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Methods  |   January 2021
Psychophysical Validation of a Novel Active Learning Approach for Measuring the Visual Acuity Behavioral Function
Author Affiliations & Notes
  • Yukai Zhao
    Center for Neural Science, New York University, New York, NY, USA
  • Luis Andres Lesmes
    Adaptive Sensory Technology, San Diego, CA, USA
  • Michael Dorr
    Adaptive Sensory Technology, San Diego, CA, USA
    Technical University of Munich, Munich, Germany
  • Peter J. Bex
    Department of Psychology, Northeastern University, Boston, MA, USA
  • Zhong-Lin Lu
    Center for Neural Science, New York University, New York, NY, USA
    Division of Arts and Sciences, NYU Shanghai, Shanghai, China
    Department of Psychology, New York University, New York, NY, USA
    NYU-ECNU Institute of Brain and Cognitive Neuroscience, Shanghai, China
  • Correspondence: Zhong-Lin Lu, Center for Neural Science, New York University, 4 Washington Place, Room 621, New York, NY 10003, USA. e-mail: zhonglin@nyu.edu 
Translational Vision Science & Technology January 2021, Vol.10, 1. doi:https://doi.org/10.1167/tvst.10.1.1
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      Yukai Zhao, Luis Andres Lesmes, Michael Dorr, Peter J. Bex, Zhong-Lin Lu; Psychophysical Validation of a Novel Active Learning Approach for Measuring the Visual Acuity Behavioral Function. Trans. Vis. Sci. Tech. 2021;10(1):1. doi: https://doi.org/10.1167/tvst.10.1.1.

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      © ARVO (1962-2015); The Authors (2016-present)

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Abstract

Purpose: To evaluate the performance of the quantitative visual acuity (qVA) method in measuring the visual acuity (VA) behavioral function.

Methods: We evaluated qVA performance in terms of the accuracy, precision, and efficiency of the estimated VA threshold and range in Monte Carlo simulations and a psychophysical experiment. We also compared the estimated VA threshold from the qVA method with that from the Electronic Early Treatment Diabetic Retinopathy Study (E-ETDRS) and Freiburg Visual Acuity Text (FrACT) methods. Four repeated measures with all three methods were conducted in four Bangerter foil conditions in 14 eyes.

Results: In both simulations and psychophysical experiment, the qVA method quantified the full acuity behavioral function with two psychometric parameters (VA threshold and VA range) with virtually no bias and with high precision and efficiency. There was a significant correlation between qVA estimates of VA threshold and range in the psychophysical experiment. In addition, qVA threshold estimates were highly correlated with those from the E-ETDRS and FrACT methods.

Conclusions: The qVA method can provide an accurate, precise, and efficient assessment of the full acuity behavioral function with both VA threshold and range.

Translational Relevance: The qVA method can accurately, precisely, and efficiently assess the full VA behavioral function. Further research will evaluate the potential value of these rich measures for both clinical research and patient care.

Introduction
Visual acuity (VA) is a measure of spatial resolution that provides the most important metric in assessing functional vision.1 As a primary clinical measure for characterizing optical and neural deficits and an important endpoint for treatment efficacy in a large number of diseases,2 VA is also the metric for specific minimum vision standards in many professions.3 For these reasons, the quality of visual acuity assessment is extremely important. Inaccurate and/or imprecise VA assessment could result in unfair classification for paralympic athletes,4 lost job opportunities,5,6 or missed diagnoses of real disease and its progression, which may lead to loss of disability benefits7 or delayed treatment.8 
VA is typically expressed as a single score in logMAR, obtained from testing vision with printed VA charts916 and computerized tests.1725 However, a single VA score alone does not describe the full visual acuity behavior of the observer; rather, an acuity psychometric function (i.e., performance in optotype recognition as a function of optotype size) is required (Fig. 1a). Typically, the function has a sigmoidal shape, is monotonically increasing with optotype size, and is well characterized by a mathematical formula with two parameters: threshold and slope (or range), with the VA threshold defined as the optotype size corresponding to a defined performance level (e.g., 67% correct) and slope (or range) quantifying the steepness of the psychometric function—that is, how fast acuity behavior changes with increasing or decreasing optotype sizes. Both VA threshold and slope may vary across individuals and disease stages.2629 For an observer described by a single acuity psychometric function, VA thresholds at different performance levels are different (Fig. 1a); we cannot directly compare estimated VA scores of an observer obtained from different instruments unless they measure VA thresholds at the same performance level. Moreover, when we compare different observers or a single observer at different disease stages described by acuity psychometric functions with different slopes (Fig. 1b), the magnitude and sign of change in VA threshold depend on the performance level. It may increase at one performance level (B–A in Fig. 1b) but decrease at another performance level (D–C in Fig. 1b). Specifications of the VA threshold with its corresponding target performance level and the slope (or range) of the acuity psychometric function are both necessary to interpret VA and VA changes. 
Figure 1.
 
Visual acuity psychometric function. (a) A single VA psychometric function; the VA thresholds at two different performance levels are different. (b) Two VA psychometric functions with different slopes; changes in the VA thresholds at two different performance levels exhibit opposite signs. The values 0.02 logMAR and 0.10 logMAR correspond to a VA change of one letter and one line on an ETDRS chart, respectively.
Figure 1.
 
Visual acuity psychometric function. (a) A single VA psychometric function; the VA thresholds at two different performance levels are different. (b) Two VA psychometric functions with different slopes; changes in the VA thresholds at two different performance levels exhibit opposite signs. The values 0.02 logMAR and 0.10 logMAR correspond to a VA change of one letter and one line on an ETDRS chart, respectively.
Unfortunately, printed VA charts916 and computerized tests,1725 except for the Freiburg Visual Acuity Text (FrACT) method,30 use operational procedures to generate VA scores without specifying the VA behavioral function, the target performance level, and the slope of the acuity psychometric function. For the same observer, VA scores obtained from different charts and/or the same chart with different heuristic termination rules can be different31 because they target different performance levels on the acuity psychometric function (Fig. 1a); yet, an unbiased estimate of VA change can only be obtained when the two VA estimates are obtained at the same performance level. For different observers measured with the same chart and heuristic termination rule, or a single observer tested at different disease stages, VA scores may correspond to different performance levels if the VA behavioral functions have different slopes (Fig. 1b). To compare VA scores obtained from these tests, the slope (or range) of the VA behavioral function is necessary for correcting differences caused by different performance levels. However, none of the existing VA instruments1725,30 measures the slope of the VA behavioral function. 
In addition, existing tests face a challenge of achieving high accuracy, precision, and efficiency. The most popular VA chart, the Snellen chart,16 is neither accurate nor precise.14,3235 Following Bailey and Lovie,10 logMAR charts14,1719,33,3638 and computerized tests1725 were developed to improve the precision of VA testing. However, the improvement has been limited because of the coarse sampling density of optotype size required to limit the library of test items to reduce testing time.14,33 This is true even for the Early Treatment Diabetic Retinopathy Study (ETDRS) chart12 (Fig. 2a) and its computerized version, the E-ETDRS method,17 with 0.02 logMAR and 0.10 logMAR corresponding to a VA change of one letter and one line on an ETDRS chart, respectively. More recently, new logMAR charts and computerized tests have been developed to either reduce testing time at the cost of precision or improve precision at the cost of testing time.14,18,30,33,39,40 However, it is still a challenge to assess VA with both high precision and high efficiency. 
Figure 2.
 
Optotype size sampling densities of (a) ETDRS (0.10 logMAR between rows) and (b) qVA (0.02 logMAR between rows).
Figure 2.
 
Optotype size sampling densities of (a) ETDRS (0.10 logMAR between rows) and (b) qVA (0.02 logMAR between rows).
Recently, a novel quantitative visual acuity (qVA) test (Patent No. US 10758120B2; Lesmes LA. IOVS. 2018;59:ARVO E-Abstract 1073)41 was developed to characterize the full VA behavioral function using a combination of Bayesian active learning and high-density sampling of optotype size (Fig. 2b; Supplementary Material, Part A). The qVA method models performance in identifying multiple optotypes in each trial with a VA behavioral function with two parameters: VA threshold at a fixed d′ performance level and VA range, corresponding to the steepness of the VA behavioral function (Supplementary Fig. S1). To sample the continuum of optotype size with fine-grained resolution and at the same time achieve high efficiency, the qVA implements an active learning approach to optimize the test stimuli in each trial.4261 Specifically, the testing algorithm begins with a joint prior probability distribution of VA threshold and range of the acuity psychometric function, selects the most informative test stimuli based on all available information before each trial, and updates the posterior distribution with Bayes’ rule. The result is a joint posterior distribution (Supplementary Fig. S2) that can be used to estimate VA threshold and VA range, as well as their uncertainties. 
In this proof-of-concept study, we used both computer simulations and a psychophysical experiment to evaluate qVA performance in terms of the accuracy, precision, and efficiency for estimating VA threshold and VA range. In addition, we compare estimated VA thresholds obtained from qVA with those from E-ETDRS and FrACT methods. The qVA method, like the E-ETDRS and the FrACT methods, is an optotype test. In this study, we applied the qVA method with three optotypes per row in both simulations and the psychophysical experiment and used the same Sloan optotype set used by the other two methods. The results suggest that qVA can estimate the full acuity behavioral function—both threshold and range—with accurate, precise, and efficient assessment. 
Simulations
Methods
Acuity Tests
The details of the qVA (Patent No. US 10758120B2; Lesmes LA. IOVS. 2018;59:ARVO E-Abstract 1073),41 E-ETDRS,17 and FrACT26,30 methods have been published in the cited references and are briefly summarized in the Supplementary Material, Parts A, B, and C and Figures S1, S2, S3, S4, and S5. Whereas the qVA and FrACT methods measure the VA threshold at 67% and 55% correct, respectively, the E-ETDRS method uses an operational procedure that can lead to VA threshold estimates at different performance levels for observers with different slopes of the acuity behavioral function. 
Simulated Observers
We conducted Monte Carlo simulations using the qVA,41 E-ETDRS,17 and FrACT26,30 methods in a 10-alternative forced-choice (10AFC) letter identification task for two observers with the same VA threshold—0.25 logMAR at d′ = 2 (i.e., 67% correct in 10AFC optotype identification)—but different ranges (0.3 and 0.6 logMAR). Specifically, the performance of each simulated observer in identifying multiple optotypes (probabilities of correctly identifying k [= 0, …, N] out of N optotypes) in each trial was modeled by composite multiple-optotype psychometric functions based on the single-optotype d′ acuity psychometric function (Supplementary Material, Part A). 
Simulation Procedures
One thousand qVA runs with 45 three-optotype rows (135 optotypes), 1000 E-ETDRS runs, and 1000 FrACT runs of 45 optotypes were simulated for each observer. 
For each method, the N optotypes presented in each trial were determined by the same test procedure for real observers (Supplementary Material, Parts AC). The only difference between the simulation and the psychophysical testing was that the response of each simulated observer was determined by an acuity psychometric function, which specifies the probability of correctly identifying k (= 0, …, N) out of N optotypes of a given size, along with a random number that was used to select the actual number of correct responses in each trial. 
Reanalysis of E-ETDRS and FrACT with Bayesian Procedure in the qVA Algorithm
We used the simulated E-ETDRS and FrACT data as the input and used the scoring algorithm in the qVA to estimate both VA threshold and VA range. In this reanalysis, the sequence of optotype sizes was selected by either the E-ETDRS or the FrACT method, and the responses were made by the simulated observers in the tests, but the data were scored with the Bayesian procedure in the qVA algorithm. 
Evaluations
The estimated parameters from VA tests reflect the contributions of both systematic and random measurement errors.62 The systematic measurement error or bias is defined as the difference between the mean of an estimated parameter across (infinitely) repeated measurements and the truth (Fig. 3). The smaller the bias is, the more accurate the estimate. In empirical studies, the truth is often unknown; the agreement between one method and a known standard method, evaluated by the 95% limits of agreement between the two methods in the Bland–Altman analysis,63,64 is typically used to provide a proxy measure of accuracy. When the two methods cannot be directly compared, correlation is computed instead. The random measurement error refers to the variability of repeated measures of an estimated parameter. It is typically quantified by the standard deviation (SD) of the estimated parameter from repeated measures with a single method or the 95% repeatability coefficient64 that is defined as 1.96 × √2SD. For continuous measurements, precision is often quantified with 1/SD; however, coarse measurements with large quantization steps may lead to artificially small SDs that do not necessarily imply good precision.65 Fractional Rank Precision (FRP) was recently introduced as a metric that takes into account the effects of measurement grain.66 In addition, the testing time and/or number of optotypes required to reach a desired accuracy and/or precision level are used to quantify the efficiency of a method. 
Figure 3.
 
Illustration of accuracy and precision. The true acuity is 0.3 logMAR. The means of the accurate (a, b) and inaccurate (c, d) measurements are 0.3 and 0.6 logMAR, respectively. The SDs of the precise (a, c) and imprecise (b, d) measurements are 0.05 and 0.15 logMAR, respectively.
Figure 3.
 
Illustration of accuracy and precision. The true acuity is 0.3 logMAR. The means of the accurate (a, b) and inaccurate (c, d) measurements are 0.3 and 0.6 logMAR, respectively. The SDs of the precise (a, c) and imprecise (b, d) measurements are 0.05 and 0.15 logMAR, respectively.
The qVA method generated a joint posterior distribution of the two parameters of the VA behavioral function (VA threshold and VA range) after each trial (Supplementary Material, Part D). The accuracy of each estimated parameter was then quantified by its bias. The precision was quantified by both cross-run and within-run variability. The cross-run variability of each estimated parameter was quantified by its SD across the 1000 simulated runs (tests). The within-run variability of each estimated parameter was quantified by the half-width of the 68.2% credible interval (68.2% HWCI) of its marginal posterior distribution.67,68 The 68.2% HWCI defines the shortest interval that contains the true value of the estimated quantity with 68.2% probability (for details, see Supplementary Material, Part D). The 68.2% HWCI is equal to the cross-run variability (SD) under two conditions: (1) observer behavior does not change across runs, and (2) the qVA procedure has converged with sufficient testing. 
The bias and precision of the estimated VA from the FrACT method were defined similarly, except that the true VAs of the simulated observers were set at 0.210 and 0.170 logMAR, respectively, corresponding to the 55% correct performance level. For the E-ETDRS method, only the precision of the estimated VA scores was evaluated. We could not compute the true VA scores and therefore the bias because the method does not specify the target performance level. 
We computed the sensitivity at 95% specificity to detect a true change of 0.15 logMAR. Given a change criterion and results from two repeated measures with a single method, specificity is defined as the probability of correctly identifying an individual who has not undergone a change, and sensitivity is defined as the probability of correctly identifying an individual who has undergone a change.69,70 Using the 95% repeatability coefficient64 (1.96 × √2SD) as the change criterion,69,70 the specificity is 95% by definition. Sensitivity at 95% specificity is equal to \({\rm{P}}\{ {\rm{X}} > 1.96\; \times {\rm{\;}}( {1 - \frac{d}{{1.96\; \times {\rm{\;}}\surd 2{\rm{SD}}}}} )\} \) where X is a random variable, normally distributed with mean = 0 and SD = 1; d = 0.15 logMAR is the true VA change. 
We chose the change criterion corresponding to specificity at 95% to compare sensitivity across different methods on an equal footing.69,70 Although the logMAR values associated with the criterion level changed with the SD of the estimated VA, the criterion itself was the same if it is normalized by the SD, and the specificity was the same (95%) across all the methods and conditions. We chose 0.15 logMAR as the magnitude of true change, because a greater-than-15-letter VA improvement (0.3 logMAR) is considered by the US Food and Drug Administration as an acceptable endpoint of a clinical trial, although a greater-than-10-letter VA improvement (0.2 logMAR) has been used when the benefits can outweigh the safety risks of the proposed method or product.71 
Results
Figure 4 shows the test histories of the qVA (Figs. 4a, 4d), E-ETDRS (Figs. 4b, 4e), and FrACT (Figs. 4c, 4f) methods in one run of simulated Observer 1. 
Figure 4.
 
(a, b, c) Example test histories of the qVA (a), E-ETDRS (b), and FrACT (c) methods in the simulation study (Observer 1). The true acuity of the simulated observer is represented by the horizontal dashed red line. The color of each dot indicates correct/incorrect responses. In a, the estimated row-by-row VA threshold and its SD are represented by the dashed black lines and shaded grey areas, respectively. In b, the vertical dashed black line separates the screening and the threshold phases. In b and c, the estimated VA threshold and its SD are represented by the light blue cross and its error bar. (d, e, f) The stimulus placement from a, b, and c relative to the acuity psychometric function; for the E-ETDRS method, only the stimuli in the threshold phase are shown in e.
Figure 4.
 
(a, b, c) Example test histories of the qVA (a), E-ETDRS (b), and FrACT (c) methods in the simulation study (Observer 1). The true acuity of the simulated observer is represented by the horizontal dashed red line. The color of each dot indicates correct/incorrect responses. In a, the estimated row-by-row VA threshold and its SD are represented by the dashed black lines and shaded grey areas, respectively. In b, the vertical dashed black line separates the screening and the threshold phases. In b and c, the estimated VA threshold and its SD are represented by the light blue cross and its error bar. (d, e, f) The stimulus placement from a, b, and c relative to the acuity psychometric function; for the E-ETDRS method, only the stimuli in the threshold phase are shown in e.
Estimated VA and Range from the qVA Method
The qVA method provided accurate (Figs. 56) and precise (Figs. 78) estimates of both VA threshold and range (Table 1). After 135 optotypes, the bias of the VA threshold and range estimates were –0.001 and 0.017 logMAR, respectively, for Observer 1, and 0.000 and –0.018 logMAR, respectively, for Observer 2. After 135 optotypes, the SDs of the VA threshold and range estimates were 0.020 and 0.060 logMAR, respectively, for Observer 1 and were 0.037 and 0.117 logMAR, respectively, for Observer 2. The 68.2% HWCI of the VA threshold and range estimates were 0.020 and 0.066 logMAR, respectively, for Observer 1 and were 0.036 and 0.119 logMAR, respectively, for Observer 2. The results suggest that the qVA provides both accurate and precise estimates of the VA behavioral function: both VA threshold and range. 
Figure 5.
 
Bias of the estimated VA from the qVA (black lines) and FrACT (green lines) methods as functions of number of optotypes for simulated Observer 1 (a) and Observer 2 (b). The horizontal dashed lines indicate zero bias.
Figure 5.
 
Bias of the estimated VA from the qVA (black lines) and FrACT (green lines) methods as functions of number of optotypes for simulated Observer 1 (a) and Observer 2 (b). The horizontal dashed lines indicate zero bias.
Figure 6.
 
Bias of the estimated range from the qVA (black lines) method as functions of number of optotypes for simulated Observer 1 (a) and Observer 2 (b). The horizontal dashed lines indicate zero bias.
Figure 6.
 
Bias of the estimated range from the qVA (black lines) method as functions of number of optotypes for simulated Observer 1 (a) and Observer 2 (b). The horizontal dashed lines indicate zero bias.
Figure 7.
 
SDs of the estimated VA from the qVA (68.2% HWCI, black dashed lines; SDs, black dotted lines), E-ETDRS (red asterisks), and FrACT (green dotted lines) methods as functions of number of optotypes for simulated Observer 1 (a) and Observer 2 (b).
Figure 7.
 
SDs of the estimated VA from the qVA (68.2% HWCI, black dashed lines; SDs, black dotted lines), E-ETDRS (red asterisks), and FrACT (green dotted lines) methods as functions of number of optotypes for simulated Observer 1 (a) and Observer 2 (b).
Figure 8.
 
SDs of the estimated range from the qVA method as functions of number of optotypes for simulated Observer 1 (a) and Observer 2 (b) (68.2% HWCI, black dashed lines; SDs, black dotted lines).
Figure 8.
 
SDs of the estimated range from the qVA method as functions of number of optotypes for simulated Observer 1 (a) and Observer 2 (b) (68.2% HWCI, black dashed lines; SDs, black dotted lines).
Table 1.
 
Bias, SDs, and 68.2% HWCI of the Estimates from the Three Methods in the Simulations
Table 1.
 
Bias, SDs, and 68.2% HWCI of the Estimates from the Three Methods in the Simulations
Comparison of Estimated VA from the Three Methods
Because the qVA provides both VA threshold and range estimates, we compared the VA estimates obtained from the qVA with those from the E-ETDRS and FrACT methods. 
The VA thresholds obtained from the qVA exhibited higher accuracy (bias = –0.001 and 0.000 logMAR for Observers 1 and 2, respectively) than those from FrACT (bias = –0.011 and –0.023 logMAR for Observers 1 and 2, respectively). We cannot compute the bias of the estimated VA from the E-ETDRS method because the cumulative letter score does not specify the target performance level. 
The estimated VA from the qVA method also exhibited higher precision (SD = 0.020 and 0.037 logMAR for Observers 1 and 2, respectively) than those from the FrACT (SD = 0.038 and 0.063 logMAR for Observers 1 and 2, respectively) and E-ETDRS (SD = 0.062 and 0.082 logMAR for Observers 1 and 2, respectively) methods (Fig. 9). At 95% specificity, the sensitivity values for detecting an estimated change of 0.15 logMAR were 100.0%, 40.2%, and 79.7% for Observer 1 and were 81.8%, 25.3%, and 39.1% for Observer 2 for the qVA, FrACT, and E-ETDRS methods, respectively (Table 2). 
Figure 9.
 
Distributions of the estimated VA threshold from the qVA (black lines), E-ETDRS (red lines), and FrACT (green lines) methods for simulated Observer 1 (a) and Observer 2 (b).
Figure 9.
 
Distributions of the estimated VA threshold from the qVA (black lines), E-ETDRS (red lines), and FrACT (green lines) methods for simulated Observer 1 (a) and Observer 2 (b).
Table 2.
 
Sensitivity at 95% Specificity to Detect an Estimated Change of 0.15 logMAR for the Three Methods in the Simulations
Table 2.
 
Sensitivity at 95% Specificity to Detect an Estimated Change of 0.15 logMAR for the Three Methods in the Simulations
The qVA method was also more efficient than the other two methods in estimating VA threshold. It required only nine and six optotypes for Observers 1 and 2, respectively, to reach the same bias (–0.011 and –0.023 logMAR) and 42 and 39 optotypes to reach the same precision (SD = 0.038 and 0.063 logMAR for Observers 1 and 2, respectively) of the FrACT method, which was based on 45 optotypes. It required 18 and 24 optotypes to reach the same precision of the E-ETDRS method (SD = 0.062 and 0.082 logMAR), which was based on 36 and 43 optotypes (on average) for Observers 1 and 2, respectively. 
Reanalysis of the E-ETDRS and FrACT Methods
The reanalysis did improve the accuracy and precision of the estimated VA in some cases (Tables 12). The SDs of the reanalyzed VA from the E-ETDRS simulations were reduced by 25.8% and 7.3% for simulated Observers 1 and 2, respectively, and the bias of the reanalyzed VA from the FrACT simulations was reduced by 54% for simulated Observer 2. On the other hand, the estimated slope (quantified by VA range) did not converge in the reanalysis, as indicated by the larger 68.2% HWCIs relative to the SDs. In general, the estimated VA from the reanalysis of the simulated E-ETDRS and FrACT data was less accurate and precise than that from the qVA when the number of optotypes was matched, especially for simulated Observer 2, who had a shallower acuity psychometric function. The estimated VA range from the simulated FrACT data exhibited the largest bias (86% and 233% greater than those from the qVA) because the FrACT test concentrated on optotype sizes near the 55% correct VA threshold. In summary, the qVA method provided the most accurate and precise estimates of both VA threshold and VA range parameters, even when the number of optotypes was matched. 
Psychophysical Evaluation
We conducted a psychophysical experiment to evaluate the precision of VA threshold and range estimates obtained from qVA and compared its estimated VA threshold with those from E-ETDRS and FrACT. Pairwise correlations between VA estimates from the three methods were used to validate the qVA method. Although all subjects had normal or corrected-to-normal vision, we degraded their vision with three different levels of Bangerter foils72 to expand the dynamic range of their measurable visual acuities. 
Methods
Apparatus
The experiment was conducted on a PC computer with Psychtoolbox 3.0.11 extensions73 in MATLAB R2013a (MathWorks, Natick, MA). The computer was used to drive a 24-inch Dell P2415Q liquid-crystal display monitor (Dell Technologies, Round Rock, TX) with a resolution of 3840 × 2160 pixels and a background luminance of 97 cd/m2. The display was viewed monocularly with natural pupil at a viewing distance of 4 meters while the other eye was covered by an opaque eye patch. A chin rest was used to stabilize the observer's head. Acuity was degraded by three levels of Bangerter occlusion foils (Ryser Ophtalmologie, St. Gallen, Switzerland) with nominal acuities of 20/25, 20/30, and 20/100. The room was dimly lit throughout the experiment. 
Stimuli
The optotype stimuli consisted of 10 black Sloan letters (C, D, H, K, N, O, R, S, V, and Z). In the qVA method, Sloan letters74 in 91 optotype sizes, equally spaced between –0.5 and 1.3 logMAR with step-size resolution of 0.02 logMAR, were used. Three letters of the same size and with a center-to-center distance of 1.75 letter width were presented in each qVA trial (one row) (Fig. 10a). In the E-ETDRS method, Sloan letters74 of 20 optotype sizes, equally spaced between –0.3 and 1.6 logMAR with a step size of 0.1 logMAR, were used. To mimic crowding effects in the ETDRS chart in which each row consists of five optotypes, flanker bars with the same stroke width as the letter and a flanker-to-center distance of 1.75 letter width were used (Fig. 10b). In the FrACT method, Sloan letters with optotype sizes sampled between –0.76 and 1.11 logMAR were used along with an anti-aliasing method.75 A two-letter-wide box was used to mimic crowding effects (Fig. 10c). 
Figure 10.
 
Illustration of test stimuli used in the qVA (a), E-ETDRS (b), and FrACT (c) methods.
Figure 10.
 
Illustration of test stimuli used in the qVA (a), E-ETDRS (b), and FrACT (c) methods.
Observers
Six naïve observers and the first author participated in the experiment (25–48 years old; mean, 35 years; all with at least a bachelor's degree). All observers had normal or corrected-to-normal visual acuity (equal to or better than 0.0 logMAR). The study was approved by the Institutional Review Board of Human Subject Research at the Ohio State University. Written consent was obtained from each observer in the beginning of the experiment. Observers (except the first author) were paid $10/hour. The research followed the tenets of the Declaration of Helsinki. 
Design
Monocular acuity behavior, with the non-tested eye covered by an eye patch, was measured for both eyes of each subject in four sessions, one for each of the three foil and no-foil conditions. Each session consisted of eight blocks, four for the left eye and four for the right eye. In each block, the subject was tested in one eye once with each of the three methods. The test order of the three methods and eyes was randomly selected for each block. This resulted in four repeated tests for each combination of subject, eye, method, and foil condition. 
Procedure
Both the qVA and E-ETDRS methods were programmed in MATLAB. In each qVA trial, three letters on a row were randomly sampled without replacement from the 10 Sloan letters, with optotype size determined by the qVA method. The observer was instructed to report the identity of the three letters verbally from left to right. Each qVA run consisted of 45 three-optotype rows each (Fig. 10a; Supplementary Material, Part A). The E-ETDRS procedure17 was used to program the E-ETDRS method (Fig. 10b; Supplementary Material, Part B). To accelerate testing, subjects were allowed to report “I don't know” in the qVA and E-ETDRS methods if a letter could not be identified, rather than forcing a response from the optotype set, and the corresponding response was scored as incorrect. FrACT software (version 3.9.8; downloaded from https://michaelbach.de/fract/) was used to test the FrACT method.30 Each FrACT run consisted of 45 one-optotype trials with the “no easy trial” option (Fig. 10c; Supplementary Material, Part C). The experimenter entered the verbal responses via a wireless keyboard for all three methods. 
Analyses
The precision of the estimated VA threshold and VA range from the qVA methods was quantified by the SD, defined as the SD of the four repeated measures of the same eye in the same condition. The average SD of each estimated parameter was defined as the square root of the mean variance across eyes. 
Because the no-foil and foil conditions simulate vision in healthy and degraded vision, the SDs of the VA estimates were calculated in the no-foil and foil conditions separately. Although the Bangerter occlusion foils are labeled with nominal degrees of acuity degradation (e.g., 20/30), the optical degradations caused by them are known to deviate from the labels.72,76 Furthermore, the irregular transparencies on the surface of the foils may lead to different degrees of visual degradation for the same observer wearing the same foil, depending on the exact alignment used at the time of testing.76 We developed a method (Supplementary Material, Part E) to correct for the systematic between-block variability introduced by foil variabilities. 
Because it provides both VA threshold and range estimates, we compared VA threshold estimates from qVA with those obtained from E-ETDRS and FrACT. To validate qVA, pairwise correlations were computed among the VA thresholds obtained from the three methods. Based on the estimated VA threshold and slope from the qVA method, we simulated the psychophysical experiment 100 times to compute the mean and SD of the pairwise correlation coefficients to estimate their upper bounds. SDs of the estimated VA from the E-ETDRS and FrACT methods were computed. Because SDs of repeated measures could be confounded by sampling resolution (or minimum step size) of the test,65 FRP66,77 was also used to evaluate test–retest precision of the estimated VA from the three methods. 
Results
Testing Time
The mean testing time for one run of each test was 186 ± 29 seconds for qVA (135 optotypes), 66 ± 16 for E-ETDRS (30 optotypes), and 112 ± 25 seconds for FrACT (45 optotypes). It took on average 1.4, 2.2, and 2.5 seconds to test one optotype in the three methods, respectively. The average testing times per optotype were significantly different across the three methods, F(2, 549) = 262.8, P < 0.001, with that of qVA being significantly shorter than that of FrACT (t = –22.63, Ptukey < 0.001) and E-ETDRS (t = –15.43, Ptukey < 0.001). By using three optotypes in a row, qVA sped up testing time per optotype. 
Estimated VA and Range from the qVA Method
The estimated VA threshold and range from the qVA method are shown in Supplementary Figures S6 and S7. The mean estimated ranges were 0.254, 0.340, 0.315, and 0.320 logMAR in the no-foil and three foil conditions, respectively, F(3, 18) = 9.111, P < 0.001). In addition, we found a significant correlation between the estimated VA threshold and range (r = 0.412, P < 0.001). 
Figures 11 and 12 and Table 3 show the precision of the VA threshold and VA range estimates as functions of number of tested optotypes. Figures 13 and 14 show the precision of the qVA estimates of threshold and range as functions of testing time. After 135 optotypes, the SD and 68.2% HWCI of the estimated VA threshold were 0.028 and 0.016 logMAR, respectively, in the no-foil condition, and 0.039 and 0.020 logMAR, respectively, in the three foil conditions. After 135 optotypes, the SDs and 68.2% HWCI of the estimated range were 0.074 and 0.051 logMAR, respectively, in the no-foil condition and 0.082 and 0.066 logMAR, respectively, in the three foil conditions. 
Figure 11.
 
SDs of the estimated VA from the qVA (68.2% HWCI, black dashed lines; SDs, black dotted lines), E-ETDRS (red asterisks), and FrACT (green dotted lines) methods as functions of number of optotypes in the no-foil condition (a) and three foil conditions (b), with between-block variability correction, in the psychophysical experiment.
Figure 11.
 
SDs of the estimated VA from the qVA (68.2% HWCI, black dashed lines; SDs, black dotted lines), E-ETDRS (red asterisks), and FrACT (green dotted lines) methods as functions of number of optotypes in the no-foil condition (a) and three foil conditions (b), with between-block variability correction, in the psychophysical experiment.
Figure 12.
 
SDs of the estimated range from the qVA (68.2% HWCI, black dashed lines; SDs, black dotted lines) method as functions of number of optotypes in the no-foil condition (a) and three foil conditions (b) in the psychophysical experiment.
Figure 12.
 
SDs of the estimated range from the qVA (68.2% HWCI, black dashed lines; SDs, black dotted lines) method as functions of number of optotypes in the no-foil condition (a) and three foil conditions (b) in the psychophysical experiment.
Table 3.
 
SDs of VA Estimates Obtained from Three Psychophysical Methods
Table 3.
 
SDs of VA Estimates Obtained from Three Psychophysical Methods
Figure 13.
 
SDs of the estimated VA from the qVA (68.2% HWCI, black dashed lines; SDs, black dotted lines), E-ETDRS (red asterisks), and FrACT (green dotted lines) methods as functions of testing time in the no-foil condition (a) and three foil conditions (b), with between-block variability correction, in the psychophysical experiment.
Figure 13.
 
SDs of the estimated VA from the qVA (68.2% HWCI, black dashed lines; SDs, black dotted lines), E-ETDRS (red asterisks), and FrACT (green dotted lines) methods as functions of testing time in the no-foil condition (a) and three foil conditions (b), with between-block variability correction, in the psychophysical experiment.
Figure 14.
 
SDs of the estimated range from the qVA (68.2% HWCI, black dashed lines; SDs, black dotted lines) method as functions of testing times in the no-foil condition (a) and three foil conditions (b) in the psychophysical experiment.
Figure 14.
 
SDs of the estimated range from the qVA (68.2% HWCI, black dashed lines; SDs, black dotted lines) method as functions of testing times in the no-foil condition (a) and three foil conditions (b) in the psychophysical experiment.
Comparison of Estimated VA from the qVA, E-ETDRS, and FrACT Methods
We also compared the VA estimates from the qVA, E-ETDRS, and FrACT methods (Supplementary Fig. S6). Table 4 shows that the VA estimates from the three methods were all highly correlated, with values very close to their upper bounds based on simulations. In both the no-foil and three foil conditions (Fig. 15Tables 3, 5), the qVA estimates exhibited higher precision (SD = 0.028 and 0.039 logMAR; FRP = 0.822 and 0.759) than the E-ETDRS (SD = 0.050 and 0.059 logMAR; FRP = 0.696 and 0.727) and FrACT (SD = 0.039 and 0.051 logMAR; FRP = 0.655 and 0.745) methods. The SDs of the estimated VA from the three methods were significantly different: F(2, 156) = 5.695, P = 0.004. Post hoc Tukey test showed that qVA was significantly more precise than E-ETDRS (t = –3.319, Ptukey = 0.003) and marginally more precise than FrACT (t = –2.190, Ptukey = 0.076). At 95% specificity, the sensitivity values for detecting an estimated change of 0.15 logMAR were 96.5%, 77.6%, and 56.4% in the no-foil condition and were 77.6%, 54.8%, and 43.6% in the three foil conditions for the qVA, FrACT, and E-ETDRS methods, respectively (Table 6). 
Table 4.
 
Correlation Coefficients of VA Estimates from Three Psychophysical Methods
Table 4.
 
Correlation Coefficients of VA Estimates from Three Psychophysical Methods
Figure 15.
 
Mean distributions of the estimated VA threshold from the qVA (black lines), E-ETDRS (red lines), and FrACT (green lines) methods for the no-foil condition (a) and three foil conditions (b) across all the observers in the psychophysical experiment.
Figure 15.
 
Mean distributions of the estimated VA threshold from the qVA (black lines), E-ETDRS (red lines), and FrACT (green lines) methods for the no-foil condition (a) and three foil conditions (b) across all the observers in the psychophysical experiment.
Table 5.
 
FRP of VA Estimated from Three Psychophysical Methods
Table 5.
 
FRP of VA Estimated from Three Psychophysical Methods
Table 6.
 
Sensitivity at 95% Specificity to Detect an Estimated Change of 0.15 logMAR for Three Psychophysical Methods
Table 6.
 
Sensitivity at 95% Specificity to Detect an Estimated Change of 0.15 logMAR for Three Psychophysical Methods
When the number of optotypes was matched, the qVA exhibited similar precision in estimating VA threshold (SD = 0.049 and 0.057 logMAR, FRP = 0.716 and 0.713 after 30 optotypes; 0.044 and 0.051 logMAR, FRP = 0.753 and 0.728 after 45 optotypes in the no-foil and three foil conditions) as the E-ETDRS (SD = 0.050 and 0.059 logMAR, FRP = 0.696 and 0.727 after 30 optotypes) and FrACT (SD = 0.039 and 0.051 logMAR, FRP = 0.655 and 0.745 after 45 optotypes) methods. 
The qVA method, however, was the most efficient because of the shortest testing time (qVA, 41 and 62 seconds for 30 and 45 optotypes, respectively; E-ETDRS, 66 seconds for 30 optotypes; FrACT, 112 seconds for 45 optotypes), and the average testing times per optotype were significantly different across the three methods, F(2, 549) = 262.8, P < 0.001, with that of the qVA being significantly shorter than that of FrACT (t = –22.63, Ptukey < 0.001) and that of E-ETDRS (t = –15.43, Ptukey < 0.001). When the testing time was matched (the average testing times for 45 and 78 optotypes in the qVA were the same as those for 30 optotypes in the E-ETDRS and 45 optotypes in the FrACT tests, respectively), the SD of the estimated VA from qVA was much smaller than the SDs from the E-ETDRS and FrACT tests, except that it was the same as that of the FrACT test in the no-foil condition (Table 3). On average, the SD of the estimated VA from qVA was 35.7% less than that from the E-ETDRS and 24.3% less than that from the FrACT. These translate into 35.6% and 21.9% increases of sensitivity at 95% specificity (Table 6). 
Discussion
In both computer simulations and a psychophysical experiment, we found that qVA estimates of the threshold and range of the VA behavioral function were accurate and precise. In the psychophysical experiment with 14 eyes and four Bangerter foil conditions, we found that qVA estimates of threshold and range were significantly correlated, and qVA threshold estimates were correlated with estimates from E-ETDRS and FrACT. Although more optotypes were needed for precise and accurate estimates of both VA threshold and range, the qVA required 38% and 45% less testing time to reach the same precision of the estimated VA threshold than the E-ETRDS and FrACT methods, respectively. The qVA can accurately, precisely, and efficiently quantify the full visual acuity behavioral function. 
Comparison of VA Estimates
As discussed in the introduction, both the threshold and range of the VA behavioral function are necessary to fully characterize and interpret acuity scores and their changes. The qVA is the only method that can estimate both parameters that define the full behavioral function. To evaluate the qVA method, we compared its performance in estimating the VA threshold (while simultaneously estimating the range) with that of the E-ETDR and FrACT methods, which were developed to estimate only VA threshold. 
We could not perform direct comparisons of the estimated VA thresholds from the three methods because (1) different stimulus configurations were used, and, more importantly, (2) they have different target threshold performance levels (in fact, not defined in E-ETDRS). On the other hand, the pairwise correlations among the estimated VA thresholds from the three methods in the psychophysical experiment suggest high consistencies. Although they are slightly lower than the upper bounds based on simulations (because only measurement errors but no observer variability were included in the simulations), the results provide sufficient validation for the VA threshold estimates from the qVA. The high correlations among the three methods indicated that the VA estimates from these methods tended to move in the same direction. A subject with a large VA estimate from one method was more likely to obtain a large VA estimate from another method, and vice versa. However, the high correlations did not indicate similar precision among the three methods (Table 3). 
When the number of optotypes was matched between qVA and the other two methods, the SDs of the estimated VA from the three methods (and thus the sensitivity at 95% specificity) were similar in the psychophysical experiment. In the simulations, the SDs of the estimated VA from the qVA method were the smallest among the three tests when the number of optotypes was matched. The different results from the simulations and human psychophysical experiment may have resulted from the additional variability related to internal noise of the patient (such as random fluctuations in visual processing, attentional state, decision) and variability of foil alignment that was only present in the psychophysical experiment. In addition, that the precision of the estimated VA from the qVA increases with the number of optotypes is an advantage of the qVA method. It allows clinicians to choose the number of tested optotypes based on their own target precision level. 
The estimated VA thresholds from the qVA method were also more accurate and marginally more precise than those from the FrACT method and were more precise than those from the E-ETDRS method. Figures 15a and 15b illustrate the distributions of estimated VA thresholds from the three methods in the computer simulations. The qVA method generated unbiased VA threshold estimates with the highest precision. The bias of the estimated VA thresholds from the FrACT method is a result of the mismatch between the fixed slope used in the method and the true slopes of the simulated observers (Lu Z-L, et al. IOVS. 2019;60:ARVO E-Abstract 3908). 
In addition, the SD (random measurement error) of the estimated VA threshold increases with the range of the VA behavioral function. Although the range and the SD of the estimated VA threshold can be correlated in a given measurement, they are different concepts. Whereas the range of the VA behavioral function is an intrinsic property of an observer that quantifies how performance changes as a function of optotype size, the SD of the estimated VA threshold quantifies the variability of a measurement and depends on the details of the measurement (e.g., instrument, number of trials). Furthermore, the random measurement error caused by binomial variability intrinsic in the task design (correct or incorrect response) and by internal noise of the patient (such as random fluctuations in visual processing, attentional state, decision) cannot be distinguished in the current study because both sources contribute to measurement error in every trial. The random measurement error of a patient with stable vision should also be distinguished from the true VA change of a patient whose visual disease progressed or was treated between two VA measurements. 
Although the reanalysis of the E-ETDRS and FrACT data did improve the accuracy and precision of the VA estimates from the two methods to some extent (Tables 12), the analysis algorithm of the qVA—the Bayesian procedure coupled with the full acuity behavioral psychometric functions—is only one advantage of the qVA method. The Bayesian active learning algorithm and high-density sampling of optotype size in the qVA are critical for optimizing information gain in each trial and obtaining accurate and precise VA threshold and range estimates. The ETDRS and FrACT methods are not optimal for measuring the full acuity behavioral function (see below). 
Range or Slope of the Acuity Psychometric Function
The qVA method was developed to measure both VA threshold and range (or slope) of the acuity psychometric function (Patent No. US 10758120B2; Lesmes LA. IOVS. 2018;59:ARVO E-Abstract 1073).41 We found a significant main effect of foil condition, F(3, 18) = 9.111, P < 0.001, for the steepness of the psychometric function (quantified by the range parameter) and a significant correlation between the estimated VA threshold and range (r = 0.412, P < 0.001), consistent with Carkeet et al.,27 who found that the steepness of the acuity psychometric function depended on the level of blur. Therefore, slope might be a potential endpoint in clinical vision with change measurable by the qVA method. 
On the other hand, the E-ETDRS and FrACT methods, like all other VA tests, provide only a point estimate of the VA threshold. Reanalysis of the data from the E-ETDRS and FrACT methods in the simulations with the Bayesian procedure in the qVA method showed that the data from them did not provide sufficient information to constrain range; its 68.2% HWCI was at least twice of its SD from repeated runs. Figure 4 illustrates that point with one simulated run of simulated Observer 1. In the qVA method (Figs. 4a, 4d), a wide range of optotype sizes is sampled to obtain an accurate and precise estimate of both VA threshold and range of the VA behavioral function. In the E-ETDRS method (Figs. 4b, 4e), although the screening phase covers a large range of optotype sizes, an insufficient number of optotypes were tested to constrain the range estimate. In the FrACT method (Figs. 4c, 4f), the sizes of the test stimuli were mostly concentrated near the 55% correct VA threshold after about 10 optotypes and were insufficient to constrain the range estimate. As a consequence, the estimated VA range from the simulated FrACT data exhibited the largest bias (86% and 233% greater than those from the qVA for Observers 1 and 2) (Table 1). 
To further demonstrate the importance of the full acuity function in estimating VA change, we simulated visual acuity behavior change for Observer 1. In this simulation, the observer's VA threshold increased by 0.15 logMAR and the VA range increased by 0.3 logMAR. We found the average estimated VA threshold changes were 0.15 logMAR, 0.09 logMAR, and 0.09 logMAR from the qVA, E-ETDRS, and FrACT methods, respectively. Without explicitly considering the range of the acuity psychometric function, both E-ETDRS and FrACT underestimated the VA change by 0.06 logMAR (3 ETDRS letters). 
Multi-Optotype Psychometric Function and Joint Distributions
Although the qVA method employs the same Bayesian adaptive framework as other methods,4261 such as QUEST+,54 the qVA method is tailored to the specifics involved in the visual acuity measurement. Specifically, it models the full acuity behavior via composite multiple-optotype psychometric functions derived from a single-optotype d′ acuity psychometric function to correct guessing behavior for acuity tests that often involve optotype sampling without replacement. To our best knowledge, this composite multiple-optotype model is the first that has been incorporated into an adaptive method to measure full VA behavior. 
Most adaptive testing algorithms4648,52,54,55 are sensitive to lapse, especially if it occurs early in the measurement. However, the qVA method uses an adaptive testing algorithm that is robust to lapse, based on many years of development on Bayesian adaptive tests by our group.4145,4951,53,5661 Most adaptive tests are sensitive to lapse because they assume that the observer's behavior obeys the underlying psychometric model and selects test stimulus to maximize the expected information gain in the next trial. However, because lapse is not considered, the model is incorrect if lapse occurs; thus, maximizing the expected information gain in the next trial would lead the algorithm astray. In our development, perturbation was added to the expected information gain to “shake” the tests so that they would not be trapped by lapse.43 
In addition, the joint posterior distribution of VA threshold and range from the qVA method provides rich information for further statistical analyses. For example, the 68.2% HWCI of the joint posterior distribution quantifies the within-run variability of the estimated VA threshold and range, which is not available from traditional tests with only point estimates. The joint posterior distributions can also be incorporated into hierarchical models to improve statistical power in hypothesis testing at both individual and group levels (Lu Z-L, et al. IOVS. 2020;61:ARVO E-Abstract 4615; Zhao Y, et al. IOVS. 2020;61:ARVO E-Abstract 4616). 
Future Development
The introduction of the composite multiple-optotype psychometric functions based on a single-optotype d′ acuity psychometric function is one of the key innovations of the qVA method. It is used to accommodate different tasks and chart designs. d′ is a task-invariant measure of an observer's ability to distinguish different stimuli along a selected stimulus dimension, whereas percent correct quantifies only the empirical performance in a task and depends on the details of the chart design. Although in this study we tested adults with only a 10AFC task using high-contrast Sloan letters, different optotypes (e.g., pictogram optotypes78,79), tasks (e.g., Yes–No task51), and/or test conditions (e.g., test distance), tailored to different ages, cognitive abilities, VA ranges, and/or disease characteristics, could be incorporated into the qVA method, especially given that some new chart designs have been shown to improve disease diagnosis, management, and/or treatment evaluations, such as the low luminance chart80 or high-pass letter chart15,81,82 for age-related macular degeneration patients and low contrast chart for multiple sclerosis.11,8385 
In this study, weakly informative prior distributions were used for each parameter based on pilot experiments. Because a strong incorrect prior would take extra trials for the algorithm to find the correct estimate,86 a weakly informative prior is a better option when very little information is available. It would avoid the extra trials needed to obtain a reliable estimate with a strong but incorrect prior. On the other hand, the accuracy, precision, and efficiency of the qVA method can be improved by using more informative priors.86,87 In addition, the correlation between VA and range could be incorporated into the prior to further improve the efficiency of qVA estimation. Prior knowledge can also be integrated into the estimation procedure in the form of hierarchical modeling.86,87 In addition, increasing the number of optotypes on a row (e.g., from 3 to 5) in a trial increases the information collected in each trial so the efficiency of the qVA estimation could be further improved. Because estimating the VA range requires more optotypes than estimating the VA threshold, increased efficiency is more critical in estimating range. The “I don't know” response option usually presented in unforced-choice tasks88 was introduced to improve patient experience in quick contrast-sensitivity function tests.44,50 We re-simulated the qVA method for simulated Observer 1 with the “I don't know” response based on its frequency in psychophysical experiment. We found that the response option introduced a relatively small 0.022 logMAR bias to the estimated VA threshold. We will incorporate the model of unforced-choice tasks88 into the qVA method to reduce the bias in the future. 
To the best of our knowledge, this is the first study to validate the novel qVA method in both simulations and psychophysical experiment. Although we used only seven subjects in the study, we collected a massive amount of data from each of them (6720 optotypes over four foil conditions). Nevertheless, the results are based on a relatively small number of subjects with normal or corrected-to-normal vision and high education level. More subjects and different patient populations are necessary to further evaluate the qVA method in the future. 
Conclusions
In both computer simulations and a psychophysical experiment, we showed that the qVA method can accurately, precisely, and efficiently quantify the full VA acuity behavior using an acuity psychometric function with two parameters: a VA threshold at d′ = 2 performance level and range. The concurrent measurement of VA and slope did not compromise the VA measurement, as supported by the relatively high precision and efficiency of the estimated VA threshold from the qVA method. 
Acknowledgments
Supported by Grants from the National Eye Institute, National Institutes of Health (EY021553, EY025658A). 
Disclosure: Y. Zhao, None; L.A. Lesmes, Adaptive Sensory Technology (E, I), Patent Nos. US 10758120B2, US 7938538, WO 201370091, PCT/US2015/028657 (P); M. Dorr, Adaptive Sensory Technology (E, I), Patent Nos. US 7938538, WO 201370091, PCT/US2015/028657 (P); P.J. Bex, Adaptive Sensory Technology (I), Patent Nos. US 7938538, WO 201370091, PCT/US2015/028657 (P); Z.-L. Lu, Adaptive Sensory Technology (I), Patent Nos. US 7938538, WO 201370091, PCT/US2015/028657 (P) 
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Figure 1.
 
Visual acuity psychometric function. (a) A single VA psychometric function; the VA thresholds at two different performance levels are different. (b) Two VA psychometric functions with different slopes; changes in the VA thresholds at two different performance levels exhibit opposite signs. The values 0.02 logMAR and 0.10 logMAR correspond to a VA change of one letter and one line on an ETDRS chart, respectively.
Figure 1.
 
Visual acuity psychometric function. (a) A single VA psychometric function; the VA thresholds at two different performance levels are different. (b) Two VA psychometric functions with different slopes; changes in the VA thresholds at two different performance levels exhibit opposite signs. The values 0.02 logMAR and 0.10 logMAR correspond to a VA change of one letter and one line on an ETDRS chart, respectively.
Figure 2.
 
Optotype size sampling densities of (a) ETDRS (0.10 logMAR between rows) and (b) qVA (0.02 logMAR between rows).
Figure 2.
 
Optotype size sampling densities of (a) ETDRS (0.10 logMAR between rows) and (b) qVA (0.02 logMAR between rows).
Figure 3.
 
Illustration of accuracy and precision. The true acuity is 0.3 logMAR. The means of the accurate (a, b) and inaccurate (c, d) measurements are 0.3 and 0.6 logMAR, respectively. The SDs of the precise (a, c) and imprecise (b, d) measurements are 0.05 and 0.15 logMAR, respectively.
Figure 3.
 
Illustration of accuracy and precision. The true acuity is 0.3 logMAR. The means of the accurate (a, b) and inaccurate (c, d) measurements are 0.3 and 0.6 logMAR, respectively. The SDs of the precise (a, c) and imprecise (b, d) measurements are 0.05 and 0.15 logMAR, respectively.
Figure 4.
 
(a, b, c) Example test histories of the qVA (a), E-ETDRS (b), and FrACT (c) methods in the simulation study (Observer 1). The true acuity of the simulated observer is represented by the horizontal dashed red line. The color of each dot indicates correct/incorrect responses. In a, the estimated row-by-row VA threshold and its SD are represented by the dashed black lines and shaded grey areas, respectively. In b, the vertical dashed black line separates the screening and the threshold phases. In b and c, the estimated VA threshold and its SD are represented by the light blue cross and its error bar. (d, e, f) The stimulus placement from a, b, and c relative to the acuity psychometric function; for the E-ETDRS method, only the stimuli in the threshold phase are shown in e.
Figure 4.
 
(a, b, c) Example test histories of the qVA (a), E-ETDRS (b), and FrACT (c) methods in the simulation study (Observer 1). The true acuity of the simulated observer is represented by the horizontal dashed red line. The color of each dot indicates correct/incorrect responses. In a, the estimated row-by-row VA threshold and its SD are represented by the dashed black lines and shaded grey areas, respectively. In b, the vertical dashed black line separates the screening and the threshold phases. In b and c, the estimated VA threshold and its SD are represented by the light blue cross and its error bar. (d, e, f) The stimulus placement from a, b, and c relative to the acuity psychometric function; for the E-ETDRS method, only the stimuli in the threshold phase are shown in e.
Figure 5.
 
Bias of the estimated VA from the qVA (black lines) and FrACT (green lines) methods as functions of number of optotypes for simulated Observer 1 (a) and Observer 2 (b). The horizontal dashed lines indicate zero bias.
Figure 5.
 
Bias of the estimated VA from the qVA (black lines) and FrACT (green lines) methods as functions of number of optotypes for simulated Observer 1 (a) and Observer 2 (b). The horizontal dashed lines indicate zero bias.
Figure 6.
 
Bias of the estimated range from the qVA (black lines) method as functions of number of optotypes for simulated Observer 1 (a) and Observer 2 (b). The horizontal dashed lines indicate zero bias.
Figure 6.
 
Bias of the estimated range from the qVA (black lines) method as functions of number of optotypes for simulated Observer 1 (a) and Observer 2 (b). The horizontal dashed lines indicate zero bias.
Figure 7.
 
SDs of the estimated VA from the qVA (68.2% HWCI, black dashed lines; SDs, black dotted lines), E-ETDRS (red asterisks), and FrACT (green dotted lines) methods as functions of number of optotypes for simulated Observer 1 (a) and Observer 2 (b).
Figure 7.
 
SDs of the estimated VA from the qVA (68.2% HWCI, black dashed lines; SDs, black dotted lines), E-ETDRS (red asterisks), and FrACT (green dotted lines) methods as functions of number of optotypes for simulated Observer 1 (a) and Observer 2 (b).
Figure 8.
 
SDs of the estimated range from the qVA method as functions of number of optotypes for simulated Observer 1 (a) and Observer 2 (b) (68.2% HWCI, black dashed lines; SDs, black dotted lines).
Figure 8.
 
SDs of the estimated range from the qVA method as functions of number of optotypes for simulated Observer 1 (a) and Observer 2 (b) (68.2% HWCI, black dashed lines; SDs, black dotted lines).
Figure 9.
 
Distributions of the estimated VA threshold from the qVA (black lines), E-ETDRS (red lines), and FrACT (green lines) methods for simulated Observer 1 (a) and Observer 2 (b).
Figure 9.
 
Distributions of the estimated VA threshold from the qVA (black lines), E-ETDRS (red lines), and FrACT (green lines) methods for simulated Observer 1 (a) and Observer 2 (b).
Figure 10.
 
Illustration of test stimuli used in the qVA (a), E-ETDRS (b), and FrACT (c) methods.
Figure 10.
 
Illustration of test stimuli used in the qVA (a), E-ETDRS (b), and FrACT (c) methods.
Figure 11.
 
SDs of the estimated VA from the qVA (68.2% HWCI, black dashed lines; SDs, black dotted lines), E-ETDRS (red asterisks), and FrACT (green dotted lines) methods as functions of number of optotypes in the no-foil condition (a) and three foil conditions (b), with between-block variability correction, in the psychophysical experiment.
Figure 11.
 
SDs of the estimated VA from the qVA (68.2% HWCI, black dashed lines; SDs, black dotted lines), E-ETDRS (red asterisks), and FrACT (green dotted lines) methods as functions of number of optotypes in the no-foil condition (a) and three foil conditions (b), with between-block variability correction, in the psychophysical experiment.
Figure 12.
 
SDs of the estimated range from the qVA (68.2% HWCI, black dashed lines; SDs, black dotted lines) method as functions of number of optotypes in the no-foil condition (a) and three foil conditions (b) in the psychophysical experiment.
Figure 12.
 
SDs of the estimated range from the qVA (68.2% HWCI, black dashed lines; SDs, black dotted lines) method as functions of number of optotypes in the no-foil condition (a) and three foil conditions (b) in the psychophysical experiment.
Figure 13.
 
SDs of the estimated VA from the qVA (68.2% HWCI, black dashed lines; SDs, black dotted lines), E-ETDRS (red asterisks), and FrACT (green dotted lines) methods as functions of testing time in the no-foil condition (a) and three foil conditions (b), with between-block variability correction, in the psychophysical experiment.
Figure 13.
 
SDs of the estimated VA from the qVA (68.2% HWCI, black dashed lines; SDs, black dotted lines), E-ETDRS (red asterisks), and FrACT (green dotted lines) methods as functions of testing time in the no-foil condition (a) and three foil conditions (b), with between-block variability correction, in the psychophysical experiment.
Figure 14.
 
SDs of the estimated range from the qVA (68.2% HWCI, black dashed lines; SDs, black dotted lines) method as functions of testing times in the no-foil condition (a) and three foil conditions (b) in the psychophysical experiment.
Figure 14.
 
SDs of the estimated range from the qVA (68.2% HWCI, black dashed lines; SDs, black dotted lines) method as functions of testing times in the no-foil condition (a) and three foil conditions (b) in the psychophysical experiment.
Figure 15.
 
Mean distributions of the estimated VA threshold from the qVA (black lines), E-ETDRS (red lines), and FrACT (green lines) methods for the no-foil condition (a) and three foil conditions (b) across all the observers in the psychophysical experiment.
Figure 15.
 
Mean distributions of the estimated VA threshold from the qVA (black lines), E-ETDRS (red lines), and FrACT (green lines) methods for the no-foil condition (a) and three foil conditions (b) across all the observers in the psychophysical experiment.
Table 1.
 
Bias, SDs, and 68.2% HWCI of the Estimates from the Three Methods in the Simulations
Table 1.
 
Bias, SDs, and 68.2% HWCI of the Estimates from the Three Methods in the Simulations
Table 2.
 
Sensitivity at 95% Specificity to Detect an Estimated Change of 0.15 logMAR for the Three Methods in the Simulations
Table 2.
 
Sensitivity at 95% Specificity to Detect an Estimated Change of 0.15 logMAR for the Three Methods in the Simulations
Table 3.
 
SDs of VA Estimates Obtained from Three Psychophysical Methods
Table 3.
 
SDs of VA Estimates Obtained from Three Psychophysical Methods
Table 4.
 
Correlation Coefficients of VA Estimates from Three Psychophysical Methods
Table 4.
 
Correlation Coefficients of VA Estimates from Three Psychophysical Methods
Table 5.
 
FRP of VA Estimated from Three Psychophysical Methods
Table 5.
 
FRP of VA Estimated from Three Psychophysical Methods
Table 6.
 
Sensitivity at 95% Specificity to Detect an Estimated Change of 0.15 logMAR for Three Psychophysical Methods
Table 6.
 
Sensitivity at 95% Specificity to Detect an Estimated Change of 0.15 logMAR for Three Psychophysical Methods
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