Studies of eccentricity-dependent sensitivity loss typically require participants to maintain fixation while making judgments about stimuli presented at a range of sizes and eccentricities. However, training participants to fixate can prove difficult, and as stimulus size increases, they become poorly localized and may even encroach on the fovea. In the present experiment, we controlled eccentricity of stimulus presentation using a simulated central scotoma of variable size. Participants were asked to perform a 27-alternative forced-choice shape-from-texture task in the presence of a simulated scotoma, with stimulus size and scotoma radius as the independent variables. The resulting psychometric functions for each simulated scotoma were shifted versions of each other on a log size axis. Therefore, stimulus magnification was sufficient to equate sensitivity to shape from texture for all scotoma radii. Increasing scotoma radius also disrupts eye movements, producing increases in fixation frequency and duration, as well as saccade length.

*E*) can be made equal to that observed at fixation by setting the size of the peripherally presented stimulus to a multiple (

*F*

_{ E }) of the size (

*S*

_{0}) of the foveal stimulus:

*F*

_{ E }) at each eccentricity required to elicit performance equivalent to a foveal standard (Levi, Klein, & Aitsebaomo, 1985). The free parameter

*E*

_{2}is so named because it indicates the eccentricity at which stimulus size must double to elicit equivalent-to-foveal performance. So, not only does magnification compensate for eccentricity-dependent sensitivity loss but also the required magnification is frequently a linear function of eccentricity.

*E*

_{2}can be used to characterize the eccentricity dependence of a particular task using assumption-free methods (e.g., Watson, 1987). If performance is measured at range of sizes and eccentricities, then the performance-vs.-size functions at each eccentricity will be shifted versions of each other if the only change with eccentricity is the local scale of the mechanisms engaged (Watson, 1987). If this is the case, then all performance-vs.-size functions can be collapsed onto a single curve by dividing stimulus size at each eccentricity by the appropriate

*F*

_{ E }= 1 +

*E*/

*E*

_{2}. The value of

*E*

_{2}providing the best fit to the data can be established using numerical methods. Once

*E*

_{2}is known, one can specify the magnification needed at each eccentricity to match foveal performance. For a wide array of tasks, such as orientation discrimination (Makela, Whitaker, & Rovamo, 1993; Sally & Gurnsey, 2003, 2004, 2007; Sally, Poirier, & Gurnsey, 2005), symmetry detection (Saarinen, 1988; Sally & Gurnsey, 2001), vernier acuity (Whitaker, Rovamo, MacVeigh, & Mäkelä, 1992), and grating acuity (Rovamo & Virsu, 1979; Rovamo, Virsu, & Nasanen, 1978), stimulus magnification is sufficient to compensate for eccentricity-dependent sensitivity loss. [It is worth noting, however, that there are many tasks in which a single magnification factor fails to compensate for eccentricity-dependent sensitivity loss (Chung, Li, & Levi, 2007; Chung, Mansfield, & Legge, 1998; Latham & Whitaker, 1996; Melmoth, Kukkonen, Mäkelä, & Rovamo, 2000; Pelli, Palomares, & Majaj, 2004; Poirier & Gurnsey, 2002, 2005; Strasburger, Rentschler, & Harvey, 1994).]

*blind spot*(henceforth

*scotoma*) can be appreciated by considering how many words on this page can be read when you place your thumb on some text and fixate it at arm's length. Studying peripheral vision in normal observers may therefore lead to insight into the expected information processing abilities in individuals with AMD.

*SfM*) and structure from texture (

*SfT*; Gurnsey, Poirier, Bluett, & Leibov, 2006) across the visual field. The conclusion from this developing body of research is that size scaling can compensate for eccentricity-dependent sensitivity loss in these tasks.

^{3}= 27 different stimuli.

*σ*) = 4 pixels. Each of the 27 surfaces was also defined as a 1024 × 1024 array. Hills, valleys, and planes were placed 241 pixels from the center of the array and separated by 120° on a notional circle. Hills and valleys were isotropic Gaussians with

*σ*= 85.33 pixels. The noise array was texture-mapped to the surface and projected orthographically to the image plane using the

*mesh*function in Matlab. The camera angle had azimuth of −37.5° and an elevation of 25°. This means that the surface was rotated about the

*z*-axis by 37.5° counterclockwise and then projected to a camera that was elevated 25° above the

*xy*plane. The stimulus shown in Figure 1 provides an example of one such stimulus.

*F*

_{ R }. Assuming that the magnification needed is a linear function of scotoma radius (

*R*), the function

*F*

_{ R }= 1 +

*R*/

*R*

_{2}expresses the necessary scaling required for each scotoma radius. (Note that

*R*

_{2}plays the same role in the present analysis that

*E*

_{2}plays in the studies described in the Introduction section.

*E*

_{2}refers to the eccentricity at which stimulus size must double to maintain equivalent to foveal performance and

*R*

_{2}refers to the scotoma radius at which stimulus size must double to maintain equivalent to foveal performance.) This leaves the two parameters of the Gaussian integral (i.e.,

*μ*and

*σ*) and

*R*

_{2}to be determined. This was done using the

*fminsearch*function in Matlab to minimize the sum of squared deviations from the predicted curve

*pc*is proportion correct,

*α*is the chance performance (in this case 1/27),

*x*is stimulus size,

*F*

_{ R }= 1 +

*R*/

*R*

_{2},

*μ*is the mean of the Gaussian, and

*σ*is its standard deviation.

*R*

_{2}value providing the best fit and the proportion of variance (

*r*

^{2}) explained by the fit. In general, the fits were very good, explaining 94% of the variability in the data on average. The average

*R*

_{2}value was 1.701 (

*N*= 5, estimated

*SEM*= 0.2874, 95% CI = 0.962 to 2.44). Therefore, size scaling eliminates most scotoma-dependent variability from the data. Of course, scotoma radius (

*R*) is a stand-in for eccentricity because the radius of the scotoma determines the eccentricity of the non-occluded part of the stimulus. It is interesting to note that Gurnsey et al. (2006) found

*E*

_{2}= 1.52 (

*N*= 6, estimated

*SEM*= 0.53, 95% CI = 0.79 to 2.25) eliminated most eccentricity-dependent variability in a similar shape-from-texture task. Therefore, whether eccentricity is controlled by a simulated scotoma or by fixating away from the stimulus, very similar effects of eccentricity are seen.

^{−1}maintained for at least 4 ms. Fixations were defined as a period of no movement for a minimum duration of 20 ms. The initial fixation in each trial was defined as the fixation following the stimulus onset; thus the fixation from the pretrial drift correction was not included in the analysis. Once fixations and saccades were defined, the following eye movement statistics were calculated: number of fixations per trial, average fixation duration, and average amplitude of saccade. Maps of the fixation position and frequency at each stimulus and eccentricity size were produced using the SR Research data viewer.

*x*-axis represents stimulus size. Moving from left to right shows results for different scotoma radii (i.e., 0 to 8°; see insets in the top row). Within each panel, the average value of the dependent variable is plotted as a function of stimulus size for the condition with 0 features (i.e., no hills or valleys; blue dots), 1 feature (green dots), 2 features (red dots), 3 features (cyan dots), and the average value over all 27 stimuli (yellow dots).

*η*

_{p}

^{2}) as an effect size measure,

^{1}and the reported p-values correspond to those obtained following the Greenhouse–Geisser correction for violations of sphericity (Greenhouse & Geisser, 1959). There were relatively large effect sizes for the main effect of scotoma radius,

*η*

_{p}

^{2}= 0.567,

*F*(5,20) = 5.245,

*p*= 0.055, and the interaction of stimulus sizes and scotoma radius,

*η*

_{p}

^{2}= 0.504,

*F*(30,120) = 4.06,

*p*= 0.029, but the effect size for the main effect of stimulus size was small,

*η*

_{p}

^{2}= 0.106,

*F*(6,24) = 0.474,

*p*= 0.622. It is the interaction between stimulus sizes and scotoma radius that is most interesting. For the two “smallest” scotomas (no scotoma and scotomas of radius 0.5°), the number of fixations is relatively independent of stimulus size. As scotoma radius increases to 1 and 2°, the number of fixations increases at small stimulus sizes but decrease as stimulus size increases. In other words, when the stimulus is small relative to the scotoma, participants make more fixations, but increases in stimulus size compensate for the loss of central vision, and consequently, fewer fixations are made. For scotomas of radius 4, the number of fixations is essentially independent of stimulus sizes, whereas for scotomas of radius 8, the number of fixations is low at small sizes and increases as stimulus sizes increase. For the 8° scotoma, very few fixations are made for the smallest stimuli, suggesting that participants simply give up because the stimuli are too small to resolve in the presence of a large central scotoma. However, more fixations are made for larger stimuli, suggesting that as stimulus size increases participants can extract usable information with peripheral vision and devote more time attempting to identify the stimulus.

*η*

_{p}

^{2}= 0.132,

*F*(5,20) = 0.61,

*p*= 0.531, and the interaction of stimulus sizes and scotoma radius,

*η*

_{p}

^{2}= 0.204,

*F*(30,120) = 1.025,

*p*= 0.399, and the effect size for the main effect of stimulus size was modest,

*η*

_{p}

^{2}= 0.422,

*F*(6,24) = 2.926,

*p*= 0.155. As shown in the second row of Figure 4, there is a modest decrease in average fixation duration with image size within each panel (representing different scotoma radii). Consistent with previous studies (Hooge & Erkelens, 1999; Jacobs & O'Regan, 1987) the more challenging (i.e., smaller) stimuli are, the longer participants will maintain fixation on a part of it. This is only partially true, however, because otherwise we would expect a larger effect of scotoma radius. Nevertheless, fixation duration seems to be largely determined by stimulus size; smaller stimuli tend to be fixated longer. It is interesting to note that fixation duration increases at small stimulus sizes between scotoma radii of 0 to 0.5. This may explain the similarity in the size-vs.-accuracy curves for these two conditions.

*η*

_{p}

^{2}= 0.645,

*F*(5,20) = 7.256,

*p*= 0.047, and stimulus size,

*η*

_{p}

^{2}= 0.647,

*F*(6,24) = 7.316,

*p*= 0.033, and the effect size for the interaction of scotoma radius and stimulus size was modest,

*η*

_{p}

^{2}= 0.293,

*F*(30,120) = 1.654,

*p*= 0.245. As shown in the third row of Figure 4, saccade amplitude increased with stimulus size for each scotoma radius and there was a general increase in saccade amplitude across scotoma radii.

*σ*= 0.5°) at the location of each fixation, combining the fixations of all participants. The summed values at frequently fixated regions will be larger than those at infrequently fixated regions. Intensity in the these maps range from red, indicating the highest frequency of fixation, to green, representing regions of few fixations. These color-coded fixation maps are superimposed on a sample stimulus in Figure 5; this stimulus has fixed size (21.84 degrees of visual angle) for this illustration. Each row depicts a different scotoma radius and each column corresponds to the number of features in the display. The results from the 0° (no scotoma) condition (first row) show that participants distribute their fixations over the image in a manner independent of the number of features in the display. Furthermore, for scotoma radii of 0 to 2° the fixations seem to be contained within roughly the same area. However, for scotoma radii of 4 and 8° (last two rows) fixations extend beyond the confines of the stimulus. Of course, in these cases participants are forced to fixate away from a feature in order to identify it because of the scotoma. It is extremely interesting to note that the distribution of fixations expands mostly above the stimulus and to the sides. Clearly, participants found it more informative and/or more natural to have the stimulus fall in the lower visual field.

*x*-axis), there was a large effect size for the main effects of stimulus size,

*η*

_{p}

^{2}= 0.651,

*F*(6,24) = 7.476,

*p*< 0.01. As the image size increases, then participants move their eyes away from the center of the stimuli, and more toward the left of the stimulus (Figure 6). There were also moderate effect sizes for scotoma radius,

*η*

_{p}

^{2}= 0.37,

*F*(5,20) = 2.347,

*p*= 0.176, and the interaction of scotoma radius and stimulus size,

*η*

_{p}

^{2}= 0.331,

*F*(30,120) = 1.978,

*p*= 0.199. As seen in the first column of Figure 6 (relative image size = 1), as the scotoma radius increases then the average location fixated moves to the left of the stimulus.

*y*-axis), there were large effect sizes for the main effects of stimulus size,

*η*

_{p}

^{2}= 0.823,

*F*(5,20) = 18.587,

*p*< 0.01, and scotoma radius,

*η*

_{p}

^{2}= 0.507,

*F*(6,24) = 4.117,

*p*< 0.1, and a moderate effect size for the interaction of scotoma radius and stimulus size,

*η*

_{p}

^{2}= 0.321,

*F*(30,120) = 1.978,

*p*= 0.199. As shown in Figure 6, as the stimuli become more difficult to perceive, either due the increase in scotoma radii or decrease in image size, then the participants tend to fixate on the top of the stimulus.

*x*and

*y*positions during a trial.

*d*was measured on each trial for each participant and these values were then averaged for each participant, thus providing a measure of fixation dispersion for each participant for each scotoma radius and image size. These measures of dispersion were submitted to a 7 (stimulus sizes) by 6 (scotoma radii) within-subjects ANOVA. There were very large effect sizes for the main effects of stimulus size,

*η*

_{p}

^{2}= 0.874,

*F*(6,24) = 27.72,

*p*< 0.001, and scotoma radius,

*η*

_{p}

^{2}= 0.702,

*F*(5,20) = 9.44,

*p*< 0.001, but the effect size for the interaction of scotoma radius and stimulus size was weak,

*η*

_{p}

^{2}= 0.109,

*F*(30,120) = 0.49,

*p*= 0.606. Generally, as scotoma radii and image size increased, there was a corresponding increase in the dispersion of fixations.

*R*

_{2}) at which stimulus size must double to match performance with no central scotoma. The mean

*R*

_{2}value was 1.701.

*R*

_{2}= 1.7 and in Gurnsey et al. by a mean

*E*

_{2}= 1.52.

*R*

_{2}of 1.7 suggests that size thresholds (sizes eliciting 64% correct in a 27 AFC task) should change by only about 60% for scotomas (or eccentricities) in the range 0 to 1°. However, this does remain a point for further investigation.

^{1}We are persuaded that the primacy of

*p*-values is a bad thing for science and that measures of effect size should take their place (e.g., Cohen, 1994; Kline, 2004). Therefore, we report effect size measures (partial eta squared,

*η*

_{p}

^{2}= SS

_{effect}/(SS

_{effect}+ SS

_{error})) before the

*F*ratios and their

*p*-values. (Partial eta squared must not be confused with eta squared

*η*

^{2}= SS

_{effect}/SS

_{total}, which represents the proportion of total variability [in the dependent variable] explained by the effect in question; see Pierce, Block, and Aguinis (2004) for examples of mistaken use.) The advantage of effect size measures is that their expected values are independent of sample size. However, for a fixed

*η*

_{p}

^{2}in the population the corresponding

*F*ratio (

*F*= SS

_{effect}/SS

_{error}* df

_{error}/df

_{effect}) increases with the number of degrees of freedom in the error term. Consequently, as oft noted, a trivially small effect size (

*η*

_{p}

^{2}) becomes statistically significant with sufficient degrees of freedom. Conversely, relatively large values of

*η*

_{p}

^{2}may not be statistically significant if there are insufficient degrees of freedom. Therefore, an estimate of the size of the effect stays closer to the data and is more meaningful. Partial eta squared is the standard effect size measure reported in SPSS for Analysis of Variance, which we used to perform our ANOVAs. Unfortunately, SPSS has not evolved to the point of providing confidence intervals for partial eta squared. We categorize our effect sizes as small (

*η*

_{p}

^{2}< 0.3), medium (0.3 <

*η*

_{p}

^{2}< 0.5), large (0.5 <

*η*

_{p}

^{2}< 0.6), and very large (

*η*

_{p}

^{2}> 0.6), but we acknowledge that this is somewhat arbitrary. However, because the effect sizes themselves are reported readers are free to judge whether they agree with our characterizations. As noted by Cohen (1988), whether a particular effect size is considered large or small depends on the relevant literature. It would be worthwhile to survey the literature in vision science journals to determine the range of effect sizes typically found. (Note,

*η*

_{p}

^{2}= 1/[1 + (

*F** df

_{effect}/df

_{error})

^{−1}], which means that the effect size can be recovered from the

*F*ratio if the degrees of freedom have been reported.)

*p*< .05). American Psychologist, 49, 997–1003.