The one-sample Kolmogorov-Smirnov test was used to determine if results from each measurement followed a normal distribution. Where the data followed a normal distribution, parametric analysis was performed, whilst nonparametric statistical analysis was used for nonnormally distributed data.
18 All defocus curve acuities were corrected for spectacle magnification (SM) according to a back vertex distance (BVD) of 12.0 mm (equation 1
10).
For each defocus curve, a best-fit polynomial regression curve was fitted to the data points with SigmaPlot 2000 (SPSS Inc., Chicago, IL) (y is the visual acuity [LogMAR] and x is the optical defocus [diopters]). Each data set (14 points) was fitted with a 4th, 5th, 6th, 7th, 8th, 9th, 10th, 11th, and 12th order polynomial. The curve-fitting process was limited to 200 iterations for each curve. Increasing the order of the polynomial to fit the defocus curves resulted in a higher
r2 and decreased the standard error of estimate up until a ninth order polynomial; fitting polynomials of 10th order displayed more variability in the standard error of estimation, as valid curves could not be fitted to all data sets within the iteration limit (
Table 2). Therefore, a ninth order polynomial function was used for all further analysis as it was the lowest-order polynomial that provided a universal best fit to all data sets (equation 2).
VA at 40 cm, measured with the near EDTRS chart, was compared with the VA with a defocus lens of −2.50 D, using Pearson's product moment correlation and Bland and Altman limits of agreement.
19
A relative range of focus was calculated for each subject by determining when the polynomial dropped below +0.04 LogMAR greater than the patient's best VA as described by Gupta and colleagues.
10 If the second multifocal peak (corresponding with the near focal point) also met these criteria, the range of defocus values meeting the criteria for both focal points was summated. The absolute criteria of 0.30 LogMAR was also used to calculate depth of focus; the Newton-Raphson method
20 was used to calculate
x when
y = 0.3. The Newton-Raphson method is used to find the roots of a function, by adjusting the polynomial function by 0.3 to find
x when
y = 0.3 (equation 3). The table of corresponding
x and
y values produced by Sigmaplot 2000 was used to determine the initial approximation
x0.
The resultant
x1 from equation 3 is a better approximation of
x when y = 0.3; however, for increased accuracy this process is repeated by taking the resultant
x1 to be
xn and putting this value through equation 4 until the percentage error (% error) is reduced to 0 (equation 5).
The Newton-Raphson method was used to determine each intersection of the curve at 0.3 LogMAR. The range of focus was calculated as the dioptric distance over which VA was better than 0.3 LogMAR.
The polynomial equations for each curve were integrated so that a new “area of focus” metric (LogMAR * m
−1) could be derived (equation 6). In accordance with the consensus of previous literature, the upper limit for depth of focus was defined as 0.3 LogMAR, corresponding with the European
14 and American
15 binocular visual acuity driving standards.
The defocus curves were divided into distance, intermediate, and near zones. The near zone was defined as between −4.00 and −2.00 D, corresponding with a 25- to 50-cm range, commonly referred to as the range of near vision.
21 The intermediate zone was defined as −2.00 to −0.50 D, from 50 cm (approximately arm's length) to 2 m. Beyond this, the distance zone was defined as the distances between −0.50 to +0.50 D. These zones were used to define the limits of integration. A two-way repeated measures ANOVA was used to determine if there was any statistically significant difference in the area-of-focus and the defocus curves between lenses. If a significant difference was found, then a one-way ANOVA was used to examine differences by applying Bonnferoni post hoc tests to determine pairwise differences.
The subjective rating of intermediate vision was correlated with the absolute, relative, and area (intermediate) defocus curve metrics, as well as with best distance-corrected intermediate VA (79 cm), by using the Spearman's rank correlation coefficient. The subjective rating of near vision was correlated with the absolute, relative, and area (near) defocus curve metrics, as well as with best distance-corrected near VA (at 40 cm), by using the Spearman's rank correlation coefficient.