# "meanA" Function. Statistics in the astigmatism analysis with the Alpins method.

This function computes the mean (standard deviation) and the median (interquartile range) from an array of astigmatism values containing the Magnitude and Direction. The mean computed without considering the axis, only describes information about the magnitude of astigmatism but not about the type of astigmatism (with-the-rule, WTR; against-the-rule, ATR; oblique). For considering not only the magnitude but also the orientation a previous decomposition in the orthogonal X and Y components over a double angle plot figure is required. Then, inferential statistics can be applied considering these orthogonal components.

Astigmatism analysis is based on vector arithmetic. The current standard method in Ophthalmology practice is the Alpins method^{1} whereas in Optometry practice is more common to use the Thibos method.^{2} In the practice, both methods result in similar outputs.

Matlab Toolbox “refractiveAnalysis” developed by Manuel Rodríguez-Vallejo for research purposes included in the version 1.0.0 the function “meanA”.

The "meanA" function computes the statistic indexes of centrality or dispersion and outputs the astigmatism in X and Y components in order to be later used in other software packages as SPSS or for performing inferential statistics in Matlab.

Below you have the help of the function. For more information about "refractiveAnalysis" Toolbox please Contact the author

```
% Computes the mean (standard deviation) and median (interquartile range)
% from a vector of astigmatism.
%
% DESCRIPTION
%
% The cartesian components of the astigmatism are computed and the mean
% and median are calculated for the X and Y vector
%
% PARAMETERS
%
% V Matrix of two columns. First column with the magnitude
% of the astigmatism and second column the orientation of the astigmatim.
%
% APPLICATIONS
%
% Computes the mean of the astigmatism considering the orientation.
%
% OUTPUTS
%
% md Contains [meanX meanY SDx SDy] in cartesian coordinates for
% the double plot.
% MD Contains [medianX medianY IQRx IQRy] in cartesian coordinates for
% the double plot.
% Mxy Components X and Y of the double plot.
%
% P Contains the mean resultant in Polar coordinates.
%
% EXAMPLE
%
% [Mxy,md,MD,P]=meanA(V)
%
% VERSION 1.0.1
```

References

- Alpins NA. A new method of analyzing vectors for changes in astigmatism. J. Cataract Refract. Surg. 1993; 19: 524–33. 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.