Article Text

Original research
Multivariate analysis of refractive state in eyes with keratoconus
  1. Elizabeth Chetty
  1. Department of Optometry, University of Johannesburg, Auckland Park, South Africa
  1. Correspondence to Dr Elizabeth Chetty; echetty{at}


Objective To demonstrate a multivariate method of analysis of the short-term variation of refractive state in keratoconus (KC) patients.

Methods and analysis In this observational study, 19 eyes with KC and 19 healthy control eyes were measured. The study included both male and female participants and the mean age was 23.6 years (range 18–34 years) and 23.2 years (range 22–26 years) for KC and control participants, respectively. Forty consecutive autorefractor measurements were taken for each participant and the short-term variation thereof was analysed using multivariate methods of analysis.

Results and conclusion Short-term variation of refractive state is greater in eyes with KC than in healthy control eyes and variation increases with severity of disease. A novel finding was that there was much more ortho-astigmatic and oblique-astigmatic variation seen in KC eyes than in control eyes which had predominately stigmatic variation. Refractive state is described by three components, namely, sphere, cylinder and axis. Although it is multivariate in nature, it is often analysed using univariate statistical methods. In diseases such as KC, where early diagnosis is crucial for a good prognosis, it is necessary that researchers endeavour to investigate the disease from different perspectives to fully understand the nature of the disease. This paper comprehensively demonstrates the multivariate statistical methods of analysis of refractive data. The implementation of this analysis provides insight into the short-term variation of refractive data in healthy and keratoconic eyes, and these findings have not been demonstrated before using univariate statistics.

  • cornea
  • optics and refraction

Data availability statement

Data are available in a public, open access repository.

This is an open access article distributed in accordance with the Creative Commons Attribution Non Commercial (CC BY-NC 4.0) license, which permits others to distribute, remix, adapt, build upon this work non-commercially, and license their derivative works on different terms, provided the original work is properly cited, appropriate credit is given, any changes made indicated, and the use is non-commercial. See:

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  • Patients with keratoconus exhibit higher and more unstable refractions than healthy eyes.


  • This study provides a more comprehensive multivariate analysis of variation of refractive state which provides insights on disease manifestation that cannot be investigated with the commonly used univariate methods of analysis of refractive data.


  • Using multivariate methods of analysis for multivariate data such as refractive and keratometric state is necessary to fully understand the data and the disease being investigated. The early diagnosis of keratoconus has long evaded clinicians; therefore, perhaps the use of multivariate statistics may help improve data analysis thereby developing a method of early diagnosis of keratoconus.


Keratoconus (KC), traditionally characterised as a chronic, bilateral (although asymmetrical), progressive non-inflammatory thinning of the cornea resulting in steepening of corneal curvature and irregular astigmatism, is an ectatic disease that once was believed to be uncommon.1 However, with the boom in tomography instrumentation providing a more detailed analysis of the cornea, it is becoming easier to diagnose the disease, even in its early stages. The multifactorial nature of this disease, with complex interactions between genetic and environmental factors, has hindered the precise determination of its aetiology. Although genetic associations have been established,2 3 the specific gene responsible for KC remains unknown. While KC primarily presents as an isolated condition, its links to various disorders, such as Down’s syndrome, Leber’s congenital amaurosis, Ehler’s-Danlos syndrome, osteogenesis imperfecta and atopy, have been documented.1 4 5 Notably, eye rubbing, prevalent in some of these disorders, is considered a significant contributor to the progression of KC. The characteristic thinning of the cornea in keratoconic eyes affects the corneal curvature, which impacts the eye’s corneal power and refractive state. A refraction measurement is composed of sphere, cylinder and axis. Often, this sort of refractive data is analysed partially or incompletely.6 7 For example, the axis of the refraction measurement is sometimes omitted, and/or only the spherical equivalent is used in data analysis. Pertinent findings may be overlooked by not using a more holistic approach by including all the components of the measurements in the analysis of this sort of data.

Between the late 70s and 80s, a combination of efforts from Long,8 Keating9–11 and Harris7 led to the formalisation of a mathematically meaningful representation of dioptric power via the concept of a symmetric, square 2×2 matrix. The matrix representation of dioptric power has enabled scientific methods of analysis to be conducted on critical optometric and ophthalmologic data for over three decades12–17 that allows dioptric power to be analysed in its entirety. The use and application of these methods to analyse dioptric data is discussed in detail elsewhere.17

There are three components to refractive error, namely, sphere, cylinder and axis. Most studies that investigate refractive data analyse each of the three components individually rather than as the holistic entity. In doing so, relevant findings may be inadvertently omitted. Thus, the aim of this study is to investigate the short-term variation of refractive state in patients with KC using multivariate statistical methods. Investigating the short-term intrasubject variation of refractive state in keratoconic and normal corneas may lead to earlier diagnosis and perhaps better treatment options for the disease which could improve quality of life for keratoconic patients. This study (which is part of a larger doctoral study) provides insight on refractive state in keratoconic eyes that has not been presented before.


Study sample

This was a non-randomised prospective observational and quantitative study that took place at the specialty contact lens clinic at the University of Johannesburg (UJ), South Africa. It was not appropriate or possible to involve patients or the public in the design, or conduct, or reporting, or dissemination plans of the research. The study sample was not randomised as every patient that attended the clinic that was eligible to participate was invited to join the study. The data analysed in this study formed part of a larger study conducted by the author for her doctoral thesis. This particular section of the study included 40 consecutive measurements on each of 38 participant eyes. The research sample consisted of 14 keratoconic participants (19 eyes with KC) and 19 healthy control participants. The mean age was 23.6 years (range 18–34 years) and 23.2 years (range 22–26 years) for KC and control participants respectively. Demographic information for participants is provided in table 1. Participant selection was not dependent on ethnicity nor gender. All patients who attended the UJ specialty contact lens clinic and had been diagnosed with KC by the results of routine preliminary tests (such as slit lamp and corneal tomography) were invited to participate in this study. Control participants were selected by convenience sampling and was made up of UJ optometry students.

Table 1

Demographic information for participants

Data collection

After receiving informed consent from participants, they underwent preliminary testing (tomography, slit lamp, ophthalmoscopy and autorefraction) which were used to determine which prospective participants would be excluded from the study. Exclusion criteria were current or recent contact lens wear, ocular pathology other than KC, recent eye surgery or any medication with ophthalmic side effects. The same exclusion criteria, as well as no ocular pathology, were applicable for control participants. Keratoconic participants with severely distorted corneas were also excluded because measurements could not be acquired on these patients.

The Nidek autorefractometer (ARK 700) was used to take 40 consecutive non-cycloplegic autorefractive measurements on each participant. Measurements were taken on both eyes in the KC group. Due to the disparate nature of KC between the right and left eye of a participant, it was decided to use data from both eyes where possible. McAlinden et al18 agree that it is unlikely that issues of laterality will affect the analysis of data for eyes with KC. Only the right eye of each participant in the control group was measured similarly. The left eye for controls was not used to avoid issues of laterality. The measurement session for KC and control eyes lasted on average 30 min and 10 min, respectively, per participant.

Data analysis

According to Harris,7 credit should go to Blendowske19 and Long8 who pioneered the idea that dioptric power could be expressed as a 2×2 matrix. For sphero-cylindrical power, this 2×2 matrix is symmetric, that is, the off-diagonal entries are equal, therefore, coincidently, there are only three distinct numbers just as there are three numbers (sphere, cylinder and axis) representing dioptric power in clinical notation.

With sphero-cylindrical power having a mathematical representation, any mathematical function possible with matrices is possible with refractive data including calculating means and variances which are two paramount statistics when comparing samples of data and making inferences for populations.20 All the statistical analyses conducted in this study are based on the dioptric power matrix. Harris, Malan and Rubin21–23 have all contributed to the development of statistical and software methods that were specifically designed (using MATLAB) to convert such data into matrix representations which could then be used for multivariate statistical analyses. For the purposes of this study, these methods were used to convert the refractive powers into matrices so that they could then be plotted in three-dimensional (3D) dioptric power space in the form of stereo-pair scatter plots. Thereafter, all the statistical functions and methods required to analyse the data were carried out on the matrix equivalents. Statistica was used for the univariate analysis of data where necessary. To view the 3D percept of the stereo-pairs, one should allow one’s eyes to drift into an exoposition to merge the stereo-pair.


As mentioned previously, the data collected were part of a larger study which had included 38 stereo-pairs for the refractive state data analysis. It is superfluous to include that many figures for the purposes of this paper, therefore, stereo-pairs of two randomly selected participants, one KC and one control participant, are provided in online supplemental figure 1a,b, respectively, to illustrate the pertinent findings of this study. The full set of figures can be viewed elsewhere.24 Normality of data was evaluated using skewness and kurtosis. Due to most data sets not being normally distributed, non-parametric statics were used for the univariate analysis. Using a graphical representation of data such as 3D stereo-pairs allows for data to be visualised without any underlying assumptions of normality being necessary to validate results. Each stereo-pair (see online supplemental figure 1) includes 40 autorefractive measurements obtained with the Nidek autorefractometer. Each dot on the stereo-pair is an actual representation of the autorefractive measurement transformed from sphere, cylinder and axis to a symmetric matrix that is plotted in 3D Euclidean space so that all components are included in the analysis. The axes I, J and K is synonymous to vectors M, J0 and J45, respectively, which is used in data analyses by Thibos et al25 and others researchers26–28 where I is the stigmatic axis, J is the ortho-astigmatic axis and K is the oblique-astigmatic axis. These stereo-pairs also include corresponding 95% distribution ellipsoids. The two scatter plots below share the same axis length (2D) and tick interval (1D). The origin is placed at the mean of the sample. Although this scale is not suitable for the control subjects (online supplemental figure 1B), it allows for a quick visual comparison between the two groups. This study deals primarily with the multivariate analysis of refractive state, however, univariate analysis on the matrix equivalent of the refraction measurements also provides a good substantiation to certain findings.

Online supplemental figure 1A depicts the results of one randomly selected eye with KC and provides an example of the typical finding for the KC group. All 19 eyes with KC show large amounts of variation in autorefractive measurements as is generally evident by the loosely dispersed data and large distribution ellipsoids. For eyes with KC, stigmatic variation with some astigmatic variation is evident as the data is dispersed along all axes. On direct comparison of the sizes and volumes of the distribution ellipsoids, the control eyes display much less variation in measurements as seen by the tighter cluster of data and smaller volumes. This can be noted on direct comparison of online supplemental figure 1A,B, and on inspection of the distribution ellipsoid volumes provided in table 2. Inspection of online supplemental figure 1B enables one to note that for the control eye, the longest axis of the distribution ellipsoid aligns closely with the stigmatic axis. This indicates that the refractive variation exhibited by the control eye is largely stigmatic in nature (probably mainly relating to changes in ocular accommodation) unlike that which is noted for the eye with KC which exhibits stigmatic as well as astigmatic variation. This is the general finding for all eyes in each group. While online supplemental figure 1 provides a visual representation of the differences in 95% distribution ellipsoids (and hence differences in spread of data) across the KC and control groups, table 2 provides the volumes of the 95% distribution ellipsoids. The 95% distribution ellipsoid volumes were used to generate box-and-whisker plots to further illustrate the differences in short-term variation of refractive state. Online supplemental figure 2 represents the box-and-whisker plots for 95% distribution ellipsoid volumes for severe (n=6), moderate (n=13) and control (n=19) eyes. For the box-and-whisker plots in online supplemental figure 2, the KC participants were graded according to the severity of the disease using the well-known CLEK method.29 In general, the greater the severity of KC, the larger the volume of the distribution ellipsoid and hence the greater the intrasubject variation in refractive state. It is also clear that the control eyes exhibit much less short-term variation in refractive state than all eyes with KC. The descriptive statistics for the box plots are summarised in table 3. These box plots further substantiate that the greater the severity of KC, the greater the short-term variation of refractive state.

Table 2

95% distribution ellipsoid volumes

Table 3

Descriptive statistics for the box-and-whisker plots for 95% distribution ellipsoid volumes (D3) for refractive state for severe (n=6), moderate (n=13) and control (n=19) eyes

According to Harris20: ‘the mean is a value around which the sample clusters while the variance is a measure of the spread or dispersion of the cluster around the mean’. The mean and variance of a sample are statistical characteristics that are pertinent to finding correlations in data and drawing conclusions about the populations that the samples represent. Saunders30 asserted that dioptric power could not be squared and hence the variance of a sample could not be calculated. Harris31–34 found this to be incorrect and showed for the first time that variance–covariance matrices could be calculated for dioptric power. Table 4 shows the variances extracted from the variance–covariance matrices for refractive state for 19 KC and 19 control eyes where SII, SJJ, SKK are relative to the stigmatic and astigmatic variances, respectively. These variances were used to generate the box-and-whisker plots in online supplemental figure 3 and 4.

Table 4

Variances (D2) extracted from the variance–covariance matrices for refractive state for 19 KC and 19 control eyes

For the KC group (online supplemental figure 3), the Kruskal-Wallis test indicated that medians were statistically different from each other (H (2, N=57) = 24.815, p=0.000). The post hoc multiple comparisons test revealed that while the median of the stigmatic variance was statistically different from the other two medians, the ortho-astigmatic and oblique-astigmatic medians were not statistically different from each other.

For the control group (online supplemental figure 4), the Kruskal-Wallis test indicated that medians were statistically different from each other (H (2, N=57)=33.448, p=0.000). The post hoc multiple comparisons test revealed that while the median of the stigmatic variance was statistically different from the other two medians, the ortho-astigmatic and oblique-astigmatic medians were not statistically different from each other. Note the large difference in scales for the y-axis for online supplemental figures 3 and 4. There is almost no or very little variation in autorefraction in the controls (see online supplemental figure 4) as compared with the KC sample (see online supplemental figure 3). Table 5 shows the associated descriptive statistics for the box-and-whisker plots in online supplemental figures 3 and 4 where mean and median stigmatic variations are largest for both KC and control eyes, however, eyes with KC exhibit much more variation overall. The mean and median astigmatic variations are also much greater in eyes with KC than for control eyes.

Table 5

Descriptive statistics for the box-and-whisker plots for variances of refractive state for KC eyes and control eyes


It was difficult to make direct comparisons between this and other (similar) studies because many factors vary between studies of this nature. There are also many different instruments and methods that can be used to take refractive measurements, all of which are not necessarily interchangeable and directly comparable. Complicating matters further, is that there are different statistical methods that could be used to assess variation (all of which could give varying results)18 and in general, the nature and grading of KC can be rather disparate between study samples. These are only a few of the problems that arise when comparing studies. While there is a paucity of literature that investigates the short-term variation of refractive state in KC eyes, many authors do, however, agree that dioptric power should be analysed in its entirety for research purposes.27 28 35–38

Inspection of online supplemental figure 1 and table 2 illustrates that the short-term variation in refractive state is much greater in KC eyes than healthy control eyes. Online supplemental figure 2 demonstrates that this variation increases as the KC progresses to more severe stages. On average, distribution ellipsoid volumes (spread of data points) for severely affected eyes with KC varied almost twice as much as moderately affected eyes with KC, which in turn varied more than 130 times greater than control eyes (table 3). Volumes of distribution ellipsoids could become important in early diagnosis of KC and possibly in monitoring disease progression. The medians for variance in the spherical, ortho-astigmatic and oblique-astigmatic components of refractive state in KC eyes are 32, 47 and 38 times greater, respectively, for control eyes (table 5). Online supplemental figures 3 and 4 show that while most of the variation in refractive state is predominantly found in the stigmatic component for both KC and control eyes, there is much more ortho-astigmatic and oblique-astigmatic variation seen in KC eyes than in control eyes.

Using the vector analysis method,25 Raasch et al35 investigated the repeatability of subjective refraction in the right eyes of 40 normal myopic patients and one randomly selected eye of 138 patients with KC. The authors used the same sample from a previous study39 where they had analysed data in the conventional form of sphere, cylinder and axis. They concluded that there is greater variation in refractive state in KC eyes than in myopic patients without KC and that the difference in medians were as much as four to six times greater in the KC group than in the myopic group. They also acknowledged that converting conventional refractive data into vector powers in 3D power space provides a more comprehensive analysis with more meaningful results.

Stereo-pair scatter plots provides a unique and more understandable visual representation of the distribution or spread of data (40 points here per sample), each of which was an actual representation of sphere, cylinder and axis transformed and thereafter plotted in 3D dioptric power space for each eye. Analysis of dioptric power is very complicated, but this type of graphical and quantitative analysis goes a long way to simplifying and promoting a deeper and clearer understanding of the questions and issues involved in the clinical and research area concerned. Together with 95% distribution ellipsoids, such stereo-pairs provided an indication of direction and nature and amount of the spread given in terms of sample variances and the orientations of the longest axes of such ellipsoids and their volumes. The ellipsoids and their volumes that were represented and calculated using multivariate methods were then extracted and further explored using univariate methods, which included the use of box-and-whisker plots. Sample normality and descriptive statistics for central tendency, dispersion and variances and covariances were also investigated in detail. As explained above, by converting dioptric power into its matrix form, it is possible to square dioptric powers and hence calculate vital statistics such as the variances of the sample concerned. Means and variances (or SD or quartile deviations) are important factors when investigating central tendency and dispersion and they provide an indication of the spread of data in relation to the sample mean. Data cannot be adequately understood without these two very basic statistics and given the trivariate nature of dioptric power, multivariate methods are essential to effectively understand and represent such data and, for example, to calculate the variance–covariance matrix for each sample of 40 consecutive measurements for each eye as analysed here. The variances extracted from the matrix concerned were further analysed using univariate methods such as box-and-whisker plots with corresponding descriptive statistics. At its simplest, samples of dioptric power need three variances and three covariances for proper understanding of data and the variance–covariance matrix is an essential element for such understanding. A great deal remains to be explored but this study provides an important stepping stone towards increasing one’s knowledge of KC and its impact on ocular parameters such as variation of refractive state.

One of the limitations of the study is sample size. On first inspection of the sample size (19 KC and 19 control eyes) one may be inclined to believe it small. However, one should consider that the primary aim of this study was to provide an alternative method to evaluate the short-term variation of refractive state using multivariate statistical methods, something which has not been investigated in sufficient detail previously in relation to both keratoconic and normal age-related control eyes. To this aim, 40 non-cycloplegic autorefractive measurements per eye were individually evaluated for an intrasubject analysis. Therefore, essentially the nature and spread of over 1500 measurements has been evaluated and presented in this study. A strength of this study is that it contributes useful and important results in KC and age-related controls due to its combination of univariate and multivariate methods to explore and interpret the data as collected. Most studies done on the anterior segment of keratoconic eyes have investigated correlations between various factors such as anterior and posterior corneal curvature,26 39 40 corneal elevation,41–43 anterior chamber depth44 45 and volume,45 46 pachymetry44 45 and aberrations47 48 to differentiate between keratoconic and normal eyes. However, this study is believed to be the first to use methodology such as stereo-pairs, variance–covariance matrices and distribution ellipsoids to compare the short-term variation of refractive state to use as a potential tool to differentiate between keratoconic and normal eyes. Hence, this paper has provided a feasible and more complete approach to the analysis of dioptric power in patients with KC.

Data availability statement

Data are available in a public, open access repository.

Ethics statements

Patient consent for publication

Ethics approval

This study involves human participants and approval to conduct the study was obtained form the Research Ethics Committee of the Faculty of Health Sciences of the University of Johannesburg, South Africa (ethical clearance number: REC-241112-035). Participants gave informed consent to participate in the study before taking part.


The larger doctoral study from which this data was taken from was under the supervision of Prof A Rubin, Department of Optometry, University of Johannesburg.


Supplementary material


  • Contributors All work was done by the author, E Chetty.

  • Funding This study was funded by the Thuthuka Grant (TTK160519165562) from the National Research Foundation in South Africa.

  • Competing interests None declared.

  • Provenance and peer review Not commissioned; externally peer reviewed.

  • Supplemental material This content has been supplied by the author(s). It has not been vetted by BMJ Publishing Group Limited (BMJ) and may not have been peer-reviewed. Any opinions or recommendations discussed are solely those of the author(s) and are not endorsed by BMJ. BMJ disclaims all liability and responsibility arising from any reliance placed on the content. Where the content includes any translated material, BMJ does not warrant the accuracy and reliability of the translations (including but not limited to local regulations, clinical guidelines, terminology, drug names and drug dosages), and is not responsible for any error and/or omissions arising from translation and adaptation or otherwise.