In ophthalmologic literature, several conflicting methods for reporting surgically induced astigmatism (SIA) have been described.1–8 Some are incorrect,8,9 but as the observer has no real sense of what result to expect, the fallacy of a specific method may be difficult to conceive. A method of assessing SIA should be optically meaningful, consistent within its own definition, and correct for both single and aggregate data.
In the present study, I reviewed the various assessment methods and then tested the methods in situations in which the expected result was obvious. When this result was not produced, I regarded the specific assessment method as falsified in a classical “Popperian” manner. Karl Popper is known for his thesis that scientific theories cannot be proved but only falsified. Rather than attempting to elaborate on a theory, the scientist should constantly set up experiments allowing for falsification of the theory, thereby paving the way for new theories.
Materials and methods
Analysis of Aggregate Data
A calibrated steel test ball (Noco Nordisk Optisk Co.) of an assigned diameter of 15 mm was placed in its holder and mounted on the chin rest of a Haag-Streit slitlamp. Keratometry of the steel ball was performed twice with a Nidek model ARK-2000S autorefractokeratometer, as previously described.10 Following a double determination, the steel ball was rotated at random and a new double measurement performed. A total of 50 double measurements were performed.
In the following, the subscripts 1 and 2 refer to measurements derived from the first and second keratometer reading. The following entities were registered:
- α1 and α2: Direction in degrees of the steeper corneal principal plane5
- φ1 and φ2: Direction in degrees of the flatter corneal principal plane, where φ = α + 90°
- M1 and M2: Difference in power, expressed in diopters (D), between the steeper and the flatter principal corneal plane5
- (M1 @ α1) and (M2 @ α2): Net astigmatisms as previously described5,6
The mean difference between the measurements was calculated by several assessment methods, and 95% confidence intervals (95% CI) were computed for univariate data. Bivariate data were analyzed with the Hotelling T2 test.8,11 In all cases, the difference between the 2 measurements was taken as
For these data, the average difference between the 2 measurements would be expected to be close to and not statistically significantly different from zero, as the examined objects would not change as a result of the measurements. Assessment methods producing average differences significantly different from zero would be regarded as falsified.
Methods of assessing changes in net astigmatism have been described.12–19
Simple Subtraction Method
This method considers only the astigmatic magnitude, disregarding the astigmatic direction.
M1alg and M2alg change signs in accordance with the direction of the steeper corneal meridian; for 45 ≤ α <135, a “+” sign is added; for all other values of α, M is denoted a minus sign. The method describes the with-the-rule (WTR) and against-the-rule (ATR) change between 2 measurements.
where Δx = x1 – x2; in this equation x1 = M1 · cos α1, x2=M2 · cos α2, and Δy = y1 – y2; in this equation y1 = M1 · sin α1, y2 = M2 · sin α2. Schematically,
As in the algebraic method, the Cravy system reports the WTR and ATR change and has an elaborate convention2,4,15,18 for determining the delta values. Cravy's ΔK yields results very close to ΔKP(90).
Astigmatic Magnitude Not Considering Axis
Several methods have been described.1,3,4,15,16,21,22 All yield identical results and have various sign conventions.
Magnitude of induced astigmatism for a single patient or observation:
Direction of induced astigmatism for a single patient or observation:
For aggregate data, Mvec is averaged directly without considering direction.8,9,23
It is now generally agreed that separate analysis of astigmatic direction is meaningless.3,8,14 Average values for φvec will therefore not be reported.
This method reports the WTR and ATR components from the values generated from astigmatic magnitude not considering axis.3,4,24
The WTR and ATR components derived from polar values:
Bivariate statistical analysis was performed as previously described.8,10
Mean difference, expressed as net astigmatism, was calculated as previously described.7–9,24
Alpins25,26 and Holladay27 Methods
Both methods use the basal entities M · cos2 φ and M · sin2 φ and are therefore identical to the Bennett24 and polar value methods, shown in equations 8 and 9, as7,8
We will therefore not perform a separate analysis of these methods.
Analysis of Data from 2 Measurements from a Single Patient
I constructed examples in which the expected results were obvious. When this result was not produced by a specific assessment method, it was regarded as falsified.
All data were processed and statistical analyses were performed with the SPSS program (Statistical Program for the Social Sciences). A 2-tailed paired t test was used for comparing double measurements on univariate data. Bivariate statistical analysis was performed with the Hotelling T2.8,11
Analysis of Aggregate Data
The various assessment methods of net astigmatisms yielded the mean differences between the 2 measurements shown in Table 1. The mean difference in astigmatism (0.02 D @ 173) was not significantly different from zero (bivariate analysis with the Hotelling T2 = 0.007, F = 0.164 with (2,48) degrees of freedom; P = .85). The simple subtraction, algebraic, Cravy, and polar value methods yielded results close to and not significantly different from zero. Conversely, astigmatic magnitude not considering axis and astigmatic decomposition both resulted in significant mean differences from zero. In this test situation for aggregate data, the vector analysis and astigmatic decomposition methods were falsified.
Analysis of Data from a Single Patient
Calculate the difference between the following 2 autokeratometry readings
As the magnitudes of the 2 net astigmatisms are identical and the directions are very close, little difference between the 2 measurements is expected. Table 2 shows the calculated values for the different methods.
The simple subtraction, Cravy, vector analysis, astigmatic decomposition, and polar value methods yielded results close to zero. The algebraic method resulted in a large difference between the 2 measurements, falsifying this approach. This is naturally caused by the change in sign at 45 degrees.
As the directions of the 2 meridians are orthogonal, the difference between measurement 1 and 2 easily calculates as (2.0 D @ 0). Results of the calculations are shown in Table 3.
This example falsifies the simple subtraction method. The difference between the 2 astigmatisms, expressed as polar values in 135 degrees, also amounts to zero, but this is as expected as both net astigmatisms amount to zero for this specific polar value in the oblique meridian.
The joint analysis of aggregate and single data has falsified the simple subtraction, algebraic, astigmatic magnitude not considering axis, and astigmatic decomposition methods.
Previous studies of SIA have emphasized the mathematics involved. A different strategy was chosen for this study. A variety of assessment methods were tested under conditions in which the correct result was known or obvious. Assessment methods failing to produce this result were regarded falsified.
Astigmatism is characterized by both direction and magnitude, and it is therefore not surprising that the simple subtraction and algebraic methods, which consider only astigmatic magnitudes, are inconsistent, as shown in examples 1 and 2. As pointed out by Olsen and Dam-Johansen,14 the Cravy method is a mathematical concept, not based on optics. This method will not be considered in the following discussion.
Jaffe and Clayman1 originally described 2 methods for analysis of SIA. In one, the astigmatic direction was considered; in the other, it was disregarded.9 This last method has been extensively used under the term “Jaffe vector analysis.” Surgically induced astigmatism is calculated correctly for individual patients, while astigmatic direction is disregarded in the analysis of aggregate data.7–9,23,26,27 For average values, a systematic error is introduced, as astigmatisms in various degrees of inclination are not cancelled out.23 This group of methods1,3,4,15,16,21,22 was recently given the more neutral term astigmatic magnitude not considering axis.9 In the present study, the difference between 2 consecutive keratometry readings on unchanging surfaces averaged 0.19, a value significantly different from zero. Vector decomposition is based on the results from vector analysis and therefore also produced a systematic error (Table 1).
The polar value methods are conceptually based on the surgically induced flattening and torque of the preoperative cylinder. Alpins25,26 and Holladay27 have used different concepts to describe similar methods. The Naeser, Alpins, and Holladay methods agree with current research in optometry.28–30 The next steps are to take advantage of these valid methods fully by using them with multivariate statistics to describe not only astigmatism and spherocylinders8 but also more complex optical phenomena.
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