Estimating the Proportion of True Null Hypotheses, with application to DNA Microarray Data

Abstract

We consider the problem of estimating the proportion of true null hypotheses, "π" 0, in a multiple-hypothesis set-up. The tests are based on observed "p"-values. We first review published estimators based on the estimator that was suggested by Schweder and Spjøtvoll. Then we derive new estimators based on nonparametric maximum likelihood estimation of the "p"-value density, restricting to decreasing and convex decreasing densities. The estimators of "π" 0 are all derived under the assumption of independent test statistics. Their performance under dependence is investigated in a simulation study. We find that the estimators are relatively robust with respect to the assumption of independence and work well also for test statistics with moderate dependence. Copyright 2005 Royal Statistical Society.

Comparison of Methods for Estimating the Proportion of Null Hypotheses π0 in High Dimensional Data When the Test Statistics is Continuous

“…Langass and Landqvist's estimator is based on the non-parametric MLE of the p-value density [13]. The authors restricted their attention to decreasing and convex decreasing densities because of their many mathematically attractive properties.…”

Annotated Minimum Volume Sets for Nonparametric Anomaly Discovery

“…A consistent estimator for this problem has been studied in [17]. Practical estimators have also been developed in recent work on multiple testing [18,11] where they are used to estimate the proportion of true null hypothesis. There the p-values of a test play a role similar to our Y i ; under a null hypothesis, p-values are uniform, just as our Y i 's are uniform under X ∼ µ.…”

Approximate Power and Sample Size Calculations with the Benjamini-Hochberg Method

“…The estimator of γ is based on the density estimators of Subsection 3.2 and, just like the estimator of Langaas et al (2005), can be seen as a generalization-and, at least asymptotically, as an improved version-of Storey's estimator (Storey (2002)). …”

“…In Section 4 we apply the convergence results of Section 3 to obtain a 'Storey-type' non-parametric estimator of γ based on density estimators (following an approach already considered by Langaas et al (2005)). In Section 5 we use the result on deconvolution estimators to prove the consistency of a 'semi-parametric' estimator of γ and of what is essentially the distribution function of the p-values associated to the false hypotheses, from which follows the consistency of our method of sample size calculation.…”

“…There are other estimators known to obviate problems at boundary points which could also be used; see the recent approach of Langaas et al (2005) to the problem of estimating γ. The beta kernel sequence of estimators of ψ is defined for x ∈ [0, 1] bŷ…”