## Introduction

Instantaneous dioptric power (a 2×2 matrix, **F**) where variation over time is momentarily ignored is fundamentally a four-variate quantity1 and any suitable analysis must take into consideration that power is quantifiable using four unique quantities or numbers. (eg, the instantaneous power of the human eye, a thick optical system, is four- dimensional).

*Refractive behaviour* or the variation over time of instantaneous refractive state (a three-dimensional quantity defined as the dioptric power of a thin lens with three df, ie, sphere, cylinder and its axis)2 is a multivariate concern. Factors such as age or visual acuity add dimensions in any analytical situation involving refractive behaviour. The logical mathematical approach for such multivariate quantities is via matrices (asymmetric or sometimes symmetric) and the various operations of linear algebra.1 2 Sometimes symmetric matrices2 are applicable and in most clinical situations such as with refractive state this simplification applies. This review will apply experimental data to highlight a few critical issues for further investigation and theoretical development in the years to come concerning dioptric power and its analysis in scientific studies.

Early attempts to understand dioptric power involved the use of optometric vectors (by Gartner3) where the angles of the cylinder axes were doubled for a more mathematically appropriate Cartesian two-dimensional space with limits of 0°–360°. Gartner limited his analysis to cylinder only but Fick published three papers4–6 in 1972–1973 that introduced the ideas of applying matrices and linear algebra to dioptric power while Long7 and Keating,8 9 working independently, provided conversion equations from clinical notation (*F*
_{s} *F*
_{c} *A*) to power matrices and also for the necessary inverse operations.

Harris10 mentions the need for a system of analysis allowing for invariance of power under sphero-cylindrical transposition and also describes methods for calculating squares of a power10 and performing mathematical operations necessary for effective determination of quantities such as means and variances when analysing and comparing samples of dioptric power.10–16 Harris *et al* describe methods for ellipsoids or surfaces of constant probability density (SCPD) when comparing distributions17 18 and for testing samples of dioptric power for variance14 and departure from normality that involve profiles of skewness, kurtosis and standardised mean deviation (see Harris and Malan19 for further developments in that area and also trajectories for dioptric power.20 This foundation for understanding distributions of power and transformations towards normality is further developed by Harris and Blackie21 22 where concepts such as Mardia’s multivariate skewness and kurtosis23 are employed including normality plots (marginal and χ^{2}) for refractive state.21 22 Hypothesis tests, integral to understanding experimental effects in the context of dioptric power, are described by Harris.12 13 15

Thibos *et al* referred to power vectors24 with three terms, namely *M*, *J*
_{0} and *J*
_{45} and an advantage of these vectors compared with those of Gartner is that they cope with the three-dimensional (3D) nature of symmetric power but not for four-dimensional asymmetric power.1 2 25 (Others such as Naeser and Hjortdal26 use KP(90) and KP(135) that are the same as *J*
_{0} and *J*
_{45}.) See Harris for a comparison of power vectors and power matrices.27 Refractive behaviour28 in Euclidean three-space can be studied using such mathematical and graphical methods with trajectories,20 comets (to link paired measurements),28 and other methods such as SCPD17 18 including also meridional profiles of various types.14 19 21 28 29 Polar plots for variance of dioptric power were first described by Harris and van Gool30 31 and van Gool used them (and much of the aforementioned methodology) for an analysis that involved ocular accommodation and leads or lags of accommodation to near stimuli.31 (Some of these methods will be illustrated later.)

Other methods to analyse and understand refractive behaviour include that of Alpins32 33 where vectors are similarly applied to, for example, investigate changes in specifically astigmatism after corneal surgery.34–36 Vector magnitudes and directions are represented using polar plots and surgical outcomes can be expressed in terms of difference vectors, and target induced and surgically induced astigmatism vectors.

Researchers have applied the concepts of a more effective analytical and scientific approach to studies of refractive behaviour and examples include Raasch,37 Kaye *et al*
38 39 and others concerned with reproducibility of measures of refractive state40 and issues relating to intraocular lens (IOL) implants.41–44 See also MacKenzie and Harris45 and MacKenzie46 for the application of multivariate methods and power matrices to intraocular implants and their uses in ameliorating refractive errors and cataract. Harris1 16 47 48 and MacKenzie,45 46 van Gool,49 Evans50 51 and others52 53 applied these theoretical ideas for linear optical systems (and ray behaviour through such systems). Since this area is the subject for another review they are not further described here.

Possible relationships between refractive behaviour and visual acuity are explored by Harris and Rubin but this multidimensional topic is not further considered here.54 55 Power vectors and power matrices and the analytical methods as explained and described mathematically below have been applied to a diverse range of clinical and experimental topics20–74 including autorefraction in adults,28–31 56 57 children,58 66 keratometric variation,59–61 comparisons of methods to measure refractive behaviour in relation to age62 63 and other factors including also luminance of targets within autorefractors65 and wavefront aberrometry,66 interocular mirror symmetry,67 cycloplegia,64 68 keratoconus,69 70 measurement error and uncertainty,64 71 72 anisometropia73 and refractive surgery.74