Crowding is a form of lateral interaction in which flanking items interfere with the detection or discrimination of a target stimulus. It is believed that crowding is a property of peripheral vision only and that no crowding occurs at fixation. If these two claims are true, then there must be a change in the nature of crowding interactions across the visual field. In three different tasks, we determined target size and flanker separation at threshold for eccentricities of 0 to 16° in the lower visual field for 7 relative separations (1.25 to 8 times target size). In all three tasks, the magnitude of crowding increases with eccentricity; there was no crowding at fixation and extreme crowding at 16°. Using a novel double-scaling procedure, we show that the non-foveal data in all three tasks can be characterized as shifted versions of the same psychometric function such that different sections of the function characterize data at each eccentricity. This pattern of results can be understood in terms of size-dependent responses to the target and distance-dependent interference from the flankers. The data suggest that the distance-dependent interference increases with eccentricity.

*s*

_{0}is stimulus size at fovea, then

*s*

_{ E }=

*s*

_{0}(1 +

*E*/

*E*

_{2}) is the stimulus size at eccentricity

*E*required to elicit equivalent-to-foveal performance;

*E*

_{2}is a task-dependent constant. The success of magnification in overcoming eccentricity-dependent sensitivity loss encourages the view that peripheral vision is simply a scaled version of foveal vision, i.e., the mechanisms available at each eccentricity are the same in all respects and differ only in scale (Gurnsey, Poirier, Bluett, & Leibov, 2006; Gurnsey, Roddy, Ouhnana, & Troje, 2008; Makela, Whitaker, & Rovamo, 1993; Rovamo & Virsu, 1979; Watson, 1987; Weymouth, 1958; Whitaker, Latham, Makela, & Rovamo, 1993; Whitaker, Makela, Rovamo, & Latham, 1992; Whitaker, Rovamo, MacVeigh, & Mäkelä, 1992). Such studies address eccentricity-dependent changes in sensitivity to isolated stimuli. In the real world, however, we are rarely confronted with isolated stimuli (i.e., on a homogenous, untextured background). Therefore, a full understanding of peripheral vision must deal with sensitivity to targets in the presence of non-target items. A large number of studies addressing this question fall in the category of “crowding studies.”

*s*

_{crit}) needed to achieve a particular threshold elevation (flanked vs. unflanked size threshold,

*T*

_{rel}) at any given eccentricity (

*E*) in terms of two free parameters:

*T*

_{rel}=

*T*

_{flanked}/

*T*

_{unflanked},

*s*

_{2}is the separation at fixation at which size (resolution) threshold was twice the unflanked threshold, and

*E*

_{2H}is the eccentricity at which the separation eliciting

*T*

_{rel}doubles. It is clear from this formulation that

*s*

_{crit}will be proportional to eccentricity when

*E*

_{2H}/

*E*is small, but the exact proportion depends on

*T*

_{rel}. The Latham and Whitaker approach provides the model for the current study, although our data deviate in interesting ways from theirs.

*s*

_{crit}increases rapidly and linearly with eccentricity.)

*s*

_{crit}were independent of target size, then, in the Latham and Whitaker study, the product of target size and relative separation at threshold should be a constant (

*s*

_{crit}= Tsize * relSep). In other words, to keep performance at 75% correct, relative separation must decrease as target size increases to keep

*s*

_{crit}constant. However, it is clear from the data of Latham and Whitaker (Figure 4) that

*s*

_{crit}increases as target size decreases. [It should be noted, however, that Latham and Whitaker defined target–flanker separation as edge-to-edge separation, whereas other authors (Levi, Hariharan, & Klein, 2002; Pelli et al., 2004, 2007) define it as center-to-center separation. When defined as center-to-center separation, the size vs. separation curves become steeper (see Figure 7).]

*s*

_{crit}was not strictly proportional to eccentricity. Rather,

*s*

_{crit}tended to increase with eccentricity. Finally, if critical spacing (however defined) represents a hardwired constraint, it is worth asking if it is task independent. Therefore, we bring together three sets of stimuli (drawn from Latham & Whitaker, 1996; Tripathy & Cavanagh, 2002; Pelli et al., 2004) within a common paradigm to address this question.

^{2}and that of the target (and flanker) regions was 0.687 cd/m

^{2}.

*relative*target–flanker (center-to-center) separation for the six eccentricities. Circles represent conditions in which relative separations were 1.25 to 8× target size. Triangles represent the unflanked condition. For purposes of illustration, data from the unflanked condition are plotted at 16 times target size. The data represent the average of four subjects.

*s*

_{ E }=

*s*

_{0}(1 +

*E*/

*E*

_{2}), where

*s*

_{0}is stimulus size at fovea,

*s*

_{ E }is the stimulus size at eccentricity

*E,*and

*E*

_{2}is a task-dependent constant. For the T-orientation task, the mean

*E*

_{2}was 1.76 (

*SEM*= 0.05; 95% confidence interval = 1.59–1.93); for the letter identification task, the mean

*E*

_{2}was 1.35 (

*SEM*= 0.11; 95% confidence interval = 1.01–1.69); for the grating discrimination task, the mean

*E*

_{2}was 1.84 (

*SEM*= 0.04; 95% confidence interval = 1.72–1.97). These values are in line with previous reports of eccentricity-dependent changes in resolution thresholds (e.g., Latham & Whitaker, 1996).

*not*shifted versions of each other but, rather, change shape with eccentricity.

_{min}is the size at which the parabola becomes parallel to the

*x*-axis (i.e., the size eventually reached in the no flanker condition), and sep

_{min}is the separation at which the parabola becomes parallel to the

*y*-axis (i.e., the separation eventually reached as target size gets very large). The parabola describes all combinations of size and separation eliciting threshold performance. By assumption, curves at any other eccentricity can be shifted onto the foveal curve by dividing size (or separation) by the appropriate scaling factor (1 +

*E*/

*E*

_{2}):

*E*

_{2V}corresponds to the vertical (downward) shift, and in Equation 4,

*E*

_{2H}corresponds to the horizontal (leftward) shift. To fit the data in Figure 2B, we used the MATLAB error minimization procedure (fminsearch) to determine the values of

*c,*size

_{min}, sep

_{min},

*E*

_{2V}, and

*E*

_{2H}that provided the best fit to the data averaged over the four subjects. (Analyses were also conducted in SPSS using the non-linear regression routines to confirm the MATLAB results and establish bootstrapped 95% confidence intervals around the parameters. The results of both analyses were identical.) Details of the data analysis strategy to fit the rectangular parabola to the data can be found in Poirier and Gurnsey (2002).

*E*

_{2H}values on the order of 10

^{−7}, meaning that the rate at which the extent of crowding regions increase with eccentricity is 10

^{7}times per degree visual angle. These implausible

*E*

_{2H}values are accompanied by sep

_{min}values also on the order of 10

^{−7}, meaning that the center-to-center separation at which size thresholds become unmeasurable involves stimuli of 10

^{−7}degrees of visual angle. Both of these failures relate to the fact that the size thresholds at fovea (blue dots in Figure 2) show no dependence on target–crowder separation and, thus, do not constrain the leftward shift of the curves. The instability of the fits is shown by the large 95% bootstrapped confidence intervals (upper and lower in Table 1) around these two parameters.

Subject | size_{min} | sep_{min} | c ^{2} | E _{2V} | E _{2H} |
---|---|---|---|---|---|

T orientation | |||||

Estimate | 0.069 | 3.5E-8 | 2.6E-10 | 1.231 | 2.0E-10 |

Standard error | 0.004 | 0.019 | 19.08 | 0.143 | 0.105 |

Lower | 0.061 | −0.038 | −77.59 | 0.942 | −0.213 |

Upper | 0.077 | 0.038 | 77.59 | 1.520 | 0.213 |

| |||||

Letter identification | |||||

Estimate | 0.060 | 1.3E-7 | 6.6E-10 | 1.488 | 9.3E-7 |

Standard error | 0.002 | 0.027 | 6.832 | 0.125 | 0.188 |

Lower | 0.055 | −0.053 | −27.78 | 1.236 | −0.380 |

Upper | 0.065 | 0.053 | 27.78 | 1.739 | 0.380 |

| |||||

Grating discrimination | |||||

Estimate | 0.069 | 4.6E-8 | 3.4E-10 | 1.772 | 1.7E-7 |

Standard error | 0.003 | 0.022 | 20.58 | 0.196 | 0.085 |

Lower | 0.063 | −0.045 | −83.70 | 1.377 | −0.172 |

Upper | 0.074 | 0.045 | 83.70 | 2.167 | 0.172 |

*F*

_{Change}statistics (

*F*

_{Change}= (

*R*

_{NL}

^{2}−

*R*

_{L}

^{2}) / (1 −

*R*

_{NL}

^{2}) * (

*N*−

*k*

_{NL}− 1) / (

*k*

_{NL}−

*k*

_{L})) associated with the change in explained variance from the five-parameter (

*k*

_{L}) linear (L) model to the six-parameter (

*k*

_{NL}) non-linear (NL) model were

*F*

_{Change}(1,42) = 191, 34, and 74, for the T, letter, and grating conditions, respectively. Table 2 shows that the fits appear,

*prima facie,*more plausible than those in Table 1, and the bootstrapped 95% confidence intervals are better behaved. From these analyses, one might argue that a linear vertical shift and a non-linear horizontal shift nicely account for the apparent differences between the curves in Figure 2B (and Figure 3). This is consistent with the notion that data from each eccentricity simply reflect different samples of the same underlying curve.

Subject | size_{min} | sep_{min} | c ^{2} | E _{2V} | α | β |
---|---|---|---|---|---|---|

T orientation | ||||||

Estimate | 0.045 | 0.025 | 0.00029 | 1.829 | 1.273 | 0.270 |

Standard error | 0.001 | 0.009 | 0.00001 | 0.152 | 0.028 | 0.107 |

Lower | 0.042 | 0.006 | 0.00014 | 1.523 | 1.216 | 0.054 |

Upper | 0.047 | 0.044 | 0.00053 | 2.136 | 1.330 | 0.485 |

| ||||||

Letter identification | ||||||

Estimate | 0.057 | 0.039 | 0.00068 | 1.554 | 1.394 | 0.800 |

Standard error | 0.002 | 0.017 | 0.00002 | 0.130 | 0.066 | 0.355 |

Lower | 0.052 | 0.006 | 0.00032 | 1.291 | 1.260 | 0.082 |

Upper | 0.062 | 0.073 | 0.00123 | 1.817 | 1.527 | 1.517 |

| ||||||

Grating discrimination | ||||||

Estimate | 0.067 | 0.062 | 0.00090 | 1.889 | 1.335 | 0.498 |

Standard error | 0.002 | 0.023 | 0.00002 | 0.179 | 0.041 | 0.200 |

Lower | 0.062 | 0.016 | 0.00040 | 1.528 | 1.253 | 0.094 |

Upper | 0.072 | 0.108 | 0.00152 | 2.251 | 1.417 | 0.902 |

*scaled size*and the corresponding

*scaled separation*. Typically, these points correspond to data from 16° eccentricity. When we express the center-to-center spacing in relative terms, we find that for the T-orientation, letter identification, and grating discrimination tasks these correspond to 9.6%, 23.3%, and 15.5% of target size. In all cases then, the predicted separation at fixation required to match the maximum observed size at threshold at 16° is much less than target size. This means that target and flankers would have to overlap, a condition that is outside the bounds of what is considered crowding and into masking territory

^{1}(Levi, 2008; Pelli et al., 2004).

*μ*

_{ E }= separation

_{ E }/size

_{ E }= 1. We then compared

*predicted*size in these conditions (

*μ*

_{ E }= separation

_{ E }/size

_{ E }= 1) with size in the unflanked conditions. For T-orientation, letter identification, and grating discrimination tasks, these correspond to increases at fixation of 27%, 29%, and 28%, respectively, over the unflanked conditions. These increases are dwarfed by the 575%, 234%, and 586% increases seen at 16° in the T, letter, and grating tasks, respectively.

^{2}

*μ*

_{ E }= separation

_{ E }/size

_{ E }= 1.

*F*

_{ EH}=

*μ*

_{ E }/

*μ*

_{0}characterizes the relative shifts on the separation axis. Therefore, if we divide all separations at eccentricity

*E*by

*F*

_{ EH}=

*μ*

_{ E }/

*μ*

_{0}, the curves at each eccentricity will shift leftward. A downward shift can be accomplished analogously. That is,

*F*

_{ EV}= unflanked

_{ E }/unflanked

_{0}expresses the unflanked threshold at each eccentricity relative to the unflanked threshold at fixation. Thus, dividing all sizes at eccentricity

*E*by

*F*

_{ EV}= unflanked

_{ E }/unflanked

_{0}shifts curves at each eccentricity downward. The results of these shifts are shown in Figure 4B.

*non-foveal*conditions seem to conform to the same function. The foveal data are isolated from the rest because there is no evidence that size thresholds increase with decreases in separation (Strasburger, Harvey, & Rentschler, 1991). We then found the best fitting rectangular parabola for these size- and separation-scaled non-foveal data. These fits are shown as continuous black lines in Figure 4B. The fits are very good. We conclude, therefore, that the curves at all eccentricities except 0° shift on to a single curve with appropriate scaling of size and separation. It is important to keep in mind that the range over which scaled size is defined differs across eccentricities.

*F*

_{ EV}= unflanked

_{ E }/unflanked

_{0}and

*F*

_{ EH}=

*μ*

_{ E }/

*μ*

_{0}, respectively, then these best fitting functions can be superimposed on the raw data. These fits are shown as the overlaid continuous black lines in the 12 panels of Figure 3. Again, this simple model matches the data very well.

^{3}

_{min}, size

_{min},

*c*

^{2}) of the fits for each subject, and the average of the four subjects, in each condition.

*E*

_{2V}, which characterizes the change in the size thresholds in the unflanked condition, as well as

*α*and

*β,*which define the necessary leftward shift (

*F*

_{H}= 1 +

*E*

^{ α }/

*β*), is also shown.

Subject | size_{min} | sep_{min} | c ^{2} | E _{2V} | α | β |
---|---|---|---|---|---|---|

T orientation | ||||||

GR | 0.0504 | 0.0652 | 0.0004 | 1.6532 | 1.3287 | 0.6179 |

P1 | 0.0527 | 0.0672 | 0.0011 | 1.6655 | 1.4737 | 1.3888 |

WC | 0.0443 | 0.0550 | 0.0007 | 1.9160 | 1.1468 | 0.5725 |

P2 | 0.0411 | 0.0525 | 0.0004 | 1.7933 | 1.4865 | 0.9249 |

Mean | 0.0472 | 0.0591 | 0.0007 | 1.7419 | 1.3704 | 0.8261 |

SEM | 0.0027 | 0.0037 | 0.0002 | 0.0617 | 0.0792 | 0.1880 |

| ||||||

Letter identification | ||||||

GR | 0.0604 | 0.0755 | 0.0011 | 1.6475 | 1.4226 | 1.2646 |

P1 | 0.0617 | 0.0908 | 0.0007 | 1.4498 | 1.3976 | 2.2557 |

WC | 0.0577 | 0.0630 | 0.0004 | 1.1850 | 1.2610 | 0.8897 |

P2 | 0.0424 | 0.0632 | 0.0002 | 1.1188 | 1.3256 | 1.0822 |

Mean | 0.0553 | 0.0761 | 0.0005 | 1.3359 | 1.4524 | 1.6775 |

SEM | 0.0045 | 0.0066 | 0.0002 | 0.1222 | 0.0366 | 0.3040 |

| ||||||

Grating discrimination | ||||||

GR | 0.0764 | 0.1162 | 0.0031 | 1.8307 | 1.1990 | 0.5374 |

P1 | 0.0794 | 0.1255 | 0.0014 | 1.9738 | 1.3307 | 1.5412 |

WC | 0.0718 | 0.0636 | 0.0004 | 1.7720 | 1.7339 | 1.3947 |

P2 | 0.0629 | 0.0703 | 0.0005 | 1.7905 | 1.5530 | 1.1178 |

Mean | 0.0730 | 0.1107 | 0.0014 | 1.8434 | 1.4323 | 1.1364 |

SEM | 0.0036 | 0.0157 | 0.0006 | 0.0457 | 0.1185 | 0.2216 |

*F*

_{ EV}= unflanked

_{ E }/unflanked

_{0}and

*F*

_{ EH}=

*μ*

_{ E }/

*μ*

_{0}in each condition and eccentricity. The magnification factors associated with the vertical shifts change rather modestly and linearly with eccentricity. As noted earlier, these magnification functions (

*F*

_{V}) are characterized by average

*E*

_{2}values of 1.76, 1.35, and 1.84, for the T, letter, and grating tasks, respectively. The magnification factors associated with the horizontal shifts change rather quickly and non-linearly with eccentricity. Because of the non-linear change with eccentricity, the concept of

*E*

_{2}does not apply. Nevertheless, the shifts are rather systematic and are nicely captured by

*F*

_{H}= 1 +

*E*

^{ α }/

*β,*as shown in columns 5 and 6 of Table 3.

*s*

_{crit,E }/

*E*) for each condition and each subject.

*s*

_{crit,E }/

*E*increases for all threshold elevations. This increase with eccentricity is a consequence of the fact that a non-linear magnification is required to shift the functions leftward onto a common function, whether one uses the double-scaling methods of Latham and Whitaker (1996) and Poirier and Gurnsey (2002) or the method summarized in Figure 4B. Second, as widely recognized,

*s*

_{crit,E }/

*E*depends on the criterion used to define

*s*

_{crit,E }; the greater the threshold elevation, the smaller

*s*

_{crit,E }will be. For one to compare critical spacing across tasks and conditions, one would need to establish a uniform definition of critical spacing.

*E*

_{2}values of 1.76, 1.35, and 1.84, for the unflanked conditions of the T, letter, and grating tasks, respectively. These results are consistent with a large literature starting with Weymouth (1958). Latham and Whitaker (1996) found average

*E*

_{2}values of 1.41 in the lower visual field, so our results are generally consistent with theirs.

*F*

_{H}≈ 1 +

*E*/0.14. If these results are applied to the representation shown in Figures 2B and 3, then the conclusion would be that linear shifts in the vertical and horizontal directions—defined by

*F*

_{V}≈ 1 +

*E*/1.41 and

*F*

_{H}≈ 1 +

*E*/0.14, respectively—are sufficient to characterize the nature of crowding and resolution limits across the visual field. The curves in Figure 5 corresponding to the horizontal magnifications (blue dots) can be approximated with a straight line with a corresponding

*E*

_{2}of 0.1 to 0.2. Therefore, our data show—like those of Latham and Whitaker (1996) and Toet and Levi (1992)—that crowding or interference zones increase far more rapidly with eccentricity than simple resolution limits. It is important to note, however, that a linear magnification does not properly characterize the changing extents of crowding zones. In fact, the needed magnifications grow exponentially with eccentricity (Figure 5). Therefore, it may be that when the needed magnification is very great it will accelerate with eccentricity.

^{4}We tested further into the periphery than did Latham and Whitaker (16° vs. 10°), and perhaps because of this, the non-linearity is revealed.

*F*

_{H}= 1 +

*E*

^{ α }/

*β*) in conjunction with a linear vertical magnification factor (

*F*

_{V}= 1 +

*E*/

*E*

_{2}) is sufficient to collapse size vs. separation data from all eccentricities onto a single function. Although this kind of double scaling technically eliminates eccentricity-dependent variance from our data set, the implied target–flanker overlap required at fixation to match the observed size threshold in the periphery is inconsistent with the consensus view of crowding, and for this reason, we reject this version of the non-linear model.

*y*-axis corresponding to small separations and large sizes), but as separation increases (and size decreases), the curve slopes and then becomes parallel to the

*x*-axis. In this latter part of the curve, size and separation trade off so that as size decreases, larger separations are needed to maintain threshold-level performance. Near fixation, the maximum threshold elevation is relatively modest, and so over most of the curve, there is a trade-off between size and separation at threshold. At further eccentricities, the maximum threshold elevation is much greater and so crowding is size independent over more of the curve. Therefore, it is not generally true that crowding is size independent. At each eccentricity, there are stimulus sizes for which crowding is size dependent. As eccentricity increases, then larger parts of the function are roughly parallel to the size axis. This region represents size independence and the range of sizes for which size independence is seen grows with eccentricity.

_{unflanked}− Acc

_{chance}) * (1 − 1/

*e*) + Acc

_{chance}. Their results for two subjects are plotted in Figure 7A. The data are somewhat noisy for subject ST, but they are quite similar to our data at 8° for the same condition. Their data differ from ours in that there is less evidence at small sizes for a shift from size independence to size dependence. Put the other way around, crowding zones do not get smaller as targets get larger as they do in our data. This difference may well be explained by the negative correlation between stimulus size and contrast (e.g., Strasburger et al., 1991). (We return to this point.)

^{5}This may be because the size-dependent increase in (internal) signal strength was offset by a decrease in contrast. This size/contrast trade-off may have held the strength of the internal response to the target constant and, hence, kept constant the separation required to produce the criterion reduction in accuracy. In fact, our experiments can be seen as differently controlled versions of theirs. They found a threshold separation between targets and flankers when contrast is inversely related to target size, whereas we found threshold separation between targets and flankers when contrast is fixed. Therefore, our results confirm their implicit assumption that the extent of crowding would be size dependent when contrast does not covary with target size.

*s*

_{crit}is independent of many differences between target and flankers, the representation of the data makes it difficult to reanalyze. We note, however, that their paradigm and ours are quite different. Their conclusions were based on the effects of flankers on contrast thresholds and ours are based on effects of flankers on size thresholds. It may be that these two paradigms reflect different aspects of neural interactions. One indication that there may be a difference between paradigms is the finding of Pelli et al. (see their Figure 16) that orientation discrimination is not subject to crowding when the gratings to be discriminated differed by 90°. We found very strong crowding effects under these conditions. Clearly, a direct test of paradigm dependence of crowding is on order. We should also note that there is growing evidence that flanker characteristics make important contributions to the degree of crowding observed (Kooi, Toet, Tripathy, & Levi, 1994; Levi & Carney, 2009; Livne & Sagi, 2010; Nazir, 1992).

*some*deviation from performance in the uncrowded condition. Just what this deviation should be is a matter of choice. We have just shown that the magnitude of crowding increases with eccentricity (e.g., Figure 6A). Therefore, if one defines the critical separation stringently (e.g., the separation at which threshold doubles from the uncrowded case), then it may be undefined at some eccentricities. For example, if critical spacing were defined as a threshold doubling, then it would be undefined for some subjects at small eccentricities in our experiments. Pelli et al. (2004) argue that any deviation from the uncrowded threshold should define critical separation. The advantage of this definition is that if crowding exists then critical spacing is defined even if the magnitude of crowding is relatively modest, i.e., does not rise to the level of some specified threshold elevation. The drawback is that in many cases the precision with which we can measure such a point is extremely coarse. Because the tail of any psychometric function approaches chance asymptotically, there is really no point that can be defined as

*the*deviation from chance. Alternatives involve fitting the sloping and asymptotic parts of the curve with straight lines and looking at the point of intersection. Such an approach can do some violence to the data and has an arbitrary quality to it.

*s*

_{crit,E }/

*E*) generally remains well below Bouma's constant of 0.5, which may be considered an upper limit on critical spacing.

*F*

_{H}= 1 +

*E*/

*E*

_{2}, then

*F*

_{H}/

*E*= 1/

*E*+ 1/

*E*

_{2}and this function drops rapidly to an asymptotic value of 1/

*E*

_{2}. If

*E*

_{2}= 0.2, then

*F*

_{H}/

*E*drops by only 8% from 2 to 16°. Therefore, one might expect crowding to be proportional to eccentricity. However, we find, as did Pelli et al. (2007), that this is not strictly the case. At 1°, critical spacing relative to eccentricity is generally greater than at 2°, but this ratio tends to increase with eccentricity thereafter (Figure 6B). Therefore, we find the notion of Bouma's constant to be a very rough approximation to the true nature of crowding.

- The maximum threshold elevation (
*μ*_{ E }) increases non-linearly with eccentricity. - Left shifting all curves on the separation axis by
*F*_{ EH}=*μ*_{ E }/*μ*_{0}and downshifting all curves on the size axis by*F*_{ EV}= unflanked_{ E }/unflanked_{0}aligns all but the foveal curve. - These scaled data can be fit with a single curve, thus providing a complete description of the relationship between size and separation at threshold across eccentricities of 0 to 16°.
- The non-linear change in the extent of crowding implies that the “critical spacing” is not proportional to eccentricity.
- Furthermore, “critical spacing” depends on the level of performance used to define it.

*unflanked*condition.

*t*

_{3}= 2.02, 1.09, and −0.76;

*p*= 0.14, 0.35, and 0.50, respectively; effect sizes = 1.02, 0.52, and −0.40, respectively). The corresponding analysis at 16° yields average increases of 467%, 171%, and 394% for the T-orientation, letter identification, and grating discrimination tasks, respectively. All of these differences are statistically significant (

*t*

_{3}= 5.24, 5.22, and 6.24;

*p*= 0.014, 0.014, and 0.008, respectively; effect sizes = 2.62, 2.61, and 3.12, respectively).

*μ*

_{ E }= separation

_{ E }/size

_{ E }= 1. When we then fit these shifted data with a single rectangular parabola, the value for

*μ*

_{ E }= separation

_{ E }/size

_{ E }= 1 on the best fitting curve will not equal those used to left shift in the first place. Inspection of Figure 3 shows that this small distortion is of little consequence.