A Clinically Applicable Bivariate Agreement Analysis of Astigmatism

Congratulations to Feng et al. [1] for what is a large and comprehensive comparative accuracy study of four flagship biometers. Of note, I commend the authors for acknowledging the bivariate nature of astigmatism by additionally analysing its orthogonal power components (J0, J45). However, I felt it relevant to comment on the statistical approach used to measure agreement, which involved a classic Bland–Altman analysis of the J0 and J45 Fourier components. Although component powers are a meridional composite of both magnitude and direction, analysing their agreement separately does not constitute a true bivariate analysis because it does not account for the extra error produced by changes in both components simultaneously. Further, knowing the agreement in component powers separately offers little clinical utility. In example, the MS-39 had a bias of +0.180 D for J0 and −0.062 D for J45 when compared to the Casia SS-1000. Though what does this mean clinically? The reader is left with two statistical parameters that indirectly represent one clinical entity. It is tempting to back-convert these powers into a hypotenuse magnitude, though this is flawed. The bivariate bias must be measured at the case level first and then aggregated at the cohort level [2]. A valid approach needs to be bivariate and produce outputs that are clinically relevant. Ultimately, surgeons are interested in the potential refractive astigmatic error attributable to bias between the biometers. This is particularly important when treating keratoconic eyes because their high magnitude of astigmatism increases the risk of residual astigmatic error. I write to draw attention to a very intelligent and possibly underappreciated agreement analysis performed by Sorkin et al. [3] The authors calculated the co-ordinate differences in the double angle component powers of astigmatism between two devices: the IOLMaster 700 (Carl Zeiss Meditech, Germany) and Eyestar 900 (Haag-Streit, Switzerland). At first glance, this statistical approach may not seem particularly useful. However, readers with an appreciation for the history and development of astigmatism analysis may recognise its clinical significance. It is well known that double angle co-ordinates are nothing more than a vector-based analysis of Stokes' double angle parallelogram [4]. However, it is important to recognise that Stokes' parallelogram is only a geometric simplification of meridional power calculus. To calculate the surgically induced astigmatism is akin to calculating the magnitude and orientation of a cylinder that was obliquely crossed with the pre-operative cylinder to produce the post-operative. To do this, the meridional power curve of the pre-operative cylinder is subtracted from the post-operative. It is the derivative of the resultant power curve that reveals the orientation and magnitude of the SIA cylinder. Stokes' parallelogram and the current vector convention can be derived from an algebraic simplification of meridional power calculus that uses the 'Sine Rule' as a trigonometric identity [5]. It is here that the utility of Sorkin's analysis is revealed: a Euclidean distance between two sets of double angle co-ordinates is equivalent to the magnitude of an obliquely crossed cylinder that will transform the first measurement into the second. In which case, the magnitude of the difference co-ordinates for each eye reveals the magnitude of the obliquely crossed cylinder that describes the bias in the measurement. The mean magnitude of the difference co-ordinates can be conceptualised as a relative (i.e., between the biometers) 'mean absolute astigmatic error'. This output is not simply the univariate difference in astigmatism magnitude between the two measurements, but rather is the astigmatic magnitude of the error cylinder. This analysis cannot discriminate between positive and negative cylindrical differences because component powers always describe astigmatism as a positive cylinder (hence 'absolute'). The standard deviation of the mean magnitude reveals the sample variability of the 'error cylinder' between the biometers. The 2-D Euclidean confidence ellipse similarly represents the probabilistic density of the error cylinder magnitudes. Not only is this analysis truly bivariate, though its output is clinically relevant: it describes the relative astigmatic prediction error when selecting one device over another. Again, I congratulate the authors for a fantastic paper. I only wish to highlight the importance of performing an agreement analysis of astigmatism that is statistically valid and clinically relevant. The author declares no conflicts of interest. Data sharing is not applicable to this article as no new data were created or analysed in this study.