**Purpose**:
Visual acuity tests are generally performed by showing eye charts to the subjects and registering their correct/incorrect identifications for the presented optotypes. We recently developed a correlation-based scoring method that significantly reduces the statistical error associated with relative letter legibility. In this paper, our purpose was to demonstrate the advantages and clinical utility of our scoring scheme compared to standard methods.

**Methods**:
We developed a new computer-controlled measurement setup aligned with the ophthalmological standard. With this system, we presented the application of our correlation-based scoring in conventional clinical environment for 25 subjects and estimated the systematic error of the obtained acuity values. A separate experiment was performed by 14 additional subjects to reveal the test-retest variability of the new scoring method.

**Results**:
The average systematic error relative to standard probability-based scoring is 0.01 logMAR over the examined subject group. Application of the correlation-based scheme when used in clinical environment with five letters per size decreases the repeatability error by ∼20% and increases diagnosis time by ∼10%.

**Conclusions**:
The new scoring scheme is directly applicable in clinical practice providing unbiased results with improved repeatability compared to standard visual acuity measurements. It reduces test-retest variability by the same amount as if the number of letters was doubled in traditional tests.

**Translational Relevance**:
Our new method is a promising alternative to conventional acuity tests in cases when high-precision measurements are required, for example evaluating implanted intraocular lenses, testing subjects with retinal diseases or cataract, and refractive surgery candidates.

^{1,2}In ophthalmic studies, clinically relevant treatment effects should be prespecified and then justified in terms of “change in visual acuity from baseline.” Most of the (re)treatment criteria are based on the measured progress of visual acuity, using different expectations in letter score increment (e.g., >5 letters or >10 letters). Both in these clinical situations and in scientific studies, a visual acuity measurement method with high accuracy and reproducibility, that is low test-retest variability (

*TRV*, standard deviation of the results of several repeated measurements), is desirable.

^{3,4}The specific motivation of our research group comes from the field of cataract surgery: we are primarily interested in improving intraocular lens (IOL) design and implantation. Comparison of different types of premium IOLs (i.e., aspheric, toric, or diffractive lenses), as well as assessment of the outcome of a surgery, require a method that enables the detailed analysis of visual acuity with greater precision than current approaches. Although these goals may not be universal, we hope our results will be beneficial for other areas of vision science, too.

^{5,6,7}Its logarithmic layout

^{6,8,9}standardizes the visual task and the effects of letter crowding.

^{10,11}Furthermore, to ensure equalized difficulty for the rows, the ETDRS protocol uses only certain combinations of the Sloan letters,

^{5,6,12}whose selection has been still further refined in the currently used ETDRS 2000 series charts.

^{13}Thus, the only significant variable that changes from one line to the next is the letter size, usually characterized by the

*α*visual angle, that is the angle that the stroke width (and smallest gap) of the optotypes subtend at the eye.

^{1,14–16}According to the line-assignment evaluation, the visual acuity value is determined by the visual angle of the smallest letter size (threshold), where the majority of the optotypes is recognized correctly.

^{12,17}It has been shown that the variance of the measurement is almost constant across a wide visual acuity range if the scaling is logarithmic.

^{9,18}Therefore, in currently used eye charts, the decrease of the letter size from line to line follows a geometric progression with a quotient of 10.

^{1/10}Correspondingly, the 𝒱 visual acuity value is usually expressed in logMAR units (i.e., the decimal-base logarithm of the minimum angle of resolution),

*α*

_{0}denotes the visual angle at the threshold line in minutes of arc. Although the theoretical probability threshold is 50%,

^{12,17}using the ETDRS charts (implemented with five letters per line) the actual threshold rises to 60%, 80%, or even 100% depending on the distribution of correctly recognized letters, which causes noticeable error in the results relative to theory. According to the literature,

^{6,14,19}the statistical error of current line-assignment-based visual acuity measurements varies between 0.6 and 1.5 line (0.06 logMAR <

*TRV*< 0.15 logMAR) for subjects with normal vision. This accuracy is sufficient for screening purposes as part of preventive health care; however, epidemiologic surveys and clinical research require higher precision and reliability as the successive measurements are to be compared to each other.

^{4,9}

^{4,14,15,20}The special design of the ETDRS chart allows the examiner to recompense the subject's visual acuity by −0.02 logMAR unit for each correctly recognized letter.

^{4,6,21}Correspondingly, the visual acuity value can be determined from the

*T*total number of correct identifications in the chart as:

_{c}*TRV*≈ 0.04 logMAR),

^{22,23}its outcome does not correspond exactly to the theoretical 50% probability threshold. It is offset by approximately half a line (i.e., +0.05 logMAR) systematic error.

^{4,21}

^{3,4,21}nonlinear (e.g., logistic) regression provides an alternative possibility in clinical research to achieve the same reduced amount of

*TRV*as single-letter-scoring:

*TRV*≈ 0.04 logMAR. In this case, the measured

*P*recognition probability values at the

*x*= log

_{10}(

*α*) letter sizes are fitted by a monotonic differentiable S-shaped curve, the so-called psychometric function of vision.

^{16,24–26}Visual acuity is determined by the

*x*

_{0}letter size at which the

*L(x)*psychometric function intersects the theoretical

*P*

_{0}= 0.5 probability threshold,

^{22,27,28}that is:

^{17}and, thus, eliminates systematic error. Because it has the lowest statistical error as well, in our former paper

^{29}we concluded that applying nonlinear regression is the best way to determine visual acuity.

*TRV*of the measurements is also influenced by the recognizability of individual characters. Despite the extensive effort put into balancing legibility,

^{12,30}certain Sloan letters are still easier to identify, whereas some are easier to confuse with others.

^{24}If the within-line legibility differences are greater than the between-line legibility differences of the chart, then an increased variability may occur in the results.

^{31,32}To overcome this issue, we formerly introduced a correlation-based scoring method to further reduce the statistical error of the measurements. In our previous studies,

^{29,33}we presented the theoretical background and the technical details of the method and verified that our correlation-based approach is practically equivalent to standard probability-based scoring for subjects with normal and supernormal vision. We also demonstrated that it reduces the statistical error by ∼28% under special high-precision laboratory conditions. However, to test our correlation-based scoring in the clinical practice over subjects from a wider range of visual quality, new measurements became necessary.

^{29}in a clinical environment, covering the −0.2 to +0.7 logMAR range. We also compared and contrasted the results of our measurements to the outcomes of standard ETDRS trials to assess the widespread applicability of the new method. Finally, we gave an estimation for the statistical error reduction of the correlation-based scoring when used in clinical practice.

^{29,33}

*P*(

*x*) recognition probability as a function of the letter size. However, to increase accuracy, examiners sometimes omit minor errors (e.g., misidentifying C as O or R as P),

^{9}acknowledging that human letter recognition is more complex than a simple binary scheme. Consequently, if the subject is able to see some features of a misidentified letter, then it is worth for characterizing how bad or good his/her guess is. Thus, we have formerly introduced a new quantity called optotype correlation (

*OC*) to characterize the physical similarity of the letters.

^{29,33}The

*OC*value of a character pair has been determined by the Pearson's cross-correlation of the binary (black/white) ideal images of the letters.

*OC*spans between −1 and +1, with higher values corresponding to more similar letter pairs, and identities (i.e., correct identifications) are represented by 1. The numerical values of the entire

*OC*matrix corresponding to a complete character set have been transformed linearly so that the expected value of two randomly selected letters (i.e., misidentifications) equals 0 (excluding identical letter pairs, i.e., accidental correct answers). In this way,

*OC*is directly comparable to the conventional binary scheme of true/false identifications.

^{29,33}we strove for high precision; thus, all characters of the extended Sloan font type

^{34}were displayed, including the complete English alphabet. Accordingly, we expected that the subjects' identifications covered the same 26-letter selection. In our new experiments, corresponding to the real composition of the ETDRS charts, the presented optotypes are selected only from the 10 original Sloan letters.

^{5,6}However, we still consider all letters of the complete English alphabet as potential guesses because the subjects are not supposed to know this restriction. Due to this difference from our former measurements,

^{29}the numerical values of the

*OC*matrix had to be recalculated for clinical use to ensure that the expected value of misidentifications still equals 0. The updated

*OC*matrix of the potential presented-perceived letter pairs is depicted in Figure 1, where rows represent the displayed Sloan characters, columns indicate the potential identifications, and the

*OC*values are color coded according to the

*Viridis*colormap. The numerical values are listed in Table A1 in the Appendix.

^{35}because character pairs with higher correlations are more likely to be mixed up. In other words, the

*OC*value of more similar letters (lighter/yellowish cells in Fig. 1), such as C and O (0.861), is larger than that of less similar characters (darker/bluish cells in Figure 1), such as H and T (−0.671).

^{29,33}we proposed a new quantity, namely the rate of recognition (

*RR*), instead of

*P*to describe visual quality. For a fixed

*x*letter size (i.e., an actual line of the ETDRS chart),

*RR*is defined as the average

*OC*value of the presented-perceived character pairs:

*RR*is directly comparable to recognition probability but provides more information about visual perception. According to our interpretation, the limiting case of

*RR*(

*x*) represents the psychometric function of vision, when the number of letters per row tends to infinity.

^{29,33}we applied decimal notation to express the letter size of the rows, but for the evaluation of clinical trials, we decided to use the more widespread logMAR units (i.e.,

*x*= log

_{10}(

*α*)). In the former case,

^{29}a super-Gaussian function

^{36}provided the most robust fit for the interpolation of the psychometric function, having only two independent parameters. However, switching the letter size expression from decimal to logMAR units, we also have to change the regression function. In this case, we can apply the

*L*(

*x*) sigmoid-shape logistic function, which is the most frequently used two-parameter curve to approximate any psychometric function.

^{22,26,27}Its mathematical formula is described by:

*x*parameter sets the midpoint position of the sigmoid, while

_{mp}*k*/4 determines the steepness of the curve at this point. To make sure that the limits of the psychometric function correspond to the theoretically expected

*RR*values, it has to be further transformed linearly as

^{22,27}:

*RR*values collected at the discrete letter sizes of the eye chart (

*x*= 1.0, 0.9, 0.8 to −0.3 logMAR). In accordance with the measurement standard,

^{17}the 𝒱 acuity value corresponds to the specific

*x*

_{0}letter size at which the value of the function equals the given threshold (

*RR*

_{0}):

*RR*

_{0}= 0.68.

^{29}Figure 2 shows the concept of curve-fitting-based evaluation by the average psychometric curves determined in our previous paper

^{29}(

*L*:

_{P}′*x*= −0.214 logMAR,

_{mp}*k*= 13.22 logMAR

^{−1}, and

*L*:

_{RR}′*x*= −0.269 logMAR,

_{mp}*k*= 17.11 logMAR

^{−1}) and emphasizes the differences between the probability- and correlation-based scoring methods.

*RR*

_{0}threshold under conventional clinical conditions, we had to perform new trials. Details of the measurements are discussed below, and their results are presented in the Results section. It is important to note that according to our experience, the difference between the visual acuity values obtained by the decimal and the logMAR notation is negligible.

^{37,38}In addition to our new correlation-based measurements, we used the same monitor to display an ETDRS chart and perform conventional trials to provide reliable reference for cross-validation. For all subjects, the ETDRS test followed by the correlation-based experiment were carried out at one sitting to ensure exactly the same conditions. In accordance with the European clinical standard,

^{6,12,17}we took our measurements from a viewing distance of 4 meters in a dimly lit exam room, with an illuminance of 150 lux. To investigate complete five-letter lines from 0.9 to −0.5 logMAR value, we used a Samsung U24E590D 4k UHD LED monitor with a diagonal size of 600 mm and a pixel pitch of 0.1358 mm, having a matte screen to reduce glare. The luminance of the monitor was set to 100 cd/m

^{2}, which fulfilled the International Council of Ophthalmology (ICO) standard (min. 80 cd/m

^{2}).

^{17}

*OC*matrix (1/26 ≈ 0.038), representing a random choice. Right after the measurement, the software also performed logistic regression to determine the visual acuity value. Figure 4 depicts the graphical result of a representative trial.

*RR*

_{0}= 0.68 value can be used in clinical measurements as well.

*OC*value of the displayed-identified letter pair drops below 0.85 (this value corresponds to the confusion of very similar letter pairs, such as C and O).

*RR*value achieved at the smallest letter size is still larger than 0.4, then the trial is supplemented with the next two smaller sizes to provide sufficient data for curve-fitting. In this stage, five optotypes are examined at each letter size. The tested character sets comply with those used in the ETDRS 2000 charts,

^{13}and are displayed together in five-letter lines. The spacing between the characters equals the letter size to take the effects of letter crowding into account.

^{6,12}As a validation, we compared the results to the visual acuity values measured by a standard ETDRS chart displayed by the same monitor to keep differences between the measurements at a minimum.

^{29,33}predicted significant statistical error reduction when using the correlation-based method instead of probability-based scoring. With this new experiment, our purpose was to precisely determine uncertainty in standard clinical environment as well. To analyze the variation of relative error reduction with respect to the number of tested letters, we examined all 10 Sloan characters per letter size.

^{39–41}we determined the monocular visual acuity value only for one eye per subject. We followed the tenets of the seventh revision of the Declaration of Helsinki (2013) in our study.

*uncorrected*visual acuity was measured, and the subjects were classified into three categories based on their refractive power error: group I, 0 to −0.5 D; group II, −0.5 to −1.5 D; and group III, −1.5 to −2.5 D. Group I had 15 members, while group II and III both contained 5 subjects each.

*best-corrected*visual acuity was measured. The refractive power error of the subjects covered the +1.5 to −6.0 D interval. The refractive power error measurements were taken by a TopCon autorefractor.

_{P}and 𝒱

_{RR}, respectively). To exclude any differences except for the scoring method, we calculated

*both*visual acuity values for each subject from the same raw data of experiment 1. 𝒱

_{P}was determined by the standard

*P*

_{0}= 0.5 recognition threshold, whereas in case of correlation-based scoring, we applied the formerly calibrated

*RR*

_{0}= 0.68 threshold.

^{29}The two acuity values were compared for each person by Bland-Altman analysis, the result of which is shown in Figure 5.

_{P}and 𝒱

_{RR}visual acuity values is −0.01 logMAR (with ±0.01 standard deviation), which gives an estimation of the systematic error of correlation-based scoring. A comparative error analysis is presented in the next subsection, revealing that the

*TRV*of the measurement equals 0.036 logMAR. From this result, we conclude that the visual acuity values determined by the two scoring methods are equal within the margin of error, which supports the applicability of our method and the adequacy of the

*RR*

_{0}threshold in the conventional clinical environment too.

_{P}′ visual acuity value as a reference. Figure 6 depicts the individual test results for all subjects, grouped by their refractive power error.

_{RR}visual acuity values to those obtained by the standard ETDRS test (𝒱

_{P}′). The result of the Bland-Altman analysis performed on the 𝒱

_{RR}and 𝒱

_{P}′ values is shown in Figure 7.

_{P}′ and 𝒱

_{RR}equals −0.008 logMAR (with ±0.044 standard deviation), which estimates the systematic error of experiment 1 relative to the standard ETDRS test. According to the

*TRV*of standard probability-based visual acuity measurements,

^{4,22,23}the values are equal within the margin of error, which means that eliminating interline crowding in our experiment 1 has no effect on the visual acuity results.

*RR*taken into consideration (test B1). Third, all 10 original Sloan optotypes were investigated by recognition probability again (test A2). Finally, all 10 letters were analyzed by taking advantage of

*RR*(test B2). To quantify the precision of the different methods, we compared them by the

*TRV*calculated from the results of 10 consecutive measurements. We performed the error analysis for each subject individually, then averaged the results, which are presented in Table 1.

*TRV*value of the methods indicates a well-observable improvement from A1→B1→A2→B2. The error of test A1 (i.e., the conventional approach: five tested letters at each letter size, evaluated by probability-based curve-fitting) is in good agreement with repeatability data presented in the literature:

*TRV*≈ 0.04 logMAR.

^{3,4}The other results demonstrate that both the increased number of tested letters and the application of

*RR*reduce the repeatability error. The former statistically decreases the error by a factor of 2

^{−1/2}as expected because tests A2 and B2 contain two times as many optotypes at each letter size as tests A1 and B1. The most important outcome of the experiments is that the application of the correlation-based scoring in itself reduces the error by 19%. Based on our results, by replacing the conventional A1 evaluation with B1, the uncertainty error decreases by 0.0083 logMAR thanks to

*RR*. This significant improvement justifies the extra requirements of the correlation-based approach in the clinical practice.

*TRV*(B1) ≈

*TRV*(A2). This confirms our former statement that the utilization of the correlation-based scoring affects the results in the same way as the duplication of the number of tested letters; however, it does not increase the duration of the test proportionally (see next subsection). In our former laboratory measurements,

^{29}we examined 26 letters per size, which predicted 28% statistical error reduction when using the correlation-based method instead of probability-based scoring. Now we see that the error reduction is 21% in case of 10 letters and it is 19% with five letters per line. From this finding, we conclude that replacing probability- with correlation-based scoring reduces the repeatability error in a larger extent when more letters are tested at a given letter size.

*L′*(

*x*). To avoid this, we constrained the maximum steepness by

*k*= 35 logMAR

_{max}^{−1}in the regression.

^{22,23}Table 3 contains the most important parameters of the average curves together with the mean visual acuity values of the three subject groups we investigated.

*x*lateral shift parameter nicely correlates to the visual acuity value. According to the definition of the logistic function described in Eq. (5) and Eq. (6), the slope of the

_{mp}*L′*(

*x*) psychometric function is 25/26/4·

*k*. Based on the above results, the slope values obtained for the probability-fits are in good agreement with those presented in the literature: 3 to 10 logMAR

^{−1}.

^{21,24,42}In addition, the shape of the psychometric function definitely flattens with increasing refractive power error, which implies a deterioration in measurement accuracy.

^{22,23}Based on the individual results, the slope of the average correlation-based psychometric curve is 1.23 ± 0.27 times larger at the 0.5 threshold value than that of the probability-based function. This means that correlation-based scoring would exhibit an even smaller

*TRV*if using

*RR*

_{0}= 0.5 as a threshold; however, that would result in a small offset relative to conventional measurements.

*RE*refractive power error expressed in diopters:

*m*= −0.34 ± 0.02 logMAR/D and

*c*= −0.13 ± 0.01 logMAR. The

*m*slope parameter that describes the alteration of visual acuity with respect to refractive power error is in good agreement with the numerical value presented in the literature (

*m*= 0.36 logMAR/D),

^{6}which also supports the relevance of the method.

^{17}Along with it, the price of the whole system is about the same as or less than that of an ETDRS chart, which can cost up to $1000.

^{43}

*OC*matrix to a smaller one containing only the letters really used as responses (down to 10 × 10) and adjusting the theoretical limits, as well as the linear transformation of the psychometric function accordingly (see Eq. [6]). However, because this problem rarely occurs, it is very hard to quantify the resulting residual error. This represents the final limitation of our method, similarly to any curve-fitting evaluation.

*OC*of the presented-answered letter pairs, the system calculates the subject's psychometric function by logistic regression and then determines their visual acuity value by thresholding. Our new method provides more detailed information about the quality of vision than standard recognition-probability-based measurements. We applied different protocols to determine the systematic error of our scoring method (experiment 1) and to demonstrate its statistical error reduction relative to the conventional tests (experiment 2).

*TRV*of the measurements (0.036 logMAR, see experiment 2). This supports the applicability of our new scoring scheme and confirms the adequacy of the formerly calibrated

*RR*

_{0}= 0.68 threshold under clinical conditions. The primary purpose of experiment 2 was to assess the formerly predicted repeatability error reduction of the correlation-based method in a standard clinical environment too. As logistic regression is the most precise currently existing method, we evaluated our data by curve-fitting. The results indicate that the application of correlation-based scoring further decreases the statistical error by ∼20% relative to standard probability-based approach, which corresponds to the same improvement of repeatability as if the number of letters was doubled in traditional tests.

**C. Fülep**, (2016). Measuring visual acuity of a client. World Intellectual Property Organization, WO/2018/020281 A1, PCT/HU2016/000050, patent pending (P);

**I. Kovács**, None;

**K. Kránitz**, None;

**Z.Z. Nagy**, None;

**G. Erdei**, (2016). Measuring visual acuity of a client. World Intellectual Property Organization, WO/2018/020281 A1, PCT/HU2016/000050, patent pending (P)

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*OC*, arranged in alphabetical order, are presented in Table A1. The matrix consists of 10 × 26 cells because the standard ETDRS chart contains 10 different characters, but the subjects are not supposed to know about this restriction, so they can theoretically identify any character of the 26-letter English alphabet. These numbers are more adequate for clinical practice than those presented in our previous paper,

^{29}where all 26 letters were involved in the measurement.