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J Cataract Refract Surg. 2010 Apr;36(4):594-602. doi: 10.1016/j.jcrs.2009.11.002.

Multivariate nonparametric techniques for astigmatism analysis.

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  • 1Department of Ophthalmology, Jules Stein Eye Institute, David Geffen School of Medicine at UCLA, Los Angeles, California 90095-7002, USA.



To describe the application of nonparametric multivariate statistical methods to the analysis of astigmatism treatment outcomes.


Jules Stein Eye Institute and Department of Ophthalmology, David Geffen School of Medicine at UCLA, Los Angeles, California, USA.


Nonparametric methods were applied to a published data set and to 12 test data sets created for test purposes. Results of 3 multivariate nonparametric tests were compared with those obtained using the Hotelling T(2), a multivariate parametric test. The nonparametric tests were the rank-based multivariate analysis of variance (MANOVA), sign-based MANOVA, and bootstrapping based on the Hotelling T(2) statistic.


Reanalysis of the published data set using the 3 nonparametric tests detected statistically significant treatment effects at all postoperative examinations. The Hotelling T(2) and 3 nonparametric tests detected differences in astigmatism outcomes for multiple test data sets that simulated normal distributions. For test data sets simulating non-normal distributions, the Hotelling T(2) test and bootstrapping based on Hotelling T(2) detected a difference in 1 test data set while rank-based and sign-based MANOVA detected differences in outcomes for multiple data sets.


Rank-based and sign-based MANOVA had comparable or slightly lower power than the Hotelling T(2) test in detecting differences in normally distributed data. For data sets in which the rectangular components of astigmatism vectors do not distribute normally in both dimensions, only the nonparametric statistical methods were valid. The sign-based MANOVA was the most sensitive in detecting differences in non-normally distributed astigmatism outcomes in the data sets.

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