Nonstandard measures of spatial vision (contrast sensitivity, low-contrast acuity, disability glare, etc.) often have numerically high correlations with standard high-contrast visual acuity suggesting a close relationship between the variables. The purpose of this article was to examine how well this allows an individual’s spatial vision function to be predicted from standard visual acuity alone, and whether measuring other spatial vision measures is clinically useful.

Despite previously published cautions about the exclusive use of regression analysis and calculation of correlation coefficients when comparing medical data, ^{1,2} this method of analysis is commonly used in vision research to validate new tests for the purposes of repeatability and to predict one measure from another. Visual acuity is the most frequently used indicator of spatial vision in clinical studies and is commonly correlated with other spatial vision measures such as contrast sensitivity, low-contrast acuity, acuity in the presence of glare, etc. The degree of correlation is often used to make inferences about the underlying mechanisms, i.e., a low correlation between acuity and contrast sensitivity may suggest that different spatial channels are detecting the targets. ^{3}

The reported degree of correlation between various spatial vision measures and standard visual acuity is quite variable in the literature. For example, the correlation coefficient between visual acuity and contrast sensitivity using Pelli-Robson charts has been reported as low as 0.27 and as high as 0.79. ^{3,4} Others have reported correlations between acuity and contrast sensitivity using Vistech charts with a similarly wide range of results; correlations range from 0.25 to 0.88. ^{5,6} What accounts for such a large range of values? It is well known that a correlation is always relative to the situation from which it was calculated. The types of population samples, the range of values, testing conditions and methods, the step size of the measurement, and the number of subjects in the samples are all factors that contribute to the variation. ^{7}

In addition to the wide range of reported correlation coefficients even using the same tests and scoring methods, the conclusions drawn from these correlations are also quite variable. When the correlations are high between acuity and other measures, some authors conclude that the other measures of spatial vision add no further information over that given by standard high-contrast visual acuity. For example, Brown and Lovie-Kitchin ^{6} state “Given that there is a low prevalence of undetected ocular disease in the general population in most Western countries and that the contrast sensitivity function (CSF) values correlate highly with visual acuity test results, there seems to be little point in using (tests of CSF) on a routine basis for screening of ocular disease.”

Hirvelä et al. ^{8} measured contrast sensitivity using two different tests (Cambridge contrast grating test and Pelli-Robson chart) as well as visual acuity in 476 older observers. They conclude that “The correlation between the three tests was good......According to these results, measurement of contrast sensitivity does not add much to the general information obtained by visual acuity measurement alone...”

Others conclude the opposite on the basis of correlational analysis. For example, the SEE study ^{9} found a correlation coefficient between visual acuity and contrast sensitivity of 0.58. The authors conclude that “In a population study, the correlation between vision tests is only moderate, suggesting several different dimensions of vision are being assessed.”

In a study correlating visual acuity (ETDRS chart) and contrast sensitivity (Pelli-Robson chart), Elliott et al. ^{3} stated “As would be expected from a technique which is claimed to measure predominantly CS at low spatial frequencies, there is no significant correlation between CS and logarithm of the minimum angle of resolution (logMAR) VA, (r = 0.27, NS).”

The purpose of this paper is to determine the intercorrelations between a series of spatial vision measures in a fairly large older population and to determine to what extent other spatial measures can be predicted on an individual basis from high-contrast visual acuity using regression analysis. In other words, what if any information is gained by measuring other spatial vision functions in addition to standard high-contrast visual acuity?

#### METHODS

##### Sample

A community-living sample of 900 older people participated in these studies. The sample has been described previously. ^{10} They ranged in age from 58 to 102 years with an average age of 75.5 years (SD = 9.3). About half (53.8%) were female. This research fol-lowed the tenets of the Declaration of Helsinki. Informed consent was obtained from the subjects after explanation of the nature and possible consequences of the study. The research was approved by the institutional review board.

##### Spatial Vision Tests

All testing was binocular with habitual correction in place. The luminance was 150 cd/m^{2}. Each subject was tested in a single session on 1 day. All tests were given in the same order for all subjects. For the acuity measures, subjects were required to read all five letters of the first line correctly and continue reading until two or fewer letters were correctly identified. Scoring was done letter by letter on all the acuity charts, and each letter represents 0.02 log units. The following vision tests were administered in the order that they are listed here.

##### High-Contrast Distance Visual Acuity.

A Bailey-Lovie high-contrast (>90% Weber contrast) chart was used at 10 feet. This chart presents five letters per line, and successive lines change size in a logarithmic fashion.

##### Low-Contrast Distance Visual Acuity.

A low-contrast version of the Bailey-Lovie chart was used at 10 feet. The contrast of the gray letters against the white background is 18% Weber contrast.

##### Contrast Sensitivity.

The Pelli-Robson Chart has letters that vary in contrast rather than size with two letter triplets per line. The contrast between successive triplets changes 0.15 log units. The chart was scored letter by letter like the acuity tests, but in this case each letter represents 0.05 log units. The first line had to be read correctly. Subjects continued reading until all three letters of a triplet were missed. The overall letter size at the 10-foot test distance is one degree, requiring a resolution of at least 20/240 (equivalent to 3.6 c/d).

##### Low-Contrast, Low-Luminance Near Visual Acuity.

The SKILL (Smith-Kettlewell Institute Low Luminance) card, consists of two near charts, one with black letters on a white background and the other with black letters on a gray background. The Weber contrast is >90% on the light side and 14% on the dark side. The luminance was 150 cd/m^{2} light side and 15 cd/m^{2} on the dark side. The test distance was 40 cm. The subject’s own bifocal correction or reading glasses were used.

##### Berkeley Glare Test-Disability Glare.

The Berkeley Glare Test consists of a low-contrast (10% Weber contrast) letter chart presented at 40 cm surrounded by a translucent panel with a rear-mounted glare source. The luminance of the white background without the glare was 80 cd/m^{2}. Observers read the chart with and without the brightest disability glare level (3300 cd/m^{2} or 3.5 log cd/m^{2}). If the largest line was not correctly identified with this glare level, a lower glare level (800 cd/m^{2}or 2.9 log cd/m^{2}) was used. Sixty-two people (6.9%) of the sample were tested with the lower glare level.

To estimate the acuity in glare for the highest glare level for those unable to see the largest line at that glare level, 0.5 log units were subtracted from their acuity for the lower glare level. If the relation between letters lost in glare and glare level was linear, we should subtract 0.6 log units because the higher glare level is 0.6 log units brighter. If the relation is instead exponential, we should subtract a larger number. We assumed conservatively that the relationship is approximately linear but subtracted only 0.5 log units. If the largest line could not be seen even at the lower glare level, the results were excluded from this analysis because these results are constrained by a ceiling effect.

#### RESULTS

Figure 1 shows the regression equation and the correlation between high-contrast visual acuity (in logarithm of the minimum angle of resolution [logMAR]) and log-contrast sensitivity measured using the Pelli-Robson chart. The correlation coefficient was 0.86, and the adjusted r^{2} was 0.73, indicating that 73% of the variation in log-contrast sensitivity was accounted for by the variation in visual acuity. The solid line is the regression equation, and the dashed line represents unit slope. The slope of the regression equation is 1.39. The correlation was highly statistically significant as expected. Eleven people were unable to read the largest line on the high-contrast acuity chart and were removed from these analyses. Note the wide range of values on the y axis for any particular x value. For example, for a logMAR of 0.1 (20/25), contrast sensitivity varied from 1 to 10% (one log unit).

The predicted value for contrast sensitivity was calculated individually on the basis of the regression equation developed for the sample as a whole. Figure 2 shows the difference between log-contrast sensitivity predicted from the regression equation with high-contrast visual acuity and the log-contrast sensitivity value measured for that individual. These differences are plotted against high-contrast visual acuity. If the correlation between acuity and contrast sensitivity were perfect, this difference should be zero for all values of acuity, and all the points should lie on a single horizontal line with an ordinate value of zero. Clearly the points do not fall on a single line. Instead, a wide range of values are found. The discrepancy can be described in several different ways. A total of 55.1% of individuals fell outside a range of ±0.1 log units (±2 letters on the Pelli-Robson chart), whereas 23.2% fell outside a range of ±0.2 log units (±4 letters). That is, if acuity is measured and contrast sensitivity is predicted from that acuity measure using the derived regression equation, in more than half of people, the prediction was wrong by more than ±0.1 log units, and in one-quarter of the people, the prediction was wrong by more than ±0.2 log units. Another way to express the same information is that less than half of the people fell within the band demarcated by the two horizontal lines in Figure 2 at y = 0.1 and y = −0.1.

The 95% confidence limit for repeatability on the Pelli-Robson chart when scored letter by letter is ±0.09 log units. ^{11} Thus, more than half the contrast sensitivity values predicted from visual acuity fall outside the 95% confidence limits for repeatability. To actually include 95% of the population, the ranges have to be expanded to ±0.36 log units. This is shown by the shaded area in Figure 2. This means that if acuity is measured and the regression equation is used to predict contrast sensitivity, the person’s actual contrast sensitivity may be 0.36 log units worse or 0.36 log units better than the prediction. The entire visible chart covers only about 2 log units, so the prediction would be good enough to say that the person’s contrast sensitivity is in the upper, middle, or lower third of the chart. There was no significant influence in the level of the acuity. In other words, the slope between the variables in Figure 2 is not significantly different from zero.

Figure 3 shows the relation between high-contrast visual acuity and low-contrast acuity, both measured using Bailey-Lovie charts. The correlation coefficient was 0.91, and the adjusted r^{2} was 0.83. The solid line is the regression equation, and the dashed line represents perfect agreement. The slope of the regression equation is 1.2. It is clear from the figure that low-contrast acuity was generally worse than high-contrast acuity and that low-contrast acuity fells off faster than high-contrast acuity. Data from those 28 subjects who were unable to see any letters on the largest line of the low-contrast chart were eliminated from these analyses. The range of values on the y axis was still quite variable. For example, for a logMAR of 0.1 (20/25), low-contrast acuity varied from 20/27 to 20/100.

Figure 4 shows the difference between low-contrast acuity predicted from the regression equation and the measured low-contrast acuity. A total of 26.2% of individuals fell outside a range of ±0.1 log units (±5 letters) whereas 4.5% fell outside a range of ±0.2 log units (±10 letters). To include 95% of the population, the ranges have to be ±0.195 log units (indicated by the shaded area). Thus, the predicted low-contrast acuity would be somewhere in a range of 3.9 lines of acuity around the predicted value. This is true even though the correlation coefficient is the highest of all.

Figure 5 shows the same type of analysis for the SKILL dark chart acuity. The correlation coefficient was 0.74, and the adjusted r^{2} was 0.55. The correlation coefficient was slightly lower but still highly statistically significant. The solid line is the regression equation, which has a slope of 0.98. The dashed line represents perfect agreement. It is clear that the SKILL card acuity was considerably worse than high-contrast acuity. The data for those 37 people who were unable to see any letters on the largest line of the SKILL card (acuity worse than 20/632) were eliminated in these analyses. Their high-contrast visual acuity varied from 20/28 to worse than 20/800. For this test, for a logMAR of 0.1 (20/25), SKILL dark acuity varied from 20/45 to 20/200.

Figure 6 shows the difference between the predicted and measured values. A total of 52% of individuals fell outside a range of ±0.1 log units (±5 letters), whereas 21.2% fell outside a range of ±0.2 log units (±10 letters). To include 95% of the population, the ranges had to be ±0.33 log units (shaded area). In this case, the predicted acuity would be somewhere in a range of 6.6 lines of acuity. The coefficient of repeatability for the SKILL card was about ±0.08 log units. ^{12} More than 61% of the low-contrast low-luminance acuity values predicted from visual acuity fell outside the 95% confidence limits for repeatability.

Figure 7 shows the relationship between high-contrast visual acuity and low-contrast acuity in the presence of glare. The solid line is the regression equation, whereas the dashed line represents perfect agreement. The slope of the regression line was 1.28. The correlation coefficient (r = 0.68; r^{2} = 0.46) was only slightly lower than that for the SKILL Dark acuity. Nevertheless, visual inspection of the figure shows much broader confidence limits. This is more obvious in Figure 8, which shows the difference between the predicted and measured values for low-contrast acuity in glare. A total of 61% of individuals fell outside a range of ±0.1 log units (±5 letters), whereas 28% fell outside a range of ±0.2 log units (±10 letters). To include 95% of the population, the ranges had to be ±0.52 log units (shaded area). In this case, the predicted acuity would be somewhere in a range of 10.4 lines of acuity (this is equivalent to the range from 20/20 to worse than 20/200;10+ lines). The coefficient of repeatability for low-contrast acuity in glare on the Berkeley Glare test was ±0.14 log units. ^{13} Nearly half of the differences exceeded this range. The 67 people who could not see the top line on the glare chart at the lowest glare level were eliminated from these analyses. Their high-contrast acuity ranged from close to 20/20 to worse than 20/800. For the glare test, people with high-contrast acuity of logMAR 0.1 (20/25) had acuity in glare that varied from 20/60 to 20/730.

Figure 7 Image Tools |
Figure 8 Image Tools |

#### DISCUSSION

##### Factors Influencing Correlation Coefficients

The results from this comparison of spatial vision measures have shown the difficulty of predicting one variable from another using the regression equations, even when the correlation coefficients are quite high. Even with correlation coefficients in the 0.8 to 0.9 range, many of the predictions fell considerably outside the acceptable ranges determined by repeatability. Why were the correlation coefficients so high when so many people fell outside reasonable ranges for predictive values? The key to this question lies in the basic nature of correlation coefficients—how they are calculated, and how they are affected by the range of values spanned by the data set under study.

The linear regression equations and the relationship to the correlation coefficient is given by the following formulas :^{25}MATHMATHwhere y′ and x′ are the predicted values, SD is the standard deviation, and m is the mean. In the case of predicting y from x, the slope of the regression equation is:MATH MATH

Equation U1 Image Tools |
Equation U2 Image Tools |
Equation U3 Image Tools |

Equation U4 Image Tools |

When the variable y′ is predicted from the regression of y on x, the correlation coefficient is directly proportional to the variability of values on the x axis (reflected in the standard deviation in the equation above). The same is true for the regression of x on y. The variability of both x and y thus affect the correlation coefficient—the higher the variability, the higher the correlation coefficient everything else being equal. In this instance, we are interested in the regression of y on x (predicting y from x). By selecting or including only a narrow range of x values, the correlation coefficient will be considerably lower than if a wide range of x values are included.

Table 1 lists studies from the literature that have correlated visual acuity with various other spatial vision measures. The table indicates the range of acuity values on the x axis and the correlation coefficient reported by the authors. As an example, Elliott et al. ^{3} measured visual acuity using an ETDRS chart and contrast sensitivity using the Pelli-Robson chart. They scored both letter by letter and tested under high-illumination conditions. Their testing methods thus were essentially identical to those that we used. They found a correlation coefficient of 0.27. The range of high-contrast acuities in their study was 0.38 log units (Table 1). The range in our study was 1.38 log units, and our correlation was 0.86. The other major difference between the studies was the number of subjects—55 in their study and 889 in ours.

But it is the range of values on the x axis that has the largest impact on the resultant correlation. This can be demonstrated more intuitively in several ways. First, if we choose a restricted range of values by including x values from the median x value and worse acuity values, the correlation coefficient between high-contrast acuity and contrast sensitivity remains fairly high (r = 0.84), but when we select x values from the median value and better acuity values, the correlation drops to r = 0.50. The range of x values in the first case is 1.08 log units but only 0.3 log units in the second case.

Second, Figure 9 shows the correlation coefficients from this study and the others in Table 1. The correlation coefficient (r) is plotted against the range of acuity values on the x axis (in log units) for each study. The larger the range of values, the higher the correlation coefficient. The correlation coefficient between (r) and the range is quite high: 0.91 (probably because the range on the x axis is large). There is an insignificant correlation between the correlation coefficients in Table 1 and the number of subjects in each study (r = 0.19, NS), even though the range of number of subjects is large (from 15 to 2520). A large range is necessary but not sufficient.

Thus, the primary factor influencing the variation in correlation coefficients between the studies is the range of acuity values measured in the particular study. High values of correlation coefficients can be obtained in a study sample that spans a wide enough range of acuity values, even though many individual values lie far from the regression line. Conversely, results spanning only a small range will result in a much lower correlation even when there is an underlying relationship between the variables. There are formulas developed for compensating for the restriction of range. ^{26} They require that the variables studied are normally distributed in the population. If restriction is produced by selection on the basis of x, it is possible to estimate what the correlation coefficient would be for a group when x is unrestricted. The standard deviations in x must be known for both the restricted and unrestricted group, and the correlation coefficient must be known in the restricted group. As an example, the correlation coefficient between high-contrast acuity and contrast sensitivity for the whole sample was 0.86, and the standard deviation of the x variable (visual acuity) was 0.207 log units. The standard deviation for the sample restricted to include values from the median acuity value and better was 0.071 log units, and the correlation coefficient for this restricted sample was 0.50. The estimated correlation coefficient for the whole sample is:MATH

This is exactly what was found for the whole sample. Other formulas are available for correcting correlation coefficients for the restricted range depending on what information about the variables in the restricted and unrestricted samples is known. ^{26}

These examples suggest that caution should be used when interpreting correlation coefficients. If the correlation coefficient is low and the range of values is low, no strong conclusions should be drawn that the variables are not related to each other. Conversely, if the correlation coefficient is high and the range of values is also large, the accuracy of prediction on an individual basis from the regression equation may still be quite poor. ^{15–24}

#### CONCLUSION

The large discrepancies between spatial vision variables predicted from high-contrast acuity and the actual measurements indicate that measurement of other spatial vision variables provides additional information to that provided by acuity. The results have shown that other vision measures cannot be predicted with a high degree of accuracy on an individual basis from high-contrast acuity. Only relatively coarse predictions are possible despite the high correlations between the variables. The very significant effect of the range of x (and y) values on the numerical values of correlation coefficients is emphasized.

#### ACKNOWLEDGMENTS

Supported by National Eye Institute grant EY09588 (National Institutes of Health, Bethesda, MD) to JAB and by the Smith-Kettlewell Eye Research Institute.