Multivariate nonparametric techniques for astigmatism analysis.
ABSTRACT 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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ABSTRACT: Hotelling's T 2 test is known to be optimal under multivariate normality and is reasonably validity-robust when the assumption fails. However, some recently introduced robust test procedures have superior power properties and reasonable type I error control with non-normal populations. These, including the tests due to Tiku & Singh (1982), Tiku & Balakrishnan (1988) and Mudholkar & Srivastava (1999b, c), are asymptotically valid but are useful with moderate size samples only if the population dimension is small. A class of B-optimal modifications of the stepwise alternatives to Hotellings T 2 introduced by Mudholkar & Subbaiah (1980) are simple to implement and essentially equivalent to the T 2 test even with small samples. In this paper we construct and study the robust versions of these modified stepwise tests using trimmed means instead of sample means. We use the robust one- and two-sample trimmed- t procedures as in Mudholkar et al. (1991) and propose statistics based on combining them. The results of an extensive Monte Carlo experiment show that the robust alternatives provide excellent type I error control and a substantial gain in power.Journal of Applied Statistics. 01/2000; 27(5):599-619.
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ABSTRACT: Asymptotic Pitman efficiencies of multivariate spatial sign and rank methods are considered in the one-sample location case. Limiting distributions of the spatial sign and signed-rank tests under the null hypothesis as well as under contiguous sequences of alternatives are given. Formulae for asymptotic relative efficiencies are found and, under multivariate t distributions, relative efficiencies with respect to Hotelling's $T^2$ test are calculated.The Annals of Statistics 01/1997; · 2.53 Impact Factor
- British Journal of Ophthalmology 01/2003; 86(12):1458-9. · 2.73 Impact Factor