Change Detection . . . 1
Supplemental Materials
To quantify VSTM capacity (K), we used a generalization of the standard model for estimating capacity from change detection performance under whole-report conditions (Pashler, 1988; Rouder, Morey, Morey, & Cowan, 2011). This generalization allowed for multiple changes on each trial instead of assuming a single change. This model uses the following terms:
K = the number of items held in memory on a given trial
g = the probability that the participant makes a “change” guess when the changed item is not one of the K items in memory or when there was no change
N = the number of items in the sample array
C = the number of items that change on a given trial
H = the hit rate (probability of responding “change” when a change is present)
F = the false alarm rate (probability of responding “change” when no change is present)
Pin = the probability that a changed item was in memory
Pout = the probability that no changed item was in memory
As in the standard model, a hit can occur when the changed item is in memory or when the changed item is out of memory, but the participant makes a lucky guess. A false alarm happens only when the participant guesses. Thus, H and F are computed as:
H = Pin + gPout
F = g
The probability that an item is in memory is always the complement of the probability that the item is out of memory:
Pin = 1 - Pout
This simple relationship between Pinand Poutis important because, when multiple items change, it is simpler to compute the probability that the changed item is not one of the items in memory (Pout) than the probability that the changed item is one of the items in memory (Pin). When there are no changes (C=0), Pin = 0 and Pout = 1. When 1 item changes (C=1), the probability that the changed item is out of memory is simply:
Pout = (N-K)/N
For trials with 0 or 1 changes, this model is equivalent to the standard model. To extend it to cases with multiple changes, we need to define Pout for cases where C > 1. In these cases, Pout is equal to the joint probability that each of the C items is not in memory. For two changes, this is the probability that one of the N items is not in memory multiplied by the probability that the other one of the remaining N-1 items is also not in memory. That is:
For C = 2, Pout = (N-K)/N x ((N-1)-K)/(N-1)
More terms are added to this equation as the number of changed items increases, accounting for the additional changed item that was not one of the K items in memory:
For C = 3, Pout = (N-K)/N x ((N-1)-K)/(N-1) x ((N-2)-K)/(N-2)
For C = 4, Pout = (N-K)/N x ((N-1)-K)/(N-1) x ((N-2)-K)/(N-2) x ((N-3)-K)/(N-3)
This process of adding terms continues indefinitely as C increases, until the number of changed items is so great that one of them must have been one of the K items in memory (i.e., when C > N – K), at which point Pout becomes 0. For example,at least one of the changed items must have been present in memory 3 of the 8 items in the display were present in memory (K = 3 and N = 8) and 6 items changed (C = 8).
The originalmodel of Pashler (1988) assumes that participants are always paying full attention and never confuse the two response buttons. These assumptions are reasonable approximations for healthy young adult human participants being tested under optimal conditions. However, they are unlikely to be good approximations for other participants, such as pigeons or individuals with impaired cognitive function. We therefore adopted the approach of Rouder et al. (2011) and added a lapse rate parameter (L) reflecting the probability that participants occasionally fail to attend to the sample array or confuse the two response keys. On this proportion of the trials, we assume that the participant makes “uninformed” guesses with a likelihood ofu. On the remaining 1-L trials, the standard model applies. Thus, when L is included in the model, the equations for H and F change to:
H = Lu + (1-L)(Pin + gPout)
F = Lu + (1-L)g
To estimate a single value for K that combines the data for all values of C, we used a nonlinear generalized reduced gradient minimization algorithm (implemented in Microsoft Excel) to determine the values of K, g, u, and L that minimize the root mean square error between the predicted and observed values for H and F across the different values of C. The estimation procedure was performed on both the group mean data and the individual-participant data.
References
Pashler, H. (1988). Familiarity and visual change detection. Perception & Psychophysics, 44, 369-378.
Rouder, J. N., Morey, R. D., Morey, C. C., & Cowan, N. (2011). How to measure working memory capacity in the change detection paradigm. Psychonomic Bulletin & Review, 18, 324-330.