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Analysis of aggregate surgically inducedrefractive change, prediction error, andintraocular astigmatism

Jack T. Holladay, MD, MSEE, John R. Moran, PhD, MD,Guy M. Kezirian, MD

ABSTRACT

Purpose: To demonstrate analytical methods for evaluating the results of keratorefractivesurgical procedures and emphasize the importance of intraocular astigmatism.

Setting: University of Texas Medical School, Houston, Texas, USA.

Methods: A standard data set, provided by an editor of this journal, comprising thepreoperative and postoperative keratometric and refractive measurements of 100 eyesthat had keratorefractive surgery was evaluated by 2 methods, vector and sphero-equivalent (SEQ) analysis. The individual and aggregate surgically induced refractivechanges (SIRCs) and prediction errors were determined from the refractive and kera-tometric measurements using both methods and then compared. The refraction vertexdistance, keratometric index of refraction, and corneal asphericity were used to makethe results calculated from refractive data directly comparable to those derived fromkeratometric data. Doubled-angle and equivalency plots as well as frequency andcumulative histograms were used to display the data. Standard descriptive statisticswere used to determine the mean and standard deviation of the aggregate inducedastigmatism after converting the polar values (cylinder and axis) to Cartesian (x and y)values.

Results: The preoperative SEQ refractive errors were undercorrected by at least 0.25diopter (D) in most cases (78%). Six percent were corrected within 6 0.24 D, and 16%were overcorrected by at least 0.25 D SEQ. The mean SEQ was 26.68 D 6 2.49 (SD)before and 20.61 6 0.82 D after surgery, reflecting a SIRC SEQ of 26.07 6 2.40 D. Thedefocus equivalent (DEQ) was 7.41 6 2.53 D before and 0.96 6 0.74 D after surgery;for a nominal 3.0 mm pupil, this corresponded to an estimated improvement in uncor-rected visual acuity (UCVA) from worse than 20/200 to better than 20/25, respectively.The predictability of the treatment decreased as the attempted refractive correctionincreased. The average magnitude of the refractive astigmatism was 1.46 6 0.61 Dbefore and 0.40 6 0.38 D after surgery. The centroid of the refractive astigmatism was10.96 3 87.9 6 0.85 D, r 5 0.43 before and 10.11 3 83.1 6 0.37, r 5 0.49 aftersurgery. The decrease in the square root of the centroid standard deviation shape factor(r1/2) indicated an 8% increase in the amount of oblique astigmatism in the population.The prevalence of preoperative keratometric irregular astigmatism in excess of 0.5 D inthis group of patients was 13%. The correlation between keratometric and refractiveastigmatism was extremely poor before (r2 5 0.26) and especially after surgery (r2 5

0.02), demonstrating the presence of intraocular astigmatism and the limitations ofmanual keratometry. The centroid of intraocular astigmatism at the corneal plane was10.48 3 178 6 0.49 D, r 5 0.59, and was compensatory.

© 2001 ASCRS and ESCRS 0886-3350/01/$–see front matterPublished by Elsevier Science Inc. PII S0886-3350(00)00796-3

Conclusions: The 2 analytical methods are complimentary and permit thorough andquantitative evaluation of SIRCs and allow valid statistical comparisons within andbetween data sets. The DEQ allows comparison of refractive and visual results. Thedecrease in refractive predictability with higher corrections is well demonstrated by theSEQ and doubled-angle plots of the SIRC. Doubled-angle plots were particularly usefulin interpreting errors of cylinder treatment amount and errors in alignment. The corre-lation between refractive and keratometric astigmatism was poor for preoperative,postoperative, and SIRC data, indicating the presence of astigmatic elements beyondthe corneal surface (ie, intraocular astigmatism). Sources of error in refractive outcomestatistics include the use of multiple lens systems in the phoropter, errors in vertexcalculations, difficulty in accurately defining the axis of astigmatism, and failure toconsider measurement errors when working with keratometric data. The analysis ofthis particular data set demonstrates the significant clinical benefits of refractivesurgery: an 8-fold increase in UCVA, an 11-fold decrease in SEQ refractive error,as well as a 9-fold and nearly a 2 1/2-fold decrease in the magnitude and distri-bution of astigmatism, respectively. J Cataract Refract Surg 2001; 27:61–79 © 2001ASCRS and ESCRS

The mathematics for calculating surgically inducedrefractive change (SIRC) was first described in

1849.1 One hundred twenty-six years later, Jaffe andClayman2 properly applied this method to analyzethe relationship between cataract surgical techniqueand the refractive result in individual patients. Sincethen, several mathematically incorrect methods havebeen reported.3–5

In 1992, to evaluate evolving corneal surgery tech-niques, we described a method for calculating the surgi-cally induced spherical and astigmatic change in anindividual patient.6 In 1999, we extended the applica-bility of this method to include aggregate data.7 Thecurrent literature indicates that significant confusionstill surrounds the question of how best to evaluateSIRCs.8,9 We appreciate the opportunity to demon-strate our methods for the 100-case data set provided byDouglas Koch, MD,9 one of the journal editors.

Materials and MethodsStatistical analyses and graph preparation were per-

formed on a personal computer using Microsoft Excel97 and SPSS SigmaPlot 3.0, respectively.

Data Preparation and Sources of ErrorVertexing spherocylinders to the corneal plane. Refrac-

tions are normally performed at the spectacle plane or inthe phoropter and not at the corneal plane. Refractivemeasurements must be vertexed to the corneal planebefore they can be compared to those obtained by kera-tometry or topography. Inaccuracies in the measure-ment of vertex distances and inappropriate applicationof the vertex formula are frequent sources of error inplanning and reporting the results of refractive surgicalprocedures.

Refractions performed in the phoropter are at anominal vertex distance of 13.75 mm when the cornealvertex is located at the large mark on the calibrationscale. Within the phoropter, multiple lenses are alignedto produce the combination lens used in refraction. Thiscombination lens has an effective vertex distance thatfrequently differs from the nominal value indicated onthe vertex scale. This difference, along with the diffi-culty of maintaining precise patient alignment, causethe vertex distances measured with the phoropter tobe unreliable, especially at higher refractive powers. Amore accurate measurement is obtained by perform-ing the refraction over a soft contact lens with a power

Accepted for publication October 30, 2000.

From University of Texas Medical School (Holladay) and Houston EyeAssociates (Holladay, Moran), Houston, Texas, and SurgiVision Con-sultants Inc. (Kezirian), Paradise Valley, AZ, USA.

None of the authors has a financial interest in any product or devicementioned.

Reprint requests to Jack T. Holladay, MD, 5108 Braeburn Drive, Bel-laire, Texas 77401, USA. E-mail: [email protected].

ANALYZING ASTIGMATISM: HOLLADAY

J CATARACT REFRACT SURG—VOL 27, JANUARY 200162

near the spheroequivalent (SEQ) of the refractiveerror.

For example, the SEQ of a 211.50 12.00 3 90refraction at the phoropter with a nominal vertex dis-tance of 13.75 mm is 210.50 diopters (D). When ver-texed to the corneal plane, the SEQ is 29.18 D.Overefraction of the same patient performed with a210.00 D soft contact lens gives 22.00 12.00 3 90.The final refraction at the corneal plane (vertex 50 mm) is then 212.00 12.00 3 90, yielding an SEQrefraction of 211.00 D at the corneal plane. The valuewith the contact lens is 21.82 D more myopic than withthe vertexed phoropter refraction. The contact lensmethod is always more accurate provided the soft con-tact lens is labeled properly. The correct vertex distancemust be entered into all laser and surgically planningsoftware programs to avoid inducing residual refractiveerrors.

Since the plus and minus cylinder notation for re-fraction represents a difference rather than an actualpower in either meridian, they must first be converted tothe cross-cylinder notation before performing vertex cal-culations. The spherocylindric refraction in this exampleis vertexed to the corneal plane as shown below:

Cross cylinder form @ spectacle: 212.00 3 180 and210.00 3 90

Vertex: 13.75 mm

Vertex formula from spectacle plane (REFs) to corneal plane(REFc)

10,11:

REFc 51000 p REFs

1000 2 REFs p Vertex (mm)(1)

Substituting the values from the example

REFC1 51000 p ~212.00!

1000 2 ~212.00! p 13.755 210.30 D

REFC2 51000 p ~210.00!

1000 2 ~210.00! p 13.755 28.79 D

Cross cylinder form @ cornea: 210.30 3 180 and28.79 3 90

Vertex: 0 mm

Plus cylinder form: 210.30 1 1.51 3 90

Minus cylinder form: 28.79 21.51 3 180

When correctly vertexed, the astigmatism at the cor-neal plane is 21.51 D, almost one half a diopter less

than the value obtained by incorrectly vertexing the cyl-inder portion of the refraction expressed in minus-cylinder notation (21.95 D). Vertexing a myopicrefraction from the spectacle to the corneal plane willalways reduce the magnitude of the astigmatism. Forhyperopia, the relationship is just the opposite.

To compare SIRCs calculated from refractive andkeratometric data, the refractive data must be vertexedto the corneal plane before the SIRC calculation is per-formed. Performing a vertex calculation on the results ofan SIRC calculation gives an incorrect answer.

Special considerations with keratometric data. Kera-tometric data are already at the corneal plane and do notrequire vertex adjustment. However, an additional issuearises with respect to the correct method of convertingthe anterior radius of the cornea to the net corneal re-fractive power. The formula used to convert radii topower is the simple spherical refracting surface formula(SSRS):

Kk 5n2 2 n1

r(2a)

The variables n1 and n2 are the indices of refraction ofthe first and second media, respectively, and r is theradius of curvature of the interface. The value for n1 is1.000 (index of refraction for air). The standardizedkeratometric index of refraction (1.3375) was chosen forn2 many years ago.12 The origin of this value for n2

remains obscure, appearing to have been arbitrarily se-lected in the 19th century so that an anterior radius ofcorneal curvature of 7.5 mm would yield a power of45.00 D.13

Kk 51.3375 2 1.000

ra5

0.3375

ra(2b)

where ra is the anterior radius of curvature of the cor-nea (m) and Kk is the standardized keratometric netcorneal power (D).

The cornea, like any meniscus lens, has a frontsurface power, a back surface power, and a net orequivalent power. To compute the surgically inducedcornea power change, one must know whether thesurgical procedure alters the front surface, back sur-face, or net cornea power. The front surface powerand net power changes are the only clinically relevantconsiderations, since there are no keratorefractive


J CATARACT REFRACT SURG—VOL 27, JANUARY 2001 63

procedures that are intended to change only the backsurface power.

For procedures such as photorefractive keratectomy(PRK), laser in situ keratomileusis (LASIK), and proba-bly radial keratotomy, the change in dioptric power ofthe cornea is almost entirely due to front surface powerchanges in the cornea. To compute front surface power,the change in media for the light rays is from air (n 51.000) to cornea (n 5 1.376), so, as Holladay and War-ing13 and Mandell14 have recommended, the correctformula for computing the power and any change inpower would be

Ka 51.376 2 1.000

ra5

0.376

ra(2c)

where ra is the anterior radius of curvature of the cor-nea (m) and Ka is the front surface corneal power (o).

The front surface power of a cornea with an anteriorradius of 7.5 mm would be 50.13 D (0.376/0.0075),5.13 D greater than the standardized keratometricpower of 45.00 D (0.3375/0.0075). Front surface pow-ers are 11.14% (0.376/0.3375) larger than keratometricvalues. When determining the change in the refractivepower of the eye for procedures that change only thefront surface of the cornea, the change computed fromkeratometry must be increased by 11.14% to compen-sate for the difference in the index of refraction.

For analyzing results in which it is believed thatboth the front and back surfaces have been changedequally, it is appropriate to use the net or equivalentcorneal index of refraction. The most common applica-tion of this conversion is in intraocular lens (IOL) powercalculations. There is still debate among investigators asto the most appropriate value for the net or equivalentindex of refraction. Binkhorst15 and Olson16 have em-pirically determined this value to be 4/3 and 1.3315,respectively. For standardization purposes,13 we haverecommended adopting the Binkhorst value, since it ishas been the one most frequently used for the past 20years. The equation to compute net or equivalent cor-neal power, using indices of refraction for air (n 51.000) and cornea (n 5 4/3), would be

Kn 54⁄3 2 1.000

ra5

1⁄3

ra(2d)

where ra is the anterior radius of curvature of the cor-nea (m) and Kn is the net corneal power (D).

The net power of a cornea with an anterior radius of7.5 mm would be 44.44 D (1/3 4 0.0075), 0.56 D lessthan the standardized keratometric power of 45.00 D.Net or equivalent corneal powers are 98.76% (1/3 40.3375) of the standardized keratometric values. Whenthe net or equivalent power of the cornea or changes inthe net corneal power are needed, the standardized kera-tometric values must be reduced to 98.76% of theiroriginal values to accurately reflect the net refractivepower change in the cornea. For net powers of the cor-nea used in IOL calculations, a 1.24% overestimate ofthe corneal power results in a 0.56 D error, which issignificant and intolerable for these calculations. How-ever, for calculating changes in corneal power producedby refractive surgical procedures, this 1.24% error isclinically negligible. Nevertheless, when reporting netcorneal power changes from standardized keratometrymeasurement, the values should be reduced by 1.24% tobe correct.

Another source of difference between keratometricand refractive measurement arises from the relative flat-tening of the central cornea by myopic refractive sur-gery.17,18 Keratometers and topographers sample theparacentral cornea (nominal diameter of 3.2 mm for a44.00 D cornea) rather than the surgically flattened op-tical center and thus tend to overestimate the cornealpower after myopic refractive surgery. This study foundan additional 10% difference between the actual centralpower change and the paracentral keratometric and to-pographic change. The refractive power of the corneachanged by a factor 1.21 times more than the apparentkeratometric change when considering the progressiveflattening error and the front surface index of refractionerror.17,18

It is important to note that the value of 1.333 forthe net corneal index of refraction may change follow-ing refractive surgical procedures that alter epithelialthickness or remove corneal tissue (ie, PRK andLASIK). The refractive indices of the epithelium andstroma are different, and there may be subtle intra-stromal differences, eg, between Bowman’s layer andthe posterior stroma.19 Procedures that alter epithe-lial thickness could change the refractive power of theepithelium, and removal of stromal tissue could alterthe net refractive index of the stroma. These changescould be important in determining the correct valueof net corneal power for IOL calculations, since, as



noted above, a change of only 1% to 2% could pro-duce unacceptably high errors.

Surgically induced refractive change calculations re-quire that the astigmatism is regular, ie, the steepest andflattest meridians must be orthogonal (90 degrees apart).In the data set used here, some cases had non-orthogonalaxes. In these cases, the steep axis was maintained andthe flat axis was assigned a value 90 degrees away. Thevector difference in the original keratometric astigma-tism and the orthogonalized values is a measure of theirregular, oblique axes (not oblique axis) astigmatism.These data are shown on the doubled-angle plot in Fig-ure 1. Thirteen eyes had magnitudes greater than the0.50 D magnitude that is considered clinically signifi-cant irregular keratometric astigmatism. All SIRC calcu-lations performed here used the orthogonalized data.

Precision of the magnitude and axis of astigmatism.The question often arises as to what is the angular errorin astigmatism that is comparable to a given magnitudeerror. Intuitively, most clinicians know that the higherthe degree of astigmatism, the more accurate the axis ofastigmatism must be; eg, a 10 degree error in the axis of

6.00 D of astigmatism is more significant than a 10degree error in the axis with 1.00 D of astigmatism. Theangular error that is equivalent to a dioptric error for agiven astigmatism magnitude is given by the followingformula:

u 5 ArcSinS0.5 pt

MD (3)

where t is the tolerance (precision, eg, 6 0.50 D), M isthe magnitude of the astigmatism, and u is the angularerror that is equivalent to the tolerance in diopters. Ta-ble 1 tabulates these values, and Figure 2 illustrates theirrelationship.

Figure 1. (Holladay) Doubled-angle plot of the non-orthogonal,keratometric irregular astigmatism. The data points represent the dif-ference vector in the original and orthogonalized keratometric astig-matism vector. The centroid is 0.09 D 3 57.1 6 0.23 D, r 5 2.86. The13 data points outside the 0.50 D radius (blue) would be consideredclinically significant irregular keratometric astigmatism of oblique axes.

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

Table 1. Angular error (6 degrees) corresponding to a magnitudeerror (D) at different measurement precisions.

Tolerance(D)

Measured Magnitude (D)

0.125 0.250 0.50 1.00 2.00 4.00 8.00

60.125 30 14 7 4 2 1 0

60.25 90 30 14 7 4 2 1

60.50 — 90 30 14 7 4 2

61.00 — — 90 30 14 7 4



Methods of CalculationCalculating prediction error from the desired and ac-

tual postoperative refraction. Excimer lasers, toric IOLs,and incisional surgery are capable of inducing sphericaland astigmatic changes in the refraction. In most cases,the goal of refractive surgery is to neutralize the sphero-cylindric correction so the final postoperative refractionis plano. When this goal is achieved, the desired SIRCand the actual SIRC are equal. Likewise, when the finalrefraction does not match the desired postoperative re-fraction, the desired SIRC and the actual SIRC must bedifferent.

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

Prediction Error 5 Desired Postop Ref

2 Actual Postop Ref (4)

The difference between the actual postoperative re-fraction and the desired or predicted postoperative re-fraction must be calculated like any other SIRC. This isusually easy to accomplish since the desired postopera-tive refraction is most often spherical, so that the solu-tion for obliquely crossed cylinders is not necessary.

A numeric example of the PE calculation may behelpful to illustrate this concept. Consider a 50-year-oldpatient with a refractive error of 25.00 13.00 3 90 atthe corneal plane (vertex 5 0). A LASIK procedure isplanned to leave the patient 20.50 D myopic. The de-sired SIRC at the corneal plane is therefore 24.5013.00 3 90, 0.5 D less than the refractive error at thecorneal plane. The patient’s actual postoperative refrac-tion at 1 month vertexed to the corneal plane was20.50 10.50 3 80.

Prediction Error 5 ~20.50 sphere! 2 ~20.50

1 0.50 3 80) 5 plano 2 0.50 3 80

The value of the difference between the desired andthe actual postoperative refraction is plano 20.50 3 80.This value indicates that the error from the desired targetwas 20.50 D of cylinder at an axis of 80 degrees. The PEand SIRC can be treated similarly when analyzing aggre-gate data.

Analyzing aggregate data. The evaluation and dis-play of aggregate data of a group of patients is performedafter the refractive measurements have been vertexed to

the corneal plane and standardized keratometric mea-surements converted to front surface or net cornealpower. Descriptive statistics of SEQ values such asmeans, standard deviations, standard error of the means,and correlation coefficients are calculated in the normalmanner.

Spheroequivalent statistics and graphs are particu-larly valuable for analysis of procedures in which noinduced astigmatism was intended, such as spherical re-fractive surgery (spherical PRK or LASIK).

Graphical display of the SEQ of the SIRC is partic-ularly useful. The SEQ of any spherocylinder is thesphere plus one half the cylinder. The formula is

Spheroequivalent 5 Sphere 11

2p Cylinder (5a)

A plot of SEQ SIRC data is shown in Figure 3A on anequivalency plot. When data points are on the diagonal(the equivalency line), the desired and actual refractionsare equal. When the result is above the diagonal, there isan overcorrection; when below the diagonal, an under-correction. The PE can be plotted versus the desiredSEQ SIRC as shown in Figure 3D. The informationdisplayed is similar to the equivalency plot, but the exactvalues for the errors are easier to see.

The magnitude of astigmatism (cylinder power inthe plus or minus cylinder form) can be analyzed in asimilar manner, but the information about the axis ofthe astigmatism is lost. It is not possible from the mag-nitude of astigmatism plots alone to infer any trendssuch as against-the-rule (ATR) or with-the-rule (WTR)astigmatism changes from the data. The most appropri-ate method for evaluating, reporting, and displayingastigmatism data requires conversion of the data fromthe usual method of describing astigmatism in polar co-ordinates (cylinder and axis) to a Cartesian coordinatesystem.

The defocus equivalent (DEQ) correlates betterwith uncorrected visual acuity (UCVA) than the SEQand is a useful measure of surgical success.20 The for-mula for the defocus equivalent is

Defocus Equivalent (DEQ) 5 Magnitude of SEQ

11

2Magnitude of Cylinder (5)

An SEQ of zero indicates that the circle of least confu-sion is located on the retina but gives no information



Figure 3A. (Holladay) Equivalency plot of SEQ SIRC data. Whendata points are on the diagonal, the desired and achieved correctionsare equal. When the result is above the diagonal, there is an overcor-rection; when below the diagonal, there is an undercorrection. Mostcases (78%) were undercorrected by at least 0.25 D, 6% were correctwithin 60.24 D, and 16% were overcorrected by at least 0.25 D. Theregression line is shown in solid red (95% confidence intervals indashed red) with R2 5 0.89 and the standard error 5 0.79 D.

Figure 3B. (Holladay) Equivalency plot of achieved and desiredSIRC sphere data in minus cylinder notation. Most cases (69%) wereundercorrected by at least 0.25 D, 14% were corrected within 60.24D, and 17% were overcorrected by at least 0.25 D. The regression lineis shown in solid red (95% confidence intervals in dashed red) withR2 5 0.91 and the standard error 5 0.73 D.

Figure 3C. (Holladay) Equivalency plot of achieved and desiredSIRC cylinder data in minus notation. Thirty-two percent of the caseswere undercorrected by at least 0.25 D, 54% were corrected within60.24 D, and 14% were overcorrected by at least 0.25 D. Theregression line is shown in solid red (95% confidence intervals indashed red) with R2 5 0.64 and the standard error 5 0.40 D.

Figure 3D. (Holladay) Prediction error plot. Since the target in all100 patients was given as plano, the PE is simply the SEQ of thepostoperative refraction. The percentages of undercorrection andovercorrection are the same as in the equivalency plot. It is evident inboth plots that the greater the attempted correction, the larger thevariability and the greater the amount of undercorrection. The Pearsoncorrelation coefficient (r) is 0.26, and the r2 is 0.07.



about its size. The DEQ is proportional to the diameterof the blur circle for a given pupil size. For example, apatient with a refraction of 20.50 11.50 3 90 has anSEQ of 10.25. A patient with a refraction of25.00 110.50 3 90 also has an SEQ of 10.25, butno one would expect the UCVAs to be equal. In thefirst case, DEQ 5 1.00 (0.25 1 0.75) and in thesecond case, DEQ 5 5.5 (0.25 1 5.25), 51⁄2 timeslarger. The UCVA in the second case would be 51⁄2times worse if the pupil sizes were the same. The rela-tionship between uncorrected Snellen visual acuity andpupil size as a function of defocus is shown in Figure 4Aand Table 2. For example, a patient with a DEQ of 2.00D and a 3.0 mm pupil will have a UCVA of 20/49.

Calculating, Reporting, and Displaying AggregateAstigmatism Data

Astigmatism data are difficult to analyze primarilybecause of the very definition of astigmatism. Theaxis of astigmatism returns to the same value when ittraverses an angle of 180 degrees. In geometry andtrigonometry, an angle must traverse 360 degrees toreturn to its same value. To apply conventional ge-ometry, trigonometry, and vector analysis to astigma-tism, the angles of astigmatism must be doubled sothat 0 degree and 180 degrees are equivalent. Oncethis transformation has been performed, all the stan-

dard formulas can be used and produce the correctsingular value for the SIRC.

Because astigmatism traverses an entire cycle in 180degrees, the most appropriate plot of aggregate astigma-tism data is a doubled-angle polar plot. The angular scaleof doubled-angle plot range from 0 degree to 180 de-grees. The radial axes are oriented with 45 degrees at 12o’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 180degree locations are the same, just as 0 degree and 180degrees are the same for the axis of refraction. Any pro-cedure that on the average is astigmatically neutral musthave the centroid of the surgically induced cylinder dataat the center of the plot. On a single-angle plot (standardpolar plot), none of these statements is true. An exampleof the doubled-angle plot is shown in Figure 6A.

Determining the mean cylinder and axis for inducedastigmatism. In our original article,6 we presented amethod for determining the average axis or meridian ofastigmatism. Since then, we have recognized that thismethod is incorrect7 because it does not appropriately

Figure 4A. (Holladay) The relationship between Snellen visualacuity, pupil size, and refractive error in the human. The DEQ plot isthe best measure of potential (no other pathology) UCVA. For eachpupil diameter and refraction, a Snellen visual acuity can be deter-mined. For example, for a 3.0 mm pupil and 3.0 D of defocus, thenormal Snellen visual acuity is 20/80 (reprinted with permission fromHolladay et al.20).

Figure 4B. (Holladay) Equivalency plot of the preoperative DEQon the x-axis versus the postoperative DEQ on the y-axis. Ideally, allthe y-values would be zero, indicating the SEQ is zero and there is noastigmatism. All 100 patients had a reduction in their DEQ. The meanDEQ postoperatively was 0.96 D. For a 3.0 mm pupil, the Snellenvisual acuity would be 20/24. Preoperatively, the DEQ was 7.4 D,which is worse than 20/200.



incorporate the magnitude and axis of the SIRC into thecalculations. For standard descriptive statistics (means,standard deviations) to be applied correctly, each datapoint must be converted to an x–y coordinate system.Descriptive statistics cannot be applied to polar coor-dinates because cylinder magnitude and axis are notindependent (orthogonal) parameters. Polar valuescan be converted to Cartesian values using equations6a and 6b.

x 5 Cylinder p Cos~2 p axis! (6a)

y 5 Cylinder p Sin~2 p axis! (6b)

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

The centroid or mean of a set of x and y values iscalculated by independently finding the mean of eachvariable. In equation form,

Mean of x 5O

i 5 1

n

x j

n(7a)

Mean of y 5O

i 5 1

n

xi

n(7b)

For example, the centroid of preoperative cylinderat the spectacle plane is x 5 2 0.956 and y 5 0.070.Converting the Cartesian values to the standard polarnotation for astigmatism,7,21

Cylinder 5 Îx2 1 y2 (8a)

Angle 51

2p Arc tanS y

xD (8b)

If x and y . 0 then Axis 5 Angle (8c)

If x , 0 then Axis 5 Angle 1 90° (8d)

If x . 0 and y , 0 then Axis 5 Angle 1 180° (8e)

and substituting values for x and y,

Cylinder 5 Î20.9562 1 0.0702 5 0.959 D

Angle 51

2p Arc tanS 0.070

20.956D5 22.1°, since x and y . 0

Axis 5 Angle 5 87.9°

gives mean value of the astigmatism as 10.959 3 87.9.An additional important descriptive statistic is the

mean absolute value.

Mean Absolute Value of r 5 MAVr 5O

i 5 1

n

uriu

n(9)

If the postoperative target refraction is plano, the meanabsolute cylinder magnitude equals the average absoluteastigmatism PE, regardless of the axes. For example, if 2PEs were equal and opposite (eg, 11.00 D 3 90 and21.00 D 3 90), the algebraic or vector average (cen-troid) would be zero. The average magnitude, however,is 1.00 D. The mean absolute value is a linear average ofthe magnitude of the variable and weighs all PEs equally.The statistics for variability, variance, and standard de-viation weigh the PEs by the square of the difference,making the larger PEs more important.

Variance (s2) is defined as

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

DEQ

Pupil Size (mm)

0.5 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0

TDL 20/36 20/18 20/09 20/06 20/04 20/04 20/03 20/03 20/02

0.0 20/36 20/18 20/10 20/09 20/10 20/10 20/11 20/11 20/11

0.5 20/36 20/22 20/13 20/15 20/19 20/24 20/28 20/30 20/31

1.0 20/36 20/27 20/19 20/24 20/33 20/44 20/52 20/56 20/58

2.0 20/37 20/33 20/36 20/49 20/68 20/95 20/121 20/130 20/135

3.0 20/38 20/39 20/60 20/83 20/117 20/168 20/214 20/230 20/239

4.0 20/39 20/47 20/95 20/132 20/182 20/252 20/307 20/330 20/343

5.0 20/40 20/56 20/140 20/190 20/258 20/348 20/428 20/460 20/478

DEQ 5 defocus equivalent; TDL 5 theoretical diffraction limit



Variance of x 5 sx2 5

Oi 5 1

n

~xi 2 x#!2

n 2 1(10a)

Although variance is an important measure of vari-ability, it is a squared parameter and does not have thesame units as the original data. Standard deviation s isdefined to be the positive square root of the variance (s2)and has the same units as the original data. The formulafor the standard deviation s is

Standard deviation of x 5 sx 5 ÎOi 5 1

n

~xi 2 x#!2

n 2 1(11a)

Standard deviation of y 5 sy 5 ÎOi 5 1

n

~yi 2 y#!2

n 2 1(11b)

The standard deviation in this data set is calculatedby independently determining the standard deviationsof the x and y variables. Since x and y are orthogonal, theyhave no statistical influence on each other. When thestandard deviation of x and y are unequal, the standarddeviation is an elliptical area surrounding the centroid asseen in Figure 6A. When the standard deviations areequal, a circle would be formed around the centroid.The formula for an ellipse is

~x 2 x#!2

sx2 1

~y 2 y#!2

sy2 5 1 (12)

When sx is greater than sy, then sx is the semi-majoraxis of the ellipse, sy is the semi-minor axis, and theellipse is oriented horizontally. When sy is greater thansx, then sy is the semi-major axis of the ellipse, sx is thesemi-minor axis, and the ellipse is oriented vertically.The semi-major and semi-minor axes of the ellipse cannever be rotated with respect to the x- and y-axes. Thiscondition is very important for determining standarderror of the means and confidence intervals.

The shape of an ellipse will vary depending on thelength of the major and minor axes. Several terms havebeen used to describe this relationship: shape factor (r),asphericity (Q), and eccentricity (e2). The mathematicalrelationship among these parameters is

sy2

sx2 5 r 5 1 1 Q 5 1 2 e2 (13)

On a doubled-angle plot, the y-axis is coincidentwith the axis of oblique astigmatism (45 degrees, 135

degrees) and the x-axis is coincident with the axis ofnonoblique (90 degrees, 0 degree WTR and ATR) astig-matism. Since most populations have WTR or ATRastigmatism, the ellipses are oriented horizontally (r ,1). In a population with a higher percentage ofoblique astigmatism, the ellipse would be orientedvertically (r . 1). In a population with uniformlydistributed oblique and nonoblique astigmatism, thestandard deviation would be circular (r 5 1). Theshape and orientation of the ellipse is therefore help-ful in determining the degree of oblique astigmatismin a population. The square root of the shape factor(r1/2) is the actual ratio of oblique to nonobliqueastigmatism.

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

Area of ellipse 5 p*sx*sy (14)

Area of circle 5 p~sc)2 (15)

Setting the 2 areas equal gives

~sc)2 5 sx p sy (16)

Therefore, for the area of the ellipse to have the samearea as a circle, the radius of the circle (sc) must be equalto the square root of the product of the semi-major andsemi-minor axes:

Radius of the equal area circle 5 sc 5 Îsx p sy (17)

In other words, the standard deviation of a popula-tion with unequal x and y components is the geometricmean of the standard deviation of the x and y compo-nents. Another way of visualizing this relationship is thatsc

2 represents the variance of a circular distribution(area) with a radius sc corresponding to the standarddeviation. The variance of the elliptical distribution(area) is sx p sy and its square root is the standard devia-tion of the ellipse (geometric mean radius sc). The stan-dard deviation of the centroid (sc) is the square root ofthe product of the individual standard deviations for x(sx) and y(sy). When comparing 2 data sets, the set withthe lower variability (“tighter data”) is the data set withthe lower variance (sx p sy) and standard deviation(=sx p sy).



Another important descriptive statistic is the stan-dard error of the mean, defined as the standard deviationof the sampling distribution of a statistic for randomsamples of size n. It is computed by taking the standarddeviation and dividing by the square root of the samplesize n. The formula for the standard error of the mean x#is

Standard error of the mean, x# 5 ex 5sx#

În(18)

The standard error of the mean is helpful in deter-mining confidence intervals and determining whether 2sample populations are statistically different at a givenprobability (usually P # .05 or 2 standard errors fromthe mean) or 95% confidence interval. These are thevalues found in a “z-table” for standard normal distribu-tions and “t table” for a Student t distribution.

ResultsThe mean age of the 100 patients was 44.69 years 6

8.26 (SD), with 54% men and 58% right eyes. Table 3summarizes the mean SEQ, mean DEQ, mean absolutemagnitude of astigmatism, and mean magnitude sphere(in minus cylinder form) preoperatively, postopera-tively, and the SIRC by refraction and keratometry. Ta-

ble 4 summarizes preoperative, postoperative, and SIRCusing vector analysis of astigmatism measured by refrac-tion and keratometry.

Figures 3A to 3C are the equivalency plot for theachieved versus attempted correction. The preoperativeSEQ refractive errors were undercorrected by at least0.25 D in most cases (78%). Six percent were correctedwithin 60.24 D, and 16% were overcorrected by at least0.25 D SEQ. Figure 3D is the PE plot. Since the targetin all 100 patients is plano, the PE is equal to the SEQ ofthe postoperative refraction. The percentages of under-correction and overcorrection are the same as the equiv-alency plot. It is evident in both plots that the greater theattempted correction, the larger the variability and thegreater the amount of undercorrection. The Pearsoncorrelation coefficient (r) is 0.26 and the r2 is 0.07.

In Figure 4B, the preoperative DEQ is plotted onthe x-axis and the postoperative DEQ on the y-axis.Ideally, all the y-values would be zero, indicating theSEQ and astigmatism equal zero. The Pearson correla-tion coefficient is 0.33 and r2 5 0.11. As in Figures 3Ato 3D, these data confirm that the higher the intendedtreatment, the greater the variability and the higher theaverage DEQ. The data indicate the higher the preop-erative DEQ, the smaller the chance of the patient

Table 3. Summary of mean SEQ, DEQ equivalent, and absolute astigmatism changes.*

Data Source SEQ (D) DEQ (D)Mean Absolute

Cylinder (D)Mean Sphere

(D)

Refraction (vertex 5 12 mm)Minus cylinder form

Preop 26.68 6 2.49 7.41 6 2.53 1.46 6 0.61 25.96 6 2.48

Postop 20.61 6 0.82 0.96 6 0.74 0.40 6 0.38 20.40 6 0.38

SIRC 26.07 6 2.40 — 1.32 6 0.67 25.41 6 2.36

Refraction (vertex 5 0 mm)Minus cylinder form

Preop 26.10 6 2.14 — 1.24 6 0.52 25.48 6 2.17

Postop 20.60 6 0.80 — 0.39 6 0.37 20.40 6 0.81

SIRC 25.51 6 2.08 — 1.12 6 0.57 24.94 6 2.07

KeratometryMinus cylinder form

Preop 44.15 6 1.56 — 1.57 6 0.71 44.94 6 1.66

Postop 40.03 6 1.87 — 0.84 6 0.51 38.77 6 1.80

SIRC 24.12 6 1.58 — 1.04 6 0.50 23.60 6 1.59

SEQ 5 spheroequivalent; DEQ 5 defocus equivalent; SIRC 5 surgically induced refractive change*Mean 6 SD



achieving plano and, consequently, the lower the aver-age potential UCVA.

Figure 5A is an equivalency plot of the magnitude ofthe preoperative keratometric astigmatism (x-axis) ver-sus the magnitude of the preoperative refractive astigma-tism at the corneal plane. There is a significant lack ofcorrelation (r2 5 0.26). This demonstrates that a largenumber of patients have intraocular astigmatism. Thedegree of corneal astigmatism is much higher than thedegree of refractive astigmatism, indicating that thelenticular astigmatism generally reduces the amountof refractive astigmatism. This finding is similar tofindings in previous studies that compare refractiveastigmatism with keratometric and/or topographicastigmatism.17,18,22

Figure 5B shows the relationship of the postopera-tive refractive cylinder to the keratometric cylinder. Thecorrelation is much worse (r2 5 0.02), but there is adramatic reduction in both refractive and keratometricastigmatism. The fact that the refractive astigmatism ismuch lower indicates that the residual corneal astigma-tism is neutralizing the lenticular astigmatism. This isalso confirmed by Figure 5C, which shows the refractiveSIRC and keratometric SIRC have the highest correla-tion. In short, in a patient with lenticular astigmatism,the refractive astigmatism can only be zero when the

Figure 5A. (Holladay) Equivalency plot of the magnitude of thepreoperative keratometric astigmatism (x-axis) versus the magnitudeof the preoperative refractive astigmatism (y-axis) at the corneal plane.The significant lack of correlation (r2 5 0.26) indicates the presenceof lenticular astigmatism. The fact that the refractive is less than thekeratometric cylinder shows that the intraocular astigmatism partiallycompensates (emmetropization) for the corneal astigmatism.

Table 4. Vector analysis of astigmatism results.

Data SourceSphere

(D)Cylinder

(D)Axis or

Meridian (°)6SD(D)

ShapeFactor

Refraction (vertex 5 12 mm)(axis notation)

Preop 27.41 10.96 3 87.9 0.85 0.43

Postop 20.81 10.11 3 83.1 0.37 0.49

SIRC 26.74 10.85 3 88.5 0.83 0.51

Refraction (vertex 5 0 mm)(axis notation)

Preop 26.72 10.81 3 88.0 0.73 0.42

Postop 20.79 10.11 3 83.1 0.36 0.49

SIRC 26.07 10.70 3 88.7 0.73 0.52

Keratometry(power notation)

Preop 43.37 11.29 @ 88.0 0.77 0.38

Postop 39.61 10.63 @ 86.5 0.52 0.48

SIRC 24.64 20.66 @ 89.4 0.71 0.70

SIRC 5 surgically induced refractive change



corneal astigmatism is exactly equal and opposite. Theratio of the refractive SIRC and keratometric SIRC was1.34 (25.51/24.12), slightly higher than the previouslyreported value of 1.21.

Figure 6A represents the preoperative refractiveastigmatism at the spectacle plane (vertex 5 12 mm).The centroid was 10.96 D 3 88 6 0.85 D, r 5 0.43,illustrating that 90% of the patients had some degree ofWTR astigmatism (left side of circle) preoperatively.This finding is expected for the relatively young averageage for the refractive surgery patient (44.7 years).

Figure 6B is the doubled-angle plot of the preop-erative keratometric cylinder at the corneal plane.The centroid of the keratometric astigmatism was11.29 D @ 88 6 0.77 D, r 5 0.38. Figure 6C showsthe 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 pres-ence of lenticular astigmatism.

Figures 7A, 7B, and 7C are the doubled-angleplots of the postoperative astigmatism from refrac-

tion, keratometry, and the composite, respectively. InFigure 7A, the centroid of the postoperative cylinderat the spectacle plane (vertex 5 12 mm) is 0.11 D 383 6 0.37 D, r 5 0.49, a dramatic reduction fromthe preoperative refractive astigmatism. The centroidof the postoperative keratometric astigmatism is10.63 3 87 6 0.52 D, r 5 0.48 is shown in Fig-ure 7B. Figure 7C illustrates the centroids and stan-dard deviations of the postoperative keratometric andrefractive astigmatism at the corneal plane, whichshow little correlation.

Figure 8A shows the relationship between preop-erative and postoperative refractive astigmatism at thecorneal plane. These data show that the refractiveastigmatism centroid is 8 times (0.81 D/0.11 D)closer to zero and the standard deviation of the astig-matism was reduced by a factor of 2.0 (0.73 D/0.36 D). The amount of improvement is demon-strated graphically on the doubled-angle plots by therelocation of the centroid closer to the origin and thecontraction of the ellipse.

Figure 5B. (Holladay) Equivalency plot showing the relationship ofthe postoperative refractive cylinder to the keratometric cylinder. Thecorrelation is much lower (r2 5 0.02), but there is a dramatic reduc-tion of both refractive and keratometric astigmatism. The fact that therefractive astigmatism is much lower indicates that the residual cor-neal astigmatism is neutralizing the lenticular astigmatism.

Figure 5C. (Holladay) Equivalency plot of the refractive and kera-tometric SIRC. The refractive and keratometric SIRC have a correla-tion that is very similar to that of the preoperative values. In short, in apatient with lenticular astigmatism, the refractive astigmatism can onlybe zero when the corneal astigmatism is exactly equal and opposite tothe lenticular astigmatism.



Figure 8B shows the relationship between preoper-ative and postoperative keratometric astigmatism. Thepreoperative centroid was 1.29 @ 88 6 0.77 D, r 50.38 and the postoperative centroid was 0.63 @ 87 60.52 D, r 5 0.48. Although there was a significantreduction in the mean keratometric astigmatism, it wasconsiderably less than that seen for the refractive datadue to the lenticular astigmatism.

Figure 8C is the doubled-angle plot of the intraoc-ular astigmatism at the corneal plane (posterior cornealastigmatism 1 lenticular astigmatism). The centroid is0.48 D 3 178 6 0.49 D, r 5 0.59. The centroid of theintraocular astigmatism is ATR while the front surfaceastigmatism is WTR. The intraocular astigmatismtherefore partially compensates for front surface astig-matism and accounts for the lower degree of refractiverelative to keratometric astigmatism seen in this andother studies.

Figures 9A shows a comparison of the range, stan-dard deviation, and mean values of the preopera-tive and postoperative astigmatism and SIRC measured

by refraction and keratometry. As expected, the refrac-tive data show a much greater reduction in astigmatism.

Figure 6A. (Holladay) Doubled-angle plot of the preoperativespectacle plane refractive astigmatism at a vertex of 12 mm. Thecentroid was 10.96 D 3 88 6 0.85 D, r 5 0.43. These data illustratethat 89% of the patients had some degree of WTR astigmatism (leftside of circle) preoperatively. This finding is expected for the relativelyyoung average age of the refractive surgery patient (ie, 44.7 6 8.26years).

Figure 6B. (Holladay) Doubled-angle plot of the preoperativekeratometric cylinder at the corneal plane. The centroid of the kera-tometric astigmatism was 11.29 D @ 88 6 0.77 D, r 5 0.38. Thedifference in magnitude and axis from the refractive plots confirms thepresence of lenticular astigmatism.

Figure 6C. (Holladay) The centroids and standard deviations ofthe preoperative corneal plane refractive and the keratometric astig-matism are shown on the same plot to emphasize the difference. Thisdifference confirms the presence of lenticular astigmatism. A 0.48 Ddifference in the centroids indicates that on the average, the intraoc-ular astigmatism is 0.48 D ATR.



Figure 9B is a histogram of the frequency distribu-tion of preoperative refractive cylinder (black) at thespectacle plane and the postoperative refractive cylinder(red) at the spectacle plane. The histogram shows a sig-nificant left shift in the data, indicating a substantial

reduction in the astigmatism. Figure 9C is a cumulativeplot of the same data. The shaded area between the stairsteps represents the improvement in the postoperativeastigmatism.

Figure 9D is a histogram of the frequency distribu-tion of preoperative keratometric cylinder (black) andthe postoperative keratometric cylinder (red) at the spec-tacle plane. The histogram shows a left shift in the data,indicating a significant reduction in the keratometricastigmatism. Figure 9E is a cumulative plot of the samedata. The shaded area between the stair steps representsthe improvement in the postoperative astigmatism.There is less improvement in keratometric astigmatism.

DiscussionTable 3 and Figure 9A show the average magnitude

of astigmatism was 1.46 6 0.61 D before and 0.40 60.38 D after surgery. This indicates that, regardless ofaxis, 50% of the patients were left with less than 0.40 Dof astigmatism and 50% were left with more than0.40 D. This is an important value that helps cliniciansand patients develop realistic expectations.

The PE plot (Figure 3D) illustrates the commonpreference of surgeons to err on the myopic side (79%).The target postoperative refraction for the patients in

Figure 7A. (Holladay) Double-angle plot of the postoperativespectacle plane refractive astigmatism. The centroid of the postoper-ative cylinder at the spectacle plane is 0.11 D 3 83 6 0.37 D, r 5 .49,a dramatic reduction from the preoperative astigmatism.

Figure 7B. (Holladay) Double-angle plot of the postoperative kera-tometric astigmatism. The centroid of the postoperative keratometricastigmatism is 10.63 @ 87 6 0.52 D, r 5 0.48. Notice the greatervariance in the keratometric than in the refractive astigmatism.

Figure 7C. (Holladay) Doubled-angle plot of the postoperativecorneal plane refractive and keratometric astigmatism. The centroidand standard deviation of the refractive and keratometric postopera-tive astigmatism on the same plot show very little correlation.



this study was plano. The mean postoperative refractionwas 20.61 6 0.82 D, suggesting the actual targets forsome patients were probably slightly myopic. The largerdegree of outcome variability as the magnitude of thetreatment increase can also be seen.

The DEQ relates refractive error to UCVA. Fig-ure 4B shows that most patients have a DEQ of ,1,indicating good uncorrected vision. The mean postop-erative DEQ was 0.96 D, which for a nominal 3.0 mmpupil corresponds to 20/24 visual acuity. However, asthe attempted treatment increased above 6.0 D of myo-pia, the variability of the outcome increased.

The lack of correlation between the preoperativekeratometry and refraction (at the corneal plane) indi-cates the presence of intraocular (posterior cornealand/or lenticular) astigmatism. Both the preoperativekeratometric and corneal plane refractive astigmatismwere WTR, but the magnitude of the keratometric cyl-inder was 0.48 D larger, 1.29 D and 0.81 D, respec-tively. The intraocular astigmatism therefore must be0.48 D and ATR to account for the net corneal planerefractive astigmatism. The value of the centroid for theintraocular astigmatism at the corneal plane is shown in

Figure 8C and is 0.48 3 178. We have found the samecompensatory relationship between extraocular (ie,front surface power) and intraocular astigmatism afterreviewing the data from other studies.17,18,22

Figure 8A. (Holladay) (A) Doubled-angle plot of the preoperativeand postoperative corneal plane refractive astigmatism. These datademonstrate that the refractive astigmatism centroid is 8 times (0.81D/0.11 D) closer to zero and the standard deviation of the astigmatismwas reduced by a factor of 2.0 (0.73 D/0.36 D). The amount of im-provement can be seen directly on the doubled-angle plots by howmuch closer the postoperative centroid is to the origin and how muchtighter the cluster (smaller the ellipse).

Figure 8B. (Holladay) The relationship between the preoperativeand postoperative keratometric astigmatism. The preoperative cen-troid was 1.29 @ 88 6 0.77 D, r 5 0.38 and the postoperativecentroid was 0.63 @ 87 6 0.52 D, r 5 0.48. Although there was asignificant reduction in the keratometric astigmatism, it is much differ-ent than the refractive values because of the lenticular astigmatism.

Figure 8C. (Holladay) Doubled-angle plot of the intraocular astig-matism (posterior corneal astigmatism 1 lenticular astigmatism). Thecentroid is 0.48 D 3 178 6 0.49 D, r 5 0.59. Notice the centroid ofthe intraocular astigmatism is ATR. The front surface being WTR andthe intraocular ATR explains why the refractive astigmatism is lessthan the keratometric.



The preoperative and postoperative doubled-angleplots (Figures 6 to 8) document the relocation of thecentroids toward the origin and contraction of their re-spective elliptical standard deviations. These changes in-dicate a significant reduction in astigmatism. To have apostoperative centroid nearly zero (0.11 D) with 95% ofpatients with less than 0.74 D of astigmatism is remark-able. Figure 7A shows these results. The shaded area onthe cumulative distribution plots, Figures 9C and 9E,demonstrate this dramatic improvement.

SummaryThe SIRC for an individual patient is a unique

value, and all valid methods of analysis will yield thesame result. Vector and SEQ analyses provide differenttypes of information that, when considered together,provide a thorough quantitative analysis of refractivechanges.

Sources of error in refractive outcome statistics in-clude the use of multiple lens systems in the phoropter,errors in vertex calculations, difficulty in accurately de-

fining the axis of astigmatism, and failure to considermeasurement errors when working with keratometricdata.

Refractive data must be adjusted for vertex distancebefore comparison to topographic or keratometric data.Descriptive statistics such as means, standard deviations,shape factor (r), standard error of the mean, and corre-lation coefficients can be calculated only after convert-ing polar to Cartesian values.

Doubled-angle formulas and plots are necessary toaccurately compute and display the results of aggregateastigmatism analysis and are particularly useful in inter-

Figure 9A. (Holladay) Temporal comparison of the range, stan-dard deviation, and mean value of the preoperative, postoperative,and astigmatism for both spectacles plane refraction and keratom-etry. The data show that refractive changes correlate, ie, the spreadand magnitude of the postoperative changes are a result of the re-fractive SIRC. The keratometric data show a reduction but not asgreat as seen with the refractive cylinder.

Figure 9B. (Holladay) Histogram of the frequency distribution ofpreoperative refractive cylinder (black) at the spectacle plane and thepostoperative spectacle plane refractive cylinder (red). The histo-grams 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 theimprovement in postoperative astigmatism.



preting errors in cylinder magnitude and alignment.The DEQ allows comparison of refractive and visualresults. The decrease in refractive predictability withhigher corrections was well demonstrated with plots ofSEQ, DEQ, and doubled-angle plots of the SIRC. Thecorrelation between refractive and keratometric astig-matism was poor for preoperative, postoperative, andSIRC data, indicating the presence of intraocular astig-matism and the limitation of manual keratometry.

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Figure 9D. (Holladay) Histogram of the frequency distribution ofpreoperative keratometric cylinder (black) and the postoperative kera-tometric cylinder (red) at the cornea. The histograms show a substan-tial reduction in the astigmatism.

Figure 9E. (Holladay) Cumulative plot of the data in Figure 9D. Thegray area between the red and black stair steps represents the im-provement in the postoperative astigmatism. The keratometric astig-matism shows less improvement than the refractive astigmatism.



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