# Stitching time onto a tree: joining models of accuracy with response times to understand cognition

Time is what we want most, but what, alas! we use worst.” —William Penn

There are many dimensions of human behavior. Consider a typical recognition memory task in which a participant is given a list of words to remember. A little while later, suppose this participant is shown the word “bear” and asked whether it was on the list (in which case it is “old”) or not (“new”). Some participants will answer “old” and some “new”; some participants will respond quickly, some slowly; some will press the response button harder than others; some participants will look at the word more; and so on. Within an experiment, a single participant will vary on all of these measures as well across time spent in the laboratory. In most experiments, researchers choose only a handful of variables to observe out of necessity: human behavior is too complex to understand all at once.

In the aforementioned memory task, accuracy is traditionally the target of analysis. But we can do better than just looking at accuracy. For example, multinomial processing tree models (MPTs) transform accuracy data into more useful form: numbers that represent different cognitive processes. Suppose, for instance, that when an “old” word (called a “target”) is presented, a person can remember that it was in the original list with some probability (called,do, where the “o” is for “old”). In this case, they’ll respond “old”. If, however, they cannot remember the word, then they do not know whether it was actually in the list but they forgot, or whether it is truly a new word. The participant is then assumed to be in a state of uncertainty, in which they could guess either way. The probability that someone guesses “old” to a word they don’t remember is called g. The basic model can be seen in the figure below, which shows a multinomial processing tree model of memory called the ‘2 high threshold’ model. Each possible response follows a branch in the tree, with probabilities of branching given by the parameters (e.g., do  and g).

Putting these two probabilities together, the probability that they correctly answer “old” to a word that was truly on the list is this: do+(1-do)g. The first term is if they say “old” to a word they remember, and the second if they say “old” when guessing. We can use a similar way of thinking about what happens when a new word (a “lure”) is presented, with a parameter dn (probability of correctly recognizing that a word is new) and the same guessing parameter g

By working backward from the observed accuracies over many words for a single participant, we can estimate do, dn, and g, and hopefully learn something about how memory works. For example, although raw accuracy might be changed by changes in guessing, do and dn would not be: they give us a view of memory “uncontaminated” by guessing.

MPTs have been used for learning about perception, memory, child cognitive development, aging, and more. Note that MPTs are only using information about what a participant responded (“old” or “new”), and not other aspects of behaviour, such as how long . We could get a more complete picture of memory by incorporating information about how long it took a participant to respond (response time). Information about response times is critical to understanding psychological processes. For instance, we might be interested in the processing speed of various stages of a decision task, such as our memory decision. Do they occur in serial or in parallel? Which stages, if any, slow down when we tire, or as we age?

In a recent article in Psychonomic Bulletin & Review, Daniel Heck and Edgar Erdfelder describe a way of joining MPT models of accuracy and response times.

The key insight behind Heck and Erdfelder’s method is that there are several ways to arrive at a particular response. In our memory experiment, for instance, we might arrive at saying “old” to a word that was on the study list by truly remembering that it was, in fact, on the list. Or, we might have arrived at that response by not remembering it, but guessing that it was. The distribution of response times for each way of arriving at an “old” response will be different. If I look at the distribution of response times for just “old” responses, it will be a mixture of response times for the two ways we could have arrived at an “old” response, and we know the probabilities of the two parts of this mixture from the MPT model. In our memory example, the “truly remember” component of the “old” response times will occur do/(do+(1-do)g) of the time, and the rest will be “guess old” responses. The trick is to “unmix” the two kinds of “old” responses to see how long they take.

Heck and Erdfelder describe the mathematical tools they use for unmixing the response time distributions, and the conditions under which it is possible. Sometimes there is not enough information in the data to unmix them. But if one can perform the unmixing, then it becomes possible to test some interesting hypotheses. Consider the following simple model of the memory task mentioned above. Whenever you see an item, you consider what you have in your memory for the list. If you find it, you say “old”. If you do not, you have to guess. Importantly, though, you only guess once you’ve exausted the possibilities in your memory.

This model makes an important prediction: the response times for saying “old” when you truly remember it will be faster than the response times for saying “old” when guessing. Moreover, they must be faster in a particular way: for any given possible response time, more of the “truly remember” responses will be faster than that time than the “guess old” responses (called stochastic dominance). This must be true, because guessing only happens after one has completely failed all attempts to remember the item. Heck and Erdfelder test this hypothesis in an experiment and find that it works well: that is, the performance in the memory task we’ve described is consistent with an account where only guesses after the memory process has a chance to run its course.

The figure below, taken from Heck and Erdfelder’s paper, shows how guessing (dashed) is stochastically slower than memory detection (solid), using response time distributions estimated with their method. This panel shows the proportion of responses (y axis) faster than a particular response time (x axis), for a single condition in their experiment.

The approach described by Heck and Erdfelder has a number of strengths compared with other methods one might consider. First, they use a non-parametric method of estimating response time distributions, based on histograms. This is free of assumptions regarding the shape of the distributions. Second, their method relies primarily on the assumptions of the MPT models, and not on assumptions about how the response times arise. Other models (for instance, the well-known drift diffusion model) depend on assumptions aboutthe specific processes by which the evidence for the responses accumulate. As long as one is satisfied with the assumptions of MPT models, Heck and Erdfelder’s approach joins response times with accuracy to provide a deeper, more complex look at memory or perception than could be achieved with either alone.

Article focused on in this post:

Heck, D. W. & Erdfelder, E. (2016). Extending multinomial processing tree models to measure the relative speed of cognitive processes. Psychonomic Bulletin & Review, 23, 1440-1465. DOI: 10.3758/s13423-016-1025-6