Only essential elements (for example, Equation 1) are included here as a transformation of refractive errors in clinical notation (S C A or F_s F_c A) to 2 × 2 symmetric power matrices F (or vectors, f or h) and analysis of a dioptric (D) power has been previously described in extensive detail.^{3,4,5,6,7,8,10} (References 3, 8, and 10 will be helpful for any readers less familiar with this topic.) F=(f11f12f21f22)=(FS+FCsin2A−FCsinA cos A−FCsinA cosAFS+FCcos2A)m−1or D[Eqn 1]

Notice that f₁₁ and f₂₂ in Equation 1 are (curvital) powers in the horizontal and vertical meridians, respectively, while f₁₂ = f₂₁ are (torsional) powers in the reference meridian (typically horizontal but not always). Cylinders in ophthalmology and optometry are measured, of course, in relation to the horizontal meridian.

From clinical notation or F, Equation 2 can be used for the scalar (or stigmatic) and antiscalar (or antistigmatic) coefficients of power (see vector f [from Harris] or t [from Thibos et al.¹⁸]): f=(FIFJFK)=(12(f11+f22)12(f11−f22)12(f21−f12))=(FS+0.5FC−0.5FC cos2A−0.5FC sin2A)=(MJ0J45)=t[Eqn 2]

The scalar or stigmatic (F_I = M) coefficient is essentially the spherical equivalent while the term antiscalar (or antistigmatic or sometimes ‘Jacksonian’) refers to powers that are Jackson Cross Cylinders (JCC). Notice that for asymmetric power matrices (where f₁₂ ≠ f₂₁), there is another vector element (F_L) that we have ignored here. Also, M = F_I, J₀ = F_J, and J₄₅ = F_K.¹⁰ These coefficients are used with basis matrices I, J, and K, respectively (1001),(100−1) and (0110) for stereo-pairs plots.

Coordinate vector f and Equation 3 are used for plots of Mahalanobis distances (MD) to identify possible outliers in distributions of refractive errors.^10,13 Such plots provide estimations of the confidence level with which one can expect any specific measurement to be an outlier: MDi=(fi−f¯)TSff−1(fi−f¯).[Eqn 3]

The 3 × 3 symmetric variance-covariance matrix S_ff provides the necessary variances and covariances for distributions of refractive errors:^{7,10,11,12,13} Sff=1n−1∑i=1N(fi−f¯)(fi−f¯)T[Eqn 4]

And Sff(SIISIJSIKSJISJJSJKSKISKJSKK)⋯D2[Eqn 5] where S_II is the stigmatic variance, S_JJ and S_KK are the ortho-antistigmatic and oblique antistigmatic variances, respectively, while S_IJ (= S_JI) is the stigmatic ortho-antistigmatic covariance, S_IK (= S_KI) is the stigmatic and oblique antistigmatic covariance, and S_JK (= S_KJ) is the ortho- and oblique antistigmatic covariance. Given that S_ff is symmetrical, only six entries are distinct; variances are always positive, but covariances can be positive or negative.

Means, variances, and covariances are essential statistics for distributions of refractive errors, and Equation 6 is the sample mean^8,10,12: F¯i=N−1∑i=1Nfi.[Eqn 6]

The normality of refractive error distributions is assessed with meridional or polar profiles using univariate Mardia’s skewness (β₁), kurtosis (β₂), and standardised mean deviation (SMD or A). The expected skewness (β₁) should be zero for a symmetric or normal data distribution, but values above or below zero are considered positively and negatively skewed, respectively.^10,13,19,20 The expected value of kurtosis (β₂) should be three (3) for mesokurtic distributions of refractive errors and values below or above three are considered platykurtic or leptokurtic.^{10,13,19,20,21,22} The expected value for SMD (A) is approximately 0.7979 (≈ 0.80). (References 23–25 are useful for some of the software implementations as applied herein.^23,24,25)

The primary purpose of this study was to investigate and compare samples of NCSR over 2 years (2018 and 2019) to better understand the prevalence, type, and stability or lack thereof of NCSR for the rural clinic concerned. (Such information might be useful for planning and budgetary purposes for rural optometric and ophthalmology clinics.)

The clinical records were randomly selected using a probability-stratified random sampling method.²¹ Sampled clinical records were spread into two strata (that is, 2018 and 2019) using the stratified formula²⁶ (= sample size for the whole study divided by population size × stratum size). The population size for this study comprised 1140 records over the 2 years. Of these, 706 records were for 2018 and 434 records were for 2019. The sample size for 2018 became 238 records, and 146 records for the 2019 stratum after using the stratified formula. A total of 200 records were added to each stratum to increase the statistical power of the study and to allow for the possible exclusion of incomplete records. So, records for 2018 increased from 238 to 438 records, and for the 2019 sample, 146 to 346 records. Records with incomplete information were excluded resulting in final sample sizes for the 2018 and 2019 samples of 279 records and 234 records, respectively. Thus, 513 clinical records in total and ≈134% above the calculated minimum for the whole sample from Cochrane’s formula (Equation 7 below)^14,26: n=Z2P(1−P)e2=(1.96)20.50(1−0.50)(0.05)2=384[Eqn 7] where n = the required minimum sample size, P = 0.5 the percentage occurrence at 50% of the refractive error condition, and e = the margin of error or risk the researcher is willing to accept, relating to factors such as missing or incomplete clinical records in the study, and Z =1.96, the probability value at a significant level of 0.05 corresponding to the level of confidence chosen (here 95%).

The NCSR and other variables of interest such as age and gender were captured in an MS Excel spreadsheet (Microsoft 365) for Windows 11 and then imported into Matlab software (The MathWorks, USA) where NCSR were transformed into the dioptric power matrices for further analysis.

Ethical approval (REC-1170-2021) was obtained from the Research Ethics Committee in the Faculty of Health Sciences (FREC) at the University of Johannesburg (SA). Permission to conduct the study at the selected hospital was granted by the Provincial Health Research and Ethics Committee in the Limpopo Department of Health, the Senior Clinical Manager, and the Chief Executive Officer (CEO) of Sekororo Hospital.

Analysis of sample normality¹⁰ and Mahalanobis distances (MD)¹⁰ for the right and left eyes of NCSR for 2018 and 2019 mainly demonstrated moderate (> 4) to severe (up to 25) leptokurtosis and mild (± 0.75) negative or positive skewing (for sample normality, kurtosis is 3 and skewness is zero). For conciseness, normality plots are not included here. Given the wide range of refractive states (see stereo-pairs in Figure 1), these results were not unexpected. Outliers were infrequent in the samples, but they also contributed to departure from sample normality. Again, for conciseness, plots of MD are not included here (such plots are available from the first author on request).

Figure 1 shows the stereo-pair plots with 95% distribution ellipsoids (DE) for the right and left eyes of 279 non-cycloplegic subjective refractive errors for the 2018 sample and another 234 non-cycloplegic subjective refractive errors for the 2019 sample. For both stratified random samples and right and left eyes (in 2018 and 2019), the plots showed vertically orientated DE along the stigmatic axes (F_II), and this implies that the distributions in both samples were mainly stigmatic, ranging from hyperopic to myopic eyes with mild (< 2 D) to moderate ([1.25–2 D]) astigmatism (points are mostly close to the stigmatic axis). Refractive errors (points) are more densely clustered near the sample means that are not far from the origin (0 D or 0 m^–1), and this indicates that there are many non-cycloplegic subjective refractive errors for the right and left eyes in both samples that are not too far off from emmetropia (reflecting the process of emmetropisation).

Figure 2 shows the same plots as in Figure 1, but rotated (0, ‒90°) so that the data are viewed along the stigmatic axis with the antistigmatic (or Jacksonian) plane viewed in the plane of the page, and the further a measurement is from the origin, the greater the antistigmatic powers (F_J and F_K), and also cylinder (F_c) or astigmatism present for any eye (The term antistigmatism is synonymous with JCC and F_J and F_K are equivalent to J₀ and J₄₅).

Figure 3 shows polar profiles for curvital variances for the refractive states (after transformation to power matrices – see Equation 1) for the right and left eyes in the 2018 and 2019 samples. The profiles represent the three variances, namely, f₁₁ and f₂₂ for the curvital coefficients of power, and f₁₂ (=f₂₁) for the torsional coefficients of power. Profiles for curvital variances (f₁₁ and f₂₂) are represented on the same profiles but shifted by 90°. The origin of each polar plot is at zero squared dioptres (0 D²) meaning no variance. So, profiles closer to the origin show smaller variation and profiles further away from the origin show greater variation. The radial scale in the polar plots shows the magnitude of variance (in D²). Variation can be either uniform (completely or partially uniform) or non-uniform across the meridians of the eyes concerned.

Figure 3A indicates that curvital variances in the samples for 2018 were ≈ 2 to 2.5 D² and less than for the 2019 samples (see Figure 3B where the range was ≈ 3.75 D² to 4.25 D² depending on meridian). In 2019, refractive errors were more variable for the right eyes than for the left eyes (compare the outer profiles in green and blue). The curvital profiles are almost uniform or constant, that is, variation is roughly similar for all meridians across the eyes concerned. Torsional variances (innermost profiles and see also Figure 4) were smaller than curvital variances and similar irrespective of the year (2018 or 2019).

In Figure 4, the profiles for the torsional variances are shown with dashed lines and they have a resemblance to ‘rabbit ears’. Irrespective of laterality (right or left eyes), the torsional variances are small (< 0.125 D²). This suggests that astigmatism for the eyes in the samples was not very variable, although cylinders (F_c) ranged from −0.25 D to −4 D. By contrast, spherical powers (F_s) ranged from −18 D to 12 D for the samples for 2018 and 2019.

The design of this study was a cross-sectional retrospective study based on historical records extracted from the clinical archive of the Sekororo Hospital, Limpopo, South Africa for the patients who consulted at the Optometry Clinic over 2 years starting from 01 January 2018 to 31 December 2019, and this design could not establish the causality of the subjective refractive errors. As a result of the COVID-19 pandemic, records for 2020 were not included in this study because of disruptions in clinic visits and booking schedules. Although the study’s sample sizes were relatively small, and cycloplegia was absent, which may have had an impact on the results, they were sufficient for the study’s aims. It is worth observing that the small sample sizes may limit the generalisability of the findings. However, given the study’s specific research aim and objectives, the results still provide valuable insights. Moving forward, it may be beneficial to consider including larger sample sizes and cycloplegia to further validate these findings. In future studies, the analysis could be augmented by including autorefraction and/or retinoscopy, in addition to the subjective method. For this study herein, methods such as retinoscopy were used before NCSR and this increases the potential reliability of the NCSR as determined for analysis. Random selection (extending beyond the limitations of a single clinical environment) would also be useful to form an improved understanding of the population distributions of URE in the geographic region concerned. Such approaches could lead to a more comprehensive understanding and knowledge about the general topic of refractive error and potentially also the probability of correctable and uncorrectable vision impairment (UVI) in relation to refractive error.

All measurements of refractive state were obtained via a single optometrist with extensive clinical experience. Randomisation was used to select participants from the larger populations that utilised the optometric refractive services at the clinic concerned. The study provides results for refractive errors in African eyes, and this article amply illustrates multivariate methods that are important for the analysis of such data. These methods have not been used previously to any great extent, particularly involving samples that include African eyes and participants only. Thus, this article provides original and important information in this field of study.

The researchers suggest that future studies relating to refractive errors be conducted in other primary high-level public (or state-owned) hospitals such as regional, provincial, and national hospitals, and private optometric facilities for comparison of results. Where possible, samples should be increased in size and cycloplegia should be incorporated in the studies involving refractive state especially those involving children.