The mathematical representation and statistical manipulation of spherocylindrical refractive errors is a recurring topic in ophthalmic literature.^{1–12} Much of this debate has revolved around the difficulties that arise when astigmatism is represented in the traditional polar form of magnitude and axis rather than the more mathematically tractable Cartesian form. In general, statistical analysis of angular data (eg, astigmatism axes, compass bearings, angle of disappearance of migratory birds over the horizon) is fundamentally different from the analysis of nondirectional data.^{13} Consequently, the inappropriate application of conventional statistical methods to directional data can give very misleading results. If, however, astigmatism is represented in rectangular vector form, conventional scalar methods can be applied to each vector component. Furthermore, standard multivariate statistics can be used to compute population means and variances, define confidence intervals, and test hypotheses.

Power vectors are a geometrical representation of spherocylindrical refractive errors in 3 fundamental dioptric components.^{12} The first component is a spherical lens with power M equal to the spherical equivalent of the given refractive error. If this spherical power is removed from the prescription, the result is a Jackson crossed cylinder (JCC) equivalent to a conventional cylinder of positive power J at axis α + 90° crossed with a cylinder of negative power −J at axis α. By convention, we describe this astigmatic component as a JCC of power J at axis α (the meridian of maximum positive power). This JCC can be further resolved into the sum of 2 other JCC lenses, one with power J _{0} at axis α = 0° = 180° and the other with power J _{45} at axis α = 45°. With this decomposition method, we are able to express any spherocylindrical refractive error by the 3 dioptric powers (M , J _{0} , J _{45} ). It is convenient to interpret these 3 numbers geometrically as the (x , y , z ) coordinates of a point in a 3-dimensional dioptric space. Thus, a power vector is that vector drawn from the coordinate origin of this space to the point (M , J _{0} , J _{45} ). The length of this vector is a measure of the overall blurring strength B of a spherocylindrical lens or refractive error.^{14,15}

The primary advantage of representing refractive errors by power vectors is that each of the 3 fundamental components of a power vector is mathematically independent of the others. In other words, a spherical lens cannot be produced by any combination of JCC lenses, a JCC lens with axis 0 degree cannot be produced by any combination of spherical lenses with JCCs at axis 45 degrees, and a JCC lens with axis 45 degrees cannot be produced by any combination of spherical lenses with JCCs at axis 0 degree. This notion of independence, which is formalized in the mathematical concept of orthogonality, simplifies practical problems involving the combination, comparison, and statistical analysis of spherocylindrical lenses or refractive errors. We demonstrate these advantages in this report by analyzing refractive data for 100 eyes treated with refractive surgery to correct myopia and astigmatism.

Materials and methods
Refractive data were supplied by Douglas Koch, MD, as manifest refractions and corneal keratometry readings before and after laser in situ keratomileusis. Manifest refractions in conventional script notation (S , C × α) were converted to power vector coordinates and overall blurring strength by the following formulas:

These formulas apply whether the script is written in positive-cylinder form or negative-cylinder form, provided C is a signed number (ie, C gt; 0 for +cyl form, C < 0 for −cyl form).^{12}

Keratometry data were recorded as the power and meridian of maximum power and the power and meridian of minimum power. These primary meridia were not always perpendicular, which suggested the presence of small amounts of measurement error and/or irregular astigmatism. We dealt with this uncertainty by averaging the 2 given meridia and assuming the actual primary meridia were oriented ±45 degrees from this average meridian.

The change in refraction caused by treatment (refractive surgery in the present case) is computed by the usual rules of vector subtraction. Since the power vectors P are given in rectangular coordinates, the difference vector is easily computed by subtracting corresponding values along each of the coordinate axes separately.

Results
We sought answers to the following questions: Was the refractive surgery successful in correcting the patient's refractive error? Was the change in manifest refraction produced by the surgery fully accounted for by the change in corneal refraction? To answer the first question, we examined the statistical distribution of manifest refractive errors before and after treatment. This was done by expressing each presurgical and postsurgical refractive error as a power vector and computing the mean and standard deviation of each vector component. We also computed the length of each power vector, and a statistical summary of the results is shown in Table 1 . To get an overall sense of the effect of surgery on manifest refractive error, we collected the values of blur strength B into the frequency histograms of Figure 1 . The results clearly demonstrate that refractive surgery compressed a wide range of presurgical refractive errors into a narrow distribution near emmetropia. This visual impression is substantiated quantitatively by the numerical data in Table 1 . For comparison, we collected histograms of the spherical equivalent M of each refractive error as shown in Figure 2 . A comparison of Figures 1 and 2 shows that the postsurgical distribution of blur strength is narrower than the distribution of spherical equivalent. This difference is due to the fact that blur strength does not distinguish between myopic and hyperopic blur.

Table 1: Statistical summary of the distribution of manifest refractive errors before and after refractive surgery, referenced to spectacle plane.

Figure 1.:
(Thibos) Reduction in overall blurring strength B of manifest refractive error as a result of refractive surgery. Values referenced to the spectacle plane.

Figure 2.:
(Thibos) Reduction in spherical equivalent M of the manifest refractive error as a result of refractive surgery. Values referenced to the spectacle plane.

To visualize the change in astigmatism caused by refractive surgery, Figure 3 shows the astigmatic component of the power vector as represented by the 2-dimensional vector (J _{0} , J _{45} ), which is the projection of the power vector into the astigmatism plane formed by the coordinate axes (J _{0} , J _{45} ). For clarity, we do not show the entire vector extending from the origin but instead show only the endpoint of the vector. Since the origin in this graph represents an eye free of astigmatism, we expected to see the cluster of points collapse around the origin following surgery, and this was the result obtained.

Figure 3.:
(Thibos) Manifest astigmatism before surgery (A) is reduced on average following surgery (B). Each data point represents the astigmatism component of a power vector for 1 eye, referenced to the spectacle plane.

A formal statistical test of the null hypothesis that the population mean of postsurgical power vectors is equal to zero can be constructed 2 ways. First, we can conduct a series of 3 monovariate t tests of the hypotheses that the mean power vector component in each of the 3 dimensions is zero. The results of these tests, conducted at the 0.01 level, indicated that neither astigmatism component is significantly different from zero, but the spherical component does differ significantly from zero. The second approach is to use the Hotelling T ^{2} test of multivariate statistics to determine whether the mean power vector is significantly different from a vector of zero length.^{16} This test generates an F statistic that can be compared with tabulated values of the Fisher F distribution. The computed F statistic for the postsurgical, 3-dimensional power vectors was 18.47. This value is highly significant compared with the tabulated F value for 3 and 97 degrees of freedom of 3.99 at the α = 0.01 significance level, which indicates that the overall refractive error was not fully corrected in this population of eyes. However, when the analysis was repeated on the 2-dimensional astigmatism component of refractive error, the computed F statistic was 2.35, which is not significant at the 0.01 level. This result leads us to conclude that the astigmatism component of refractive error was well corrected by surgery in this population of eyes.

To answer the second question, we transferred manifest refractive errors from the spectacle plane to the corneal plane (assuming a vertex distance of 14 mm) so they could be directly compared with keratometry data.^{17} These refractive errors were then converted to power vectors to facilitate calculation of the change in manifest refractive error and the change in corneal refractive error produced by the surgical treatment. Thus, for each eye, we have a pair of power vectors representing the change in manifest refraction and the change in corneal refraction, as illustrated schematically in Figure 4 . If the change in manifest refraction was due solely to the change in corneal power, the 2 power vectors should be parallel and of equal length, so their difference should be zero.

Figure 4.:
(Thibos) Power vector test of the hypothesis that the change in manifest astigmatism was due to the change in corneal astigmatism produced by surgery. For this hypothesis to be true, the 2 astigmatism vectors with large arrowheads shown in the upper diagram should be parallel and the same length. Filled and open symbols represent astigmatism before and after surgery, respectively. The difference between the 2 change vectors is shown by the vector with a small arrowhead. The distribution of this latter quantity is displayed for a sample of 100 eyes in the scattergraph below.

To test this prediction, we computed the difference between pairs of change vectors for each eye, and the results are shown in Figure 4 . As expected, the population of difference vectors is tightly clustered about the origin. In this sample of eyes, 40% of all points are within 0.25 diopter (D) of the origin and 80% of all points are within 0.50 D of the origin. Such small discrepancies are within normal clinical tolerances for measurement of refractive error, so we conclude that the change in manifest refraction following surgery can be accounted for by the change in corneal refraction. This conclusion was substantiated by the results of a statistical T ^{2} test, which indicated that the mean of the data in Figure 4 is not significantly different from zero at the α = 0.05 level.

The statistical summary of the change in astigmatism produced by refractive surgery reported in Table 2 indicates that the manifest change in spherical equivalent M was 1.3 D greater than the corneal change in M on average. In other words, the manifest degree of myopia was reduced 1.3 D more than could be accounted for by the change in corneal curvature on average. This discrepancy may indicate an inappropriate choice of refractive index used by the keratometer for converting corneal curvatures to refractive power.^{17}

Table 2: Statistical summary of the change in astigmatism produced by refractive surgery, referenced to corneal plane. Manifest refractive errors were transferred from spectacle plane to corneal plane prior to this statistical analysis.

Discussion
One conceptual advantage of power vectors is that they represent a complicated entity (a spherocylindrical lens or refractive error) as a simple point in a 3-dimensional space. If the optical characteristics of an eye change over time due to surgery, contact lens wear, other forms of treatment, injury, or disease, the trajectory of this point graphically depicts the resulting changes in refractive error. In the present study, we followed the trajectory of 100 such points to gain insight into the optical outcome of refractive surgery. We found that the change in manifest refractive error was well correlated with the change in corneal refractive error. Although the difference between these 2 changes was small, it was greater than could be accounted for by statistical variability. Perhaps the postsurgical corneas were more aberrated than the presurgical corneas,^{18} which would have rendered keratometry less reliable. To pursue this suggestion would require a more comprehensive optical assessment before and after surgery of the higher-order aberrations of the eye, such as that obtained with a Shack–Hartmann aberrometer.^{19}

Power vectors were conceived as a way of transforming a conventional spherocylindrical refractive error into independent, orthogonal components better suited to mathematical and statistical analysis. In a broader sense, all the classical optical aberrations (eg, coma, spherical aberration), as well as the higher-order aberrations that are loosely grouped together by the term “irregular astigmatism,” can be similarly transformed into orthogonal components that are mutually independent. The traditional basic functions used for such a decomposition are the Zernike polynomials.^{20} This series of analytical functions includes 3 polynomials of the second degree that describe (1) the spherical wavefront aberrations produced by defocus, (2) the toric wavefront aberrations produced by with-the-rule or against-the-rule astigmatism, and (3) the toric wavefront aberrations produced by astigmatism with axes at 45 degrees or 135 degrees. The weighting coefficients applied to each of these basic forms of wavefront aberration required to describe any particular eye correspond to the 3 components of a power vector. Thus, another advantage of the power vector approach is that it creates a natural link from current clinical practice to an optical theory powerful enough to provide a comprehensive description of all the refractive imperfections of an eye. This link will become increasingly important as clinicians aim to go beyond the correction of defocus and astigmatism to achieve total vision correction and a perfect retinal image.^{21}

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