Traditionally, calculating the mean of refractive data usually requires a change in format, such as converting from standard notation into a matrix formalism and then converting back to standard notation. This paper provides a method that does not require data transformation. That is, the data are entered and retained in conventional form. This also allows a real-time analysis, or visualization of a moving average as each additional data point is entered.

When analyzing changes in astigmatism after anterior segment surgery, the change in overall refractive power must be considered. That is, the change in astigmatism must be assessed in terms of the change in overall refractive power. We have previously shown, for example, that there is a relationship between the degree of myopia and astigmatism^{1} —the greater the myopia, the greater the astigmatism.

Assessing the results of refractive surgery depends on characterizing and comparing preoperative and postoperative data and providing indices that reflect the success of surgery. Calculation of mean refractive power,^{2,3} refractive surgical effect (RSE), and preoperative (K1 ), postoperative (K3 ), and surgically induced astigmatism (SIA) (K2 ) can be achieved using vectorial analysis or Long's matrix formalism.^{4} Harris^{5–10} has developed a method for the statistical analysis of dioptric power, which we have developed to compare paired preoperative and postoperative data.

Indices of outcome as a percentage of success can also be applied to refractive data. For example, a global index of correction (GIC) can be used to measure the overall success of refractive surgery. As such an index does not distinguish between a change in the spherical and astigmatic components, an index for the spherical component (spherical index of correction [SIC])^{11} and astigmatic component or surgical accuracy (SA)^{11} is provided.

Conventional keratometry provides measurements that describe the curvature of the anterior corneal surface and tear film. Extrapolation to refractive power is then made by assuming a standard refractive index for the cornea. In essence, keratometry reflects, very approximately, the refractive power of the corneal surface, while refraction indicates what is needed to bring the power of the eye back to zero or emmetropia. Although keratometry as such provides incomplete data on corneal curvature, it does provide additional information to assess treatment results. It is appropriate, therefore, to characterize and provide a method for analyzing and comparing keratometry data to refractive data.

A composite method for the analysis of astigmatism after refractive surgery is presented. The method incorporates our previous work^{1,11} and that of Long,^{4} Keating,^{2,3} and Harris.^{5–10} The method of each analysis is presented in turn and applied to the results in a group of patients who had laser in situ keratomileusis (LASIK). Each component method of analysis is illustrated by a result, a summary of which is given in a group-representative table.

Patient population
The database, provided by Douglas D. Koch, MD, contained refractive and keratometric data of 100 eyes of 46 women and 54 men (mean age 45 years) before and 3 months after LASIK performed by 1 surgeon.

Characterization of preoperative and postoperative refractive data
Astigmatism and Myopia
As shown previously,^{1} there was a significant linear correlation preoperatively between the degree of refractive astigmatism and the spherical component of myopia (r = 0.22, P = .03). The amount of refractive astigmatism showed a linear increase with increasing myopia.

Surgical astigmatism
In calculating K1 , K2 , and K3 and the overall refractive power using vectorial analysis, the nomenclature of Jaffe and Clayman,^{12} Kaye and Patterson,^{1} and Kaye and coauthors^{11} was adhered to, in which K1 ^{1,11} is the preoperative astigmatism, K3 ^{1,11} the postoperative astigmatism, and K2 ^{1,11} the induced astigmatism. K2 and the axis of K2 can be easily calculated using the method of Kaye and Patterson^{1} and Kaye and coauthors^{11} (Figure 1 ). That is, K2 can be calculated algebraically without the need for double-angle diagrams or the ambiguity of extracting square roots. As shown previously,

Figure 1.:
(Kaye) Representation of preoperative astigmatism (K1 at angle α), postoperative astigmatism (K3 at angle θ), astigmatism due to surgical effect (K2 , surgically induced astigmatism at angle β) (reproduced with permission, Br J Ophthalmol 1992; 76:739).

where K2 is at angle β, K3 at angle θ, and K1 at angle α, and β = ½ arctan[(K1 sin 2α − K3 sin 2θ)/(K1 cos 2α − K3 cos 2θ)].

If K2 is negative, conversion to a positive cylinder can be made by adding or subtracting 90 degrees to or from β, if β is greater than or less than 0 degrees. The proof^{11} for this can be seen by taking the second differential (d ) of K2 with respect to β, that is

If K2 < 0, then d ^{2} K2 /d β^{2} gt; 0 and K2 is a minimum for that value of β. Adding or subtracting 90 degrees from β will make K2 gt; 0 and a maximum.

Calculating the Mean Preoperative, Postoperative, and Surgically Induced Astigmatism
As K1 , K2 , and K3 are vectors, they are vectorially summed before calculating the respective means. The vectorial sum can be achieved by solving K3 for each addition. That is,

K3 is thus the resultant vector (K _{R} ). The next subject then becomes K2 in equation 4 and is added to K _{R} , which becomes K1 in equation 4 , giving a new resultant vector K _{R+1} . Thus, if each new subject that is added is called K_{new} and K _{R+1} is the resultant vector of this addition, then for each new vector (K _{New} ) that is added,

This iteration is repeated n times, where n is the number of patients, giving a final resultant vector, K _{R+n} or K _{Final} . The average vector is then K _{Final} /n . The advantage of this approach is that the data can be kept in conventional coordinates (S –C axis) and as each new entry is added, the resulting moving average can be conceptually visualized or displayed on a diagram.

Result.
The means for K1 , K2 , and K3 were K1 +0.96 × 87.91, K2 +0.87 × 178.51, and K3 +0.095 × 82.41 (Table 1 ). These results can also be obtained using Long's matrix formalism^{4} below and are highlighted in Figure 2 , which is a scatterplot of the preoperative, surgical, and postoperative astigmatism in the 100 eyes.

Table 1: Mean^{∗} refractive data. Note the similarity between the axes of refraction and the normalized or regular keratometric astigmatism for the preoperative, postoperative, and surgically induced astigmatism. A meridional difference varying from 90 degrees reflects the departure from regular keratometric astigmatism.

Figure 2.:
(Kaye) Scattergraph of the results in 100 eyes before and after LASIK (k-pre = preoperative regular keratometric astigmatism; k-post = postoperative regular keratometric astigmatism; kSE = keratometric surgical effect; K1 = preoperative refractive astigmatism; K2 = surgically induced refractive astigmatism; K3 = postoperative refractive astigmatism; GIC = global index of correction; SIC = spherical index of correction; SA = surgical accuracy). The mean for the 100 eyes (large open circle) and patient 85 (large diamond) are highlighted.

Refractive Power and RSE
The mean preoperative and postoperative refraction and RSE can also be calculated using the means calculated by the vectorial analysis for K1 , K2 , and K3 above. That is, if the nearest equivalent sphere^{3,6} (NES) preoperatively is −6.68 D, where NES = Sphere (S ) + ½ cylinder (C ), then, the mean preoperative refraction = (mean NES − ½ mean K1 ), mean K1 at axis; that is, mean refraction = (−6.68 D − 0.48 D) + 0.96 D × 87.91 = −7.16 +0.96 × 87.91. Likewise, the mean postoperative refraction is −0.66 +0.095 × 82.41.

The mean RSE can similarly be calculated from the preoperative refraction + RSE = postoperative refraction; that is,

That is, RSE = (−0.61 D − −6.68 D) − 0.435,+ 0.87 × 178.51 = +5.64 +0.87 × 178.51. The same analysis can be applied to the keratometric data (below).

Development of Long's Matrix Formalism and Keating's Method for RSE
The mean refractive indices, S , C , and axis (a ), preoperatively and postoperatively, can also be calculated according to the method of Keating,^{2,3} as well as by the vectorial analysis already given. The dioptric power of each subject, that is S /C × axis, is first converted into Long's dioptric power matrix^{4}

where

The means of f _{11} , f _{12} , f _{21} , and f _{22} are calculated, and then S , C , and a are solved. Keating^{2} provided an easy method for solving S , C , and a . That is, if the trace (t ) of the dioptric power matrix is (f _{11} + f _{22} ) and the determinant (d ) is (f _{11} f _{22} − f _{12} f _{21} ), then C = (t ^{2} − 4d )^{1/2} , S = (t − C )/2, and tan a = (S –f _{11} )/f _{12} .

Result.
The mean preoperative and postoperative refraction was

and

which is the same result as that obtained using a vectorial analysis.

Refractive Surgical Effect
The RSE can also be calculated using a matrix analysis. Thus, from equation 8 , if the preoperative refraction + RSE = postoperative refraction, then

Result.
This method of analysis gives an RSE of +5.64/+0.87 × 178.51, which is the same result as that obtained using the vectorial analysis.

Global Index of Correction
It is useful to have a single index that reflects the overall accuracy of the refractive change. Although it is possible to calculate the difference between the preoperative refraction and the postoperative refraction as shown above, it becomes more problematical to express this difference as a percentage change or index. While the ratio of preoperative – postoperative/preoperative can be expressed as a matrix; that is,

this does not provide an easily interpretable index. An alternative method that is meaningful and easy to interpret is to write the refractive powers in the form

where S is the spherical refractive power and C is the refractive cylinder, at meridian (a ).

Adapting Jaffe and Clayman's^{12} vectors of flattening and steepening, consider the preoperative refraction: S _{0} @ a _{0} /(S _{0} + C _{0} ) @ a _{0} + 90, and postoperative refraction S _{1} @ a _{1} /(S _{1} + C _{1} )@a _{1} + 90. If in the first instance there is no change between the preoperative and postoperative meridians, a _{0} = a _{1} , then the effectiveness in achieving emmetropia is simply

(If ametropia is desired, then one divides by S _{0} and S _{0} + C _{0} minus or plus the amount of ametropia required, analogous to Alpin's target induced astigmatism.^{13} )

Thus, for example, if the preoperative refraction is −6.00 × 180/−3.00 × 90 and the postoperative refraction is −2.00 × 180/−1.00 × 90, the index of correction is ½ (−6–−2)/−6 + (−3 − −1)/−3) = ⅔, or 0.67 for emmetropia or 0.90 if the aim for the postoperative was −1.00 D spherical.

If as is usual, however, the preoperative and postoperative meridians or axes differ, that is a _{0} differs from a _{1} , it is necessary to transform the refractive meridians to reflect this. That is, it is necessary to calculate both S _{0} and S _{0} + C _{0} in the postoperative meridian, that is S _{0} ′ and (S _{0} + C _{0} )′, and to calculate both S _{1} and S _{1} + C _{1} in the preoperative meridian, that is S _{1} ′ and (S _{1} + C _{1} )′. With reference to Figure 3 ,

Figure 3.:
(Kaye) Preoperative (left) and postoperative (right) optical crosses, and preoperative and postoperative refractions at meridians a_{0} and a_{1} .

A GIC can then be defined as

Clearly, if a_{0} = a_{1} , the GIC reduces to ½ {(S _{0} − S _{1} )/S _{0} + [(S _{0} + C _{0} ) − (S _{1} − C _{1} )/(S _{0} + C _{0} )]}; that is, equation 15 .

Result.
The GIC is illustrated using the mean preoperative and postoperative refractions. The mean preoperative refraction was −7.16 +0.96 × 87.91, and the mean postoperative refraction was −0.66, +0.095 × 82.4. From equations 16 through 19 , S _{0} ′ = −7.15 D, (S _{0} + C _{0} )′ = −6.23, S _{1} ′ = −0.66, and (S _{1} + C _{1} ) = −0.57. Then, GIC = −7.16 − (−0.66)/−7.16 + −6.21 − (−0.57)/−6.21 + −7.15 − (−0.66)/−7.15 + −6.22 − (−0.57)/−6.22 = 0.9, or 91% of correction toward emmetropia. The actual mean GIC for the group of 100 patients was 0.90 (0.13), which reflects an overall 90% success rate of refractive correction toward emmetropia.

Although the GIC provides an overall index of the results of surgery, it is necessary to determine the effectiveness in the reduction in the spherical component and in the astigmatic component of the refraction. These components can be determined as previously shown by the SIC^{11} and the SA.^{11}

Spherical Index of Correction
The SIC^{11} is as previously defined using the following principle: preoperative sphere − postoperative sphere/preoperative sphere. This can be calculated using S , provided no transposition is made.

Result . Using the mean refractive powers, the SIC for the mean of the 100 patients was SIC = −7.16 − −0.66/−7.16 = 0.90, or 90% of spherical correction. As the NES^{3} is invariant,^{5,6} it is perhaps more appropriate to calculate the SIC using the NES mean preoperative and postoperative results. Thus, the mean NES SIC for the above results is (−6.69) − (−0.61)/−6.69 = 0.91, or a 91% correction of the nearest equivalent sphere.

Because the GIC provides an overall effectivity index and the SIC indicates the effective spherical change, it is necessary to have an index that reflects the effectiveness of treating astigmatism—both magnitude and direction. This is given by the SA index.

Surgical Accuracy
The ideal surgical meridian of K2 is at 90 degrees to K1 (K1 and K2 both deemed positive or negative). If K2′ represents the net effect of K2 in the desired meridian, that is, K2 a′ − K2 b′ (Figure 4 ), then SA can be defined as K2′ /K2 + K3 . The value of this definition of SA^{1,11} is that it provides a single index that reflects the accuracy of both the magnitude and direction of the SIA. The SA ranges from −0.5 to +1.0, where a negative value indicates surgical misalignment by more than 45 degrees with a net increase in the astigmatism in the preoperative axis.

Figure 4.:
(Kaye) The net effect of K2 in the ideal axis; that is K2′. K2b′ is the effect of K2 in the direction of K1 (preoperative astigmatism) and K2a′, the effect of K2 in the ideal surgical axis, which is at 90 degrees to K1 . K2′ is, therefore, the net effect of K2 in the ideal axis; that is, K2a′ − K2b′.

Clearly, if K2 is placed nearer K1 than the desired meridian, K2′ and therefore SA are less than 0. If however, K2 = K1 and is placed in the desired meridian, then K2 = K2′ , K3 = 0, and SA = 1; that is, a perfect result if sphericity was intended.

An overcorrection of astigmatism occurs when the postoperative axis is more than 45 degrees from the preoperative axis and an undercorrection, when the postoperative axis is within 45 degrees of the preoperative axis and K3 < K1 . Potential overcorrection occurs when K2 is greater than K1 , and actual overcorrection when K2′ gt; K1 , resulting in K3 being more than 45 degrees from K1 . The importance of the distinction between potential and actual overcorrection is seen in the situation in which K2 gt; K1 but K2 is misaligned; K2′ will be less than K2 and may be less than K1 , so an overcorrection may not occur. The degree of K2 misalignment is seen in the ratio K2′ /K2 which lies in the interval +1 gt; = K2′ /K2 < = −1.

Result.
Although the alignment of K2 in treating the preoperative astigmatism was good as seen in the ratio K2′ /K2 (0.96), the magnitude of the surgical vector (K2 ) was less than K1 (0.87 × 1.78 versus 0.96 × 87.91). This is reflected in the mean SA of 0.75 (0.27) for the refractive data and 0.49 (0.30) for the keratometric data. The refractive SA data for right and left eyes were similar as were the data for women (0.74) and men (0.75).

Statistical Comparison of Preoperative and Postoperative Refractive Data
Statistical analysis was based on an addition to the method of Harris^{5–10} and Long's dioptric power matrix formalism as defined in equations 8 through 11 using the multivariate test statistic^{7}

where w is the multivariate test statistic equivalent to the univariate statistic t ^{2} . A significant difference between samples using the F distribution at α = 5% occurs if w gt; F _{0.05,3,} _{n} _{−3} at 3 and n − 3 degrees of freedom.^{7}

where

Then ā _{1} and ā _{2} are the respective means^{7} of the samples of dioptric powers; that is,

matrix

S ^{−1} the inverse of the variance matrix S , given by^{7}

A development of this method was applied to compare the paired data presented as follows. Equivalence of variance was assessed by comparing the ratio of the variances to the identity matrix [I], that is, Sa _{1} /Sa _{2} , which is equivalent to Sa _{1} Sa _{2} ^{−1} .

As the data were paired, equivalence of variance covariance could be assumed so that the pooled variance (S ) was used; that is,

and as n _{1} = n _{2} , this simplifies to

Result.
The means ā _{1} and ā _{2} of the preoperative and postoperative data, given above. The pooled covariance was

There was a significant difference between the mean preoperative and postoperative refractions (P < .001).

Keratometry
Normalizing the Keratometry
Regular astigmatism has the steepest and flattest meridians at 90 degrees to each other. If during keratometry, the presumed meridians are found not to be orthogonal, normalization of the keratometry data is required to compare keratometric dioptric power to refractive power. That is, because regular astigmatism implies that the steepest and flattest meridians are orthogonal, it is necessary to calculate the meridional power at 90 degrees to either the presumed flattest or steepest meridian. This assumption is reasonable, as there was a good correlation (r = 0.81, P < .001) between the keratometry and refractive axes. (If neither k1 nor k2 is a maximum or minimum, it is not possible to determine the steepest and flattest meridians.) Whether the presumed steepest or flattest meridional power is chosen, the difference between them (ie, the keratometric astigmatism) will be the same. With reference to Figure 5 , k2 is at meridian (a _{2} ) and k1 is at some other meridian a _{1} , where

Figure 5.:
(Kaye) Normalization of keratometric data. If k1 and k2 are not orthogonal, the power of k1 orthogonal to k2 can be calculated so that the difference between k1 and k2 (keratometric astigmatism) is regular and a maximum. Whether k2 is presumed a maximum and k1 normalized, or whether k1 is presumed a minimum and k2 normalized, the resultant or regular keratometric astigmatism is the same.

For regular astigmatism, there are 2 possibilities: k2 is a maximum and k1 is not a minimum, or k2 is not a maximum, but k1 is a minimum. When k2 is a maximum, to determine k1min,

If k1 is a minimum,

Whether k2 is a maximum or k1 a minimum, the difference between k2 − k1min = k2max − k1, which is the maximum regular or normalized keratometric astigmatism.

Result.
The average keratometry before and after normalization, both preoperatively and postoperatively, is shown in Table 1 . The mean keratometry meridional difference between k1 and k2 was 84.95 degrees and 87.47 degrees preoperatively and postoperatively. Normalization of k1 resulted in keratometric astigmatism 1.16 × 177.41 preoperatively and a normalized value of 0.64 × 86.72 postoperatively with a mean keratometric surgical effect of 0.66 × 179.69 (Figure 5 ). The effect of this is apparent when one considers an individual, such as patient 51 (Table 2 ). Here, the preoperative keratometry meridians are 65 degrees apart, with a difference between k2 and k1 of 1.25 D. Normalizing the keratometry (the meridians of which lie 65 degrees apart) results in a normalized k1 of 41.98 D as opposed to a nonorthogonal or irregular k1 of 42.25 D. The calculated regular keratometric astigmatism is 1.52 D, which is closer to the refractive astigmatism preoperatively (1.50 D) than the keratometric astigmatism (1.25 D).

Table 2: Refractive data of patient 51, a 23-year-old man. Note the closeness in the magnitude of the refractive astigmatism and the normalized keratometric astigmatism.

Correlation Between Keratometric Component of Astigmatism and Spherical Component
As with the refractive data, there was a trend toward increasing keratometric astigmatism (k2 − k_{n} 1) and mean keratometry (k2 + kn_{n} 1)/2 using normalized keratometric data (r = 0.17, P = .08).

The GIC, SIC, K2, and SA for Keratometry
The GIC, SIC, K2, and SA for keratometry are calculated using the same method as that for the refractive data. The results are shown in Tables 1 and 2 .

Correlation between refractive and keratometric data
Correlation Between Mean Keratometry and Nearest Equivalent Sphere
There was no significant linear correlation between the mean keratometry and NES preoperatively (r = 0.17, P = .09). There was, however, a highly significant correlation between the changes in NES and the change in the mean normalized keratometry (r = 0.94, P = 5.9 × 10^{−47} ) (Table 3 ).

Table 3: Linear regression analysis.

Correlation Between Regular Keratometric Astigmatism and Refractive Astigmatism
The refractive astigmatism was calculated in the plane of the cornea using the given back-vertex distance of 12.5 mm. The back-vertex power was calculated using Keating's matrix calculations of effectivity.^{2} Essentially, S and S + C are calculated in the corneal plane and the difference between them is the refractive astigmatism at the corneal plane.

Result . There was a significant linear correlation between the normalized keratometric astigmatism and the refractive astigmatism preoperatively (r = 0.50, P = 1.4 × 10^{−7} ). There was no significant correlation postoperatively (r = 0.13, P = .19), largely because there was little postoperative astigmatism.

Correlation Between Change in Refractive and Keratometric Astigmatism
There was a significant correlation between K2 and the change in keratometric astigmatism preoperatively and postoperatively (r = 0.50, P = 1.3 × 10^{−7} ).

Discussion
The data used in this study pertain to the results of LASIK performed by 1 surgeon and cannot necessarily be generalized. The analysis of refractive data after anterior segment surgery provides the surgeon with information regarding the effectiveness of the surgical technique in inducing a change in the refractive status of the eye. Combined with keratometry, it also reflects how the change in refraction has been achieved. For example, the high correlation between the change in keratometry and the change in myopia indicates that at least 80% of the effect of LASIK in reducing myopia in this study is through its effect on the central anterior corneal surface, rather than some other unexpected ocular effect. It is thus essential to have a reliable and representative method for analyzing and correlating changes in refraction and keratometry. The analysis of astigmatism is a continually evolving process, and there are numerous published approaches, each with its own merits. We developed a comprehensive method for the analysis of astigmatism following surgery that incorporates our previous work and that of other research in thefield.^{2–10,12}

Mean dioptric power, surgical effect, and preoperative, postoperative, and surgically induced astigmatism can all be reliably calculated and represented, either as a vectorial system or a matrix formalism. Both methods are essentially the same (as are many other reports), differing only in their representation of dioptric power. Representing refractive powers as vectors in an optical cross is conceptually and visually meaningful to the surgeon and relates directly to the recording of retinoscopy. Surgeons are able to visualize the refractive effects of their surgery. Thus, the iterative method provided for the calculation of the mean refraction provides the surgeon with a continuous-moving average—a real-time analysis—without the need to change the format of the data from conventional notation.

The advantage of Long's matrix formalism^{4} is that by providing a similarity transformation, dioptric powers can be represented in the x –y frame. It is essential to be able to analyze refractive powers statistically. Based on Long's matrix formalism, Harris^{5–10} has provided a significant advance in this area. We have used this analysis with an additional modification for the comparison of paired data. This has shown, for example, that the preoperative data are significantly different from the postoperative data.

Indices of success are equally important as outcome measures. The global index of correction (GIC) provides an index that characterizes the overall success of refractive surgery in achieving emmetropia or another desired goal. For the above data, the GIC of 0.9 indicates a success rate of 90% for LASIK. A further breakdown into the spherical index of correction (SIC) of 0.9 for the spherical component and a mean surgical accuracy (SA) of 0.75 for the astigmatic component of refraction identifies that the overall success of the surgery is largely the result improvement in the spherical correction. The index used for SA is particularly useful for assessing effectiveness in the treatment of astigmatism. As a single index, it provides information both for the magnitude and the direction of the surgical effect. The mean SA of 0.75 (scale −0.5 to +1.0), indicates, for example, that both the orientation and alignment of the laser and the magnitude of its effect were largely correct. The correlation between SA and preoperative refraction implies that LASIK is more accurate in treating higher than lower amounts of astigmatism within the given range.

Although conventional keratometry has been largely superseded by computerized corneal topography, which lends itself to the analysis of surfaces (eg, by a Fourier series), keratometry provides useful information. Keratometry principally measures curvature along the presumed steepest and flattest meridians, which for regular astigmatism are orthogonal. If the recorded measurements are not on orthogonal meridians, normalization of the data is needed before a comparison with refractive powers can be made. This allows inferences to be made between refractive and keratometric data. For example, the absence of a correlation between mean keratometry and nearest equivalent sphere is in keeping with axial length being the main determinant of myopia. Conversely, the strong correlation between keratometric astigmatism and refractive astigmatism, both in terms of power and meridian, indicates that within the parameters of this study, the corneal surface is the main determinant of astigmatism.

The comprehensive method presented for the analysis of astigmatism after anterior segment surgery allows a meaningful and representative interpretation of both the refractive and keratometric data. In particular, the iterative method used for calculating the mean astigmatism allows the data to be retained in conventional form, with easy visualization of the moving average as each new entry is added.