ORIGINAL ARTICLE. Corneal and Refractive Error Astigmatism in Singaporean Schoolchildren: a Vector-Based Javal s Rule

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1 /01/ /0 VOL. 78, NO. 12, PP OPTOMETRY AND VISION SCIENCE Copyright 2001 American Academy of Optometry ORIGINAL ARTICLE Corneal and Refractive Error Astigmatism in Singaporean Schoolchildren: a Vector-Based Javal s Rule LOUIS TONG, MBBS, FRCS, ANDREW CARKEET, BAppSc(Optom)(Hons), PhD, SEANG-MEI SAW, MBBS, PhD, and DONALD T. H. TAN, MBBS, FRCS, FRCO, FAMS Singapore National Eye Centre (LT, DTHT), Singapore Eye Research Institute (AC, DTHT), Department of Community, Occupational and Family Medicine, National University of Singapore (SMS) ABSTRACT: Background. Traditional approaches to Javal s rule do not use data from subjects with oblique astigmatism and have not been used to make predictions about subjects with oblique astigmatism. Vector approaches to analyzing refractive error can circumvent these problems. Methods. Subjects were 993 Singaporean schoolchildren. We performed linear regression of refractive error astigmatism on corneal astigmatism, using J 0 vectors to describe with-therule and against-the-rule astigmatism and J 45 vectors to describe oblique astigmatism. Results. We obtained the following statistically significant regression relationships: RJ CJ and RJ CJ , where R and C denote refractive error astigmatism and corneal astigmatism, respectively. Conclusion. Our vectorbased Javal s rule gives closer predictions of refractive astigmatism than the original Javal s rule and the simplified Javal s rule and can be applied in cases of corneal oblique astigmatism. (Optom Vis Sci 2001;78: ) Key Words: astigmatism, vectors, Javal s rule It has been known for more than a century that in humans, corneal astigmatism can be used to make clinically useful predictions about refractive error astigmatism. Based on his clinical patients, Javal 1 postulated an approximately linear relationship between refractive error astigmatism and corneal astigmatism: specifically that in the general population, refractive astigmatism could be calculated by multiplying corneal astigmatism by a constant p (which was approximately 1.25) and then adding an extra constant K (0.5 D against-the-rule) astigmatism. This rule has subsequently become known as Javal s rule. Javal s original rule was not based on a rigorous examination of data, but merely on Javal s own clinical impressions, and he noted that the values for coefficients p and K had not been definitively established. Later studies, performing more rigorous analyses, found a shallower gradient to the relationship. For example, from their empirical studies of 1058 eyes, Grosvenor et al. 2 determined the following regression equations: TA 0.76CA 0.40 D for myopic children; TA 0.84CA 0.32 D for patients at a university clinic; and TA 0.87CA 0.43 D for patients in an optometry practice; where TA is the total (or refractive error) astigmatism (the cylindrical component in the refractive error) and CA is the corneal astigmatism. In these relationships, positive values represent with-the-rule (WTR) astigmatism (in which the negative cylinder axis is oriented at 180 ) and negative values represent against-the-rule (ATR) astigmatism (in which the negative cylinder axis is oriented at 90. Thus, these relationships only apply to what we have termed Cartesian astigmatism: forms of astigmatism with their principal axes at 180 and 90. On the basis of these findings, Grosvenor et al. 2 devised what they termed their simplified Javal s Rule: TA CA 0.5 D. Other authors have shown similar linear relationships between corneal astigmatism and refractive error astigmatism in different populations, with slopes varying around 1 and slightly negative y intercepts. 3 9 There are, however, some interpretational problems with all of the previous Javal s rule studies, and they relate to the practicalities of determining Cartesian astigmatism. Patients seldom have corneal or refractive cylinder axes that lie exactly on 180 or 90, and thus definitions for WTR or ATR astigmatism need to encompass a wider range of axes than the horizontal and vertical. To address this issue, researchers often set boundary criteria on axes for WTR and ATR astigmatism, and these boundary criteria can vary for different authors. For example, in their research work, Grosvenor et al. 2 classify all negative cylinder axes between 150 and 30 (inclusive) as WTR and those be-

2 882 Vector-Based Javal s Rule Tong et al. tween 60 and 120 (inclusive) as ATR. All other axes are classified as oblique. This definition is also commonly found in textbooks, 10 although Rosenfield 11 defines WTR and ATR astigmatism more narrowly (negative cylinder axes 160 to 20 as WTR; 70 to 110 as ATR; and 20 to 70 and 110 to 160 as oblique). Other studies of Javal s rule 4, 5, 8 use even narrower boundaries for WTR and ATR, i.e., negative cylinder axes of 165 to 15 (inclusive) for WTR and 75 to 105 (inclusive) as ATR, with all other axes being oblique. In their research work on the etiology of myopia, Gwiazda et al. 12 adopted definitions whereby negative cylinder axes of were designated as ATR, designated as WTR, and all other axes designated as oblique. Setting boundary conditions in this way does present some problems for research into Javal s rule. First, when formulating the rule, subjects with axes classified as oblique (for cornea or refractive error) must be dropped from the analysis, thereby reducing the subject pool. Second, because most subjects do not have axes exactly on 90 or 180, most results included in analysis will be contaminated by some oblique cylinder component. Third, it is unclear how Javal s rule should be applied in practice for predicting refractive cylinder. Should it be applied to oblique astigmatism? If the corneal cylinder axis does not line up exactly with 180 or 90, should the refractive error cylinder be aligned with the Cartesian axes or left at the original cylinder axis? One way around these interpretational problems is to make use of vector methods advocated by Thibos et al. 13, which originated in the methods of Gartner 14 and Deal and Toop 15 and which are closely analogous to the methods used by McKendrick and Brennan 16 for analyzing refractive error and corneal astigmatism. This is an approach in which any refractive error can be expressed as a combination of three orthogonal components or thin lenses M, J 0, and J 45.Incommon clinical terms, M is equal to the spherical equivalent and is relatively unimportant for the purposes of studying astigmatism. The J 0 component can be thought of as a Jackson cross-cylinder (JCC) with its axes at 180 and 90. A positive value for J 0 denotes that the negative cylinder axis of the JCC is at 180, and a negative value denotes that the negative cylinder axis of the JCC is at 90. Thus, J 0 can be used for quantifying Cartesian astigmatism. A patient with a large amount of WTR astigmatism would have a high positive J 0, and, likewise, a large amount of ATR astigmatism would manifest as a highly negative J 0. McKendrick and Brennan 16 made similar observations about their vector i 1 which is equal to 2 J 0. Thus, regression of J 0 for refractive error against J 0 for the cornea could be one way of deriving equations similar to Javal s rule. All subjects in a sample can be included in such an analysis because a J 0 value can be calculated for any refractive error or corneal cylinder. The J 45 component can be thought of as a JCC with its axes at 45 and 135. A positive value for J 45 denotes that the negative cylinder axis of the JCC is at 45, and a negative value denotes that the negative cylinder axis of the JCC is at 135. Thus J 45 might be thought of as quantifying the amount of oblique astigmatism in a refractive error. McKendrick and Brennan 16 used a similar vector i 2 which is equal to 2 J 45. From a clinical perspective, oblique astigmatism is often treated as a single category, but it is possible to split into two categories (in the same way that ATR and WTR are a dichotomy). Negative cylinder axes that are closest to 45 (i.e., positive J 45 ) we term levo-oblique (LO) astigmatism, and those that are close to 135 (i.e., negative J 45 ) we term dextro-oblique DO astigmatism. Although it has never been done before, it is possible to frame rules like Javal s rule for oblique astigmatism (i.e., by regression of refractive error J 45 against corneal cylinder J 45 ). The purpose of this study was to apply this vector approach to Javal s rule in a sample of Singaporean schoolchildren. Because the vector approach allows for the analysis of data from all subjects and also provides information about oblique astigmatism, we expected that it would provide better and more general predictions of refractive error astigmatism than the traditional Javal s rule approaches. METHODS Subjects Some initial cross-sectional results of the Singapore Cohort Study of the Risk Factors of Myopia (SCORM), 17, 18 an ongoing longitudinal observational study that commenced in 1999, are reported here. This study was approved by the ethics committee of the Singapore Eye Research Institute, and all procedures adhered to the Declaration of Helsinki. Informed written consent was obtained from the subjects parents. All schoolchildren in grades I to III in a school in Eastern Singapore and a school in Northern Singapore were recruited for this study. Subjects with established eye pathology or amblyopia detected before the commencement of the study or with known allergy to eyedrops were excluded. The participation rate was 62%. We were able to obtain sufficient data for our analyses from 1004 subjects (505 females and 499 males) aged from 7 to 13 years (mean, ). Data from the right eyes of these subjects are reported in this study. Data Collection The measurements reported in this study were taken under cycloplegia, which was accomplished with three drops of topical 1% cyclopentolate in each eye, each drop instilled at 5-min intervals, after instillation of 0.5% proparacaine. Autorefraction and keratometry measurements were performed 30 min after the last drop instillation. Measurements were taken objectively using one of two Canon RK-5 autorefractor/autokeratometers (Canon, Tochigiken, Japan). For each subject, five refractive error measurements were taken, each converted from their sphero-cylinder notation to vector notation using the following equations 13 : M S C 2 (1) J 0 C cos2 (2) 2 J 45 C sin2 (3) 2 where S is sphere, C is cylinder power, and is cylinder axis. Results reported for individual subjects are the average of five sets of vectors. Corneal astigmatism was calculated based on the autokeratometry reading and assumes a corneal refractive index of We expressed corneal astigmatism as the correcting cylinder that would be combined with the corneal power to give zero astigmatism. Cylinder power was calculated as F min F max where F max and F min are the maximum and minimum corneal powers, respectively. Cyl-

3 Vector-Based Javal s Rule Tong et al. 883 inder axis was set at the corneal meridian along which power was minimum. Corneal astigmatism was then expressed as J 0 and J 45 vectors using the same equations listed above. Data Analysis We performed linear regression of refractive error astigmatism against corneal astigmatism for three different sets of variables. First, we regressed refractive error J 0 against corneal J 0. Second, we performed a similar regression using refractive error J 45 and corneal J 45. Finally, we performed the traditional Javal s rule analysis whereby refractive error cylinder power was regressed against corneal cylinder power, with ATR astigmatism being assigned negative values and WTR astigmatism being assigned positive values and subjects with oblique astigmatism in cornea or refractive error being excluded from the analysis. Our definitions of WTR and ATR astigmatism were, respectively, negative cylinder axes of inclusive and inclusive, with the remaining axes being classified as oblique, thereby matching criteria laid down by other researchers. 4, 5, 8 Occasionally, autokeratometry can provide cylinder power measurements that are both extremely inaccurate and extremely high, so we followed the example of previous researchers 5 and excluded very high corneal cylinder powers from our analyses. Our analyses were therefore performed using corneal cylinder powers of 4 D (i.e., 3 SD from the mean). This resulted in 11 cases being removed from the analyses, leaving a total of 993 subjects (494 male and 499 female) aged 7 to 13 years (mean, ). The cases that were excluded from the analysis had cylinder powers ranging from 4.13 to D (mean, D). RESULTS Our results showed strong linear relationships between refractive error astigmatism vectors and corneal astigmatism vectors, relationships that are very similar to Javal s rule. This is illustrated in Fig. 1, a plot of refractive error J 0 against corneal J 0, and in Fig. 2, a similar plot for J 45 vectors. In Fig. 1, the regression equation for refractive error J 0 (RJ 0 ) against corneal J 0 (CJ 0 ) was: FIGURE 1. Refractive error J 0 plotted against corneal J 0. The regression line shown represents Equation 4. ATR, against-the-rule astigmatism; WTR, with-therule astigmatism. determining coefficients for Javal s rule. In Fig. 3, refractive error cylinder power (TA) is plotted against corneal cylinder power (CA), WTR astigmatism is given a positive sign, and ATR astigmatism is given a negative sign. Oblique axes (more than 15 from the horizontal or vertical) were excluded in this plot. Under this criterion, 327 subjects had oblique astigmatism for refractive error, 110 subjects had oblique astigmatism for corneal cylinder, and 367 subjects had oblique astigmatism under either criterion (70 of RJ CJ D (4) The correlation coefficient was 0.835, which indicated a significant linear relationship (p 0.001). For the gradient in this relationship, 95% confidence limits were and For the y intercept, 95% confidence limits were and D. In Fig. 2, the regression equation for refractive error J 45 (RJ 45 ) against corneal J 45 (CJ 45 ) was RJ CJ D (5) The correlation coefficient was 0.691, which was significantly different from zero (p 0.001). For the gradient in this relationship, 95% confidence limits were and For the y intercept, 95% confidence limits were and D. When combined, Equations 4 and 5 can be used to predict refractive error astigmatism based on corneal astigmatism. We have termed this combination the vector-based Javal s rule. Fig. 3 shows our data analyzed using the traditional approach to FIGURE 2. Refractive error J 45 plotted against corneal J 45. The regression line shown represents Equation 5. DO, dextro-oblique astigmatism; LO, levo-oblique astigmatism.

4 884 Vector-Based Javal s Rule Tong et al. these were oblique for both the corneal and refractive error axes) and were excluded, leaving 626 subjects for this analysis. The regression equation for Fig. 3 is similar but not identical to that for Fig. 1 and is TA 0.985CA D (6) This relationship has a significant correlation coefficient of (p 0.001), with 95% confidence limits for the gradient of and For the y intercept, 95% confidence limits were and D. Thus, the gradient in Equation 4 is slightly flatter than in Equation 6. It should also be noted that the scales differ between Figs. 1 and 3. Fig. 1 is plotted in terms of Jackson cross-cylinders (as is Fig. 2), which have half of the dioptric value of the conventional cylindrical power used for plotting Fig. 3. When expressed in normal cylinder power (i.e., doubled), the confidence limits for the y intercept in Fig. 1 becomes D with 95% confidence limits of and D, overlapping the mean and confidence limits in Fig. 3. DISCUSSION Our vector-based Javal s rule represents an improvement on previous approaches to Javal s rule because in formulating it: (1) we did not exclude subjects whose axes did not lie close to Cartesian axes and (2) we included information about oblique axes and Cartesian axes in its formulation. Likewise, the vector-based Javal s rule can be used to make predictions about refractive astigmatism regardless of orientation of the subject s corneal cylinder axis, predictions that will provide information about oblique and Cartesian components of refractive error astigmatism. The slopes and intercepts in Equations 4 and 6 are not significantly different (after matching scales); thus, for our subjects, our FIGURE 3. Refractive error cylinder magnitude plotted against corneal cylinder magnitude. Negative cylinder axes outside and were excluded from the analysis. With-the-rule astigmatism (WTR) astigmatism was designated positive, and against-the-rule (ATR) astigmatism was designated negative. The regression line shown represents Equation 7. vector-based Javal s rule gives similar results to the traditional approach to Javal s rule. In addition, the slopes and intercepts are also similar to those found by previous studies. We note that McKendrick and Brennan 16 performed a very similar correlation analysis for corneal and total astigmatism using their vectors i 1 and i 2 (scale equivalents of J 0 and J 45, respectively). Their correlation coefficients (r) were slightly lower than ours: 0.69 for the Cartesian meridians and 0.59 for the oblique meridians (right eye only), compared with our values of and for corresponding meridians. This difference can be accounted for by the larger population variance of refractive error astigmatism in our study. Correlation can be expressed in terms of the coefficient of determination, r 2, which is equal to the proportion of variance of one variable that can be explained in a linear model by variance of the second variable. Sample variance in our refractive error J 0 data was D 2, and for McKendrick and Brennan, this was D 2. Thus, based on r 2 values, D 2 of our refractive error J 0 variance can be predicted from variance of corneal J 0, compared with D 2 for McKendrick and Brennan. However, the residual (uncorrelated with cornea) variance in refractive error J 0 of 0.04 D 2 for our study was almost identical with D 2 for McKendrick and Brennan. This residual variance can be thought of as a measure of the variance of the data above and below the regression line, i.e., noise in the refractive error data J 0, that isn t explained by corneal J 0. Similar calculations, using r 2 and the refractive error J 45 sample variance of D 2 for our study and D 2 for McKendrick and Brennan, yield residual variance for refractive error J 45 of D 2 for both studies. Thus, correlation leaves similar amounts of unexplained refractive error astigmatism variance for both our study and that of McKendrick and Brennan. The difference in correlation coefficients for the two studies can therefore be explained in terms of the larger overall variance in refractive error astigmatism in our study. This difference may be due to the different ethnic groups and ages of subjects in each study, our subjects being Asian children and McKendrick and Brennan using young Australian adults from a broad range of socioeconomic backgrounds. 16 To our knowledge, we are the first group to describe the regression relationship between the oblique components of corneal and refractive error astigmatism. The small slope and intercept of Equation 5 imply that the oblique components of refractive astigmatism are usually less in magnitude than the oblique components of corneal astigmatism. This is true, with the average difference between magnitudes of corneal J 45 and refractive error J 45 being D (t , p 0.001) This result seems paradoxical, given that in our traditional Javal s rule analysis we excluded only 110 subjects with oblique corneal cylinders, but 367 subjects with oblique refractive error cylinders. (This trend can be confirmed by reviewing axis orientation; 664 subjects had refractive error axes that were more oblique than corneal cylinder axes, and 315 subjects showed the reverse trend.) So why does a shift to smaller refractive error J 45 lead to a greater tendency to oblique refractive error axes? This interesting effect arises because most cases of corneal astigmatism have negative cylinder axes that lie close to, but not exactly on, 180 and are greater than approximately 0.25 D in magnitude. For such cylinders, obliquely crossing them with the approximately 0.5 D ATR astigmatism implied by the intercepts in Equations 4 and 6 results in a more oblique final axis. Thus, these more oblique refractive error axes are a consequence of an againstthe-rule (i.e., nonoblique) shift and might have been predicted

5 Vector-Based Javal s Rule Tong et al. 885 based on previous versions of Javal s rule. To our knowledge, though, we are the first to observe that one of the consequences of Javal s rule is a tendency for refractive error axes to be more oblique than corneal astigmatism axes. If corneal astigmatism was the sole cause of refractive error astigmatism, the gradients in Equations 4, 5, and 6 would be equal to 1, and the intercepts would be equal to zero. Previous researchers have suggested a number of reasons why this does not occur in practice, including misalignment of the corneal or crystalline lens optic axis with the line of sight, crystalline lens astigmatism, vertex distance effects, and inappropriate refractive index being used for keratometry conversions. 2, 6 Because of the particular ethnic composition of this study population, which consisted primarily of Chinese children (75%) with a smaller proportion of Malay (20%) and Indian (5%) representation, our results may not necessarily be applicable to other populations. It should be noted, though, that Javal s rule seems to apply reasonably well across different ethnic groups, 2, 4 6, 8 although ethnic variability in the prevalence of astigmatism has been reported 16 20, with average levels of astigmatism in the Native American population being higher than other populations in the United States All these epidemiological studies have not, however, compared the vector components of the astigmatism between ethnic groups. Because SCORM is a longitudinal study, we intend to present the change in the astigmatism in these children using similar vector methods. Clinical Utility of Vector-Based Javal s Rule We have developed a vector-based Javal s rule, which allows information about corneal Cartesian and oblique astigmatism to FIGURE 4. Accuracy of predictions based on different versions of Javal s rule. Relative frequency histograms show residual refractive error cylinder power (difference between predictions and obtained values) for different versions of Javal s rule. For reference, refractive error cylinder magnitude is plotted in the top frame. be used to predict refractive error astigmatism. To assess the clinical utility of this vector-based Javal s rule, for each subject we calculated how much residual cylindrical power (C res ) would remain uncorrected if the vector-based Javal s rule was used to calculate astigmatism (i.e., if a cylinder was prescribed using our vector-based Javal s rule, C res would be the cylinder power to be added at the appropriate axis to give the actual refractive error cylinder). Predicted RJ 0 and RJ 45 (PRJ 0 and PRJ 45 ) were found by modifying Equations 4 and 5: PRJ CRJ D (7) PRJ CRJ D (8) The predicted refractive error minus cylinder power (PC) and negative cylinder axis (PA) are given by PC 2 PRJ 0 2 PRJ 45 2 ) (9) PA 1 2 tan 1 PRJ 45 PRJ 0 (10) (using a four-quadrant tan 1 function in Equation 10). C res was calculated as follows: C res 2 PRJ 0 RJ 0 ) 2 PRJ 45 RJ 45 ) 2 ) (11) If the vector-based Javal s rule was used to estimate refractive error cylinder, the results would differ from the measured refractive error by cylinders of 0.25 D in 36% of cases, 0.50 D in 77% of cases, and 0.75 D in 92% of cases (mean, D; median, 0.32 D). We compared these results with predictions based on Javal s original rule 1 and the simplified Javal s rule. 2 Our approach to these rules differs from previous researchers in that we decided to apply them to cases of oblique corneal astigmatism as well as the traditional Cartesian axes. This version of the simplified Javal s rule might be expressed as the following: TA can be calculated by crossing a cylinder of 0.50 D 90 with corneal astigmatism. This is easily accomplished with the following equations: PRJ 0 CRJ D and PRJ 45 CRJ 45, with C res calculated using Equation 11. The original Javal s rule might be expressed as the following: TA can be calculated by multiplying corneal cylinder by 1.25 (axis unchanged) and crossing this with a cylinder of 0.50 D 90. This can be accomplished numerically as PRJ CRJ D and PRJ CRJ 45, again with C res calculated using Equation 11. Residual cylinders for the simplified Javal s rule are slightly and significantly worse (t , p 0.001) than for the vector-based Javal s rule, with only 28% being smaller than 0.25 D, 70% being smaller than 0.50 D, and 89% being smaller than 0.75 D (mean, D, median 0.37 D). The original Javal s rule gives significantly worse residual cylinder powers than the vector-based Javal s rule (t , p 0.001) and the simplified Javal s rule (t , p 0.001), with 15% being smaller than 0.25 D, 44% being smaller than 0.50 D, and 73% being smaller than 0.75 D (mean, D; median, 0.54 D). For visual comparison of the different rules, relative frequency histograms of C res values are plotted in Fig. 4 along with the residual cylinders if no attempt was made to correct astigmatism.

6 886 Vector-Based Javal s Rule Tong et al. Thus, it appears that the vector-based Javal s rule and the simplified Javal s rule have some clinical usefulness because for a large percentage of cases, they can provide a close match for refractive error astigmatism. Although autorefractors are becoming increasingly common in ophthalmic practices, they are far from ubiquitous. Thus, astigmatism estimates based on keratometry can prove useful starting points for subjective refraction. When applying the vector based Javal s rule to left eyes, the y intercept should be set to 0.01 in Equation 8 to take into account midline asymmetry, or the small intercept could be ignored completely for left and right eyes. We recognize that using Equations 4, 5, 7, 8, 9, and 10 to estimate refractive error astigmatism might prove cumbersome in a clinical setting. However, the same operations can be accomplished in seconds using graphical techniques described in the Appendix. CONCLUSION We have developed a vector approach to Javal s rule, in which Cartesian astigmatism is represented by J 0 vectors and oblique astigmatism is represented by J 45 vectors. This approach is closely analogous to the traditional approach to Javal s rule, but the ability to make predictions about oblique astigmatism makes the vector-based rule more general and more accurate. In our sample of 993 Singaporean schoolchildren, we found that our vector-based rule could be used to predict cylinder powers to within 0.32 D in 50% of cases and to with 0.75 D in 92% of cases and thus may provide clinically useful information. APPENDIX Graphical Approaches to Vector-Based Javal s Rule and Simplified Javal s Rule Implementing the vector-based Javal s rule involves using Equations 4, 5, 7, 8, 9, and 10. Even in the case of the simplified Javal s rule Grosvenor et al. 2 felt that the rule should be confined to Cartesian meridians, in part to avoid the mathematics of obliquely FIGURE 5. Graphical method for applying the vector-based Javal s rule. See description in the Appendix (along with a method of applying the simplified Javal s rule to oblique meridians). FIGURE 6. Examples of graphical method for Javal s rule. Example A (hatched circles) shows calculations for the vector-based Javal s rule. Example B (x s) shows calculations for the simplified Javal s rule.

7 crossed cylinders. Although modern calculators, computers, and spreadsheets make such mathematics feasible in a clinical setting, graphical techniques can also provide rapid solutions to these problems. Fig. 5 and the examples in Fig. 6 show the framework for such graphical techniques. They can be implemented as follows. Vector-Based Javal s Rule The corneal cylinder and negative cylinder axis is plotted on the double polar plot in the upper part of Fig. 5, and the corneal J 0 and J 45 vectors are then read off the grid plot (interpolation on plotting and reading vectors will improve accuracy). These J 0 and J 45 vectors are then replotted on the double polar plot in the lower part of Fig. 5. (The scale changes and shifting of these vector coordinates have been calculated from Equations 7 and 8, although we have ignored the very small intercept from Equation 8.) The estimate of refractive error cylinder power and negative cylinder axis can be read directly from the double polar plot (again with interpolation). This is shown in example A, plotted in Fig. 6 using the hatched circles. From K readings of D at 15 and D at 105, the corneal cylinder is (specified in terms of the cylinder required to correct corneal astigmatism). This cylinder power and negative cylinder axis is plotted on the upper double polar diagram, and the J 0 and J 45 vectors are read off as approximately 0.37 and 0.62 D, respectively. These J 0 and J 45 vectors are then plotted on the lower diagram, and the estimate of refractive error astigmatism can be read off the double polar plot, giving approximately 0.75 D 20. This is close to the numerically calculated value of Simplified Javal s Rule Again, the corneal cylinder is plotted on the double polar plot in the upper part of Fig. 5. This point is then replotted one grid line to the left on the same diagram (this corresponds to 0.5 D cylinder ATR shift, which is equivalent to a 0.25 D change in the J 0 component of the astigmatism), and the refractive error astigmatism is read from the same double polar diagram. Again, interpolation at each of these stages will improve the accuracy of the final result. Example B, which is plotted as the x s in Fig. 6, shows how to use a simplified Javal s rule with a double polar diagram. For K readings of D at 160 and D at 70, corneal cylinder is This cylinder power and axis is plotted on the upper double polar diagram (rightmost x). This point is then replotted one grid line to the left (it is probably not necessary to read any J 0 or J 45 vectors off the graph, but rather just measure or estimate the distance of one grid line). The estimate of refractive error astigmatism can be read off the double polar plot, giving approximately 0.75 D 145. This is close to the numerically calculated value of The simplified Javal s rule may be quicker to implement graphically than the vector-based Javal s rule but, as shown in the main body of the text, may be slightly less accurate ACKNOWLEDGMENT This study was supported by Singapore Eye Research Grant SERI/MG/97-04/0005. Received May 29, 2001; revision received August 22, REFERENCES Vector-Based Javal s Rule Tong et al Javal E. Memoires d Ophthalmometrie. G. Masson, ed. Paris: Librarie de l Academie de Medecine, 1890: Grosvenor T, Quintero S, Perrigin DM. Predicting refractive astigmatism: a suggested simplification of Javal s rule. Am J Optom Physiol Opt 1988;65: Heard T, Reber N, Levi D, Allen D. The refractive status of Zuni Indian children. Am J Optom Physiol Opt 1976;53: Dobson V, Miller JM, Harvey EM, Tyszko RM. The relation between corneal and refractive astigmatism in a preschool population with a high prevalence of astigmatism. 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