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. 2015 Jan;16(1):189-204.
doi: 10.1093/biostatistics/kxu029. Epub 2014 Jun 23.

Bias and Variance Reduction in Estimating the Proportion of True-Null Hypotheses

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Free PMC article

Bias and Variance Reduction in Estimating the Proportion of True-Null Hypotheses

Yebin Cheng et al. Biostatistics. .
Free PMC article

Abstract

When testing a large number of hypotheses, estimating the proportion of true nulls, denoted by π(0), becomes increasingly important. This quantity has many applications in practice. For instance, a reliable estimate of π(0) can eliminate the conservative bias of the Benjamini-Hochberg procedure on controlling the false discovery rate. It is known that most methods in the literature for estimating π(0) are conservative. Recently, some attempts have been paid to reduce such estimation bias. Nevertheless, they are either over bias corrected or suffering from an unacceptably large estimation variance. In this paper, we propose a new method for estimating π(0) that aims to reduce the bias and variance of the estimation simultaneously. To achieve this, we first utilize the probability density functions of false-null p-values and then propose a novel algorithm to estimate the quantity of π(0). The statistical behavior of the proposed estimator is also investigated. Finally, we carry out extensive simulation studies and several real data analysis to evaluate the performance of the proposed estimator. Both simulated and real data demonstrate that the proposed method may improve the existing literature significantly.

Keywords: Effect size; False-null p-value; Microarray data; Multiple testing; Probability density function; Upper tail probability.

Figures

Figure 1.
Figure 1.
Plots of MSEs as functions of formula image for various formula image and formula image values, where “1” represents the bootstrap estimator formula image, “2” represents the average estimate estimator formula image, “3” represents the convex estimator formula image, “4” represents the parametric estimator formula image, and “5” represents the proposed new estimator formula image.
Figure 2.
Figure 2.
Density estimates of formula image for formula image, where the short dashed line represents the bootstrap estimator formula image, the dash-dotted line represents the average estimate estimator formula image, the dotted line represents the convex estimator formula image, the long dashed line represents the parametric estimator formula image, and the solid line represents the proposed new estimator formula image.
Figure 3.
Figure 3.
Density estimates of formula image for formula image, where the short dashed line represents the bootstrap estimator formula image, the dash-dotted line represents the average estimate estimator formula image, the dotted line represents the convex estimator formula image, the long dashed line represents the parametric estimator formula image, and the solid line represents the proposed new estimator formula image.
Figure 4.
Figure 4.
Histograms of formula image-values for the three data sets, where (A), (B), and (C) correspond to the formula image-values for the first, second, and third data set, respectively.

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