**We used a delayed-estimation paradigm to characterize the joint effects of set size (one, two, four, or six) and delay duration (1, 2, 3, or 6 s) on visual working memory for orientation. We conducted two experiments: one with delay durations blocked, another with delay durations interleaved. As dependent variables, we examined four model-free metrics of dispersion as well as precision estimates in four simple models. We tested for effects of delay time using analyses of variance, linear regressions, and nested model comparisons. We found significant effects of set size and delay duration on both model-free and model-based measures of dispersion. However, the effect of delay duration was much weaker than that of set size, dependent on the analysis method, and apparent in only a minority of subjects. The highest forgetting slope found in either experiment at any set size was a modest 1.14°/s. As secondary results, we found a low rate of nontarget reports, and significant estimation biases towards oblique orientations (but no dependence of their magnitude on either set size or delay duration). Relative stability of working memory even at higher set sizes is consistent with earlier results for motion direction and spatial frequency. We compare with a recent study that performed a very similar experiment.**

*N*simultaneously or sequentially presented motion directions, with set size

*N*being 3, 5, 7, or 9 (Blake et al., 1997). They found a significant effect of set size on average error, but no significant effect of delay duration (0, 10, or 30 s) and no significant interaction. In a two-item spatial-frequency delayed-discrimination task, Magnussen and Greenlee (1999) similarly found a strong effect of set size (one or two) and no effect of delay duration (1, 3, or 10 s).

^{2}.

^{2}. We did not gamma-correct the monitor but because the only variable of interest was orientation, we do not believe this mattered.

^{2}. The Gabor stimuli had a Gaussian envelope of standard deviation 0.25 dva, a cosine modulation with a spatial frequency of 2.28 cycles/dva, and a peak luminance of approximately 240 cd/m

^{2}. The imaginary circle had a radius of 6.27 dva.

*R*of the unit vectors corresponding to all errors. Circular standard deviation is then

*R*, which is between 0 and 1.

*π*,

*π*] by ∈

*, where*

_{i}*i*is the trial index, the circular mean

*n*is the number of trials.

*p*values lower than 10

^{−8}, effects of delay duration with

*p*values of 0.017 and lower; the significance of the interaction depends on the measure. Averaged across trials, delay duration, and set sizes, absolute error was 19.3° ± 1.0° (mean ± standard error of the mean across subjects).

*t*test with Bonferroni-Holm correction (

*p*values = 0.062, 0.13, 0.024, and 0.082). Repeated-measures ANOVAs did not reveal a significant effect of set size on forgetting slope,

*F*(3, 12) = 3.39,

*p*= 0.054.

*p*values in a repeated-measures ANOVA are greater than 0.08. For circular kurtosis, we find a significant effect of set size,

*F*(3, 12) = 73.5,

*p*< 10

^{−7}, but no significant effect of delay duration,

*F*(3, 12) = 2.42,

*p*= 0.12, and no significant interaction,

*F*(9, 36) = 2.08,

*p*= 0.058.

- Pure von Mises model: The error follows a von Mises distribution:
Display Formula \(p(\epsilon ) = {1 \over {2\pi {I_o}(\kappa )}}{e^{\kappa \cos \epsilon }}\). Here,*κ*is called the concentration parameter, and*I*_{0}is the modified Bessel function of the first kind of order 0. We define precision*J*as the Fisher information, which amounts toDisplay Formula \(J = \kappa {{{I_1}(\kappa )} \over {{I_0}(\kappa )}}\) (van den Berg et al., 2012), where*I*_{1}is the modified Bessel function of the first kind of order 1. The pure von Mises model is generally not considered a good description of the error distribution (van den Berg et al., 2014; Zhang & Luck, 2008). - Mixture model: Zhang and Luck (2008) proposed that the error distribution is a mixture of a con Mises distribution and a uniform distribution (corresponding to guesses):
Display Formula \(p(\epsilon ) = {\lambda \over {2\pi }} + {{1 - \lambda } \over {2\pi {I_0}(\kappa )}}{e^{\kappa \cos \epsilon }}\), whereDisplay Formula \(\lambda \in [0,1]\) is the weight given to the uniform component. We again define precision as Fisher information. To constrain the fits of the model, we assume that*λ*is shared across all set sizes and all delay durations. - Variable-precision model with shared scale parameter: It has been proposed that precision
*J*itself varies across trials (and items; Fougnie et al., 2012; van den Berg et al., 2012). Such variation could be caused by a variety of factors, including stimulus differences (Bae, Olkkonen, Allred, Wilson, & Flombaum, 2008; Pratte et al., 2016), variability in decay (Fougnie et al., 2012), Poisson fluctuations in spike count (Bays, 2014), and fluctuations in attention (Cohen & Maunsell, 2010; Goris, Simoncelli, & Movshon, 2014). We have previously parametrized the variable-precision model by assuming that precision*J*follows a gamma distribution with meanDisplay Formula \(\bar J\) and scale parameterDisplay Formula \(\tau :p(\epsilon ) = \int_0^\infty {{1 \over {2\pi {I_0}(\kappa )}}} {e^{\kappa \cos \epsilon }}{\rm{Gamma}}(J;{{\bar J} \over \tau },\tau )dJ\), where Gamma(*J*;*k*,*τ*) is the gamma distribution with shape parameter*k*and scale parameter*τ*(the mean of the gamma distribution is*kτ*). The parameter of interest is mean precisionDisplay Formula \(\bar J\) as a function of set size and delay duration. To constrain the fits of the model, we assume that*τ*is shared across all set sizes and all delay durations. - Variable-precision model with shared shape parameter: We previously also considered an alternative parametrization of the variable-precision model, in which the shape parameter
*k*rather than the scale parameter*τ*is shared across conditions (van den Berg et al., 2014). Thus, we useDisplay Formula \({\rm{Gamma}}\ (J;k,{{\bar J} \over k})\) instead ofDisplay Formula \({\rm{Gamma}}(J;{{\bar J} \over \tau },\tau )\). In a single condition, the two versions of the variable-precision model are equivalent; however, the different constraints across conditions make the two versions different.

*F*(8, 32) = 8.52,

*p*< 10

^{−5}; using 10° bins like Pratte et al. (2016), the corresponding result is

*F*(17, 68) = 8.26 with

*p*< 10

^{−10}. In Figure 7B, we show bias for 10° bins.

*net oblique bias*as the circular mean of the weighted errors, where the weight is 1 if the target orientation is counterclockwise from the closest oblique orientation (±45°), and −1 if the target orientation is clockwise from the closest oblique orientation (Figure 7D). Note that the calculation of this metric does not involve binning. We show net oblique bias as a function of delay duration and set size in Figure 7E. A two-way repeated-measures ANOVA shows no effect of delay duration,

*F*(3, 12) = 0.21,

*p*= 0.89, or set size,

*F*(3, 12) = 1.54,

*p*= 0.25, but a significant interaction,

*F*(9, 36) = 2.90,

*p*= 0.011; indeed, a bit of a cross-over effect is visible. Bias as a function of target orientation is shown separately for each delay duration and set size in Figure 20.

*N*– 1 distractor orientations on that trial. The histograms of these distances look nearly flat at all set sizes and all delay durations (Figure 8). To quantify this, we followed Bays et al. (2009) and van den Berg et al. (2014) and fitted separately for each subject, delay duration, and set size, a mixture model consisting of a von Mises distribution centered at the target orientation, a von Mises distribution at each of the distractor orientations (with equal weights), and a guessing rate (weight to a uniform component); all von Miseses had the same concentration parameter. Of interest is the total weight to the von Mises distributions corresponding to the distractors. Averaged across set sizes (excluding

*N*= 1) and delay durations, we find this weight to be 0.0284 ± 0.0060. Thus, the role of nontarget reports was small in this experiment. The fitted parameters of this mixture model are shown by condition in Figure 21.

*t*test,

*p*= 0.9). As in Experiment 1, the error distribution is strongly affected by set size but only weakly by delay duration (Figure 9). The measures of error dispersion confirm a strong effect of set size and a much weaker effect of delay duration (Figure 10). Repeated-measures ANOVAs (Table 5) show effects of set size with

*p*values lower than 10

^{−8}and effects of delay duration with

*p*values of 0.025 and lower, except for interquartile range. No interaction were significant.

*t*test with Bonferroni-Holm correction (

*p*values = 0.12, 0.043, 0.040, and 0.13). Repeated-measures ANOVAs did not reveal a significant effect of set size on forgetting slope,

*F*(3, 15) = 1.46,

*p*= 0.26.

*p*values in a repeated-measures ANOVA are greater than 0.12. For circular kurtosis, we find a significant effect of set size,

*F*(3, 15) = 99.2,

*p*< 10

^{−9}; a significant effect of delay duration,

*F*(3, 15) = 8.03,

*p*= 0.0020; and no significant interaction,

*F*(9, 45) = 0.708,

*p*= 0.70.

*F*[8, 40] = 6.94,

*p*< 10

^{−5}; 10° bins:

*F*[17, 85] = 7.29,

*p*< 10

^{−9}). A two-way repeated-measures ANOVA shows no effect on net oblique bias (Figure 14E) of delay duration,

*F*(3, 15) = 2.05,

*p*= 0.15; set size,

*F*(3, 15) = 0.67,

*p*= 0.58; or interaction,

*F*(9, 45) = 1.61,

*p*= 0.14.

*N*= 1) and delay durations, the weight to the nontargets was 0.053 ± 0.013. The fitted parameters of this mixture model are shown by condition in Figure 27.

*IEEE Transactions on Automatic Control*, 19, 716–723.

*Nature*, 205 (10), 1218–1220.

*SIAM Journal on Optimization*, 17, 188–217.

*Journal of Vision*, 14 (4): 7, 1–14, doi:10.1167/14.4.7. [PubMed] [Article]

*Journal of Neuroscience*, 34, 3632–3645.

*Scientific Reports*, 6, 19203.

*Journal of Vision*, 9 (10): 7, 1–11, doi:10.1167/9.10.7. [PubMed] [Article]

*Journal of Experimental Psychology: Human Perception and Performance*, 23 (2), 353–369.

*arXiv*, 1012, 2599. Available from https://arxiv.org/abs/1012.2599

*Journal of Neuroscience*, 45, 15241–15253.

*Vision Research*, 32 (6), 1033–1042.

*Nature Communications*, 3, 1229.

*Journal of Vision*, 13 (9): 19, doi:10.1167/13.9.19. [Abstract]

*Nature Neuroscience*, 14, 926–932.

*Nature Neuroscience*, 17, 858–865.

*Journal of Neuroscience*, 15 (3 Pt 2), 2287–2300.

*From pigments to perception: Advances in understanding visual processes*(pp. 451–454). New York: Plenum.

*Journal of Neuroscience*, 13 (6), 2565–2574.

*Nature*, 458 (7238), 632–635.

*Human memory and cognitive capabilities*(pp. 173–187). Amsterdam, the Netherlands: Elsevier.

*PLoS Computational Biology*, 9 (2), e1002927.

*Psychological Research*, 54, 233–239.

*Vision Research*, 36 (14), 2159–2166.

*Psychological Bulletin*, 88 (3), 785–820.

*Trends in Cognitive Science*, 17 (8), 391–400.

*Nature Neuroscience*, 17, 347–356.

*Journal of Experimental Psychology: Learning, Memory, and Cognition*, 18 (1), 151–156.

*Psychological Research*, 62 (2–3), 81–92.

*European Journal of Cognitive Psychology*, 2 (4), 345–362.

*Vision Research*, 31 (7–8), 1213–1219.

*Journal of Experimental Psychology: Human Perception and Performance*, 22 (1), 202–212.

*Journal of Experimental Psychology: Human Perception and Performance*, 24 (3), 707–718.

*Perception*, 14 (3), 265–273.

*Memory and Cognition*, 36 (1), 43–52.

*Directional statistics*. New York: John Wiley & Sons.

*Journal of Experimental Psychology: Human Perception and Performance*, 7 (1), 141–150.

*Perception & Psychophysics*, 44 (4), 369–378.

*Nature Reviews Neuroscience*, 6, 97–107.

*Journal of Experimental Psychology: Learning, Memory, and Cognition*, 43, 528–536.

*Metrika*, 60, 25–32.

*Perception & Psychophysics*, 16 (2), 283–290.

*Attention, Perception and Psychophysics*, 46, 6–17.

*Journal of the Optical Society of America A*, 2 (4), 619–621.

*Journal of Mathematical Psychology*, 56, 356–374.

*IEEE Transactions on Automatic Control*, 6, 461–464.

*Neuroimage*, 63 (1), 166–178.

*Psychological Monographs*, 74, 1–29.

*Psychological Review*, 121, 124–149.

*Proceedings of the National Academy of Sciences*, 109, 8780–8785.

*Nature Neuroscience*, 18, 1728–1730.

*Journal of Experimental Psychology: Human Perception and Performance*, 12 (2), 115–132.

*Nature Neuroscience*, 18 (10), 1509–1517.

*Journal of Neuroscience*, 32 (33), 11228–11240.

*Journal of Vision*, 4 (12): 11, 1120–1135, doi:10.1167/4.12.11. [PubMed] [Article]

*Nature*, 453, 233–235.

*Psychological Science*, 20, 423–428.

*Journal of Vision*, 11 (14): 2, 1–18, doi:10.1167/11.14.2. [PubMed] [Article]