Behavioral analyses

We analyzed our behavioral data in Matlab R2014a (MathWorks Inc., Natick, MA; code available upon request) and SPSS Statistics Version 20.0 (IBM, Armonk, NY), using repeated-measures analyses of variance (ANOVAs). Only meaningful trials without missing responses were included in any analysis. Distributions of localization responses were computed for visibility categories with at least five trials per subject. Objective working memory performance was quantified via two complementary measures. The rate of correct responding was defined as the proportion of trials within two positions (i.e., ±36°) of the actual target location and served as an index of the amount of information that could be retained. Because 5 out of 20 locations were counted as correct, chance on this measure was 25%. The precision of working memory was estimated as the dispersion (standard deviation) of spatial responses. In particular, we modeled the observed distribution of responses D(n) as a mixture of a uniform distribution (random guessing) and an unknown probability distribution d (‘true working memory’):

where p refers to the probability that a given trial is responded to using random guessing; N to the number of target locations (N = 20); and n is the deviation from the true target location. We assumed that d(n) = 0 for deviations beyond a fixed limit a (with a = 2). This hypothesis allowed us to estimate p from the mean of that part of the distribution D for which one may safely assume no contribution of working memory:

where the model is designed in such a way as to ensure that p^=1 if D is a uniform distribution (i.e., 100% of random guessing) and p^=0 if D vanishes outside the region of correct responding (i.e., 0% of random guessing). There needs to be at least chance performance inside the region of correct responding, so

which ensures 0≤p^≤1. This is the reason why, when computing precision, we included only subjects whose rate of correct responding for unseen trials, collapsed across all experimental conditions, significantly exceeded chance performance (i.e., 25%) in a χ²-test (p<0.05). An estimate of d,d^, can then be derived in two steps from Equation 1 as

We note that the distribution δ has residual, yet negligible, positive and negative mass (due to noise) outside the region of correct responding. In order to obtain d^, we therefore restricted the distribution δ to [−a, a],  set all negative values to 0, and renormalized its mass to 1. The precision of the representation of the target location in working memory was then defined as the standard deviation of that distribution.