Bivariate analysis of surgically induced regular astigmatism. Mathematical analysis and graphical display

Abstract

Astigmatisms may conveniently be symbolized as an astigmatic direction and magnitude, but are actually composed of refractive powers in the form of polar values. We are operating with two different entities, a net astigmatism and a power vector in the form of polar values. There is an unequivocal point-to-point correlation between these entities. Mathematical conversions can only be performed with polar values, but never by using net astigmatisms. All net astigmatisms must be converted to their appropriate refractive powers and the relevant calculations performed with these entities. The final result, such as an average of several astigmatisms, variances or confidence areas, may be point-to-point reconverted to and symbolized by a net astigmatism. These principles allow for exact description and comparison of surgical methods, but may be employed to describe and analyze any other population of astigmatisms, such as subjective cylinders and spectacle corrections.

Univariate and bivariate polar value analysis of corneal astigmatism measurements obtained with 6 instruments

“…19 Finally, bivariate normal ellipses and confidence ellipses of 2 polar values were produced. [20][21][22][23] Statistical analysis was performed using SAS software (version 9.2, SAS Institute, Inc.). Differences were considered statistically significant when the P values were less than 0.05.…”

Higher-order aberrations and visual function in pseudophakic eyes with a toric intraocular lens

“…The surgically induced corneal astigmatism determined using the polar coordinate system was compared between groups using a multivariate analysis of variance (ANOVA). 21,22 Categorical variables were compared between the 3 groups using the chi-square goodness-of-fit test. When a statistically significant difference was found between the 3 groups, the difference between each combination of 2 groups was further compared using the Mann-Whitney U test for continuous variables and the chi-square test for discrete variables with the Bonferroni adjustment for multiple comparisons.…”

“…The keratometric cylindrical power and axis were determined for each cornea and used to calculate the surgically induced corneal astigmatism. The surgically induced corneal astigmatism was calculated using the polar value analysis described by Naeser et al 21,22 In addition, the polar coordinate system was converted to the vector coordinate system (change vector magnitude and meridian). The UDVA and CDVA were examined using decimal charts and converted to a logMAR scale for statistical analysis.…”

Calculations of actual corneal astigmatism using total corneal refractive power before and after myopic keratorefractive surgery

“…For any mathematical conversion such as subtraction or averaging of astigmatisms, a net astigmatism is transformed into two orthonormal polar values, the curvital and torsional powers, with units in diopters separated by an arch of 45°, KP(Φ) and KP(Φ + 45), respectively. [22–24] …”

“…Bivariate analysis of polar values was performed with Hotelling’s T 2 multivariate t test. [22] where t 1 and t 2 are the paired t-values for univariate analysis of KP(Φ) and KP(Φ + 45). r is a correlation coefficient between ΔKP(Φ) and ΔKP(Φ + 45).…”