Double Angle Plot for Astigmatism Representation
Updated on version: 1.0.4


Double angle plot for describing mean, standard deviation and the 95% confidence ellipses from the centroid and from the datasets described in the JCRS editoral (Abulafia et al.2018;44(10):1169-1174)


Double Angle Plot for Astigmatism Representation


The "doublePlot" function draws, in a double angle plot, the data from an astigmatism sample. Two plots are possible, the old one which represents the mean and standard deviation by means of a square and an ellipse, respectively. The new one contains the mean and the covariance ellipses with 95% confidence ellipse of the centroid (in red) and 95% confidence ellipse of the dataset (in blue)

Astigmatism analysis is based on vector arithmetic. The current standard method in Ophthalmology practice is the Alpins method1 whereas in Optometry practice is more common to use the Thibos method.2 In practice, both methods result in similar outcomes. The figure shows an example of the new representation with the standard deviation along the axis of displacement of the centroid and perpendicular to it (covariance ellipse).3 The old representation that you can find in some papers4 refers to the standard deviation in the axis 0º-180º and the axis 45º-135º (obsolete).  In this new figure you have two ellipses:

  • Red ellipse: 95% confidence ellipse of the centroid. If the center of the coordinates from the graph is within this ellipse, the magnitude of mean astigmatism may be due to chance, so it should not be considered.
  • Blue ellipse: 95% limits of agreement of the centroid. Estimates the variability in the sample in such a way that 95% of the sample is inside the blue ellipse. It is an indicator of the bias of your astigmatism results, the smaller the size of the ellipse, the lower the variability.

The difference between the two ellipses is analogous to the concept of 95% Confidence Interval and 95% Agreement Limits (Bland-Altman). If you want to know more about the difference between both you can access to our Statistics Online Course.


  1. Alpins NA. A new method of analyzing vectors for changes in astigmatism. J. Cataract Refract. Surg. 1993; 19: 524–33. 2
  2. Thibos LN, Wheeler W, Horner D. Power vectors: an application of Fourier analysis to the description and statistical analysis of refractive error. Optom Vis Sci. 1997; 74: 367–75
  3. Abulafia A, Koch DD, Holladay JT, Wang L, Hill W. Pursuing perfection in intraocular lens calculations: IV. Rethinking astigmatism analysis for intraocular lens-based surgery: Suggested terminology, analysis, and standards for outcome reports. J Cataract Refract Surg. 2018;44(10):1169-1174
  4. Fernández J, Rodríguez-Vallejo M, Martínez J, Tauste A, Piñero DP. Prediction of surgically induced astigmatism in manual and femtosecond laser-assisted clear corneal incisions. Eur J Ophthalmol. 2018;28(4):398–405. 

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