This retrospective study was conducted at a tertiary care hospital. The study protocol was approved by the Institutional Review Board of the University of Pittsburgh (Protocol 19010171), and it adhered to the principles outlined in the Declaration of Helsinki. Clinical data were obtained by reviewing case records. Corneal tomography and wavefront data were acquired using Scheimpflug tomography (Pentacam, Oculus, Wetzlar, Germany). The criteria for disease progression were defined as an increase in central keratometry of 1.00 D or maximum keratometry (Kmax) of 1.50 D in at least two visits, with a minimum interval of six months between visits [3]. Cases with a history of previous ocular surgery, including corneal cross-linking, poor-quality scans, corneal scars, severe dry eyes, post-excimer ectasia, and pellucid marginal degeneration, were excluded.

Tomographic data was extracted from the device's csv files (load files, Pentacam, Oculus, Wetzlar, Germany). Corneal wavefront data was computed at a radius of 8 mm. The data were entered into an Excel worksheet (Microsoft, Richmond, VA) and subsequently analyzed using SPSS 16.0 (SPSS Inc., Illinois).

The fractional change is defined as the difference in the parameter being evaluated [e.g., Kmax, central corneal thickness (CCT)] between the follow-up visit and the initial visit, divided by the absolute value (modulus) at the initial visit and the time elapsed in months.

The rate of fractional change (Rx) was mathematically expressed as:

$$\mathrm= (}_- }_) / (|}_|}_})$$

where, x1 = value at follow-up visit; x0 = value at initial visit; |x0|= modulus of value at initial visit; tm = time in months (days between visits/30).

In terms of intuition, this fractional rate of change is the change occurring per unit change in the parameter as well as per unit change in time, thus creating a mathematically level playing field for comparison. Using the absolute value for the denominator ensured that the directionality of the change is not lost due to algebraic reduction. Furthermore, when asymmetric aberrations were involved, laterality (right or left) was considered in all calculations.

The concept of fractional change is derived from the concept of creating a normalized magnification scale. For example, a change of 1 unit with a base denominator of 10 is not the same as the change of 2 units with a based denominator of 50. Comparing an actual change normalized to the base value results in a fair comparison for the parameters being evaluated. The result is a ‘fraction’ and hence the fractional rate of change when this parameter is divided by the amount of time elapsed.

The distribution of the Rx was found to be non-normal (Kolmogorov–Smirnov test, P < 0.05 for all parameters). Therefore, nonparametric tests were employed. Descriptive data were presented using measures of central tendency in the format of mean ± standard deviation and median values. The difference in means between observations was assessed using the Kruskal–Wallis test. Nonparametric post hoc analysis was conducted using the Rank Sum test. The effect size for the nonparametric test was calculated using the formula:

$$\mathrm=\mathrm / (\mathrm^)$$

where S represents the effect size, z is the Z-score from the Rank Sum test, and n is the total sample size. An effect size of 0.5 was considered large, 0.3 was considered medium, and 0.1 was considered small [10, 11].

A-priori sample size estimation: Pilot data for 10 cases was used to calculate the estimated sample size for an alpha of 0.05, beta of 0.2 (power of 0.8), the standard deviation of change as 4.3 and r within as 0.97. The estimated minimal sample size was 31 cases. We included 40 cases to ensure sufficient post hoc power.

The analysis was conducted in two steps:

Step 1: Pachymetric, keratometric, and 8 mm anterior corneal higher-order aberrations root mean square (HOARMS) related variables were compared to select the ones exhibiting growth rates.

Step 2: The selected variables were compared between different groups to assess those with the greatest rate of fractional change.

Furthermore, the rate of fractional change was compared between two age groups (≤ 30 years and > 30 years) and between genders to evaluate the impact of age on progression.