Csilla Fülep, Illés Kovács, Kinga Kránitz, and Gábor Erdei, "Correlation-based evaluation of visual performance to reduce the statistical error of visual acuity," J. Opt. Soc. Am. A 34, 1255-1264 (2017)
Ophthalmologists evaluate visual acuity tests by the number of correctly recognized optotypes (usually letters) in the different lines of an eye chart. This probability-based scoring results in significant statistical error that can only be decreased by the time-consuming analysis of a larger number of optotypes. In this paper, we present a new, more precise correlation-based scoring method that takes the degree of misidentification into consideration too, rather than the mere fact of it. According to our experimental results, this new method decreases the uncertainty error by 28% if using the same number of optotypes at a given letter size or requires half the optotype number to produce the same error as that of probability-based scoring.
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Statistical Parameters of Pearson’s Correlation Distributions for the Investigated Extended 26-Letter Font Types: Sloan, Snellen, and Eyechart
Font Type
Average
Standard Deviation
Skewness
Sloan
0.53
0.17
0.25
Snellen
0.65
0.11
−0.16
Eyechart
0.38
0.18
0.54
Table 3.
Summary of Our Measurement Results: Pupil Diameters, Probability-Scoring-Based Visual Acuity Values () at Threshold, Rate of Recognition Values Calculated at , and New Visual Acuity Values () at the Average Rate of Recognition Applied as a Threshold ()a
Subject
Pupil Diameter [mm]
Kl. Mi.
6.0
1.27
0.63
1.25
Ku. Ma.
8.0
1.42
0.69
1.43
R. I.
5.0
1.46
0.75
1.50
P. B.
5.0
1.54
0.65
1.52
M. T.
5.6
1.63
0.72
1.67
S. O.
5.2
1.75
0.66
1.74
G. T.
6.2
1.77
0.73
1.82
S. T.
4.6
1.79
0.63
1.79
G. A.
5.0
1.87
0.73
1.90
U. F.
3.8
2.08
0.64
2.07
Visual acuity values are expressed in decimal units.
Table 4.
Parameters of the Average Best-Fit Super-Gaussian Interpolation Curves in Probability- and Rate-of-Recognition-Based Scoringsa
Scoring Method
[]
[-]
; [-]
[]
; [arcmin]
by
1.85
4.22
0.50
1.72
−0.83
by RR
2.05
5.40
0.68
1.72
−0.85
For the definitions of slopes and , see Section 4.
Table 5.
Measurement Error (Uncertainty) Estimated for the Four Evaluation Methods (A1, B1, A2, and B2)a
Test ID
Number of Tested Letters
Scale Grade [logMAR]
(A)
(B)
1
5
0.10
2
26
0.05
The reported values represent standard deviation.
Table 6.
Visual Acuity Errors of the Four Evaluation Methods Obtained by Two Different Calculations
Single Measurement
Test–Retest Variability
Test ID
[]
[logMAR]
[]
[logMAR]
A1
0.159
0.075
0.166
0.079
B1
0.116
0.053
0.126
0.058
A2
0.054
0.024
–
–
B2
0.039
0.017
–
–
Table 7.
Optotype Correlation Values between the 10 Letters of the ETDRS Chart and All 26 Letters of the English Alphabet (Extended Sloan Font)a
C
D
H
K
N
O
R
S
V
Z
A
−0.477
−0.529
−0.325
0.008
−0.290
−0.496
−0.015
−0.205
−0.372
−0.104
B
0.258
0.707
0.489
0.152
0.283
0.348
0.697
0.806
−0.177
0.349
C
1.000
0.458
−0.408
−0.290
−0.201
0.866
−0.004
0.186
−0.398
−0.033
D
0.458
1.000
0.127
−0.036
0.205
0.584
0.377
0.470
−0.462
0.205
E
0.035
0.517
0.412
0.245
0.200
0.160
0.582
0.644
−0.179
0.453
F
−0.239
0.249
0.482
0.243
0.182
−0.101
0.676
0.360
−0.297
0.051
G
0.888
0.463
−0.376
−0.214
−0.096
0.833
0.028
0.198
−0.329
−0.052
H
−0.408
0.127
1.000
0.159
0.599
−0.247
0.541
0.205
−0.327
−0.202
I
−0.225
0.126
−0.381
−0.033
−0.202
−0.246
−0.038
0.200
−0.314
0.600
J
−0.003
0.300
−0.077
−0.641
−0.024
0.196
−0.357
0.018
−0.537
−0.134
K
−0.290
−0.036
0.159
1.000
0.347
−0.316
0.369
−0.025
−0.391
−0.169
L
−0.070
0.369
0.012
0.073
0.022
−0.078
−0.073
0.028
−0.575
0.021
M
−0.092
0.226
0.555
0.049
0.460
0.023
0.286
0.154
0.413
−0.020
N
−0.201
0.205
0.599
0.347
1.000
−0.066
0.392
0.021
−0.324
−0.213
O
0.866
0.584
−0.247
−0.316
−0.066
1.000
0.054
0.241
−0.426
−0.064
P
−0.101
0.324
0.484
0.196
0.186
−0.034
0.831
0.394
−0.265
0.056
Q
0.798
0.552
−0.192
−0.223
0.036
0.917
0.117
0.250
−0.387
−0.026
R
−0.004
0.377
0.541
0.369
0.392
0.054
1.000
0.455
−0.280
0.117
S
0.186
0.470
0.205
−0.025
0.021
0.241
0.455
1.000
−0.170
0.316
T
−0.382
−0.215
−0.604
−0.249
−0.451
−0.415
−0.179
−0.086
−0.218
0.200
U
0.180
0.589
0.296
−0.194
0.310
0.343
0.003
0.174
−0.405
−0.125
V
−0.398
−0.462
−0.327
−0.391
−0.324
−0.426
−0.280
−0.170
1.000
−0.116
W
−0.095
0.226
0.556
0.290
0.458
0.023
0.480
0.149
−0.310
−0.021
X
−0.439
−0.435
−0.442
0.083
−0.011
−0.470
−0.191
−0.232
−0.056
0.200
Y
−0.520
−0.549
−0.640
−0.293
−0.338
−0.548
−0.486
−0.334
0.087
−0.036
Z
−0.033
0.205
−0.202
−0.169
−0.213
−0.064
0.117
0.316
−0.116
1.000
The table arrangement is alphabetical.
Tables (7)
Table 1.
Most Important Systematic and Random Sources of Visual Acuity Measurement Errors
Statistical Parameters of Pearson’s Correlation Distributions for the Investigated Extended 26-Letter Font Types: Sloan, Snellen, and Eyechart
Font Type
Average
Standard Deviation
Skewness
Sloan
0.53
0.17
0.25
Snellen
0.65
0.11
−0.16
Eyechart
0.38
0.18
0.54
Table 3.
Summary of Our Measurement Results: Pupil Diameters, Probability-Scoring-Based Visual Acuity Values () at Threshold, Rate of Recognition Values Calculated at , and New Visual Acuity Values () at the Average Rate of Recognition Applied as a Threshold ()a
Subject
Pupil Diameter [mm]
Kl. Mi.
6.0
1.27
0.63
1.25
Ku. Ma.
8.0
1.42
0.69
1.43
R. I.
5.0
1.46
0.75
1.50
P. B.
5.0
1.54
0.65
1.52
M. T.
5.6
1.63
0.72
1.67
S. O.
5.2
1.75
0.66
1.74
G. T.
6.2
1.77
0.73
1.82
S. T.
4.6
1.79
0.63
1.79
G. A.
5.0
1.87
0.73
1.90
U. F.
3.8
2.08
0.64
2.07
Visual acuity values are expressed in decimal units.
Table 4.
Parameters of the Average Best-Fit Super-Gaussian Interpolation Curves in Probability- and Rate-of-Recognition-Based Scoringsa
Scoring Method
[]
[-]
; [-]
[]
; [arcmin]
by
1.85
4.22
0.50
1.72
−0.83
by RR
2.05
5.40
0.68
1.72
−0.85
For the definitions of slopes and , see Section 4.
Table 5.
Measurement Error (Uncertainty) Estimated for the Four Evaluation Methods (A1, B1, A2, and B2)a
Test ID
Number of Tested Letters
Scale Grade [logMAR]
(A)
(B)
1
5
0.10
2
26
0.05
The reported values represent standard deviation.
Table 6.
Visual Acuity Errors of the Four Evaluation Methods Obtained by Two Different Calculations
Single Measurement
Test–Retest Variability
Test ID
[]
[logMAR]
[]
[logMAR]
A1
0.159
0.075
0.166
0.079
B1
0.116
0.053
0.126
0.058
A2
0.054
0.024
–
–
B2
0.039
0.017
–
–
Table 7.
Optotype Correlation Values between the 10 Letters of the ETDRS Chart and All 26 Letters of the English Alphabet (Extended Sloan Font)a