Anyhow, using LogMAR is the only way to calculate visual acuity means or medians. Visual perception, as all other senses I know, follow a logarIthmic scale. Consequently, it is logical to accept SD in terms of LogMAR visual acuity. Anything that calculates linear visual acuity data overestimates irrelevant differences in the upper range of VA scale.

You are raising an interesting question!

First off, I agree that averaging logMAR can be a useful way for averaging acuity measures. My reasoning would not be that MAR follows a logarithmic scale, however. By definition,

VA = 1 /MAR,

(where VA stands for visual acuity)

so log(VA) = log(1/MAR) = - logMAR.

Thus, if you average logMAR, there is a simple relationship to visual acuity. And the latter is the more common measure for visual resolution.

The (wide spread) assumption that MAR follows a logarithmic scale (i.e. a Fechner scale) is not backed by data, I believe. Indeed, the reduction of acuity in peripheral vision, e.g., is a *linear* increase of MAR (not of logMAR). So logMAR is kind of a strange beast.

I thus think it is perfectly OK to average MAR. And, no, differences in the upper range of VA are not overestimated by averaging MARs: The non-linearity is in the VA scale, so it is not a good idea to average VAs.

The issue with SD that you raise is a funny thing. Usually, standard deviations (SDs) are in the same physical units as the values that are averaged. However, in the case you describe (where all acuities are the same), the value of 1.0 does not correspond to a visual acuity, but to something like the ratio of those acuities. So it makes no sense to convert the SD of logMAR values to MAR.

Dear Heinrich and Hans, big thanks for your answers!

I took up the problem in more detail and found an article in which this question is being solved. https://www.ncbi.nlm.nih.gov/pubmed/21205268

It looks like that transformation of averages and standard deviations from one scale to another is a nontrivial task, which can not be solved with standard (naive) formulas for the trans scale transformation of visual acuity. My question was irrelevant, because I used the wrong approach to recounting SD. The formula proposed in the article give more correct estimate SD = 0 in decimal scale. All statistical calculations are best to carried out within the same scale.

About overestimates of averages in differenr ranges.

Let's calculate the average in the decimal scale first directly, and then through the transformation each VA to LogMAR and mean of LogMARs back.

l - is VA in LogMAR scale

d - is VA in decimal scale

l=-log(d);

d=10^(-l);

First, straight way:

Md1=E(d)/n;

i.e. arithmetic mean of d.

Second, convoluted way:

Ml=E(l)/n=E[-log(d)]/n = -E[log(d)]/n=-log[(Пd)^(1/n)];

Md2=10^-Ml=10^-[-log[(Пd)^(1/n)]]=10^log[(Пd)^(1/n)]=(Пd)^(1/n);

Md2=(Пd)^(1/n);

i.e. geometric mean of d

But arithmetic mean always greater then geometric mean, except the case when all VAs are equal. Here AM and GM are equal too. https://en.wikipedia.org/wiki/Inequality_of_arithmetic_and_geometric_means.

So decimal means almost always, not only in upper VA range, give higher estimates.

@Strasburger "The (wide spread) assumption that MAR follows a logarithmic scale (i.e. a Fechner scale) is not backed by data"

I'm not sure about this. Psychometric functions for acuity tend to be Gaussian on a log-scale (e.g., Anderson: https://www.osapublishing.org/josaa/abstract.cfm?uri=josaa-13-4-697). This implies that thresholds ('Detection Limens') will be log-distributed, making logarithmic correction appropriate. (Or is the argument that this holds for resolution acuity, but not recognition acuity?)

@ Siplivy "But 0 in decimal scale is 1.0 (or 20/20) that seems too high estimate for the zero variability."

As others have pointed out, the problem is that you cannot (it makes no sense to) transform the standard deviation back to the original scale. Saying that your logMar SD is 0 does *not* implies that the SD in decimal acuity would be 1.0. See, for example, this note on the subject from eminent statistician Martin Bland: http://www.bmj.com/content/312/7038/1079

Good point.

I don’t know off hand what the psychometric functions vs. acuity look like. The paper you link to (Anderson et al. 1996) is not a good example, though: The x-axis is not acuity but spatial frequency (so the link to acuity is somewhat indirect), and importantly the SF range of these psychometric functions is rather narrow, so that the difference in PF shape between a log x scale and linear x scale will be small.

Small variation on the acuity axis (or SF axis) is what is to be expected, however (because these are within-subject data), so I expect other publications will show similar results.

Generally, I believe psychometric scaling issues are very different from threshold related issues. Psychophysics communities appear sharply divided between the two areas (myself I am in the threshold camp).

Vladimir, If I understand your original question, you are asking whether you can translate std dev in logMAR units to Snellen units "For example, if all persons in a group have equal VA, SD of VA is 0 in LogMAR scale. But 0 in decimal scale is 1.0 (or 20/20) that seems too high estimate for the zero variability.". It is best to look at this in terms of the confidence intervals. Because of the log10 transformation, standard deviations transformed back to Snellen are asymmetric. In your example, the confidence intervals would be the average. So, the std dev is +/- zero (actually, zero in both scales). But that is an unusual case. Thinking that zero is 20/20 is misconception when dealing with logMAR units. It is only 20/20 when transforming back to Snellen. Not when dealing with a difference.

The average of log(x) is the geometric mean of x.

Another thing to keep in mind is that if you have people with impaired vision, it is really common to have a skewed distribution across the group, even in the logMAR scale, while the healthy eyes have a normal distribution. This is not limited to VA, and is commonly found in both psychophysical and biological measures (in that some people with the disease have it really "badly" and create a long tail). That is a bit of a tangent to your question, but is an important issue when using averages to describe (certain) samples and can have an impact on various parametric statistical methods (that compare means).

Dear Russel!

When carrying out Ttest for comparison mean visual acuity of several groups LASIK patients, I faced with strange problem. Calculations in the decimal scale were more powerful then in LogMAR scale. In decimal scale I received significant differences more often than in LogMAR scale. But nowdays it is recommended to use only a logarithmic scale for statistical calculations!

Trying to understand this problem, I tried to find an extreme group giving the strangest result in the transition from one scale to another. It seemed to me that the most strange thing was the transformation of the standard deviation in a monotonous group. This caused me some doubts about the applicability of standard statistical formulas in the LogMAR scale. Of course all answers was immediately pointed to my error.

More over Pete Jones in the answer to my other question (www.researchgate.net/post/Is_LogMAR_is_only_proper_way_to_convey_visual_acuity) pointed to a fact that I had not noticed before. Logarithmic correction for positive-skew data can normalize it, but not for negative-skew.

I guess the solution to this problem is as follows.

Visual acuity in the decimal scale has two limits lower =0 and upper, about 2.5-3.0 (due to diffraction on the circular pupil). If the average visual acuity in the group is not very high, the lower limit has a greater impact, and a positive -skew appears in the distribution. In this case, the use of a logarithmic scale should lead to a decrease this skew and an increase the power of parametric statistical methods.

If the average visual acuity in the group is large, the influence of the upper limit is strengthened. A negative skew appears. This effect I observed not only on my data, but also in some other publications. In this case, the transition to a logarithmic scale only enhances this skew and decrease the power of parametric statistical methods, for example Ttest problem I faced in the beginning. Probably it is necessary to refuse the statistical calculations in the LogMAR scale for groups with high visual acuity, despite the prevailing point of view.

I will be gracious to you for any indication of an errors in my reasoning.

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/topics?ev=nav_discussions&_tp=eyJjb250ZXh0Ijp7ImZpcnN0UGFnZSI6ImhvbWUiLCJwYWdlIjoiaG9tZSIsInByZXZpb3VzUGFnZSI6InF1ZXN0aW9uIiwicG9zaXRpb24iOiJnbG9iYWxIZWFkZXIifX0

Hi Vladimir, I think there is a misunderstanding here: The antilog of 0 std dev of logMAR is not 1.0 visual acuity but is a factor of 1.0. In other words, every value of acuity in the sample differs by a factor of 1 from the mean, i.e., is equal to the mean.

Hans

Hello Hans!

Thank you for your reply!

It all started with my attempt to understand exactly this.

In the answers to my 2018 year question, one of the main justifications for using LogMAR was that visual acuity in it tends to have a Gaussian distribution ( Pete R Jones , Russell L Woods ). I found this argument in the literature, recommended me for reading. Last year, I had the time and opportunity to test this argument together with a data analyst. We analyzed our own data (100 healthy volunteers) and reconstructed data from three large studies (ETDRS, AREDS2 and CIEMS). We checked how the distribution looks like in decimal and LogMAR, and how it changes with the Tukey power transformation. In all four cases, the distribution in decimal notation was closer to Gaussian distribution than in LogMAR.

It appears that the argument that visual acuity data tend to be Gaussian in LogMAR is incorrect, and decimal (Snellen) notation appears to be better suited for parametric statistics than LogMAR.

Our conclusion clearly does not correspond to the generally accepted one, so I started a discussion on this topic.

I would be extremely grateful for your opinion!

It looks like that I incorrectly posted the link to the discussion:

https://www.researchgate.net/post/Decimal_representation_of_visual_acuity_seems_more_suitable_for_parametric_statistics_than_LogMAR

Sorry for the long text!

With best regards

Vladimir Siplivy