let me answer with another question...

why do you want to change the size of the stimulus?

will this not represent a confounding factor in your analysis?

why not simply takinga reasonable large stimulus, say a 20/200 equivalent (0.1decimal or 1logMAR) across the field...

but maybe I misunderstood the question, the problem or your setup

Thanks, Samuel Arba Mosquera . My experiement investigates translation tolerance (i.e., the ability to recognise objects at different retinal locations). It is necessary to increase stimuli size with eccentricity because performance in same-different tasks is known to decrease with eccentricity. I would like to show that, if you control for loss of acuity (by increasing size of stimulus at larger eccentricities), performance is equivalent at each eccentricity (i.e., highly translation tolerant).

But as I outline in my initial Q, I don't know if there is a (simple) psychophysical rule for increasing stimulus size with eccentricity?

This recent paper by J Denniss is doubly relevant: https://iovs.arvojournals.org/article.aspx?articleid=2702938

First, it has a nice equation for predicting acuity in the central 10 degrees (macula). Second, it shows that acuity isn't terribly predictive, even of something as simple as differential light sensitivity (i.e., ability to detect transient flashes of light). I mean even foveal acuity is barely predictive of acuity! (compare resolution acuity vs recognition acuity)

So while controlling for acuity would be a 'nice touch', and one that I'm sure reviewers will appreciate, I'm not sure how effective it would really be at 'controlling' for eccentricity (i.e., for decreasing ganglion density) on a high-level task.

All that said, it would certainly be interesting --- to some --- to see if/how your results differ with/without scaling (i.e., using the Denniss equation, or any other reasonable model of the Hill of Vision).

Best, pete

EDIT: Given that your stimuli are essentially 'crowded', it may make as much or even more sense to use Ricco's Area as your control index, rather than acuity. No doubt there is some horrendously detailed map of Ricco's area as a function of retinal eccentricity published in JOSA or similar. See here for possible references: Article The Effect of Age on the Area of Complete Spatial Summation ...

You could even start thinking about whether variations in (unscaled) performance are better predicted by the expected rate-of-change in ricco's area or by the expected rate-of-change in acuity. But this may be very far from what you are actually interested in.

rather than increasing stimulus-size with eccentricity, perhaps consider manipulating luminance-contrast instead. The advantage here is that you can hold stimulus size constant, which is useful because stimuli out at 12 degrees will have to be huge to recognize. At least in the central fovea, the relationship between recognition acuity and contrast is quite predicable, although the nonlinear function describing that relationship shifts up or down depending upon a number of factors (e.g., age, stimuilus complexity)

lots of papers on this topic --e.g., Poynter, D. (1992). Contrast sensitivity and image recognition: applications to the design of visual displays. Displays: Technology and Applications, 13 (1), 35-43.

Hi Ryan,

Scaling stimulus size with eccentricity (M scaling) is, in my mind, nearly alwas a good idea. The effects of eccentricity are nearly always the strongest of all and without scaling, more subtle effects often go unnoticed. I don’t think you introduce a confounding variable; rather, a particularly strong effect is controlled, i.e. is accounted for.

Now, as to what exact scaling function to use is, to some degree, a matter of taste. Many authors have used the M-scaling proposed by Rovamo & Virsu (and I am fine with that except that I find the small third-order term overkill). Yet a more modern way would be to choose a fully linear function with a more current approximation of the cortical-magnification-factor-M-vs.-eccentricity function. Up-to-date functions are by Schira, Tyler et al. (2009); see Table 2 in my paper:

https://www.biorxiv.org/content/10.1101/621458v1

Schira et al. give an E2 value of 0.33. So the equation will simply be:

S/S0 = 1 + E/E2,

where S is scaled size, S0 is the size at the center, E2 is 0.33, and E is eccentricity in degrees (that is eq. 2 in that paper). Simple, isn't it?

The rationale for using M-scaling rather than some visual performance scaling (like acuity, perceptual field size, Ricco area, etc.) is that it reflects what we know about the primary visual-cortical architecture and makes no assumptions on perceptual functions coming into effect. As an aside, acuity and the M-factor scale very similarly (because they are both “low level”), so it does not make much difference. In any case, the most important point is to document the function so that experiments can be repeated.

By the way, the rate of size increase with M scaling is highly overestimated by most everybody (even textbook authors) – scaled stimuli at 10° will not “be huge”. The increase is rather modest, indeed. Check out my latest preprint on “myths”:

https://peerj.com/preprints/27353

Pete R Jones has pointed out above that crowding will most probably play a far more important role. I definitely agree with that. However, it need not bother you too much: The idea behind stimulus scaling is to get rid of a trivial factor, not to eliminate all eccentricity-dependent effects.

All the best,

Hans

Many thanks, Hans Strasburger . I still am unsure about application of this formula though:

Let’s say I use the formula: S/S0 = 1 + E/E2,

and S0=4° (based on fact previous same-different tasks use roughly this size), and stimuli is shifted to 10° eccentricity (i.e., E=10°).

So that gives me: 1 + 10/0.33 = 31.33? i.e., my stimuli should increase by 31.33% (i.e., my 4° stimulus becomes 5.25°?). Is that the correct application of the formula?

A related question is whether Schira’s (2009) E2 estimate is appropriate for my stimuli. Although I am only after a 'rough and ready' estimate for controlliing acuity, it seems like stimuli matters: my stimuli are letter-like stimuli (e.g., Chinese characters which are novel to participants), so I'd be tempted to take E2 from a letter identification experiment. Looking at Table 4 in Strasburger et al. (2011), letter identification uses E2 = 3.3, and this would give me a much lower estimate for S (i.e., an increase of just 4.03% after a 10° shift) Have I misinterpreted something?

Hi @Ryan,

Sorry for the late answer. You got me thinking there!

First off, 1 + 10/0.33 = 31.33 is correct (actually it is 31.30) -- but that does not translate to percent. The equation has the ratio S/S0 on the left, so if you solve for S you get S=31.3*S0. That is a huge increase by 31-fold at 10° ecc. (or 3000%). With your 4° foveal stimulus that would be 4°*31.3=125° size at 10° ecc., which is of course ridiculous.

I can think of three reasons why M scaling does not work here:

(1) Even though the size of the cortical area is predictive for the resolution of small stimuli, like Landolt rings at threshold, it tells us little about pattern recognition. As @Pete R Jones has pointed out above, the crowding effect is much more predictive for that.

(2) Acuity -- which follows M -- increases only very moderately with eccentricity (whereas crowding increases steeply). So I was under the impression that, with M scaling, the sizes also increase only very moderately. I forgot, though, that the rate of increase (the slope) for M-scaling depends also on the foveal size. The foveal size of a Landolt ring at threshold is 5 minutes of arc = 0.08°, whereas the foveal stimulus in your experiment has a size of 4°. I.e., it is 48 times as large, and the slope is thus also 48 times as large! So it’s really, really steep.

(3) Four degrees size does not sound much but is huge in cortical terms. A rough estimate shows that the cortical projection area for that is about 7x7 cm² in V1 (imagine that at the back of your head)! That is already a sizeable portion of the visual cortex, and we don’t know what rules to apply at those sizes.

That leaves the question, how a sensible scaling looks like. Since even the shallowest low-level scaling (E2=3.3) is probably too steep, and there does not seem to be a principled answer, I agree with your initial suggestion to investigate it empirically (psychophysically). It’s called a parametric approach; when you cannot predict the influence of a certain parameter, introduce it as an additional parameter. However, I would not choose sizes arbitrarily but would do that according to the linear scaling rule. For example, use 3 or 4 scaling regimes: (1) no scaling (E2 infinite), (2) E2=20°, (3) E2=10°, (4) E2=5°. (As a calculation example, for a foveal size of 4°, E2=10° corresponds to a size of 2*4°=8° at 10° ecc).

You might like to look at the classical paper of Watson (1987) for a discussion of general eccentricity scaling.

All the best,

Hans

Watson, A. B. (1987). Estimation of local spatial scale. Journal of the Optical Society of America A, 4, 1579–1582.