**After viewing a scene, people often remember having seen more of the world than was originally visible, an error referred to as boundary extension. Despite the large number of studies on this phenomenon, performance has never been considered in terms of signal detection theory (SDT). We report two visual memory experiments that allowed us to explore boundary extension in terms of SDT. In our experiments, participants first studied pictures presented as close-up or wide-angle views. At test, either the identical view or a different view (a closer or wider angle version of the same scene) were presented and participants rated the test image as being the same or different than before on a 6-point scale. We found that both discrimination sensitivity and bias contributed to the boundary extension effect. The discrimination sensitivity difference was at least 28%, and its presence refuted the hypothesis that boundary extension was due solely to participants' response bias to label test pictures as more wide-angled. Instead, our results support the idea that participants' responses reflect false memory beyond the view (i.e., a more wide-angle view of the world).**

*boundary extension*). Discovered in the context of long-term memory for scenes (Intraub & Richardson, 1989), boundary extension can also occur rapidly enough to be present across a saccadic eye movement (e.g., Intraub & Dickinson, 2008). This constructive memory error is interesting for two reasons. First, participants remember seeing information that had no visual-sensory correlate in the stimulus. Second, although an error with respect to the photograph, boundary extension anticipates the continuation of the scene, predicting upcoming layout in the world (Gottesman, 2011; Intraub, 1997).

*d*′, defined as the intercenter distance between two Gaussian distributions of equal standard deviation

*σ*, and normalized by this

*σ*. Naturally, in order for

*d*′ to be definable, the two distributions (noise and signal) need to be Gaussians and their variances need to be identical. While it is difficult to verify whether or not the two distributions are indeed Gaussian, it is possible to verify a weaker version of this assumption. Namely, if the two distributions are assumed to be Gaussian, then the receiver operating characteristic (ROC) in the Z coordinate space is a straight line. If the two distributions, in addition, share the same variance, then this straight ROC line in the Z coordinate space has a slope of one (Wickens, 2001).

*σ*(Wickens, 2001). For example, when measurement errors in the data are considerable, this bias can lead to the rejection of the equal variance assumption when in fact the assumption is appropriate. Such systematically biased parameter estimation is known as the attenuation bias. In nonlinear models the direction of the bias is more complex.

*x*-dimension are assumed exact or measured free of error, the residual error only represents the distance along the

*y*-dimension between a datum point and the fitted curve. However, in the total least square method, a residual represents the distance between a datum point and the fitted curve measured along a direction in both

*x*- and

*y*-dimensions. In fact, if both variables are measured with the same unit, then the residual error is the shortest distance between the datum point and the fitted curve. That is, the residual vector is perpendicular to the tangent of the curve. This is called two-dimensional Euclidean regression (Stein, 1983). In our case, it is legitimate to assume that the hit and false alarm rates are measured in the same units (in the Z-space or hit and false alarm rate space). This is because the labeling of noise and signal is arbitrary in our case, so the labeling of a hit and a correct rejection is also arbitrary. Given that the false alarm rate = 1 – correct rejection rate, the hit and false alarm rates can be reasonably assumed to share the same units. As a result, the total least square fitting provides an appropriate method for fitting the data.

*n*) is subtracted from 1 or added to 0 in order for the corresponding Z-values to be definable (where 2

*n*is usually the total number of trials in the experiment). However, this correction is arbitrary because there is no principled reason as to why this correction factor should be 1/(2

*n*), but not 1/n or 1/(4

*n*) for example.

*d′*or area under ROC, was different between those two conditions. There was also a corresponding bias change. Both the sensitivity difference and bias difference were consistent with and therefore contributed to the boundary extension effect.

**Figure 1**

**Figure 1**

*d′*for the wider study condition was smaller than for the closer study condition.

*d′*may not be definable, because the signal and noise distributions may not be Gaussian, or they may be Gaussian but having different variances. That was why a rating experiment, rather than a binary old–new experiment, was used to obtain a full ROC function. In this way, the Gaussian and equal variance assumptions could be separately verified.

*d′*difference between the wider and closer study conditions could not be due to any particular images, so long as the number of participants was reasonably large.

**Figure 2**

**Figure 2**

**Figure 3**

**Figure 3**

*d′*and area under ROC) by assuming that the noise and signal distributions were both Gaussians. Without loss of generality, we assumed that the W-W distribution was

*N*(0, 1), and the W-C distribution was

*N*(

*μ*,

*σ*). Similarly, we assumed two distributions for the C-C and C-W. The question was whether the two discrimination sensitivities thus separately obtained, measured in either

*d′*or area under ROC, were the same. It should be noted that whether the W-W and C-C distributions were identical in shape is unknown, but was irrelevant here. This is because in SDT calculations, the noise distribution is always normalized to be

*N*(0, 1).

*R*

^{2}for the linear fitting was 0.90. With the quadratic term added, 6% additional variance could be accounted for. Given that the linear fitting accounted already for 90% of the variance, we concluded that linearity was acceptable for the 24 participants' data. The average

*σ*calculated from the linear slope for the C-W distribution was 1.17, which was only marginally different from one,

*t*(23) = 1.88,

*p*= 0.07). The average

*σ*for the W-C distribution was 1.18, which was also only marginally different from one,

*t*(23) = 1.83,

*p*= 0.08; all

*t*tests in this paper were two-tailed. Because of the marginal significance, we decided to also compute the area under the ROC in addition to

*d′*to calculate discrimination sensitivities. The areas were 0.63 and 0.68 for close and wide study image conditions, respectively. The difference was statistically significant,

*t*(23) = 2.52,

*p*< 0.02. If we assume that

*d′*was definable and ignore the marginal difference above, the

*d′*values were 0.43 and 0.77, giving rise to a significant difference between them,

*t*(23) = 2.73,

*p*= 0.009. In other words, the drop of sensitivity

*d′*that was presumably due to boundary extension was 44%. Figure 3 illustrates these two pairs of distributions.

**Figure 4**

**Figure 4**

*t*(23) = 3.45,

*p*= 0.002, or

*t*(23) = 2.91,

*p*= 0.008, in that a wider test image was more likely to be considered as the same as the closer studied image, in agreement with the boundary extension effect.

*t*(23) = 1.48,

*p*= 0.15, or

*t*(23) = 1.12,

*p*= 0.27. It is interesting to note that the criterion locations in the two cases were very similar to each other (Z = 0.63 and 0.65). This makes sense because the two conditions were randomly mixed, so that it was perhaps impossible to hold two separate criteria. From this single criterion perspective, boundary extension amounted to a relative shift of the signal to the noise distribution in the condition of close study photos.

*d′*and area under the ROC as discrimination sensitivity measures. As in Experiment 1, the unbiased decision criterion should be in the middle of the six-scale rating, and we calculated the

*d′*for each participant and for close and wide study images, separately. The mean

*d′*(0.76) for close study images was smaller than that for wide study images

*d′*(0.92),

*t*(164) = 2.56,

*p*= 0.011. The decrease of

*d′*in the C-W condition relative to the W-C condition, presumably due to boundary extension, was 28%. The mean bias for the C-W condition was 0.35, and that for the W-C condition was 0.13, and the difference was highly significant,

*t*(164) = 7.40,

*p*= 6.54E-12. Figure 5 shows the results.

**Figure 5**

**Figure 5**

*t*(164) = 3.70,

*p*= 0.0003. This result was qualitatively consistent with the

*d′*analysis above, indicating that the boundary extension effect was in part accompanied by discrimination sensitivity change. The

*d′*analysis also indicated that bias was also partially responsible for the boundary extension effect.

*Nature Reviews Neuroscience*, 5, 617–629, doi:10.1038/nrn1476.

*Spatial Vision*, 10, 433–436.

*Journal of Vision*, 12 (9): 1392, doi:10.1167/12.9.1392. [Abstract]

*Society of Industrial and Applied Mathematics Journal on Numercial Analysis*, 17, 883–893.

*Journal of Experimental Psychology: Human Perception & Performance*, 37, 382–395, doi:10.1037/a0021434.

*Signal detection theory and psychophysics*. Huntington, NY: Robert E. Krieger Publishing.

*Cognitive Psychology*, 58 (2), 137–179.

*Econometrica*, 38 (2), 368–370.

*Quarterly Journal of Experimental Psychology (Hove)*, 63 (8), 1467–1494.

*Trends in the Cognitive Sciences*, 1, 217–221, doi:S1364-6613(97)01067-X.

*The psychology of learning and motivation*(Vol. 52, pp. 231–264). Burlington: Academic Press.

*Wiley Interdisciplinary Reviews: Cognitive Science*, 3 (1), 117–127.

*Psychological Science*, 19, 1007–1014.

*Journal of Experimental Psychology: Learning, Memory and Cognition*, 15, 179–187.

*Detection theory: A user's guide*. Mahwah, NJ: Lawrence Erlbaum Associates.

*La perception de la causalite*. Louvain: Publications Universitaires de Louvain.

*Psychonomic Bulletin and Review*, 21 (2), 370–375.

*Neuron*, 54 (2), 335–342.

*Spatial Vision*, 10, 437–442.

*Two dimensional Euclidean regression*. Paper presented at the Conference on Computer Mapping, Herzelia, Israel.

*Elementary signal detection theory*. New York: Oxford University Press.