Hi Samuel 

I think you are quite right with what you are saying. However, most studies compare different variables between different refractive groups (e.g. myopes vs. hyperopes) and thus provide a single SE for each of the groups separately. However, this measurement (i.e. SE) doesn't provide any information regarding the magnitude and axis of astigmatism. The most well accepted method to analyze refractive changes is to use a power vector approach, which breaks the refractive error into M, J0 and J45 power vectors) - see "Thibos LN, Wheeler W, Horner D. Power vectors: an application of Fourier analysis to the description and statistical analysis of refractive error. Optom Vis Sci 1997;74:367-75."

Hope this helps!

Jac

One other potential problem using a mean value is that it is well known that the refractive error (correction) data is leptokurtotic and negative skewed and a median measure may be more appropriate if you want to look at the average refractive error.  That being said the mean spherical equivalent value is commonly used in studies wanting to find the average error in a population.  Here "0" means something, i.e. emmetropia.  An approach others have used with spherical equivalent data is to use an absolute value of the error to find an overall amount of refractive error (away from emmetropia).

I agree with Jac that the best approach if you are looking at "change" data (e.g. pre- post- intervention or longitudinal) and/or you do not want to lose information about amount and axis of astigmatism  is the use of a vector analysis.

Another issue is that myopia is much more tolerated by patients than hyperopia, especially if the patient was myopic preoperatively. Uncorrected myopic patients have a focus range (or, depth of field) in which they can see an acceptably sharp image while uncorrected hyperopic patients, by definition, cannot see a sharp image at any distance. Refractive data should be always presented and interpreted in its clinical context.

I agree with above remarks but unless new matrix or outcome measure is validated and published this notation would not change. Better would be start putting other outcome measures into literature using original studies, letters to editors for published studies. Over time the terms would get embedded in the literature. 

Agree the more data the better for publication, thus should be included in initial protocol

Hi Samuel

You are absolutely correct.

It is mathematically and scientifically unsound to aggregate or combine data sets that are logically mutually exclusive data sets.

The aggregation of positive and negative spheres into one data set for the purpose of summation and calculation of an arithmetic mean or average value results in data suppression.

The errors resulting from this unscientific and statistically invalid methodology are easily demonstrated: all spherical powers ( including spherical equivalents and mean spheres) are modelled by one dimensional vectors displaced along the rational number line with the magnitudes representing the distance from zero. The positive magnitudes are displaced to the right and the negative magnitudes are displace to the left; I.e. into two opposite and mutually exclusive directions.

For example: Positive Spheres 0,+1,+2,+3 ... Average 6/4 = +1.5

Negative Spheres ... -3,-2,-1,0 Average -6/4 = -1.5

If the data are combined into one data set:

....-3,-2,-1,0,+1,+2,+3 ...... Average 0/7 = 0

That is a major degree of data suppression has occurred and the average does not model the data accurately.

If the Absolute Values are used: 3,3, 2,2,1,1, 0 Average 12/7 = 1.5, however this results in loss of knowledge about the relative numbers of myopes or hyperopes 

This method lead to an over and under estimation of the actual mean value calculated by the two data sets if the data sets are of unequal size e.g. -3,-2,-1,0 Average -1.5 and +3 , +1,0 average 1.33 , while the absolute value calculation 3,3,2,1,1,0,0,is +1.42.

This separation of the mean spheres or spherical equivalents into negative , positive and plano values is consistent with the clinical categories of astigmatism:

Myopic Astigmatism e.g. -3.00/+2.00 x 180 SE = -2.00D

Hyperopic Astigmatism e.g. +1.00/+2.00 x 180 SE = +2.00D

Mixed Astigmatism e.g. -1.00/+2.00 x 180 SE = 0.00D

That is there are 3 logically mutually exclusive categories of astigmatism that need to be taken into account for the purpose of any analysis of surgical correction of astigmatism. 

Jacinto is in fact correct when he highlights Thibos et al methodology; all other methods of astigmatism data analysis are in different ways mathematically flawed and illogical.

A similar argument applies for the analysis of the postop angle of error (AE). There two logically mutually exclusive categories of rotation i.e. anticlockwise(positive) and clockwise(negative).

For example: anticlockwise degrees +30,+20,+10,0 Average 60/4 = +15 degrees

clockwise degrees 0,-10,-20,-30.. Average -60/4 = -15 degrees

If the data are combine into one data set +30,+20,+10,0, -10,-20,-30 Average 0/7=0

In other words total or partial data suppression results when two mutual exclusive data sets are combined for analysis.

If the absolute values are combined into one data set: 30,30,20,20,10,10,0,0 the Average 120/8 = 15 , which is an under estimation of the true value of the average or arithmetic mean and doesn't indicate in which direction, anticlockwise or clockwise, the error is located. This is data suppression at its worst and results in the loss valuable information and doesn't accurately model the clinical  data.

The only accurate analysis that models the data accurately is to separate the data into the logically mutually exclusive mathematical categories of anticlockwise and clockwise before calculating the average or arithmetic mean.

A major refractive surgery journal has recently proposed that these methods be adopted which is leading to great confusion to the journals readers, so your observation is very relevant and more letters to editors outlining this situation should be written.

Regards

Dr Lee Lenton

There are several possibilities to report. The more information is included the easier it is to compare different studies. Distribution curve of spherical equivalent, prevalence of myopia, emmetropia hyperopia, and astigmatism all are informative and could be included.

I have encountered similar problems several times, especially (say) when almost 70% population has one type of refractive error and the rest 30% comes to nullify the magnitude, which is definitely not reasonable. 

Probably one idea to deal with this problem is to plot scatter plots between refractive errors (Myopia vs Hyperopia) and magnitude of the error which will show relative proportion of the distribution. And providing the SE for different classes (eg. age groups) would be more justifiable I assume. 

Yes, myopia and hyperopia to be reported separately based on spherical equivalent conversion. However, in a case of significant mixed astigmatism (causing vision impairment), spherical equivalent conversion may result to emmetropia. Indeed, this cannot be reported as emmetropia. Hence, in a RESC study, we reported such cases in a different category ‘Emmetropia with astigmatism’. This is defined as ‘cases of vision impairment because of astigmatism (-0.75D or more) but had emmetropia when spherical equivalent conversion used.

Paudel P et al. Prevalence of vision impairment and refractive error in school children in Ba Ria – Vung Tau province, Vietnam. http://onlinelibrary.wiley.com/doi/10.1111/ceo.12273/full

RE studies often report astigmatism separately. Hence, it is clearly a reporting issue. Agree with previous comments, new matrix or outcome measure to be validated and published to clarify the issue of standard reporting of spherical and astigmatic data in refractive error studies.

Vector analysis is usually reported in different perspectives (e.g. pre and post changes) not in the prevalence studies.

Lee and others,

Have you seen the similar treatment given to aggregation of clinical measures of spherical aberration?  The 'average' values of positive and negative results are reported with a mean close to 'zero' spherical aberration.  In a similar way this under reports the amount of spherical aberration and its effect in vision in the population. As a first approximation I take the sign free, absolute values, and take the mean of these as the amount of spherical aberration and its effect in vision in the population.

I have a question with this approach.  Is it appropriate to treat the magnitude positive and negative spherical aberration as equal?

You might take a look at converting a clinical notation into "Power Vector format" This allows a Fourier analysis of refraction, allowing a way to express sphere, cylinder and axis statistically, in contrast to "spherical equivalent". The power of these three component lenses may be interpreted as (x,y,z) coordinates of a vector representation of the power profile. Advantages of this power vector representation of a sphero-cylinder lens for numerical and graphical analysis of optometric data are described for problems involving lens combinations, comparison of different lenses, and the statistical distribution of refractive errors.

Is it right to report a single mean spherical equivalent (SE) refractive error value in a study population comprising of both myopes and hyperopes? - ResearchGate. Available from: https://www.researchgate.net/post/Is_it_right_to_report_a_single_mean_spherical_equivalent_SE_refractive_error_value_in_a_study_population_comprising_of_both_myopes_and_hyperopes#570e5e2dcbd5c235ec1c4f0b [accessed Apr 13, 2016].