Modern VA charts use a geometric progression of letter size. Either log MAR or log decimal acuity can be used to specify VA for stat analyses.

Agreed with Dr. Massof. If you want to do statistical analysis, you need to use logarithmic acuities.

Acuity is a complex concept (detection acuity, recognition acuity, resolution acuity, etc.), and precisely what you are measuring can vary depending on both on the stimuli that you use, and also where on the retina it is presented (Larry Thibos has some good papers on this). Exactly what you report and how you report it may depend on your precise circumstances.

In general though, I see nothing wrong with performing a logarithmic correction for positive-skew where it is appropriate (and I have yet to see evidence that it isn't appropriate in the case of MAR -- for more, see my comment to your previous question).

Even more generally, I would suggest you might get more useful responses if you describe a specific situation/problem. It doesn't make much sense to say X or Y is the "right" or "wrong" way to report data in the abstract.

An excellent question, I find! Forgive me thus if my answer is somewhat lengthy.

Yes, everybody is ready to agree that specification and statistical analysis should be done on logMAR. Yet I agree with you that there should be a rationale for the logarithmic scaling on first principles: And there is none as far as I am aware of!!

As to the Fechner scale, Fechner (in his book 1860) made a distinction between *extensive* and *intensive* qualities. If I remember correctly he says he will concern himself mostly with the intensive qualities (like brightness or loudness). For these, the log scale works well (even though Stevens thought that his scale is much better). In contrast, the sensation of length, i.e. of an extensive quality, appears to be approximately linear, and thus not logarithmic. MAR is a prototypical example of an *extensive* (visual) scale. (Length is the standard example in chapters on the Stevens scale in biopsychology text books; how the scale for perceptual length is derived I don’t know, but I assume that it does not work for distance away from the observer; see below).

The argument in Holladay (2004) and by many others is circular: Because steps on the acuity charts are spaced logarithmically, calculations should be done on a logarithmic scale. However, there is no logical reason why one implies the other (except that it is practical). The real question is why steps were chosen logarithmically in the first place.

The reason, I believe, is that the VA scale is intuitively non-linear: The perceptual decrease from 0.2 to 0.1 is obviously bigger in effect than one from 1.0 to 0.9. But I believe that has nothing to do with a log scale and the intuition is based on the fact that VA is the *inverse* of MAR. I.e. the relation between the two is nonlinear and ratios are preserved in the transformation. Intuitively, ratios are thus relevant. A typical example is that 0.2/0.1 is a factor of 2, whereas 1.0/0.9 is a factor of 1.1, even though the distance on the linear scale is the same. Going to the log realm means that ratios are converted to differences, which are then preserved. That does not imply, however, that ratios are the proper distance measure in the first place! In psychological / psychometric language the question is: which scale is on an interval-scale level?

Your argument that VA is linearly related to viewing distance is intriguing because it relies on first principles. It comes down to saying that seeing something from twice the distance can be considered twice as good in some sense. That makes sense. However, distance is not coded directly in the visual system, and the retinal measure is, of course, visual angle. The latter’s lower limit is given by cone spacing, which in turn is closely linked to MAR.

The discipline concerned with scaling (e.g. such that intervals are equal, thus creating an interval scale) is ‘psychometrics’ or ‘psychometric scaling’. It provides sophisticated methods of scaling (e. g. Thurstone scaling). Yet I am not aware of any psychometric-scaling attempts for acuity. So I believe there is as yet no answer what constitutes equal intervals on an acuity scale!

The only perceptual rule I am aware of is that MAR varies linearly with eccentricity in the visual field. This is very well established (Weymouth, 1958; he was the one who advocated MAR in the first place, but people have forgotten that). And this is why I personally prefer MAR over logMAR. For details see my paper in JOV:

http://www.journalofvision.org/content/11/5/13

When doing statistical analyses based on MAR, I see no necessity of going to a log scale, btw. If somebody shows me distributions of MAR that are heavily skewed (thus preventing standard least-squares statistics) that would convince me otherwise. I think people use logMAR mostly because everybody else does (and because it is equivalent to logVA, and doing statistics with VA is not a good idea).

Michael Bach btw. advocates logVA (that is, minus logMAR), because logMAR goes the wrong way: High logVA means high acuity.

Best,

Hans

I have always been involved in visual rehabilitation in patients with severe retinal compromises at the Low Vision Research Centre of Milan. Measuring visual acuity is essential to know the patient's rehabilitative possibilities. The LogMar is a unit of measurement that has statistical advantages but is an expression of an angular vision. Useful but insufficient. Furthermore, if a maculopathy has damaged the fovea the LogMar may not change after therapy even if it improves the overall visual sensitivity of the patient. The visual ability of a subject, in my opinion, can be better assessed if supplemented by the knowledge of the patient's ability to read, as this process implies the function of a functional area on the retina. We also use microperimetry that better analyzes the perception of the environment. For the reading capability we use the typographics points o pts, for the microperimetry the dB.

The evaluation of these last parameters is more indicative of the visual function than just the evaluation of the angular vision. The analysis of the three parameters gives us a complete idea.

(You might be interested in the Macular Mapping Test then: http://hans-strasburger.userweb.mwn.de/psy_soft.html#mmt)

Dear Robert Massof and Mark E Wilkinson!

I have no doubt that modern charts with geometrical progression of size like ETDRS chart provide incredible accuracy of measurements. They allow to take into account each letter and have a logarithmic report. However, this does not mean directly that the statistical analysis must also be carried out in logarithmic units. I'm just trying to explain this uncertainty to myself. Can you advise me a paper, explaining this issue in detail?

Dear Pete Jones!

Visual acuity has an upper physical limit. This is due to the size of the retinal cone mosaic and light diffraction on the pupil. I found different estimates of this limit:

- 1,6 (20/12; -0.204 logMAR) for "small pupil" and 4 (20/5; -0.602 logMAR) for dilated pupil (https://emedicine.medscape.com/article/1222586-overview#showall);

- 2,5 (20/8; -0.398 logMAR)

(http://www.carlomasci.it/biblio/aberrazioni_3.pdf)

- 1,6 (20/12; -0.204 logMAR)

(http://www.drdrbill.com/downloads/optics/physical-optics/Circles.pdf)

This fact suggests, that distribution of visual acuity data (people with good visual acuity, for example after LASIK patients) may have not positive, but negative skew. More over, I found signs of it here: http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0022756 on Figure 1.

In this situation logarithmic correction can only increase this skew.

I was faced with this problem when analyzing the visual acuity data of patients after LASIK. Statistics in the decimal scale was more adequate than in the generally accepted logarithmic scale.

MAR is a measure in physical units. Any monotonic transform can be used as an alternative. The choice of measure depends on the statistical assumptions being made. The important point is that modern acuity charts have stimuli graded on an equal interval log MAR scale, thus it would be appropriate to specify the stimuli that way. If your statistics assume Gaussian distributions, then it would be best to check your observations, and perhaps transform the data, or use a different statistic if that assumption is violated. The rationale for a log progression is based on the psychophysical JND and the psychometric function for acuity, which are consistent with a log normal distribution.

Sloan LL. Measurement of visual acuity. A critical review.Arch Ophthalmol. 1951;45:704-725.

Bailey IL, Lovie-Kitchin JE. Visual acuity testing. From the laboratory to the clinic. Vis Res 2013;90:2-9.

Elliott DB. The good (logMAR), the bad (Snellen), and the ugly (BCVA, number of letters read) of visual acuity measurement. Ophthalmic Physiol Opt 2016;35:355-358.

Dear Mr. Strasburger!

Thank you for the detailed response. You raised questions that I had not even thought before!

Dear Mr. Massof!

Thanks for the answer! I will try to find a solution in papers you suggested. I hope you will not refuse answers to my further questions.

After a long break, I had the opportunity to return to this topic. I researched the problem and obtained results that contradicted accepted practice. Please read it in this discussion. https://www.researchgate.net/post/Decimal_representation_of_visual_acuity_seems_more_suitable_for_parametric_statistics_than_LogMAR