Refractive data are usually written as sphere, cylinder, and axis. This conventional format may characterize a single refraction but is not suited to statistical analysis. There are no problems with analysis of the spherical component; the difficulties reside with the astigmatism. An astigmatism is characterized by a magnitude expressed in diopters and a direction reported in degrees. For statistical analysis, these incommensurable entities must be converted to polar values or similar entities.^{1} The polar values^{1â€“5} were specifically developed for analysis of the astigmatic component of refractive surgery.

In this paper, we present clinical data of 100 eyes that had laser in situ keratomileusis (LASIK). We will describe the polar value methods and their use for both single and aggregate data. The principles for their univariate^{2â€“5} and bivariate^{1} statistical analyses will be reported. Subsequently, we will use these methods to describe the surgically induced astigmatism (SIA) following LASIK. Finally, we will briefly outline the perspectives for using the same methods in mathematical and statistical analyses of spherocylinders and irregular optical components.

The clinical material
The clinical data were supplied by Douglas D. Koch, MD. The material consisted of 100 eyes that had LASIK. There were 54 male and 46 female eyes. The mean patient age was 44.7 years Â± 8.3 (SD) (range 23 to 62 years). Manifest refraction and simulated keratometry based on videokeratography were performed before and 3 months after surgery. In all cases, the intended postoperative refraction was plano, ie, zero for both sphere and cylinder. The database was edited and transformed to a SPSS file; all subsequent calculations were performed in this statistical program.

We will demonstrate the data format by showing the preoperative and postoperative measurements from patient 6, a 51-year-old man who had LASIK in September 1998. These data will be used in the subsequent examples.

Preoperative keratometry:K1: 42.25 D; axis1: 180Â°

K2: 43.0 D; axis2: 90Â°

Preoperative manifest refraction: (âˆ’9.5 sph. () + 1.0

cyl axis 83Â°)

Postoperative keratometry:K1: 36.5 D; axis1: 180Â°

K2: 37.0 D; axis2: 90Â°

Postoperative manifest refraction:âˆ’0.5 sph. () + 0.5 cyl axis 172Â°

Looking through the file, it is apparent that K2 and axis2 represent the magnitudes and directions, respectively, of the steeper or more powerful keratometer readings. It is also clear that K1 and K2 are not orthogonal in all cases. The polar value methods^{1â€“4} assume orthogonality of axes, ie, regular astigmatism. Therefore, in the present clinical material, the direction of the net astigmatism is taken as the value of axis2, and the flatter meridian is assumed to be orthogonal to this value.

Irregular astigmatism is, however, an important concept. Using suitable mathematical methods,^{6} it is possible to segregate corneal surfaces of any irregularity into a number of components, among these a regular astigmatic entity, which can be further analyzed with polar values. These mathematical principles will be addressed in the Discussion.

Polar value analysis of individual data
Net Astigmatism
The net astigmatism is the basal variable in polar value analysis. A net astigmatism^{2â€“4} K = (M @ Î±) consists of an astigmatic magnitude M expressed in diopters (D) and an astigmatic direction Î± expressed in degrees. For keratometry or simulated keratometry, M is the absolute difference between the higher and the lower keratometer reading, while Î± is the meridian of the higher keratometer reading.

Example 1.
From the 51-year-old patient, we obtain the following preoperative keratometer readings:

K1 = first keratometer reading = 42.25 D in the meridian 180Â°

K2 = second keratometer reading = 43.0 D in the meridian 90Â°

M = 43.0 âˆ’ 42.25 = 0.75 D

Î± = direction of A2 = 90Â°

Preoperative net astigmatism = K_{preop} = (0.75 D @ 90)

Similarly, the postoperative net cylinder K_{postop} = (0.5 D @ 90)

A net cylinder may also be derived from refractive data. Refractive data in conventional format are given by

where S = spherical value of the spherocylinder expressed in diopters, M = magnitude of cylinder expressed in diopters, Î¦ = axis of cylinder expressed in degrees. From these data we derive the spherically equivalent refraction,

We are dealing with a plus-cylinder axis format. The axis Î¦ of a refractive cylinder is, by definition, the direction of zero cylinder power. Conversely, the meridian Î± of a refractive cylinder is the direction of the maximal cylinder power M . The correlation between Î¦ and Î± is given by

To maintain consistency with the original definition of net astigmatism,^{2,3} based on keratometer readings and demonstrated in example 1, we will define a spherocylinder on the basis of the cylinder meridian, ie,

where the net cylinder is (M @ (Î¦ + 90)Â°).

Example 2.
The SER and the net cylinder of the preoperative manifest refraction of our patient is

SER_{preop} : âˆ’9.5 + Â½ Â· 1.0 = âˆ’9.0 D

K_{preop} = (1.0 D @ (83 + 90)Â°) = (1.0 D @ 173Â°)

The postoperative values amount to

SER_{postop} : âˆ’0.25 D

K_{postop} : (0.5 D @ 82)Â°

Transformation of Net Cylinders to Polar Values
The general expression^{3} for the polar value method is

where Î© and (Î© + 90) symbolize the 2 orthogonal planes under investigation. By varying Î©, several different optical qualities can be scrutinized, but for the present analysis of refractive surgery, Î© will consistently indicate the direction of the preoperatively steeper meridian. We hereby derive an expression for the astigmatic polar value (similar to equation 7 of reference ^{3} ):

The plane inclined 45 degrees against the clock relative to Î© amounts to (Î© + 45). The polar value of this oblique meridian^{4} calculates as

Any net cylinder is uniquely characterized^{1,4} by these 2 polar values, separated by an arch of 45 degrees.

Polar Values of Preoperative and Postoperative Net Cylinders
For the preoperative net cylinder (A @ a ) and the postoperative net cylinder (B @ b ), the preoperative astigmatic polar values^{3,4} are calculated by insertion in equations 6 and 7 :

The preoperative astigmatic polar values are very easy to calculate, as AKP is always identical to the nominal magnitude of the preoperative cylinder, while the oblique component always amounts to zero.

To calculate the postoperative astigmatic polar values, the preoperative steeper meridian is consistently used as a reference.^{3} Inserting in equations 6 and 7 ,

Example 3.
A. Keratometer readings: From example 1, we obtained the following net astigmatisms:

Conversion to astigmatic polar values:

B. Manifest refraction: From example 2 we obtained the following net astigmatisms:

Conversion to astigmatic polar values:

Surgically Induced Change in Keratometry and Refraction
Surgically induced astigmatism expressed as polar values is the difference^{2,3} between the postoperative and preoperative polar values.

A positive Î”AKP indicates a steepening in the surgical meridian, a with-the-incision or WTI change.^{3} A negative Î”AKP signifies a flattening of the surgical meridian, an against-the-incision or ATI change. It is expressed as

A positive and negative Î”AKP (+45) indicates an induced anticlockwise and clockwise torque,^{4} respectively.

An isolated analysis of astigmatic magnitude may be of some interest, as a small astigmatism is always preferable in refractive surgery.^{5} Surgically induced change in astigmatic magnitude:

Surgically induced change in SER^{4} :

From this value and equation (2) , we can calculate the surgically induced change in sphere, Î”S .

Example 4.
From example 3, we obtain the values for SIA.

A. Keratometer readings:

B. Manifest refraction:

We obtain the following values for the surgically induced change in spherical equivalent refraction:

Surgery induced a flattening of 0.25 D based on keratometer readings, while the flattening amounted to 1.5 D for refractive data. There was no induced rotation for keratometer readings, and the refractive data disclosed a minimal induced anticlockwise torque of 0.02 D. There was therefore a discrepancy in induced flattening between the results derived from keratometry and refraction. There was a satisfactory 8.75 D reduction in myopia.

Dioptric Error of the Surgical Procedure
For each treatment and for each patient, we have an intended keratometry and refraction. The dioptric error of the treatment is simply the difference between the actual postoperative and the intended keratometry/refraction.

A positive value for for AKP _{error} indicates an undercorrection, a negative value an overcorrection. A positive value for AKP (+45)_{error} results from an overtly anticlockwise torque, while a too strong clockwise torque is revealed by a negative error.

Any intended postoperative result may be selected. In the present clinical series, the intended keratometry/refraction was plano, ie, zero for each refractive component. The dioptric error is therefore identical to the postoperative refractive result and emerges as

Example 5.
The dioptric error for our patient, illustrated in example 3, emerges as:

A. Keratometer readings:

B. Manifest refraction:

From the keratometry, we must conclude that surgery undercorrected the astigmatism by 0.5 D. Conversely, the refractive data indicate an overcorrection by 0.5 D. No significant error in induced rotation was observed by any of these modalities. The error in spherical equivalent refraction was minimal.

Reconversion to Conventional Notation
It is possible to reconvert any pair of AKP and AKP (+45) to a conventional net cylinder notation using the following equations^{1,4} :

Any root will do, but to remain consistent with previous definitions of polar values as positive entities, we will take the plus root and calculate the astigmatic direction as

The direction Î± is the inclination in relation to the preoperative steeper meridian, which in the polar plot is given the direction 90 degrees. As previously mentioned,^{1} there are always 2 solutions, separated by an arch of 180 degrees, for net astigmatism in the interval from zero to 360 degrees. This principle is illustrated in Figures 1 and 3 .

Figure 1.:
(Naeser) Keratometry; SIA; individual values and 95% tolerance area. In the upper figure, each point represents a single induced astigmatism uniquely characterized by a pair of polar values in the form of Î”AKP and Î”AKP (+45). Any point (= combination of polar values) inside the ellipse is within the 95% distribution of the observations. In the lower figure, the same phenomenon is displayed as net astigmatisms in a polar plot. For each pair of polar values, there are 2 solutions of net astigmatisms: 1 in the interval from zero to 180 degrees and 1 achieved by a 180 degree rotation around the origio and depicted in the interval from 180 to 360 degrees. The dashed line connecting the single point of polar values to the 2 equivalent net astigmatisms below illustrates this phenomenon. Net astigmatisms are symbols, and no calculations can be performed with these entities. The confidence perimeter is constructed by point-for-point retransformation from the ellipse in the upper figure.

Figure 3.:
(Naeser) Refractive data; SIA; individual values and 95% tolerance area. This figure is equivalent to

Figure 1 .

Example 6.
Surgically induced astigmatism expressed in conventional notation is derived from the data in example 4.

A. Keratometer readings:

Surgically induced astigmatism expressed as a net cylinder emerges as (0.25 D @ 0Â°) or (0.25 D @ 180Â°).

B. Manifest refraction:

The surgically induced change in spherocylinder,

From equation 2 , it follows that the surgically induced change in sphere Î”S = 8.75 âˆ’ Â½ Â· 1.5 = 8.0 D. Resultant spherocylinder: (+8.0 DS () 1.5 DC @ 179.6Â°) = (+8.0 DS () 1.5 DC axis 89.6Â°).

Statistical polar value analysis of aggregate data
In statistical analysis, it is necessary to have expressions for both average and spread^{5} of the variables. For polar values, this analysis is performed in a univariate and bivariate manner.

Univariate Analysis
This analysis is performed in the usual manner with calculation of univariate means and paired t tests.

Bivariate Analysis
Bivariate analysis of polar values are executed with the multivariate t test, Hotelling T ^{2} .^{7} The confidence areas are displayed as ellipses in 2-dimensional space. These methods have been meticulously described in a recent paper.^{1} Here, we will show only the final results.

Results
Keratometer Reading
Univariate analysis.
The mean results are summarized in Table 1 . The paired t test naturally compares the preoperative and the postoperative data. Surgery induced a statistically significant flattening of the surgical meridian but no significant average torque. The astigmatic magnitude was also significantly reduced. Using equations (24) and (25) on the polar values, the SIA may be expressed as the net cylinder: (1.27 DC @ 179.3).

Table 1: Univariate analysis of keratometer readings.

The error of the procedure (= the postoperative keratometry) was examined with a paired t test. The undercorrection averaged 0.29 D and differed significantly from zero (P = .001). The was no significant error in the mean torque component (P = .46). The relative small average values and the large standard deviations naturally signify a considerable spread with a large number of positive and negative changes.

Bivariate analysis.
The combined mean polar values for SIA differed statistically significantly from zero (Hotelling T ^{2} = 2.04; P < .001). The dioptric error differed significantly from zero (Hotelling T ^{2} = 0.137; P = .002).

Individual pairs of polar values and the 95% tolerance ellipse for SIA are demonstrated in Figure 1 . The combined means and their 95% confidence ellipses for the preoperative, postoperative, and surgically induced values are shown in Figure 2 . The postoperative values represent the error of the procedure. All pairs of mean polar values differed significantly from zero at the 5% level.

Figure 2.:
(Naeser) Keratometry. Mean values and 95% confidence areas of the combined means, expressed as polar values. This is equivalent to the use of standard errors of the mean in univariate analysis. The means and confidence areas are shown as follows: preoperative values = line segment to the right; postoperative values (= the dioptric error of the procedure) = ellipse in the middle; SIA = ellipse to the left. The confidence area for the preoperative keratometries emerges as a line segment because the standard deviation for AKP (+45)_{preop} is zero. The bivariate statistical significance can be read directly from the figure. Confidence regions containing the point (0,0) signify mean values not differing significantly from zero at the specified confidence level. Common areas between two confidence regions reveal a lack of significant difference between the 2 involved combined means. All mean values differed significantly from zero and from one another.

Manifest Refraction
Univariate analysis.
The results are summarized in Table 2 . Surgery flattened or reduced the correction needed in the preoperatively steeper meridian a statistically significant 1.28 D but did not cause significant rotation.

Table 2: Univariate analysis of manifest refraction.

There was a significant undercorrection (paired t test; P < .001) amounting to 0.18 D. There was no significant error in torque (P = .47).

The spherically equivalent refraction was significantly reduced by 6.08 Â± 2.41 D from the preoperative level of âˆ’6.69 Â± 2.49 D to the postoperative value of âˆ’0.61 Â± 0.82 D. This error differed significantly from zero (P < .001). The average surgically induced change in refraction therefore amounted to

Using equations 2, 24, and 25 , this change can be reported as the average spherocylinder (5.44 DS () 1.28 DC @ 179.6) or (5.44 DS () 1.28 DC axis 89.6)

Bivariate analysis.
The combined means for SIA differed statistically significantly from zero (Hotelling T ^{2} = 3.3; P < .001). The error likewise differed significantly from zero (Hotelling T ^{2} = 0.191; P < .001).

Figures 3 and 4 are similar to Figures 1 and 2 , respectively.

Figure 4.:
(Naeser) Refractive data. This figure is equivalent to

Figure 2 . The confidence areas are smaller than the figures derived from keratometry because all standard deviations were smaller for the refractive data. The means and confidence areas are shown as follows: Preoperative values = line segment to the right; postoperative values equivalent to the dioptric error = ellipse in the middle; SIA = ellipse to the left. All mean values differed significantly from zero and from one another.

Evaluating the Refractive Result of Lasik
We observed an average flattening of the preoperative steeper meridian of 1.27 to 0.29 D with keratometry, while the flattening of the corrective refractive cylinder amounted to 1.28 D and reached a postoperative value of 0.18 D. These errors of approximately a quarter of a diopter were statistically, but not clinically, significant. The spherical equivalent refraction amounted to approximately half a diopter of myopia after 3 months, ie, a small, but clinically rather convenient, undercorrection. The clinical refractive results are therefore highly satisfactory.

We observed similar results for SIA in both keratometric and refractive data. A more intense investigation or comparison of these results would require further information, such as the distance from the corneal center to the measured area, the used corneal refractive index, assumptions over the possible posterior corneal changes, and pupil diameter. These issues are exciting but somewhat beyond the scope of this study.

Discussion
In the analysis of refractive data, it is imperative to use data suitable for mathematical and statistical analysis. When a specific entity does not possess this quality, it must be transformed to a suitable variable. A net astigmatism expressed as direction and magnitude may characterize a single cylinder but cannot be subjected to statistical analysis. For that purpose, each net astigmatism must be transformed to suitable power vectors.^{8,9} One such power vector is represented by the polar value system, using the general equation 5 . By varying Î©, several situations can be examined, but in practice, the following 3 are the most commonly encountered:

For Î© equal to 90 and 135, we obtain the polar values KP(90)^{2} and KP(135),^{3â€“5} used for analysis of populations of refractive data, spectacle refractions, and superior incisions^{1} consistently placed in the 90 degree meridian. We obtain the following equations:

For analysis of an incision consistently placed in the preoperative steeper meridian or indeed any procedure, such as LASIK, aiming at reducing a steep meridian, we use the astigmatic polar values^{3} AKP and AKP (+45), shown in equations 6 and 7 .

Another popular incision for cataract surgery is the temporal approach at zero or 180 degrees, requiring the following polar values:

For each of these possibilities, the meridional polar valuesâ€”KP (90), AKP, KP(0) â€”express the flattening or steepening of the surgical meridian, while the polar values of the oblique meridian^{4} â€”KP (135), AKP (+45), KP (45)â€”signify the surgically induced torque of the cylinder.

For the present study, we could have obtained additional useful information using the polar values KP (90) and KP (135) for comparison of the preoperative and postoperative population of refractions. To avoid confusion, we restricted the analysis to the astigmatic polar values AKP and AKP (+45). This system yields the most relevant information about the quality of refractive surgery, where the key issues remain precision and the question of undercorrection or overcorrection, which are given by the signs and magnitudes of AKP _{error} and AKP (+45)_{error} . For keratometry, astigmatic polar values express the change in power along the preoperatively steeper corneal meridian. For manifest refraction, astigmatic polar values express the change in corrective or refractive power along the direction of the preoperative steeper corrective cylinder meridian, ie, the reduction in spectacle dependence.

Polar values can be used for any type of refractive surgery and are very simple to use in practice. The surgeon should consider several surgical methods and establish their effect, expressed as polar values, on SIA. In planning the surgery, the surgeon should, for each eye, elect a method with a known flattening effect similar to the intended effect. The interpretation of the surgical result is similar for single and aggregate data. A positive induced meridional polar value indicates a steepening of the surgical meridian, a negative polar value a flattening. A positive oblique polar value is synonymous with an anticlockwise torque, a negative value a clockwise torque. The polar values concept is in accord with current research in optometry.^{8,9}

Each pair of polar values separated by an arch of 45 degrees uniquely characterizes an astigmatism. All types of mathematical calculations and conversions can be performed with these entities. It is possible to perform a point-for-point retransformation to conventional notation, as shown in equations 24 and 25 and Figures 1 and 3 . It is important to understand that net astigmatisms cannot be used for calculations but only as symbols^{1} for singular refractive data. In our experience, reconversion to net cylinders is not really helpful. When the surgeon becomes familiar with the use of polar values, the combined effect of flattening and torque is much more illustrative.

The 2 paired polar values may be subjected to statistical analysis separately, yielding information over the surgically induced flattening and torque. However, the 2 polar values may also be analyzed simultaneously with bivariate statistical methods. As the astigmatism is fully characterized by the 2 polar values, the bivariate analysis is superior to the univariate approach and always yields the correct statistical result.^{1}

In principle, a spherocylinder is a 3-dimensional entity. We have developed a method for multivariate analysis of spherocylinders,^{10} which may be used in future studies of refractive surgery.

Videokeratography measures corneal power along a 360 degree circumference, usually in 256 steps. At each distance from the center, these power measurements can be displayed in a polar plot of a more or less irregular configuration. Using Fourier analysis,^{6} the polar plot can be disintegrated into harmonic sinusoidal curves of increasing frequency. Each sinusoidal curve represents different optical characteristics, such as sphere, decentration, regular astigmatism, and irregular astigmatism of increasing orders. In some videokeratographers, such as those in the present study, the software does not yield the correct value for the regular astigmatism. However, Fourier analysis, already installed in some commercially available videokeratographers, always yields the correct regular astigmatic component. This regular astigmatism can be analyzed with polar values, exactly as described in this study. Furthermore, the irregular optical component can be analyzed statistically with bivariate and multivariate analyses.^{1}

In conclusion, the polar value system yields a comprehensive analysis of SIA for both single and aggregate data. It is possible to use the same principles, in combination with multivariate statistics, for analysis of spherocylinders and irregular optical entities.