Evaluating and reporting astigmatism is complex. Astigmatism is the difference between the principal powers of a toric surface, lens, and refraction. It is therefore not considered a primary variable but a secondary variable. Because it is a secondary variable, calculations such as vertexing from the spectacle to the corneal plane must be conducted with the primary variables to avoid errors.

Astigmatism may be expressed with either axis or power, and the principal powers are always 90 degrees apart (orthogonal). For axis, we will use an “×” and for power we will use an “@”. A +1.00 × 90 is identically equal to +1.00 @ 180. An astigmatic spectacle prescription may be expressed in 1 of 3 forms using spheres and cylinders: (1) plus spherocylinder, (2) minus spherocylinder, and (3) cross-cylinder. An example using axes would be +1.00 + 2.00 × 90, +3.00 – 2.00 × 180, and +1.00 × 180 and +3.00 × 90, respectively. The astigmatism is the cylinder in the spherocylindric form and the difference between the cylinders in the cross-cylinder form.

The distribution of the magnitude of keratometric astigmatism in the cataract population is summarized in Table 1 , which was originally measured in 0.25 diopters (D) steps.^{1} Currently, optical biometers, keratometers, topographers, and tomographers measure to a precision of 0.01 D, so the values have been adapted to show the percentage of astigmatism magnitudes equal to or below each dioptric value to 0.01 D. The median is 0.79 D, where 50.0% of this population is below this value and 31.1% of the population have ≤0.50 D and need no intervention to correct the astigmatism.^{2} The remaining 68.9% above 0.50 D would benefit in their uncorrected visual acuity with some type of astigmatism correction.

Table 1. -
Distribution of Astigmatism in Cataract Population

Astigmatism (D)
Cumulative (%)
≤0.25
13.1
≤0.50
31.1
≤0.75
47.6
≤0.79
50.0
≤1.00
60.6
≤1.25
70.3
≤1.50
77.5
≤1.75
82.9
≤2.00
86.9
≤2.25
89.9
≤2.50
92.2
≤2.75
94.0
≤3.00
95.4
≤3.25
96.5
≤3.50
97.3
≤3.75
98.0
≤4.00
98.5

The prevalence of the orientation of the astigmatism is related to the magnitude. Using the convention recommended by Abulafia et al., equal quadrants of 45 degrees for against-the-rule (ATR) (157.5 to 22.5 degrees), oblique 45 (22.5 to 67.5 degrees), with-the-rule (WTR) (67.5 to 112.5 degrees) and oblique 135 (112.5 to 157.5 degrees) astigmatism, we can show the relationship of the orientation as a function of magnitude (Figure 1 ).^{3} As magnitude of astigmatism increases, the percentage with ATR is nearly constant, varying from 31% to 40%; WTR, by contrast, increases from 31% to 64%, taking all the increase from oblique astigmatism. Oblique 45 (superonasal) and oblique 135 (superotemporal) decrease dramatically from 20% to 3% and are not significantly different from one another.

Figure 1.: The prevalence of the orientation of the astigmatism is a function of the magnitude. ATR is nearly constant varying from 26% to 31%, WTR, by contrast, increases from 26% to 64% taking all the increase from oblique astigmatism. Oblique 45 (superonasal) and oblique 135 (superotemporal) decrease dramatically with increasing magnitude and are not significantly different from one another. ATR = against-the-rule; WTR = with-the-rule

Scalar Considerations
The mean absolute value of the magnitude of the astigmatism is a scalar value:(1) $\text{Meanabsoluteastigmatism}=\frac{{{\displaystyle \sum}}^{\text{}}\left|\text{Astigmatism}\right|}{n}$

The mean absolute astigmatism is the scalar average of the magnitudes of the astigmatism. We use mean absolute astigmatism to avoid confusion with M _{Mean} ($\overline{M}$ ), which is the magnitude of the average astigmatism vectors (Eq 6 ) and will have an axis or meridian (Eq 7 ). The scalar value of mean absolute astigmatism is particularly helpful to surgeons in evaluating their postoperative residual astigmatism because it is completely independent of the axis and correlates best with uncorrected visual acuity and patient satisfaction.^{4,5}

Although the refractive results are typically reported at the spectacle plane, it is often desirable to report them at the corneal plane. The cross-cylinder form is the only form that can be used to vertex a prescription from the spectacle to the corneal plane, since it is the only form that expresses the principal powers of the prescription.^{6} This calculation is specific for a given object distance and is therefore different for a distant and near object. This difference is the reason for patients with myopia and early presbyopia having problems at near when going from spectacles to contact lenses, whereas their hyperopic counterpart becomes better at near. A vertex calculation can only be performed for the first lens in the optical system (spectacles). For example, it is not possible to do a vertex calculation for an intraocular lens that is behind the cornea as suggested by Goggin et al. and refuted by Simpson and Holladay because it is not the first lens in the system.^{7–9}

Vector Considerations
Unfortunately, the nomenclature in the literature for corneal astigmatism is not only confusing but also ambiguous due to changes in measuring devices over time. Net astigmatism, total astigmatism, and total keratometric (TK) astigmatism refer to the actual vector SUM of the FRONT and BACK SURFACE ASTIGMATIC POWER at the corneal vertex using actual indices of refractions for air (1.000), cornea (1.376), and aqueous (1.336), respectively. STANDARD KERATOMETRY uses the FRONT SURFACE RADIUS ONLY but uses the standardized index of refraction (1.3375) and not the corneal index of refraction (1.376). It reports STANDARDIZED KERATOMETRIC ASTIGMATISM. All these are astigmatism vectors when the angle is doubled and may be reduced to scalars when only using the magnitudes.

Astigmatism is not considered a vector. Although it has magnitude, an axis or meridian has 2 directions. When we state the axis is 30 degrees, we mean the 30- and 210-degree axis and use the lower value as a shorthand notation. Because an axis has 2 angles, it is not considered a Euclidean vector. A Euclidean vector must have one and only one direction. When representing a vector in a Euclidean polar coordinate system, the angle is zero degrees at 3 o'clock, 90 degrees at 12 o'clock, 180 degrees at 9 o'clock, 270 degrees at 6 o'clock, and 360 degrees is back to 3 o'clock. The directions (polar angles) are considered semiaxes or semimeridians.

Because astigmatism is not a vector, vector algebra may not be used to add obliquely crossed cylinders directly. Stokes recognized the problem and in 1849 made a presentation at the British Association for the Advancement of Science in which he used graphical vector analysis with all astigmatic angles doubled.^{10} For the 30- and 210-degree axis, the doubled angles would become 60 and 420 degrees, and the 2 angles become the same angle (420 degrees is the same as 60 degrees). The double-angle astigmatism vector satisfies all our requirements for a Euclidean vector.

There are 2 additional methods besides the graphical method: the analysis of the law of sines and cosines and the rectangular coordinate method, which decomposes the vector into components called eigenvectors.^{11–15} Naeser's 2008 thesis fully explains in detail the assessment and statistics of various aspects of astigmatism.^{16} We will further explore the concepts of the statistics, evaluation of surgically induced astigmatism, and prediction error (PE).

Gartner first described the optical decomposition of a cylinder in vector analysis, which are identical to C0 and C45 decomposition by Bennet and Rabbets.^{13,17} Any spherocylinder may be decomposed into a spherical equivalent (SEQ) and 2 cross-cylinders at (0 degree/90 degrees) and (45 degrees/135 degrees).

Definitions
For a cylinder of magnitude M and axis Θ, the 2 vector components are as follows:(2) $x=M\mathrm{cos}2\theta $ (3) $y=M\mathrm{sin}2\theta $

The x and y values are identical to Bennet’s C0 and C45 decomposition. The x value is relative to the horizontal cardinal axis (x axis), and the y value is relative to the vertical oblique axis (y axis). The centroid of the data is the arithmetic mean of x and y :(4) $\text{Meanof}x=\overline{x}=\frac{{{\displaystyle \sum}}^{\text{}}x}{N}$ (5) $\text{Meanof}y=\overline{y}=\frac{{{\displaystyle \sum}}^{\text{}}y}{N}$

In mathematics and physics, the centroid or geometric center of a plane (2D) figure ($\overline{\text{x}}\text{and}\overline{\text{y}}$ ) is the arithmetic mean position of all the points in the figure. When the centroid is not at the origin, it represents a constant offset of the aggregate data. If an adjustment is made (as with the improvement of a toric calculator) such that the centroid is moved to the origin without changing the total variance of the data, there will be improvement in the outcomes. In the double-angle plot figures (eg, Figure 5 ), the $\overline{\text{x}}$ and $\overline{\text{y}}$ values for the centroid have been converted back to their polar equivalent. The resulting polar equivalent vector is the magnitude of the average astigmatism at the meridian or axis of the average astigmatism. The calculation of magnitude and direction (polar coordinates) of the mean astigmatism vector from x and y is as follows:(6) ${M}_{\text{Mean}}=\overline{M}\text{}=\text{}\sqrt{{\left(\frac{{{\displaystyle \sum}}^{\text{}}x}{N}\right)}^{2}+{\left(\frac{{{\displaystyle \sum}}^{\text{}}y}{N}\right)}^{2}\text{}}=\sqrt{{\overline{\text{x}}\text{}}^{2}+{\overline{\text{y}}\text{}}^{2}\text{}}$ (7) $\theta =\mathrm{Arctan}\frac{\overline{M}-\text{}\frac{{{\displaystyle \sum}}^{\text{}}x}{N}}{\frac{{{\displaystyle \sum}}^{\text{}}y}{N}}=\text{}\mathrm{Arctan}\frac{\overline{M}-\text{}\overline{\text{x}}}{\overline{\text{y}}}$

Note that the mean absolute astigmatism in Equation (1) is a scalar measure of astigmatism, and the mean astigmatism vector in Equations (6 and 7) is a vector. For example, if we had 2 datapoints of 1 D @ 90 and 1 D @ 180, the mean absolute astigmatism would be 1 D and the mean astigmatism vector would be zero, because the vectors are in opposite direction.

The variances and standard deviations (SDs) of the astigmatic vectors are computed as follows:(8) $\text{Varianceof}x={\sigma}_{x}^{2}=\frac{1}{N}{\displaystyle \sum}_{i=1}^{N}{({x}_{i}-\overline{x})}^{2}$ (9) $\text{Varianceof}y={\sigma}_{y}^{2}=\frac{1}{N}{\displaystyle \sum}_{i=1}^{N}{({y}_{i}-\overline{y})}^{2}$

The SDs are the square roots of the respective variances of x and y (σ_{x} and σ_{y} ):(10) $\text{Totalvariance}={\sigma}_{x}^{2}+{\sigma}_{y}^{2}$ (11) $\text{TotalStandardDeviation}=\sqrt{\text{TotalVariance}}=\sqrt{{\sigma}_{x}^{2}+{\sigma}_{y}^{2}}$

The total variance and total SD are independent of the rotation and translation of the coordinate system for the confidence ellipses to be discussed, so they can be calculated from the original vector components. Note that total SD is not equal to the sum of the SDs of x and y (it is the square root of the sum of the squares of the SDs of x and y ).

METHODS
Datasets
Three datasets were available for retrospective analysis. Each dataset was from a trial sponsored by Alcon Laboratories, Inc. approved prospectively by an institutional review board, and adhered to the tenets of the Declaration of Helsinki and its statement of ethical principles for medical research involving human subjects. Informed written consent was obtained from all patients. Patients were required to have normal eye examinations except for cataract and no other history of eye disease. All IOLs were in the AcrySof series (Alcon Laboratories, Inc.).

Dataset 1 (Low Preoperative Astigmatism)
The first dataset was from a study from 2012 to 2013. It comprised 265 eyes that received a toric monofocal IOL with 1.0 D of toricity at the IOL plane (SN6AT2 IOL). Patients were required to have between 0.50 D and 1.0 D of preoperative keratometric astigmatism; the mean value was 0.72 D. Figure 2, A shows the distribution of preoperative keratometric astigmatism in the toric group: 82 (31%) had ATR astigmatism, 60 (23%) oblique astigmatism at 45 degrees, 81 (31%) WTR astigmatism, and 42 (16%) oblique astigmatism at 135 degrees. All patients were required to be seen for a follow-up at 6 months, at which time manifest refraction was performed.

Figure 2.: The distribution of the preoperative keratometric astigmatism is displayed for the 3 toric datasets: (A ) dataset 1 with low astigmatism (265) between 0.50 D and 1.00 D, (B ) dataset 2 with medium astigmatism (210) between 0.75 D and 2.88 D, (C ) dataset 3 with high astigmatism (385) between 1.68 D and 8.50 D.

Dataset 2 (Medium Preoperative Astigmatism)
The second dataset was from a study from 2002 to 2005 (ClinicalTrials.gov identifier: NCT01214863B). It comprised 209 eyes that received a toric monofocal IOL (SA60T3, SA60T4, or SA60T5 IOL of toricity 1.50 D, 2.25 D, and 3.0 D, respectively). These eyes had between 0.75 D and 2.88 D of preoperative keratometric astigmatism as measured by manual keratometry. Figure 2, B shows the distribution of preoperative keratometric astigmatism: 81 (39%) had ATR astigmatism, 21 (10%) oblique astigmatism at 45 degrees, 98 (47%) WTR astigmatism, and 9 (4%) oblique astigmatism at 135 degrees. All patients were required to be seen for a follow-up at 6 months, at which time manifest refraction was performed. In this study, no attempt was made to enroll a uniform distribution of keratometric meridians, and keratometry was performed with a manual keratometer. As a result, the cohort had few eyes with oblique astigmatism.

Dataset 3 (High Preoperative Astigmatism)
The third dataset was from a postapproval study from 2012 to 2015 (ClinicalTrials.gov identifier: NCT01601665C). It comprised 385 eyes receiving a toric monofocal IOL (SN6AT6, SN6AT7, SN6AT8, or SN6AT9 IOL of toricity 3.75 D, 4.50 D, 5.25 D, and 6.0 D, respectively). The preoperative keratometric astigmatism ranged from 1.68 to 8.05 D. Figure 2, C shows the distribution of preoperative keratometric astigmatism in this toric group: 63 (28%) had ATR astigmatism, 10 (4%) oblique astigmatism at 45 degrees, 142 (63%) WTR astigmatism, and 11 (5%) oblique astigmatism at 135 degrees. All patients were required to be seen for a follow-up at 3 months, at which time manifest refraction was performed. In this study, no attempt was made to enroll a uniform distribution of preoperative keratometric meridians, and the preoperative keratometry was performed with a manual or optical keratometer. Few eyes had oblique astigmatism.

SURGICAL TECHNIQUE
Approximately 20 surgeons were involved in each of the 3 studies. All patients received a manual temporal near-clear corneal incision. The incision size decreased over time, starting from a mean of approximately 3.1 mm in the earliest trial to approximately 2.5 mm in the latest trial. The surgeons were requested to use their standard technique for orienting the toric IOL, which was to place it at the preoperative steep keratometric meridian.

SURGICALLY INDUCED ASTIGMATISM
The basis for a toric IOL calculation is to calculate the necessary powers in the principal meridians for a given refractive target, and the difference between them determines the ideal toricity. Whether one uses Snell’s law or Gaussian vergence equations makes no difference for axial or paraxial powers. Although IOL formulas may differ in the SEQ power, if the same K readings are used, the difference between the IOL powers in the principal meridians is nearly the same, so the toric difference from the various formulas is minimal.^{3} The major differences in toric IOL calculation formulas are how the K readings are modified using various components that contribute to the postoperative refractive astigmatism.

Surgically induced astigmatism (SIA) is especially important to the ophthalmic surgeon. It has been used to determine the effect of arcuate corneal incisions, cataract incisions for toric IOLs, and many other astigmatic procedures. Unfortunately, the terminology has become confusing in recent years due primarily to our advancements in technology.

ΔK (SIA_{ΔK} )
The landmark article by Jaffe and Clayman in 1975 reported the K vector difference (ΔK Method) between the postoperative and preoperative Ks at 5 to 7 weeks postoperatively for 1557 cataract extractions performed using a 150-degree (5 o'clock hours) superior limbal cataract incision with various incision designs, sutures, and suture techniques.^{18} The prevalence of preoperative astigmatism in their study using a keratometer accurate to 0.25 D was 42.5% ATR, 30.0% WTR, 25.8% with no astigmatism, 1% oblique at 45 degrees, and 0.7% oblique at 135 degrees. The mean or median astigmatism was not reported but is likely near 0.79 D from Table 1 ; the prevalence of astigmatism types is consistent with Figure 1 . The K method reflects anterior surface changes only and uses a ring rather than an area to measure the astigmatism. The result is more variability when irregular astigmatism is present, especially postoperatively.

Back-Calculated Total SIA
Back-calculated total SIA (Back-Calculated SIA_{Total} ) is the term used to account for all factors that contribute to the difference between the preoperative K reading and the ideal, back-calculated K reading that results in no PE because it is derived from the actual postoperative refraction. The back-calculated total SIA method may still be thought of as a ΔK, but the difference from the preoperative K is calculated from the ideal, back-calculated K. The Back-Calculated SIA_{Total} will therefore include any posterior corneal surface effects and other contributions, such as IOL tilt or decentration, refractive changes in the anterior and posterior corneal surfaces from the cataract incisions, and systematic differences in measured keratometric vs actual corneal refractive astigmatism.^{9,19,A} Theoretically, these factors in the cornea and IOL could be measured and added together vectorially, resulting in a Back-Calculated SIA_{Total} , but, unfortunately, the accuracy of these measurements in the cornea and the inability to measure the contributions of IOL tilt and decentration to the astigmatism at the corneal plane are inaccurate or not available.

The more accurate way to determine the Back-Calculated SIA_{Total} is from the back-calculated K using the Gaussian vergence formula.^{20,21} The vector calculation is tedious, but the concept is simple. One needs to know the ideal K reading that is consistent with the actual postoperative refraction and the observed axis, toricity, and spherical equivalent of the toric IOL. The vector difference in the ideal, back-calculated K and the one that was used preoperatively is the Back-Calculated SIA_{Total} vector that must be added vectorially to the preoperative K reading that was measured to achieve the actual outcome. The actual equations (12–16) , are given further.

Equations 12 and 13 are the Gaussian vergence formula solved for the customary IOL power and the refraction, respectively^{22} :(12) $\text{IOL}=\frac{1336}{{\text{AL}}_{\text{o}}-{\text{ELP}}_{\text{x}}}-\frac{1336}{\frac{1336}{\frac{1000}{\frac{1000}{\text{DPostRx}}-\text{V}}+{\text{K}}_{\text{r}}}-{\text{ELP}}_{\text{x}}}$ where IOL = intraocular lens power (D); AL _{o} = optical axial length; ELP _{x} = expected lens position (mm) (distance from corneal vertex to principal plane of the thin IOL); DPostRx = desired postoperative refraction in diopters; V = vertex distance (mm) of refractions and K _{r} = net corneal power (D).

K_{r} varies depending on the value of the standardized index used to convert anterior corneal power to net corneal power, and this varies among various IOL formulas. We use the original Binkhorst net index of refraction for K _{r} of 4/3, and the measured standardized keratometric index of refraction (K_{K} ) is 1.3375.^{23} Therefore, K_{r} = K_{K} * (4/3−1)/(1.3375−1), is approximately 0.56 D less than the measured power for 45 D standardized keratometry.^{22} AL_{o} is also the Binkhorst modification, which is the measured AL_{m} plus 0.2 mm.^{24} Although this should not be necessary with optical biometry (interferometry), because it includes retinal thickness, the first optical biometers (and therefore all subsequent biometers) were calibrated using immersion A-scan to preserve the lens constants, so the adjustment is still valid. We can rearrange equation (12) , solving for DPostRx , which allows us to predict the exact refraction from available IOL powers (Eq 13 ) as follows:(13) $\text{DPostRx}=\frac{1000}{\frac{1000}{\frac{1336}{\frac{1336}{\frac{1336}{{\text{AL}}_{\text{o}}-{\text{ELP}}_{\text{x}}}-\text{IOL}}+{\text{ELP}}_{\text{x}}}-{\text{K}}_{\text{r}}}+\text{V}}$

The determination of the Back-Calculated SIA_{Total} for a toric IOL requires solving the vergence equation for K_{r} , rather than IOL power or refraction:(14a) ${\text{K}}_{\text{r}}=\frac{1336}{\frac{1336}{\frac{1336}{{\text{AL}}_{\text{o}}-{\text{ELP}}_{\text{x}}}-\text{IOL}}+{\text{ELP}}_{\text{x}}}-\frac{1000}{\frac{1000}{\text{APostRx}}-\text{V}}$

This equation uses the APostRx and actual implanted IOL power (IOL ) to determine the back-calculated K_{r} . It is the back-calculated K_{r} that: (1) is consistent with the other 5 primary variables and (2) predicts the actual postoperative refraction for a given IOL, AL _{o} and ELP _{x} .

Although the vergence equations are usually written with scalar values (SEQ powers), they may also be written as vectors for K_{r} , toric IOL, and actual postoperative refraction using doubled angles for the meridians or axes^{3,25} :(14b) $\overrightarrow{{\text{Actualback}-\text{calcK}}_{\text{r}}}=\frac{1336}{\frac{1336}{\frac{1336}{{\text{AL}}_{\text{o}}-{\text{ELP}}_{\text{x}}}-\overrightarrow{\text{IOL}}}+{\text{ELP}}_{\text{x}}}-\frac{1000}{\frac{1000}{\overrightarrow{\text{APostRx}}}-\text{V}}$

The actual ELP_{x} (not predicted) is back calculated using the SEQ postoperative refraction, SEQ IOL power, axial length, and SEQ K.^{22} When the actual toric IOL (magnitude, toricity, and observed orientation) and actual postoperative refraction are aligned (same axis or 90 degrees apart), then the calculation may be performed in both principal meridians using the scalar values in each meridian. In most cases, however, the 2 vectors are not aligned, and the calculation must be performed using a cross-cylinder calculation.^{12} Once the Actual Back-Calculated K_{r} vector is determined, the Actual Back-Calculated K_{k} is simply the magnitudes of the K_{r} vector multiplied by [1.3375 −1.000)/(4/3 −1)] = 0.3375/(1/3), the inverse of the Net K calculation: (15) $\overrightarrow{{\text{Back}-\text{calcK}}_{\text{k}}}=\overrightarrow{{\text{back}-\text{calcK}}_{\text{r}}}\times 337.5/(1000/3)$

It is important to note that the Actual Back-Calculated K_{k} is derived from the actual observed axis of the toric IOL (not the intended) and the actual postoperative refraction. It is the only K_{k} that is consistent with these values. Once the Actual Back-Calculated K_{k} vector is determined, the Actual Back-Calculated SIA_{Total} is the vector difference between the Actual Back-Calculated K_{K} and the preoperative K_{K} (Δ K). The Back-Calculated SIA_{Total} is analogous to a Jackson cross-cylinder, that is, it normally has an SEQ of zero and the cylinders are orthogonal, equal power, and opposite sign. The change in SEQ is zero because the SEQ power of the cornea should not change due to the Law of Gaussian Curvature.^{26} This finding is also confirmed with a coupling ratio of −1, indicating an equal and opposite amount of orthogonal steeping and flattening, maintaining the SEQ K^{12} :(16a) $\overrightarrow{\text{Back}-\text{CalculatedTotalSIA}}=\text{}\overrightarrow{\text{Back}-\text{Calculated}{K}_{k}}\text{}-\text{}\overrightarrow{\text{Preop}{K}_{k}}$

When a dataset of values is available (including the observed axis of the IOL), the patients may be grouped by magnitude and meridian of the preoperative keratometry, so that the centroid (mean vector) may be calculated for each group, thus generating a Back-Calculated SIA_{Total} vector, specific for every patient.^{20,21} The patient-specific Back-Calculated SIA_{Total} is then added to the measured preoperative K for a new patient (Eq 16b ) and then used in the forward toric IOL calculation (Eq 12 ) for the meridional IOL powers:(16b) $\text{Predicted}\overrightarrow{\text{Back}-\text{Calculated}{K}_{k}}=\text{}\overrightarrow{\text{Preop}{K}_{k}}+\overrightarrow{\text{Back}-\text{CalculatedTotalSIA}}$

The Back-Calculated SIA_{Total} is the vector that when added to the preoperative K_{K} predicts the Back-Calculated K_{K} and in turn the predicted ideal meridional alignment of the toric IOL and predicted postoperative refraction. Postoperatively, the difference in the observed axis and the ideal axis is how much the toric IOL must be rotated to achieve the minimum residual astigmatism.^{21} The residual refraction at the ideal axis may be determined from Equation 13 and does not require a vector calculation since it is assumed the toric IOL will be aligned with the steep meridian of the Back-Calculated K_{k} , and a scalar calculation in each principal meridian is sufficient.

The Back-Calculated SIA_{Total} for a small temporal incision is shown in Figure 3 for 4 control datasets (nontoric IOLs): 222 T0 controls (blue), 253 T2 controls (green), 202 T3–T5 IOLs (orange), and 233 T6–T9 IOLs (red).^{20} The centroid of the ellipses (or circles) shifts approximately 0.25 D ATR from T0 (blue) to T6–T9 (red). The radius of the circle and semidiameters for the ellipses increase with the magnitude of the preoperative K astigmatism. Note that the ellipses are rotated and not parallel to the x -axis and y -axis.

Figure 3.: The Back-Calculated SIA_{Total} for a small temporal incision is shown for 4 control datasets (nontoric IOLs): 222 T0 controls (blue ), 253 T2 controls (green ), 202 T3–T5 (yellow ), and 233 T6–T9 (red ). The centroid of the ellipses (circles ) shifts approximately 0.25 D ATR from T0 (blue ) to T6–T9 (red ). The radius of the circle and semidiameters for the ellipses increase with the magnitude of the preoperative keratometric astigmatism. ATR = against-the-rule; SIA = surgically induced astigmatism

The variation in Back-Calculated SIA_{Total} is due to several factors: preoperative magnitude and meridian of astigmatism, variations in posterior corneal astigmatism, the astigmatic response of the cornea as a function of the preoperative steep meridian, the decentration and tilt of the IOL, and perhaps other factors. The accuracy for WTR preoperative astigmatism (3 o'clock) and ATR (9 o'clock) is good, but the variability of the T3–T5 and T6–T9, superiorly and inferiorly, is due to the limited number of cases in the oblique meridians for medium and high amounts of astigmatism as shown in Figure 1 . The Holladay Toric Formula uses the Back-Calculated SIA_{Total} that is derived from a large composite dataset from multiple manufacturers.^{20,21} The centroid and size of the perimeters would also change as a function of the location of the primary incision. For example, if the small incisions were superior, the perimeters are shifted approximately 0.50 D ATR (to the right). For inferior and oblique primary incisions, the change is more complex.

The open access toric IOL calculators using the Back-Calculated SIA_{Total} calculation for a temporal primary incision are available on the Alcon Toric Calculator Website (https://www.myalcon-toriccalc.com/ ) and the Holladay IOL Consultant Website (https://www.hic-soap.com/calc/preop ).

SCALAR MEASURES OF RESIDUAL ASTIGMATISM
We have previously discussed the scalar analysis of the SEQ PE and will now discuss the scalar analysis of residual absolute astigmatism.^{27} The scalar analysis of residual astigmatism is particularly important because it correlates directly with the uncorrected visual acuity (dioptric blur or defocus equivalent) and patient satisfaction.^{4,5,16,28} The dioptric blur from 1 D of astigmatism is the same for any axis. Residual astigmatism is the only parameter for which scalar (arithmetic) analysis may be performed. Analysis of scalar PE can lead to incorrect results and erroneous conclusions.

To compare the accuracy of multiple toric IOL formulas in predicting residual refractive astigmatism, multiple independent datasets must be used in place of a single, dependent dataset. The reason for this is that, with 1 dataset, the final refractive astigmatism will of course be the same regardless of the formula used in the toric calculation. Therefore, when comparing 3 toric IOL formulas, a study should have 3 arms, and the cases should be randomized to each arm with the only variable being the toric IOL formulas. Because the datasets are independent, the size of the study must be much larger with multiple datasets (1 for each arm) to have the same statistical power as a dependent study with only 1 dataset.

Scalar calculations involving residual astigmatism deal only with the magnitude and ignore the angle. Formulas to perform calculations can be found on the website of The R Project for Statistical Computing at https://www.r-project.org/ . One can compare the mean (oph.astig.indepcom ), SD (oph.ind.comvar ), or the percentage of cases that are within various dioptric intervals (±0.25, ±0.50, ±0.75, ±1.00, etc.) (oph.astig.indepintervals ). These values are all scalar measures of residual astigmatism.

Toric IOL Calculation Formulas
In this study, 3 methods of calculating toric IOL power are used for the purpose of illustrating the analytical methodology that we recommend: (1) Zero toric IOL calculation based on anterior K readings alone without any modification for the posterior cornea or other variables such as IOL tilt. (2) The Barrett toric IOL calculator: The values for the Barrett toric calculation came directly from the values used in the 3 U.S. Food and Drug Administration studies. The lens shape factor was used specifically for the Alcon anterior biconvex IOLs. It is unknown as to whether any of the cases in the 3 toric datasets were used to develop the Barrett toric formula. (3) The Holladay toric IOL calculator used the Back-Calculated SIA_{Total} that were derived from a large composite dataset from multiple manufacturers.

ASTIGMATIC PE
Analyzing and reporting the Astigmatic PE is more complex than SEQ PE because it involves both scalar and vector calculations. The vector form of the PE is the actual postoperative refraction minus the predicted postoperative (PO) refraction (Ref).(17) $\text{}\overrightarrow{\text{PredictionError}}=\text{}\overrightarrow{\text{ActualPORef}}\text{}-\text{}\overrightarrow{\text{PredictedPORef}}$

Measures of Vector Astigmatic PE
Vector Magnitude Absolute Prediction
The astigmatic PE vector is the vector difference between the actual postoperative refractive astigmatism minus the predicted refractive astigmatism (Eq 17 ). The refractions are normally at the spectacle plane. Our first analysis will evaluate the magnitudes of the PE vectors. This is a scalar measure but is different from taking the scalar difference of the actual and predicted postoperative astigmatism because we first determine the vector difference and then take the magnitude. For example, suppose the actual postoperative refraction was plano + 1.00 × 90 and the predicted was plano + 1.00 × 180. The vector absolute magnitude difference is the actual minus the predicted is 2.00 × 90 (the axis is relevant). We will call this the vector magnitude absolute PE. The vector magnitude absolute PEs parameters are summarized in Table 2 for each dataset and each toric formula, along with the mean, SD, standard error, median, and other parameters.

Table 2. -
Vector Mean Absolute Prediction Error Parameters.

Underline parameter
Low
Low
Low
Medium
Medium
Medium
High
High
High
Parameter
Zero
Barr
Holl
Zero
Barr
Holl
Zero
Barr
Holl
N
265
265
265
210
210
210
385
385
385
Mean
0.44
0.49
0.39
0.56
0.67
0.60
0.79
0.71
0.70
SD
0.30
0.33
0.25
0.40
0.38
0.37
0.53
0.48
0.50
STD ERR
0.02
0.02
0.02
0.03
0.03
0.03
0.03
0.02
0.03
Median
0.35
0.45
0.34
0.50
0.59
0.52
0.69
0.62
0.58
Min
0.03
0.01
0.01
−2.50
0.01
0.03
0.03
0.01
0.04
Max
1.45
1.80
1.61
1.90
1.99
1.82
3.83
3.88
4.07
KURT
1.04
1.27
2.63
1.04
1.02
0.74
4.05
6.94
7.24
Skew
1.13
1.03
1.38
1.03
0.93
0.96
1.45
1.97
2.06
Shaprio-Wilk
0.91
0.93
0.90
0.93
0.95
0.94
0.91
0.86
0.84
Shapiro P value
8.9E-12
1.1E-09
4.3E-12
1.0E-08
6.7E-07
5.4E-08
1.5E-14
2.2E-16
2.2E-16

Barr = Barrett toric formula; Holl = Holladay toric formula; KURT = kurtosis; STD ERR = standard error; Zero = Zero Toric Formula

When comparing the various dioptric intervals (±0.25, ±0.50, ±0.75, ±1.00, up to ±2.00) of the scalar magnitude absolute PEs for dependent datasets, the McNemar test is used to determine the P values. It compares the pairs of data for each formula for each interval. Because we conducted multiple comparisons on the same dependent variable with 3 formulas, the P values have been adjusted using the Holm correction.^{29}

The percentages within a specific interval (±0.25 to ±2.0 D in 0.25 D steps) are presented in Figure 4 and Table 3, A and the P values using the McNemar test are summarized in Table 3, B , respectively, for the vector absolute PEs. Shaded values are statistically significant. In the low astigmatism dataset, the Zero formula (shaded in blue) was better than the Barrett formula at the ≤0.25 D interval, and the Holladay formula (shaded in green) was better than the Zero and Barrett formulas at the 0.5 D, 0.75 D, and 1.00 D intervals. In the medium astigmatism dataset, the Zero formula (shaded in blue) was better than the Barrett and Holladay formulas, and in the high astigmatism dataset, the Barrett (shaded in yellow) and Holladay (shaded in green) formulas were better than the Zero formula and the Holladay formula (shaded in green) better than the Barrett formula for <0.75 D (68% vs 63%, adjusted P value = .026).

Figure 4.: Cumulative percentages of vector absolute magnitude prediction errors for 0.25 D intervals from ±0.25 to ±2.00 D for the low (0.50 to 1.00 D), medium (1.00 to 2.00 D), and high (>2.00 D) astigmatism datasets for the Zero, Barrett, and Holladay formulas.

Table 3A. -
Percentages of Vector Magnitude Absolute Prediction Errors Below Specific Intervals.

Interval (D)
Low
Low
Low
Medium
Medium
Medium
High
High
High
Zero
Barr
Holl
Zero
Barr
Holl
Zero
Barr
Holl
(%)
(%)
(%)
(%)
(%)
(%)
(%)
(%)
(%)
≤ 0.25
35.09
26.79
33.21
24.76
10.48
16.67
12.73
10.91
12.99
≤ 0.50
64.91
59.62
78.11
50.95
38.10
46.67
32.47
36.62
38.96
≤ 0.75
86.04
82.26
90.57
76.19
65.24
70.00
55.32
63.12
68.05
≤ 1.00
93.96
90.57
97.36
87.62
81.90
85.71
71.69
81.30
80.00
≤ 1.25
97.36
96.60
99.25
93.81
91.90
93.81
82.34
87.79
87.79
≤ 1.50
100.00
98.49
99.62
96.19
97.14
97.14
91.95
92.99
93.51
≤ 1.75
100.00
99.62
100.00
99.05
98.10
99.52
95.58
96.88
96.10
≤ 2.00
100.00
100.00
100.00
100.00
100.00
100.00
97.14
97.66
97.40

Barr = Barrett toric formula; Holl = Holladay toric formula; Zero = Zero toric formula

Table 3B. -
P Values (With Holm Correction) Comparing Formula Pairs From

Table 3A for Specific Intervals.

Barr = Barrett toric formula; Holl = Holladay toric formula; Zero = Zero toric formula

Bivariate Analysis and Double-Angle Plots
Using the astigmatic PE vector, we can use the methods and formulas for performing a bivariate analysis. These analyses traditionally use a Hotelling transformation (t test) and then the Fisher F-test to compute the value of T for various P values (.50, .95, and .99). The use of these techniques has 2 major requirements: (1) the 2 variables must each have a Gaussian (normal) distribution, and (2) the 2 variables must be independent. Unfortunately, we will find that the distributions of x and y are rarely Gaussian. Because the x and y values are determined from a sine and cosine function, they are orthogonal and considered eigenvectors. We will see that they are not actually independent and that the correlation coefficient R determines the rotational orientation of the confidence ellipse.

The best test for normality is the Shapiro-Wilk test. Details for downloading the open-access software from The R Project for Statistical Computing can be found at https://www.r-project . The Shapiro-Wilk test in R is named “shapiro.test”. In our 3 datasets using the 3 formulas, none of the 9 distributions for x or y are normal (Table 4, A, B and C ). Therefore, the Hotelling transformation is not appropriate, and the 95% confidence ellipses are not accurate or reliable. Nevertheless, because its use is so profuse in the literature and, in rare occasions, the data may be in fact be Gaussian, it is worth discussing.

Table 4A. -
Low Astigmatism

x - and

y - Vector Prediction Error Parameters.

Barr = Barrett toric formula; Holl = Holladay toric formula; Zero = Zero toric formula

Table 4B. -
Medium Astigmatism

x - and

y - Vector Prediction Error Parameters.

Parameter
Vector
Vector
Vector
Vector
Vector
Vector
Zero
Zero
Barr
Barr
Holl
Holl
Act—pre
Act—pre
Act—pre
Act—pre
Act—pre
Act—pre
x
y
x
y
x
y
N
210
210
210
210
210
210
Mean
0.128
−0.037
−0.264
−0.052
−0.107
−0.050
SD
0.488
0.459
0.515
0.505
0.514
0.468
STD ERR
0.034
0.032
0.036
0.035
0.035
0.032
Median
0.092
−0.013
−0.261
−0.027
−0.093
−0.107
Min
−2.500
−2.500
−1.834
−1.369
−1.472
−1.378
Max
1.878
1.477
1.592
1.477
1.686
1.441
KURT
1.596
1.459
1.333
0.388
0.771
1.038
Skew
0.344
0.230
0.381
0.191
0.287
0.264
Shapiro–Wilk
9.77E-01
9.69E-01
9.82E-01
9.89E-01
9.89E-01
9.77E-01
Shapiro P value
1.86E-03
1.60E-04
8.12E-03
1.04E-01
9.25E-02
1.78E-03
Total variance
0.449
0.520
0.484
Total SD
0.670
0.721
0.695

Act-pre = actual minus preoperative; Barr = Barrett toric formula; Holl = Holladay toric formula; KURT = kurtosis; STD ERR = standard error; Zero = Zero toric formula

Table 4C. -
High Astigmatism

x - and

y - Vector Prediction Error Parameters.

Parameter
Vector
Vector
Vector
Vector
Vector
Vector
Zero
Zero
Barr
Barr
Holl
Holl
Act—Pre
Act—Pre
Act—Pre
Act—Pre
Act—Pre
Act—Pre
x
y
x
y
x
y
N
385
385
385
385
385
385
Mean
0.333
−0.068
−0.300
−0.061
−0.165
−0.068
SD
0.663
0.582
0.561
0.579
0.603
0.587
STD ERR
0.034
0.030
0.029
0.030
0.031
0.030
Median
0.338
−0.043
−0.314
−0.035
−0.172
−0.082
Min
−3.775
−1.996
−3.819
−2.356
−4.037
−2.216
Max
3.060
2.714
1.483
3.050
2.411
2.892
KURT
4.389
2.965
4.423
3.124
5.348
3.001
Skew
−0.461
0.417
−0.463
0.124
−0.455
0.360
Shapiro–Wilk
9.63E-01
9.58E-01
9.56E-01
9.65E-01
9.52E-01
9.60E-01
Shapiro P value
2.66E-08
5.00E-09
2.39E-09
6.01E-08
7.28E-10
1.07E-08
Total variance
0.779
0.650
0.707
Total SD
0.883
0.806
0.841

Act-pre = actual minus preoperative; Barr = Barrett toric formula; Holl = Holladay toric formula; KURT = kurtosis; STD ERR = standard error

Normal (Gaussian) Distributions
When the x and y components of the astigmatism are both normal, the Hotelling computations are performed. The complete derivation and process for determining the 95% confidence ellipses for the centroid and the dataset are shown by Naeser and Hjortdal.^{30,31} These 2 confidence ellipses are analogous to the use of standard errors of the mean and SDs, respectively, for univariate data. The double-angle plots for the 3 datasets for the 3 formulas are shown in Figure 5 , with the Hotelling 95% confidence ellipses for the centroid (red) and the dataset (blue).

The Double-Angle Plot Tool from ASCRS (http://ascrs.org/tools/astigmatism-double-angle-plot-tool ) is available so that investigators can generate the DAP graph and the 95% confidence ellipses of the centroid and dataset.^{B}

In Figure 6 , it should be noted that the total SD (Eq 11 ) is the hypotenuse (green) formed by ${\sigma}_{x}$ and ${\sigma}_{y}$ and not the area within the ellipse. The vertices of the ellipse always form a rhombus (4 equal sides), which is proportional to the length of the perimeter of the ellipse. In Figure 6 , we have generated a circle (Figure 6, A ) and ellipse (Figure 6, B ), which have the same total SD (4.24 D), yet the area of the ellipse has less than half the area of the circle. The length of the perimeters of the ellipses is a much better measure than the areas for visually comparing 2 total SDs.

Figure 5.: The double-angle plots for the 3 datasets for the 3 formulas (A–I ) are shown with the Hotelling 95% confidence ellipses for the centroid and the dataset. The small ellipses show the 95% confidence ellipses (red ) for the centroid and the larger ellipses, the 95% confidence ellipse for the astigmatic prediction error vectors of the dataset for each formula (blue ) and the 95% confidence convex polygon for the astigmatic prediction error vectors of the dataset for each formula (green ).

Figure 6.: 95% confidence circle (A ) and ellipse (B ) for 2 datasets that have the same total SD = 4.24 D. The areas are quite different and therefore do not provide a good visual comparison for the total SD, but the length of the perimeters are the same and proportional to the total SD.

In Figure 7 , we see that the relationship between the multiplier “t” (x -axis) from the t test for the univariate case (0.52 to 3.29) vs the Hotelling T (y -axis) for the bivariate case (1.00 to 3.72) is linear (dotted red line) between confidence ellipse probabilities from 39% and 99.9%, with a constant difference of 0.43 to 0.52 of the multipliers at each probability for large sample sizes (>200). For the 95% univariate CI, the SD is multiplied by the familiar 1.96, and, for the 95% bivariate ellipse, we multiply the marginal SDs (x and y ) by 2.45.

Figure 7.: The relationship between the multiplier t test for the univariate case (0.52 to 3.49) vs the Hotelling T for the bivariate case (1.00 to 3.72) is shown and is linear (dotted red line ) between confidence ellipse probabilities from 39% and 99.9%, with a constant difference of approximately 0.43 to 0.52 of the multipliers at each probability for large sample sizes (≥200).

It is also important to note that the Shapiro-Wilk test for normality is not infallible and may not always detect a nonnormal distribution. It is always safest to perform the analysis of nonnormal (non-Gaussian) distributions using a nonparametric measure of the depth of a point in a data cloud, described further, to see the difference in the Gaussian and non-Gaussian 95% confidence boundaries.

Analysis of Nonnormal (Non-Gaussian) Distributions Using a Nonparametric Measure of the Depth of a Point in a Data Cloud
We must now discuss the more common and general case where the bivariate distributions for x and y components of the astigmatism vector are not Gaussian. When this occurs, the shape of the confidence boundary is no longer elliptical. The analysis we will use is a nonparametric measure of the depth of a point in a data cloud; it does not rely on the distribution being Gaussian.^{32}

The 95% confidence boundary we used for non-Gaussian distributions is a nonparametric method for finding the points most tightly clustered together without making any assumptions about the type of distribution or the number of sides of the convex polygon that is needed. The center of the data is calculated using the marginal means (summarized in Table 4, A –C ). For the first point in the data cloud, all points are projected onto the line connecting this point to the center of the cloud. This process is repeated for each point in the cloud. For each projection line, the 95% confidence point is determined, and a boundary is formed using these points. All points forming internal angles >180 degrees (reflex angle) are removed leaving a convex polygon containing the 95% of the points closest to the center (oph.astig.datasetconvexpoly.mean ).

There are multiple methods for determining the 95% of points on the projection line that are closest to the marginal means, but all of them result in a convex polygon. The method we have used determines the center of the data using the marginal means. The number of vertices is determined by the specific data; no trimming occurs, and outliers are included. Other methods that remove outliers use bootstrap techniques, rank-based multivariate analysis of variance, sign-based multivariate analysis of variance, and bootstrapping based on the Hotelling T^{2} statistic will affect the 95% confidence point used on the projection line and therefore will usually reduce the boundary of the 95% confidence convex polygon.^{33,34} We used the marginal means with no trimming and outliers included because it retains all original data in the dataset and provides the best comparison to the 95% confidence ellipse.^{32}

In Figure 5 , in addition to the 95% confidence ellipses (blue) are the 95% confidence convex polygons (green) using a nonparametric measure of the depth of the points described earlier. The 95% confidence convex polygon (interior angles <180 degrees) is irregular, and the center (reference) is located at the x and y mean values (Table 2 ). The R function that performs this nonparametric method is named oph.astig.datasetconvexpoly.mean and requires the file Rallfun-v39.txt to execute. The output provides the x and y coordinates of the vertices of the 95% confidence convex polygon and the x and y mean values of the dataset. Details of the input and output format can be found in the README_ASTIGMATISM_CALC_MULTIPLE_DATASETS_CONVEX_POLYGONS File at https://osf.io/nvd59/quickfiles .

The total variance and total SD at the bottom of Table 4, A –C are measures of the spread of the data, with the centroid as the reference, so the spread alone is not an accurate indication of the effect on visual performance, and the magnitudes of each point are the vector magnitude difference, not the residual astigmatism.

For astigmatism induced by a small temporal corneal incision, stability after cataract surgery and the Back-Calculated SIA_{Total} in 3 control datasets (nontoric IOLs for astigmatism from 0 to 4.5 D) show that the measurements before 3 months are variable. There is a continuing ATR drift of approximately 0.25 D during the first year, which is more variable with larger amounts of preoperative astigmatism and greater for ATR than WTR. It should also be noted that there is a mean 0.25 D ATR change in astigmatism per decade with or without cataract surgery, as found in the 20-year longitudinal study by Hayashi et al.^{35}

The sample size for statistical significance from our data indicate that approximately 250 to 400 cases are necessary to show a statistical significance (P ≤ .01) for percentage differences ≥2.5% at intervals from 0.25 to 2.0 D.

DISCUSSION
As we mentioned initially, astigmatism is a secondary variable (the difference in the principal meridional powers), which means that analysis of the random variables must be performed carefully (vertexing, etc.). Scalar calculations for astigmatism such as mean values, medians, and SDs are calculated the same way as other scalar values, such as spherical equivalent IOL power PE. Vector computations of mean vectors (centroids), mean absolute astigmatism, SDs of x and y , and total SD are also clear.

SIA is still confusing. In ΔK (SIA_{ΔK} ), the original value was the difference in the anterior Ks, which seems appropriate because it should reflect the change in corneal power from the surgical incision. The introduction of TK (ΔTK and SIA_{ΔTK} , which adds the posterior corneal astigmatism) is a step forward but still does not fully account for changes in the front and back surface from the incision or other factors such as the decentration and tilt of the IOL.

For SIA, the vector we want is the one that can be added to the preoperative Ks (anterior or TK) to give the exact actual postoperative refraction, when implemented in the Gaussian vergence formula or Snell's law. We struggled with naming this vector but finally settled on Back-Calculated SIA_{Total} because it accounts for all causes of astigmatism that need to be added to the original Ks. The term is not perfect but unique and present in the literature for more than 2 years. The method of calculating is clear, and the formulas (Back-Calculated SIA_{Total} ) are available in a standard form and matrix form.^{21,36} Refinements in the values for the Back-Calculated SIA_{Total} will continue to improve toric IOL outcomes as they are determined for various primary and secondary incision locations and preoperative astigmatism magnitude and meridian and as we acquire more data on components such as lens decentration and tilt.

Residual astigmatism more than 1 D is primarily due to large measurement errors, and sophisticated formulas that improve the percentage within ±0.25 D and ±0.50 D intervals may also increase the percentage above 1 D, making the differences in the mean residual astigmatism less distinguishable. The clinical benchmark of percentages ≤0.50 D for postoperative refractive error is excellent because it reflects the percentage of patients who are satisfied without correction.^{2,5,37} The percentages of residual astigmatism decrease with increasing preoperative astigmatism, and this should be kept in mind when discussing patients expectations.

Vector analysis is especially valuable (1) to the investigator, as it allows one to develop algorithms within the toric calculator that reduces the vector PE and in turn reduces the scalar PE, and (2) for the clinician, to discern features such as leaving a patient routinely with residual WTR vs ATR astigmatism. In addition, as the components of SIA are better understood, it is important to know any bias in a toric formula, which vector analysis will reveal. For example, in Figure 5, A –C (low astigmatism dataset in Table 4, A ) that the centroids for Zero and Barrett formulas are 0.22 D and 0.25 D ATR, respectively. Since the drift is ATR over time, it would be prudent to target for WTR if using these 2 formulas to anticipate the change with time, as Koch has recommended, especially when preoperative keratometry WTR is present.^{38} It is also apparent that the 95% confidence boundaries for the high astigmatism (Figure 5, G , H , and I ) are longer vertically (oblique axis) than horizontally (cardinal axis, WTR, and ATR), indicating that the predictability in the oblique axes is less with all formulas. In Tables 2 , 3, A and B , 4, A –C and Figure 4 , the vector magnitude PEs and percentages for the Holladay formula are superior to the Zero and Barrett formulas.

In summary, for standard reporting of toric formula PE differences, we recommend scalar and vector analysis: (1) for residual astigmatism scalar outcomes, provide the mean values, SDs, and percentages within specific intervals (Table 2 ), and (2) for vector analysis provide the information in Tables 2 , 3, A and B , 4, A –C , along with Figures 4 and 5 . In Figure 5 , the 95% confidence convex polygon should always be shown, since it is the general boundary that has no requirement for the type of distribution, namely Gaussian. It is not necessary to show the 95% confidence ellipse unless one is trying to show the error with the ellipse.

Details for downloading the open-access software from The R Project for Statistical Computing can be found at https://www.r-project.org/ , and instructions for how to implement the 14 functions used in the article (Table 5 ) are in the README files at https://osf.io/xhe8u/ and require the file Rallfun-v39.txt to execute. The Shapiro-Wilk test for normality in R is named the shapiro.test . In Excel (Microsoft Corp.), the following functions are named: Mean is “AVERAGE”, SD of the population is “STDEV.P”, Median is “MEDIAN”, Kurtosis is “KURT”, and Skew is “SKEW”.

WHAT WAS KNOWN

Changes in postoperative and preoperative keratometry have been used for analysis of astigmatic changes for many years.
The addition of posterior corneal astigmatism to these calculations has improved the analysis and toric IOL outcomes.

WHAT THIS PAPER ADDS

Back-Calculated SIA_{Total} includes all factors including any posterior corneal surface effects, IOL tilt or decentration, refractive changes in the anterior and posterior corneal surfaces from the cataract incisions, and systematic differences in measured keratometric vs actual corneal refractive astigmatism.
The Hotelling transformation for the 95% confidence ellipse requires that the x - and y -components of astigmatism have Gaussian distributions, which is rarely the case. Using the convex polygon eliminates this requirement and provides a 95% confidence boundary that is accurate for any distribution.

Table 5. -
R Functions for Evaluation of Astigmatism.

Function
Description
oph.astig.depcom
Comparison of the dependent mean absolute scalar or vector magnitude of residual astigmatism or PEs
oph.astig.indepcom
Comparison of the independent mean absolute scalar or vector magnitude of residual astigmatism or PEs
oph.dep.comvar
Comparison of the dependent variances of absolute scalar or vector magnitudes
oph.ind.comvar
Comparison of the independent variances of absolute scalar or vector magnitudes
oph.astig.mcnemar
Determine the P values for the percentages of scalar magnitude absolute prediction error below specific intervals analysis for dependent datasets
oph.astig.indepintervals
Determine the P values for the percentages of scalar magnitude absolute prediction error below specific intervals analysis for independent datasets
oph.astig.datasetconvexpoly.mean
Coordinates of the convex polygon boundary and bivariate mean on a double-angle plot for a dataset
oph.astig.datasetconvexpoly.median
Coordinates of the convex polygon boundary and bivariate median on a double-angle plot for a dataset
oph.astig.meanconvexpoly
Coordinates of the convex polygon boundary for a bivariate mean
oph.astig.medianconvexpoly
Coordinates for the convex polygon boundary of the bivariate median
oph.astig.depbivmeans
Comparison of the dependent bivariate means
oph.astig.indepbivmeans
Comparison of the independent bivariate means
oph.astig.indepbivmarg.totvars
Comparison of the independent bivariate marginal and total variances
oph.astig.depbivmarg.totvars
Comparison of the dependent bivariate marginal and total variances