A comparison of the box-cox transformation method and nonparametric methods for estimating quantiles in clinical data with repeated measures

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Abstract
In this paper we studied the problem of estimating the quantiles of a distribution when the observations are not independent. The type of data we considered was repeated measurements on independent units. Four methods of quantile estimation were compared, one parametric and three nonparametric. Box- Cox power transformations were used to transform a set of repeated measures to multivariate normality and a quantile estimate was obtained from the inverse transformation of the quantile on the transformed scale. The first nonparametric estimate was obtained by taking a weighted average of two order statistics from a sample consisting of one observation per independent unit, the second nonparametric estimate was a bootstrap of the first estimate, and the third was a bootstrapped version of the Harrell-Davis estimate. Comparisons were made with simulated data from Gaussian, Student's Tlog-normal, chisquared, contaminated normal, and mixed distributions of the first, fifth, ninety-fifth and ninety-ninth quantiles. When sampling from Gaussian data, the Box-Cox likelihood method was least biased, and the bootstrapped Harrell-Davis estimate had smallest MSE. When the data was not Gaussian, the Box-Cox likelihood method was the least biased for estimating the 90% clinical range, and the simple percentile was least biased for the 98% clinical range; the bootstrapped percentile estimate had smallest MSE. The four methods were used to estimate the clinical range in clinical pulmonary data.

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