propose that W is an important individual difference parameter that can explain a good deal of the variance in performance on this task (both across conditions and from subject to subject).

Results and preliminary simulations

The observed data reveal a large effect of number of strings on the dependent variable proportion of trials recalled correctly. There is also a main effect of rate of presentation such that participants show better recall on slower trials than on faster trials. This may seem counter-intuitive because slower trials imply a longer delay to recalling the final digits. However, if participants rehearse the final digits at the end of each string (as our model does), the slower trials will allow more time for rehearsals. Finally, the number of digits per string did not have a large effect on recall performance.

The plot below shows the observed data (plotted by condition) as filled squares and the simulated data (also plotted by condition) as open squares. The simulated data were obtained by using the default parameters proscribed by the ACT-R theory and running our model once for each subject (N=26).

test1

This simulation (without individual differences!) provides an adequate fit to the data, R^2 = 0.88. However, there are some systematic deviations (the best-fitting line is Observed = 0.71 *Predicted + 0.16), and the simulated data have much smaller standard error bars than the observed data.

Incorporating Individual Differences into Our Model

To incorporate individual differences into our model, we used two approaches. Our first approach involved running simulations in which the W parameter was not fixed but rather varied from simulation to simulation. That is, we maintained the other parameters at their previous values but took W as drawn from a normal distribution centered at the default value. This provided more variability in our simulation results, and it also changed the mean level predictions of the model. The plot below shows the observed data as above with the model's predictions--taking into account individual differences. Note that the overall fit is much improved (R^2 = 0.92; the best-fitting line is Observed = 0.95 * Predicted + 0.02), and the simulations' standard error bars are now comparable to the observed data. The difference in the simulation results between this plot and the previous plot emphasize the importance of incorporating variability in models of working memory: When there are non-linearities in a computational model, adding indiviudal differences can not only increase the variability of the model's predictions but affect the mean level of its predictions as well.

Our second approach to incorporating individual differences into our modeling efforts involved fitting the W parameter to individual subject's data. This approach is rarely taken but it necessary to ensure that one's theoretical or computational model can accurately capture the individual differences exhibited by particular subjects. Each panel below shows an individual subject's proportion of trials correct (for the four trial types specified by the "number of strings" factor). With each subject's observed data is a set of predictions from the model, for a corresponding value of the W parameter. Notice that even at the individual subject level, our model is capturing performance quite well.

For more information, email Marsha Lovett at lovett+@cmu.edu