The mathematics for calculating surgically induced refractive change (SIRC) was first described in 1849.^{1} One hundred twenty-six years later, Jaffe and Clayman^{2} properly applied this method to analyze the relationship between cataract surgical technique and the refractive result in individual patients. Since then, several mathematically incorrect methods have been reported.^{3–5}

In 1992, to evaluate evolving corneal surgery techniques, we described a method for calculating the surgically induced spherical and astigmatic change in an individual patient.^{6} In 1999, we extended the applicability of this method to include aggregate data.^{7} The current literature indicates that significant confusion still surrounds the question of how best to evaluate SIRCs.^{8,9} We appreciate the opportunity to demonstrate our methods for the 100-case data set provided by Douglas Koch, MD,^{9} one of the journal editors.

Materials and methods
Statistical analyses and graph preparation were performed on a personal computer using Microsoft Excel 97 and SPSS SigmaPlot 3.0, respectively.

Data Preparation and Sources of Error
Vertexing spherocylinders to the corneal plane.
Refractions are normally performed at the spectacle plane or in the phoropter and not at the corneal plane. Refractive measurements must be vertexed to the corneal plane before they can be compared to those obtained by keratometry or topography. Inaccuracies in the measurement of vertex distances and inappropriate application of the vertex formula are frequent sources of error in planning and reporting the results of refractive surgical procedures.

Refractions performed in the phoropter are at a nominal vertex distance of 13.75 mm when the corneal vertex is located at the large mark on the calibration scale. Within the phoropter, multiple lenses are aligned to produce the combination lens used in refraction. This combination lens has an effective vertex distance that frequently differs from the nominal value indicated on the vertex scale. This difference, along with the difficulty of maintaining precise patient alignment, cause the vertex distances measured with the phoropter to be unreliable, especially at higher refractive powers. A more accurate measurement is obtained by performing the refraction over a soft contact lens with a power near the spheroequivalent (SEQ) of the refractive error.

For example, the SEQ of a −11.50 +2.00 × 90 refraction at the phoropter with a nominal vertex distance of 13.75 mm is −10.50 diopters (D). When vertexed to the corneal plane, the SEQ is −9.18 D. Overefraction of the same patient performed with a −10.00 D soft contact lens gives −2.00 +2.00 × 90. The final refraction at the corneal plane (vertex = 0 mm) is then −12.00 +2.00 × 90, yielding an SEQ refraction of −11.00 D at the corneal plane. The value with the contact lens is −1.82 D more myopic than with the vertexed phoropter refraction. The contact lens method is always more accurate provided the soft contact lens is labeled properly. The correct vertex distance must be entered into all laser and surgically planning software programs to avoid inducing residual refractive errors.

Since the plus and minus cylinder notation for refraction represents a difference rather than an actual power in either meridian, they must first be converted to the cross-cylinder notation before performing vertex calculations. The spherocylindric refraction in this example is vertexed to the corneal plane as shown below:

Cross cylinder form @ spectacle: −12.00 × 180 and −10.00 × 90 Vertex: 13.75 mm Vertex formula from spectacle plane (REF_{s} ) to corneal plane (REF_{c} )^{10,11} :

Substituting the values from the example

Cross cylinder form @ cornea: −10.30 × 180 and −8.79 × 90 Vertex:0 mm Plus cylinder form:−10.30 + 1.51 × 90 Minus cylinder form:−8.79 −1.51 × 180
When correctly vertexed, the astigmatism at the corneal plane is −1.51 D, almost one half a diopter less than the value obtained by incorrectly vertexing the cylinder portion of the refraction expressed in minus-cylinder notation (−1.95 D). Vertexing a myopic refraction from the spectacle to the corneal plane will always reduce the magnitude of the astigmatism. For hyperopia, the relationship is just the opposite.

To compare SIRCs calculated from refractive and keratometric data, the refractive data must be vertexed to the corneal plane before the SIRC calculation is performed. Performing a vertex calculation on the results of an SIRC calculation gives an incorrect answer.

Special considerations with keratometric data.
Keratometric data are already at the corneal plane and do not require vertex adjustment. However, an additional issue arises with respect to the correct method of converting the anterior radius of the cornea to the net corneal refractive power. The formula used to convert radii to power is the simple spherical refracting surface formula (SSRS):

The variables n _{1} and n _{2} are the indices of refraction of the first and second media, respectively, and r is the radius of curvature of the interface. The value for n _{1} is 1.000 (index of refraction for air). The standardized keratometric index of refraction (1.3375) was chosen for n _{2} many years ago.^{12} The origin of this value for n _{2} remains obscure, appearing to have been arbitrarily selected in the 19th century so that an anterior radius of corneal curvature of 7.5 mm would yield a power of 45.00 D.^{13}

where r _{a} is the anterior radius of curvature of the cornea (m) and K _{k} is the standardized keratometric net corneal power (D).

The cornea, like any meniscus lens, has a front surface power, a back surface power, and a net or equivalent power. To compute the surgically induced cornea power change, one must know whether the surgical procedure alters the front surface, back surface, or net cornea power. The front surface power and net power changes are the only clinically relevant considerations, since there are no keratorefractive procedures that are intended to change only the back surface power.

For procedures such as photorefractive keratectomy (PRK), laser in situ keratomileusis (LASIK), and probably radial keratotomy, the change in dioptric power of the cornea is almost entirely due to front surface power changes in the cornea. To compute front surface power, the change in media for the light rays is from air (n = 1.000) to cornea (n = 1.376), so, as Holladay and Waring^{13} and Mandell^{14} have recommended, the correct formula for computing the power and any change in power would be

where r _{a} is the anterior radius of curvature of the cornea (m) and K _{a} is the front surface corneal power (o).

The front surface power of a cornea with an anterior radius of 7.5 mm would be 50.13 D (0.376/0.0075), 5.13 D greater than the standardized keratometric power of 45.00 D (0.3375/0.0075). Front surface powers are 11.14% (0.376/0.3375) larger than keratometric values. When determining the change in the refractive power of the eye for procedures that change only the front surface of the cornea, the change computed from keratometry must be increased by 11.14% to compensate for the difference in the index of refraction.

For analyzing results in which it is believed that both the front and back surfaces have been changed equally, it is appropriate to use the net or equivalent corneal index of refraction. The most common application of this conversion is in intraocular lens (IOL) power calculations. There is still debate among investigators as to the most appropriate value for the net or equivalent index of refraction. Binkhorst^{15} and Olson^{16} have empirically determined this value to be 4/3 and 1.3315, respectively. For standardization purposes,^{13} we have recommended adopting the Binkhorst value, since it is has been the one most frequently used for the past 20 years. The equation to compute net or equivalent corneal power, using indices of refraction for air (n = 1.000) and cornea (n = 4/3), would be

where r _{a} is the anterior radius of curvature of the cornea (m) and K _{n} is the net corneal power (D).

The net power of a cornea with an anterior radius of 7.5 mm would be 44.44 D (1/3 ÷ 0.0075), 0.56 D less than the standardized keratometric power of 45.00 D. Net or equivalent corneal powers are 98.76% (1/3 ÷ 0.3375) of the standardized keratometric values. When the net or equivalent power of the cornea or changes in the net corneal power are needed, the standardized keratometric values must be reduced to 98.76% of their original values to accurately reflect the net refractive power change in the cornea. For net powers of the cornea used in IOL calculations, a 1.24% overestimate of the corneal power results in a 0.56 D error, which is significant and intolerable for these calculations. However, for calculating changes in corneal power produced by refractive surgical procedures, this 1.24% error is clinically negligible. Nevertheless, when reporting net corneal power changes from standardized keratometry measurement, the values should be reduced by 1.24% to be correct.

Another source of difference between keratometric and refractive measurement arises from the relative flattening of the central cornea by myopic refractive surgery.^{17,18} Keratometers and topographers sample the paracentral cornea (nominal diameter of 3.2 mm for a 44.00 D cornea) rather than the surgically flattened optical center and thus tend to overestimate the corneal power after myopic refractive surgery. This study found an additional 10% difference between the actual central power change and the paracentral keratometric and topographic change. The refractive power of the cornea changed by a factor 1.21 times more than the apparent keratometric change when considering the progressive flattening error and the front surface index of refraction error.^{17,18}

It is important to note that the value of 1.333 for the net corneal index of refraction may change following refractive surgical procedures that alter epithelial thickness or remove corneal tissue (ie, PRK and LASIK). The refractive indices of the epithelium and stroma are different, and there may be subtle intrastromal differences, eg, between Bowman's layer and the posterior stroma.^{19} Procedures that alter epithelial thickness could change the refractive power of the epithelium, and removal of stromal tissue could alter the net refractive index of the stroma. These changes could be important in determining the correct value of net corneal power for IOL calculations, since, as noted above, a change of only 1% to 2% could produce unacceptably high errors.

Surgically induced refractive change calculations require that the astigmatism is regular, ie, the steepest and flattest meridians must be orthogonal (90 degrees apart). In the data set used here, some cases had non-orthogonal axes. In these cases, the steep axis was maintained and the flat axis was assigned a value 90 degrees away. The vector difference in the original keratometric astigmatism and the orthogonalized values is a measure of the irregular, oblique axes (not oblique axis) astigmatism. These data are shown on the doubled-angle plot in Figure 1 . Thirteen eyes had magnitudes greater than the 0.50 D magnitude that is considered clinically significant irregular keratometric astigmatism. All SIRC calculations performed here used the orthogonalized data.

Figure 1.:
(Holladay) Doubled-angle plot of the non-orthogonal, keratometric irregular astigmatism. The data points represent the difference vector in the original and orthogonalized keratometric astigmatism vector. The centroid is 0.09 D × 57.1 ± 0.23 D, ρ = 2.86. The 13 data points outside the 0.50 D radius (blue) would be considered clinically significant irregular keratometric astigmatism of oblique axes.

Precision of the magnitude and axis of astigmatism.
The question often arises as to what is the angular error in astigmatism that is comparable to a given magnitude error. Intuitively, most clinicians know that the higher the degree of astigmatism, the more accurate the axis of astigmatism must be; eg, a 10 degree error in the axis of 6.00 D of astigmatism is more significant than a 10 degree error in the axis with 1.00 D of astigmatism. The angular error that is equivalent to a dioptric error for a given astigmatism magnitude is given by the following formula:

where t is the tolerance (precision, eg, ± 0.50 D), M is the magnitude of the astigmatism, and θ is the angular error that is equivalent to the tolerance in diopters. Table 1 tabulates these values, and Figure 2 illustrates their relationship.

Table 1: Angular error (± degrees) corresponding to a magnitude error (D) at different measurement precisions.

Figure 2.:
(Holladay) Cartesian plot of the angular error (degrees) that corresponds to a given magnitude error (diopters) at different levels of precision. When the tolerance is equal to the magnitude, the angular error is 30 degrees.

Methods of Calculation
Calculating prediction error from the desired and actual postoperative refraction.
Excimer lasers, toric IOLs, and incisional surgery are capable of inducing spherical and astigmatic changes in the refraction. In most cases, the goal of refractive surgery is to neutralize the spherocylindric correction so the final postoperative refraction is plano. When this goal is achieved, the desired SIRC and the actual SIRC are equal. Likewise, when the final refraction does not match the desired postoperative refraction, the desired SIRC and the actual SIRC must be different.

The definition of prediction error (PE) is given by the following equation:

The difference between the actual postoperative refraction and the desired or predicted postoperative refraction must be calculated like any other SIRC. This is usually easy to accomplish since the desired postoperative refraction is most often spherical, so that the solution for obliquely crossed cylinders is not necessary.

A numeric example of the PE calculation may be helpful to illustrate this concept. Consider a 50-year-old patient with a refractive error of −5.00 +3.00 × 90 at the corneal plane (vertex = 0). A LASIK procedure is planned to leave the patient −0.50 D myopic. The desired SIRC at the corneal plane is therefore −4.50 +3.00 × 90, 0.5 D less than the refractive error at the corneal plane. The patient's actual postoperative refraction at 1 month vertexed to the corneal plane was −0.50 +0.50 × 80.

The value of the difference between the desired and the actual postoperative refraction is plano −0.50 × 80. This value indicates that the error from the desired target was −0.50 D of cylinder at an axis of 80 degrees. The PE and SIRC can be treated similarly when analyzing aggregate data.

Analyzing aggregate data.
The evaluation and display of aggregate data of a group of patients is performed after the refractive measurements have been vertexed to the corneal plane and standardized keratometric measurements converted to front surface or net corneal power. Descriptive statistics of SEQ values such as means, standard deviations, standard error of the means, and correlation coefficients are calculated in the normal manner.

Spheroequivalent statistics and graphs are particularly valuable for analysis of procedures in which no induced astigmatism was intended, such as spherical refractive surgery (spherical PRK or LASIK).

Graphical display of the SEQ of the SIRC is particularly useful. The SEQ of any spherocylinder is the sphere plus one half the cylinder. The formula is

A plot of SEQ SIRC data is shown in Figure 3A on an equivalency plot. When data points are on the diagonal (the equivalency line), the desired and actual refractions are equal. When the result is above the diagonal, there is an overcorrection; when below the diagonal, an undercorrection. The PE can be plotted versus the desired SEQ SIRC as shown in Figure 3D . The information displayed is similar to the equivalency plot, but the exact values for the errors are easier to see.

Figure 3A.:
(Holladay) Equivalency plot of SEQ SIRC data. When data points are on the diagonal, the desired and achieved corrections are equal. When the result is above the diagonal, there is an overcorrection; when below the diagonal, there is an undercorrection. Most cases (78%) were undercorrected by at least 0.25 D, 6% were correct within ±0.24 D, and 16% were overcorrected by at least 0.25 D. The regression line is shown in solid red (95% confidence intervals in dashed red) with R ^{2} = 0.89 and the standard error = 0.79 D.

Figure 3D.:
(Holladay) Prediction error plot. Since the target in all 100 patients was given as plano, the PE is simply the SEQ of the postoperative refraction. The percentages of undercorrection and overcorrection are the same as in the equivalency plot. It is evident in both plots that the greater the attempted correction, the larger the variability and the greater the amount of undercorrection. The Pearson correlation coefficient (r ) is 0.26, and the r ^{2} is 0.07.

The magnitude of astigmatism (cylinder power in the plus or minus cylinder form) can be analyzed in a similar manner, but the information about the axis of the astigmatism is lost. It is not possible from the magnitude of astigmatism plots alone to infer any trends such as against-the-rule (ATR) or with-the-rule (WTR) astigmatism changes from the data. The most appropriate method for evaluating, reporting, and displaying astigmatism data requires conversion of the data from the usual method of describing astigmatism in polar coordinates (cylinder and axis) to a Cartesian coordinate system.

The defocus equivalent (DEQ) correlates better with uncorrected visual acuity (UCVA) than the SEQ and is a useful measure of surgical success.^{20} The formula for the defocus equivalent is

An SEQ of zero indicates that the circle of least confusion is located on the retina but gives no information about its size. The DEQ is proportional to the diameter of the blur circle for a given pupil size. For example, a patient with a refraction of −0.50 +1.50 × 90 has an SEQ of +0.25. A patient with a refraction of −5.00 +10.50 × 90 also has an SEQ of +0.25, but no one would expect the UCVAs to be equal. In the first case, DEQ = 1.00 (0.25 + 0.75) and in the second case, DEQ = 5.5 (0.25 + 5.25), 5½ times larger. The UCVA in the second case would be 5½ times worse if the pupil sizes were the same. The relationship between uncorrected Snellen visual acuity and pupil size as a function of defocus is shown in Figure 4A and Table 2 . For example, a patient with a DEQ of 2.00 D and a 3.0 mm pupil will have a UCVA of 20/49.

Figure 4A.:
(Holladay) The relationship between Snellen visual acuity, pupil size, and refractive error in the human. The DEQ plot is the best measure of potential (no other pathology) UCVA. For each pupil diameter and refraction, a Snellen visual acuity can be determined. For example, for a 3.0 mm pupil and 3.0 D of defocus, the normal Snellen visual acuity is 20/80 (reprinted with permission from Holladay et al.^{20} ).

Table 2: Snellen visual acuity as a function of pupil size and DEQ.

Calculating, Reporting, and Displaying Aggregate Astigmatism Data
Astigmatism data are difficult to analyze primarily because of the very definition of astigmatism. The axis of astigmatism returns to the same value when it traverses an angle of 180 degrees. In geometry and trigonometry, an angle must traverse 360 degrees to return to its same value. To apply conventional geometry, trigonometry, and vector analysis to astigmatism, the angles of astigmatism must be doubled so that 0 degree and 180 degrees are equivalent. Once this transformation has been performed, all the standard formulas can be used and produce the correct singular value for the SIRC.

Because astigmatism traverses an entire cycle in 180 degrees, the most appropriate plot of aggregate astigmatism data is a doubled-angle polar plot. The angular scale of doubled-angle plot range from 0 degree to 180 degrees. The radial axes are oriented with 45 degrees at 12 o'clock, 90 degrees at 9 o'clock, 135 degrees at 6 o'clock, and 180 degrees back at 3 o'clock. The 0 degree and 180 degree locations are the same, just as 0 degree and 180 degrees are the same for the axis of refraction. Any procedure that on the average is astigmatically neutral must have the centroid of the surgically induced cylinder data at the center of the plot. On a single-angle plot (standard polar plot), none of these statements is true. An example of the doubled-angle plot is shown in Figure 6A .

Determining the mean cylinder and axis for induced astigmatism.
In our original article,^{6} we presented a method for determining the average axis or meridian of astigmatism. Since then, we have recognized that this method is incorrect^{7} because it does not appropriately incorporate the magnitude and axis of the SIRC into the calculations. For standard descriptive statistics (means, standard deviations) to be applied correctly, each data point must be converted to an x –y coordinate system. Descriptive statistics cannot be applied to polar coordinates because cylinder magnitude and axis are not independent (orthogonal) parameters. Polar values can be converted to Cartesian values using equations 6a and 6b .

In the formulas, the angle of the axis of astigmatism is doubled to give the correct x and y values.

The centroid or mean of a set of x and y values is calculated by independently finding the mean of each variable. In equation form,

For example, the centroid of preoperative cylinder at the spectacle plane is x = −0.956 and y = 0.070. Converting the Cartesian values to the standard polar notation for astigmatism,^{7,21}

and substituting values for x and y ,

gives mean value of the astigmatism as +0.959 × 87.9.

An additional important descriptive statistic is the mean absolute value.

If the postoperative target refraction is plano, the mean absolute cylinder magnitude equals the average absolute astigmatism PE, regardless of the axes. For example, if 2 PEs were equal and opposite (eg, +1.00 D × 90 and −1.00 D × 90), the algebraic or vector average (centroid) would be zero. The average magnitude, however, is 1.00 D. The mean absolute value is a linear average of the magnitude of the variable and weighs all PEs equally. The statistics for variability, variance, and standard deviation weigh the PEs by the square of the difference, making the larger PEs more important.

Variance (s ^{2} ) is defined as

Although variance is an important measure of variability, it is a squared parameter and does not have the same units as the original data. Standard deviation s is defined to be the positive square root of the variance (s ^{2} ) and has the same units as the original data. The formula for the standard deviation s is

The standard deviation in this data set is calculated by independently determining the standard deviations of the x and y variables. Since x and y are orthogonal, they have no statistical influence on each other. When the standard deviation of x and y are unequal, the standard deviation is an elliptical area surrounding the centroid as seen in Figure 6A . When the standard deviations are equal, a circle would be formed around the centroid. The formula for an ellipse is

When s _{x} is greater than s _{y} , then s _{x} is the semi-major axis of the ellipse, s _{y} is the semi-minor axis, and the ellipse is oriented horizontally. When s _{y} is greater than s _{x} , then s _{y} is the semi-major axis of the ellipse, s _{x} is the semi-minor axis, and the ellipse is oriented vertically. The semi-major and semi-minor axes of the ellipse can never be rotated with respect to the x - and y -axes. This condition is very important for determining standard error of the means and confidence intervals.

The shape of an ellipse will vary depending on the length of the major and minor axes. Several terms have been used to describe this relationship: shape factor (ρ), asphericity (Q), and eccentricity (ε^{2} ). The mathematical relationship among these parameters is

On a doubled-angle plot, the y -axis is coincident with the axis of oblique astigmatism (45 degrees, 135 degrees) and the x -axis is coincident with the axis of nonoblique (90 degrees, 0 degree WTR and ATR) astigmatism. Since most populations have WTR or ATR astigmatism, the ellipses are oriented horizontally (ρ < 1). In a population with a higher percentage of oblique astigmatism, the ellipse would be oriented vertically (ρ gt; 1). In a population with uniformly distributed oblique and nonoblique astigmatism, the standard deviation would be circular (ρ = 1). The shape and orientation of the ellipse is therefore helpful in determining the degree of oblique astigmatism in a population. The square root of the shape factor (ρ^{1/2} ) is the actual ratio of oblique to nonoblique astigmatism.

To compare the standard deviations between data sets, the area and mean radius of the standard deviations must be computed. For the 2 data sets to have equivalent variance, the areas of their ellipses must be equal. The area of an ellipse is π times the product of the semi-major and semi-minor axes.

Setting the 2 areas equal gives

Therefore, for the area of the ellipse to have the same area as a circle, the radius of the circle (s _{c} ) must be equal to the square root of the product of the semi-major and semi-minor axes:

In other words, the standard deviation of a population with unequal x and y components is the geometric mean of the standard deviation of the x and y components. Another way of visualizing this relationship is that s _{c} ^{2} represents the variance of a circular distribution (area) with a radius s _{c} corresponding to the standard deviation. The variance of the elliptical distribution (area) is s _{x} ∗ s _{y} and its square root is the standard deviation of the ellipse (geometric mean radius s _{c} ). The standard deviation of the centroid (s _{c} ) is the square root of the product of the individual standard deviations for x (s _{x} ) and y (s _{y} ). When comparing 2 data sets, the set with the lower variability (“tighter data”) is the data set with the lower variance (s _{x} ∗ s _{y} ) and standard deviation (Symbol ).

Another important descriptive statistic is the standard error of the mean, defined as the standard deviation of the sampling distribution of a statistic for random samples of size n . It is computed by taking the standard deviation and dividing by the square root of the sample size n . The formula for the standard error of the mean Symbol is

The standard error of the mean is helpful in determining confidence intervals and determining whether 2 sample populations are statistically different at a given probability (usually P ≤ .05 or 2 standard errors from the mean) or 95% confidence interval. These are the values found in a “z -table” for standard normal distributions and “t table” for a Student t distribution.

Results
The mean age of the 100 patients was 44.69 years ± 8.26 (SD), with 54% men and 58% right eyes. Table 3 summarizes the mean SEQ, mean DEQ, mean absolute magnitude of astigmatism, and mean magnitude sphere (in minus cylinder form) preoperatively, postoperatively, and the SIRC by refraction and keratometry. Table 4 summarizes preoperative, postoperative, and SIRC using vector analysis of astigmatism measured by refraction and keratometry.

Table 3: Summary of mean SEQ, DEQ equivalent, and absolute astigmatism changes.^{∗}

Table 4: Vector analysis of astigmatism results.

Figures 3A to 3C are the equivalency plot for the achieved versus attempted correction. The preoperative SEQ refractive errors were undercorrected by at least 0.25 D in most cases (78%). Six percent were corrected within ±0.24 D, and 16% were overcorrected by at least 0.25 D SEQ. Figure 3D is the PE plot. Since the target in all 100 patients is plano, the PE is equal to the SEQ of the postoperative refraction. The percentages of undercorrection and overcorrection are the same as the equivalency plot. It is evident in both plots that the greater the attempted correction, the larger the variability and the greater the amount of undercorrection. The Pearson correlation coefficient (r ) is 0.26 and the r ^{2} is 0.07.

Figure 3B.:
(Holladay) Equivalency plot of achieved and desired SIRC sphere data in minus cylinder notation. Most cases (69%) were undercorrected by at least 0.25 D, 14% were corrected within ±0.24 D, and 17% were overcorrected by at least 0.25 D. The regression line is shown in solid red (95% confidence intervals in dashed red) with R ^{2} = 0.91 and the standard error = 0.73 D.

Figure 3C.:
(Holladay) Equivalency plot of achieved and desired SIRC cylinder data in minus notation. Thirty-two percent of the cases were undercorrected by at least 0.25 D, 54% were corrected within ±0.24 D, and 14% were overcorrected by at least 0.25 D. The regression line is shown in solid red (95% confidence intervals in dashed red) with R ^{2} = 0.64 and the standard error = 0.40 D.

In Figure 4B , the preoperative DEQ is plotted on the x -axis and the postoperative DEQ on the y -axis. Ideally, all the y -values would be zero, indicating the SEQ and astigmatism equal zero. The Pearson correlation coefficient is 0.33 and r ^{2} = 0.11. As in Figures 3A to 3D , these data confirm that the higher the intended treatment, the greater the variability and the higher the average DEQ. The data indicate the higher the preoperative DEQ, the smaller the chance of the patient achieving plano and, consequently, the lower the average potential UCVA.

Figure 4B.:
(Holladay) Equivalency plot of the preoperative DEQ on the x-axis versus the postoperative DEQ on the y -axis. Ideally, all the y -values would be zero, indicating the SEQ is zero and there is no astigmatism. All 100 patients had a reduction in their DEQ. The mean DEQ postoperatively was 0.96 D. For a 3.0 mm pupil, the Snellen visual acuity would be 20/24. Preoperatively, the DEQ was 7.4 D, which is worse than 20/200.

Figure 5A is an equivalency plot of the magnitude of the preoperative keratometric astigmatism (x -axis) versus the magnitude of the preoperative refractive astigmatism at the corneal plane. There is a significant lack of correlation (r ^{2} = 0.26). This demonstrates that a large number of patients have intraocular astigmatism. The degree of corneal astigmatism is much higher than the degree of refractive astigmatism, indicating that the lenticular astigmatism generally reduces the amount of refractive astigmatism. This finding is similar to findings in previous studies that compare refractive astigmatism with keratometric and/or topographic astigmatism.^{17,18,22}

Figure 5A.:
(Holladay) Equivalency plot of the magnitude of the preoperative keratometric astigmatism (x -axis) versus the magnitude of the preoperative refractive astigmatism (y -axis) at the corneal plane. The significant lack of correlation (r ^{2} = 0.26) indicates the presence of lenticular astigmatism. The fact that the refractive is less than the keratometric cylinder shows that the intraocular astigmatism partially compensates (emmetropization) for the corneal astigmatism.

Figure 5B shows the relationship of the postoperative refractive cylinder to the keratometric cylinder. The correlation is much worse (r ^{2} = 0.02), but there is a dramatic reduction in both refractive and keratometric astigmatism. The fact that the refractive astigmatism is much lower indicates that the residual corneal astigmatism is neutralizing the lenticular astigmatism. This is also confirmed by Figure 5C , which shows the refractive SIRC and keratometric SIRC have the highest correlation. In short, in a patient with lenticular astigmatism, the refractive astigmatism can only be zero when the corneal astigmatism is exactly equal and opposite. The ratio of the refractive SIRC and keratometric SIRC was 1.34 (−5.51/−4.12), slightly higher than the previously reported value of 1.21.

Figure 5B.:
(Holladay) Equivalency plot showing the relationship of the postoperative refractive cylinder to the keratometric cylinder. The correlation is much lower (r ^{2} = 0.02), but there is a dramatic reduction of both refractive and keratometric astigmatism. The fact that the refractive astigmatism is much lower indicates that the residual corneal astigmatism is neutralizing the lenticular astigmatism.

Figure 5C.:
(Holladay) Equivalency plot of the refractive and keratometric SIRC. The refractive and keratometric SIRC have a correlation that is very similar to that of the preoperative values. In short, in a patient with lenticular astigmatism, the refractive astigmatism can only be zero when the corneal astigmatism is exactly equal and opposite to the lenticular astigmatism.

Figure 6A represents the preoperative refractive astigmatism at the spectacle plane (vertex = 12 mm). The centroid was +0.96 D × 88 ± 0.85 D, ρ = 0.43, illustrating that 90% of the patients had some degree of WTR astigmatism (left side of circle) preoperatively. This finding is expected for the relatively young average age for the refractive surgery patient (44.7 years).

Figure 6A.:
(Holladay) Doubled-angle plot of the preoperative spectacle plane refractive astigmatism at a vertex of 12 mm. The centroid was +0.96 D × 88 ± 0.85 D, ρ = 0.43. These data illustrate that 89% of the patients had some degree of WTR astigmatism (left side of circle) preoperatively. This finding is expected for the relatively young average age of the refractive surgery patient (ie, 44.7 ± 8.26 years).

Figure 6B is the doubled-angle plot of the preoperative keratometric cylinder at the corneal plane. The centroid of the keratometric astigmatism was +1.29 D @ 88 ± 0.77 D, ρ = 0.38. Figure 6C shows the centroids and standard deviations on the same plot, with the refractive data vertexed to the corneal plane. The difference in magnitude and axis confirms the presence of lenticular astigmatism.

Figure 6B.:
(Holladay) Doubled-angle plot of the preoperative keratometric cylinder at the corneal plane. The centroid of the keratometric astigmatism was +1.29 D @ 88 ± 0.77 D, ρ = 0.38. The difference in magnitude and axis from the refractive plots confirms the presence of lenticular astigmatism.

Figure 6C.:
(Holladay) The centroids and standard deviations of the preoperative corneal plane refractive and the keratometric astigmatism are shown on the same plot to emphasize the difference. This difference confirms the presence of lenticular astigmatism. A 0.48 D difference in the centroids indicates that on the average, the intraocular astigmatism is 0.48 D ATR.

Figures 7A, 7B, and 7C are the doubled-angle plots of the postoperative astigmatism from refraction, keratometry, and the composite, respectively. In Figure 7A , the centroid of the postoperative cylinder at the spectacle plane (vertex = 12 mm) is 0.11 D × 83 ± 0.37 D, ρ = 0.49, a dramatic reduction from the preoperative refractive astigmatism. The centroid of the postoperative keratometric astigmatism is +0.63 × 87 ± 0.52 D, ρ = 0.48 is shown in Figure 7B . Figure 7C illustrates the centroids and standard deviations of the postoperative keratometric and refractive astigmatism at the corneal plane, which show little correlation.

Figure 7A.:
(Holladay) Double-angle plot of the postoperative spectacle plane refractive astigmatism. The centroid of the postoperative cylinder at the spectacle plane is 0.11 D × 83 ± 0.37 D, ρ = .49, a dramatic reduction from the preoperative astigmatism.

Figure 7B.:
(Holladay) Double-angle plot of the postoperative keratometric astigmatism. The centroid of the postoperative keratometric astigmatism is +0.63 @ 87 ± 0.52 D, ρ = 0.48. Notice the greater variance in the keratometric than in the refractive astigmatism.

Figure 7C.:
(Holladay) Doubled-angle plot of the postoperative corneal plane refractive and keratometric astigmatism. The centroid and standard deviation of the refractive and keratometric postoperative astigmatism on the same plot show very little correlation.

Figure 8A shows the relationship between preoperative and postoperative refractive astigmatism at the corneal plane. These data show that the refractive astigmatism centroid is 8 times (0.81 D/0.11 D) closer to zero and the standard deviation of the astigmatism was reduced by a factor of 2.0 (0.73 D/0.36 D). The amount of improvement is demonstrated graphically on the doubled-angle plots by the relocation of the centroid closer to the origin and the contraction of the ellipse.

Figure 8A.:
(Holladay) (A) Doubled-angle plot of the preoperative and postoperative corneal plane refractive astigmatism. These data demonstrate that the refractive astigmatism centroid is 8 times (0.81 D/0.11 D) closer to zero and the standard deviation of the astigmatism was reduced by a factor of 2.0 (0.73 D/0.36 D). The amount of improvement can be seen directly on the doubled-angle plots by how much closer the postoperative centroid is to the origin and how much tighter the cluster (smaller the ellipse).

Figure 8B shows the relationship between preoperative and postoperative keratometric astigmatism. The preoperative centroid was 1.29 @ 88 ± 0.77 D, ρ = 0.38 and the postoperative centroid was 0.63 @ 87 ± 0.52 D, ρ = 0.48. Although there was a significant reduction in the mean keratometric astigmatism, it was considerably less than that seen for the refractive data due to the lenticular astigmatism.

Figure 8B.:
(Holladay) The relationship between the preoperative and postoperative keratometric astigmatism. The preoperative centroid was 1.29 @ 88 ± 0.77 D, ρ = 0.38 and the postoperative centroid was 0.63 @ 87 ± 0.52 D, ρ = 0.48. Although there was a significant reduction in the keratometric astigmatism, it is much different than the refractive values because of the lenticular astigmatism.

Figure 8C is the doubled-angle plot of the intraocular astigmatism at the corneal plane (posterior corneal astigmatism + lenticular astigmatism). The centroid is 0.48 D × 178 ± 0.49 D, ρ = 0.59. The centroid of the intraocular astigmatism is ATR while the front surface astigmatism is WTR. The intraocular astigmatism therefore partially compensates for front surface astigmatism and accounts for the lower degree of refractive relative to keratometric astigmatism seen in this and other studies.

Figure 8C.:
(Holladay) Doubled-angle plot of the intraocular astigmatism (posterior corneal astigmatism + lenticular astigmatism). The centroid is 0.48 D × 178 ± 0.49 D, ρ = 0.59. Notice the centroid of the intraocular astigmatism is ATR. The front surface being WTR and the intraocular ATR explains why the refractive astigmatism is less than the keratometric.

Figures 9A shows a comparison of the range, standard deviation, and mean values of the preoperative and postoperative astigmatism and SIRC measured by refraction and keratometry. As expected, the refractive data show a much greater reduction in astigmatism.

Figure 9A.:
(Holladay) Temporal comparison of the range, standard deviation, and mean value of the preoperative, postoperative, and astigmatism for both spectacles plane refraction and keratometry. The data show that refractive changes correlate, ie, the spread and magnitude of the postoperative changes are a result of the refractive SIRC. The keratometric data show a reduction but not as great as seen with the refractive cylinder.

Figure 9B is a histogram of the frequency distribution of preoperative refractive cylinder (black) at the spectacle plane and the postoperative refractive cylinder (red) at the spectacle plane. The histogram shows a significant left shift in the data, indicating a substantial reduction in the astigmatism. Figure 9C is a cumulative plot of the same data. The shaded area between the stair steps represents the improvement in the postoperative astigmatism.

Figure 9B.:
(Holladay) Histogram of the frequency distribution of preoperative refractive cylinder (black) at the spectacle plane and the postoperative spectacle plane refractive cylinder (red). The histograms show a substantial reduction in the astigmatism.

Figure 9C.:
(Holladay) A cumulative plot of the data in Figure 9B. The gray area between the red and black stair steps represents the improvement in postoperative astigmatism.

Figure 9D is a histogram of the frequency distribution of preoperative keratometric cylinder (black) and the postoperative keratometric cylinder (red) at the spectacle plane. The histogram shows a left shift in the data, indicating a significant reduction in the keratometric astigmatism. Figure 9E is a cumulative plot of the same data. The shaded area between the stair steps represents the improvement in the postoperative astigmatism. There is less improvement in keratometric astigmatism.

Figure 9D.:
(Holladay) Histogram of the frequency distribution of preoperative keratometric cylinder (black) and the postoperative keratometric cylinder (red) at the cornea. The histograms show a substantial reduction in the astigmatism.

Figure 9E.:
(Holladay) Cumulative plot of the data in Figure 9D. The gray area between the red and black stair steps represents the improvement in the postoperative astigmatism. The keratometric astigmatism shows less improvement than the refractive astigmatism.

Discussion
Table 3 and Figure 9A show the average magnitude of astigmatism was 1.46 ± 0.61 D before and 0.40 ± 0.38 D after surgery. This indicates that, regardless of axis, 50% of the patients were left with less than 0.40 D of astigmatism and 50% were left with more than 0.40 D. This is an important value that helps clinicians and patients develop realistic expectations.

The PE plot (Figure 3D ) illustrates the common preference of surgeons to err on the myopic side (79%). The target postoperative refraction for the patients in this study was plano. The mean postoperative refraction was −0.61 ± 0.82 D, suggesting the actual targets for some patients were probably slightly myopic. The larger degree of outcome variability as the magnitude of the treatment increase can also be seen.

The DEQ relates refractive error to UCVA. Figure 4B shows that most patients have a DEQ of <1, indicating good uncorrected vision. The mean postoperative DEQ was 0.96 D, which for a nominal 3.0 mm pupil corresponds to 20/24 visual acuity. However, as the attempted treatment increased above 6.0 D of myopia, the variability of the outcome increased.

The lack of correlation between the preoperative keratometry and refraction (at the corneal plane) indicates the presence of intraocular (posterior corneal and/or lenticular) astigmatism. Both the preoperative keratometric and corneal plane refractive astigmatism were WTR, but the magnitude of the keratometric cylinder was 0.48 D larger, 1.29 D and 0.81 D, respectively. The intraocular astigmatism therefore must be 0.48 D and ATR to account for the net corneal plane refractive astigmatism. The value of the centroid for the intraocular astigmatism at the corneal plane is shown in Figure 8C and is 0.48 × 178. We have found the same compensatory relationship between extraocular (ie, front surface power) and intraocular astigmatism after reviewing the data from other studies.^{17,18,22}

The preoperative and postoperative doubled-angle plots (Figures 6 to 8 ) document the relocation of the centroids toward the origin and contraction of their respective elliptical standard deviations. These changes indicate a significant reduction in astigmatism. To have a postoperative centroid nearly zero (0.11 D) with 95% of patients with less than 0.74 D of astigmatism is remarkable. Figure 7A shows these results. The shaded area on the cumulative distribution plots, Figures 9C and 9E , demonstrate this dramatic improvement.

Summary
The SIRC for an individual patient is a unique value, and all valid methods of analysis will yield the same result. Vector and SEQ analyses provide different types of information that, when considered together, provide a thorough quantitative analysis of refractive changes.

Sources of error in refractive outcome statistics include the use of multiple lens systems in the phoropter, errors in vertex calculations, difficulty in accurately defining the axis of astigmatism, and failure to consider measurement errors when working with keratometric data.

Refractive data must be adjusted for vertex distance before comparison to topographic or keratometric data. Descriptive statistics such as means, standard deviations, shape factor (ρ), standard error of the mean, and correlation coefficients can be calculated only after converting polar to Cartesian values.

Doubled-angle formulas and plots are necessary to accurately compute and display the results of aggregate astigmatism analysis and are particularly useful in interpreting errors in cylinder magnitude and alignment. The DEQ allows comparison of refractive and visual results. The decrease in refractive predictability with higher corrections was well demonstrated with plots of SEQ, DEQ, and doubled-angle plots of the SIRC. The correlation between refractive and keratometric astigmatism was poor for preoperative, postoperative, and SIRC data, indicating the presence of intraocular astigmatism and the limitation of manual keratometry.