A GENERAL DECISION THEORETIC FORMULATION OF PROCEDURES CONTROLLING FDR AND FNR FROM A BAYESIAN PERSPECTIVE

A general decision theoretic formulation is given to multiple testing, al- lowing descriptions of measures of false discoveries and false non-discoveries in terms of certain loss functions even when randomized decisions are made on the hypothe- ses. Randomized as well as non-randomized procedures controlling the Bayes false discovery rate (BFDR) and Bayes false non-discovery rate (BFNR) are developed. These are applicable in any situation, unlike the corresponding frequentist pro- cedures that control the BFDR or BFNR, but do so under certain dependence structures of the test statistics. Even in the presence of such dependence, as simu- lations show, the proposed procedures perform much better than the corresponding frequentist procedures. They provide better control of the BFDR or BFNR than those for which control is achieved through local FDR or local FNR.

[1]  S. Sarkar.  Some Results on False Discovery Rate in Stepwise multiple testing procedures , 2002 .

[2]  S. Sarkar,et al.  Modified Simes’ critical values under positive dependence , 2006 .

[3]  Y. Benjamini,et al.  Resampling-based false discovery rate controlling multiple test procedures for correlated test statistics , 1999 .

[4]  S. Chib,et al.  Bayesian analysis of binary and polychotomous response data , 1993 .

[5]  John D. Storey,et al.  Empirical Bayes Analysis of a Microarray Experiment , 2001 .

[6]  R. Tibshirani,et al.  Empirical bayes methods and false discovery rates for microarrays , 2002, Genetic epidemiology.

[7]  K. Kwong,et al.  A more powerful step-up procedure for controlling the false discovery rate under independence , 2002 .

[8]  A. Cohen,et al.  Characterization of Bayes procedures for multiple endpoint problems and inadmissibility of the step-up procedure , 2005, math/0504506.

[9]  Y. Benjamini,et al.  A step-down multiple hypotheses testing procedure that controls the false discovery rate under independence , 1999 .

[10]  H. Finner,et al.  On the False Discovery Rate and Expected Type I Errors , 2001 .

[11]  B. Efron.  Robbins, Empirical Bayes, And Microarrays , 2001 .

[12]  Y. Benjamini,et al.  Controlling the false discovery rate: a practical and powerful approach to multiple testing , 1995 .

[13]  S. Dudoit,et al.  Multiple Hypothesis Testing in Microarray Experiments , 2003 .

[14]  Y. Benjamini,et al.  THE CONTROL OF THE FALSE DISCOVERY RATE IN MULTIPLE TESTING UNDER DEPENDENCY , 2001 .

[15]  Christopher R. Genovese,et al.  A Stochastic Process Approach to False Discovery Rates , 2003 .

[16]  John D. Storey.  The positive false discovery rate: a Bayesian interpretation and the q-value , 2003 .

[17]  L. Wasserman,et al.  A stochastic process approach to false discovery control , 2004, math/0406519.

[18]  Y. Benjamini,et al.  On the Adaptive Control of the False Discovery Rate in Multiple Testing With Independent Statistics , 2000 .

[19]  Siu Hung Cheung,et al.  A modified Benjamini–Hochberg multiple comparisons procedure for controlling the false discovery rate , 2002 .

[20]  L. Wasserman,et al.  Operating characteristics and extensions of the false discovery rate procedure , 2002 .

[21]  P. Müller,et al.  Optimal Sample Size for Multiple Testing , 2004 .

[22]  Jie Chen,et al.  A Bayesian determination of threshold for identifying differentially expressed genes in microarray experiments , 2006, Statistics in medicine.

[23]  John D. Storey,et al.  Strong control, conservative point estimation and simultaneous conservative consistency of false discovery rates: a unified approach , 2004 .

[24]  John D. Storey.  A direct approach to false discovery rates , 2002 .

[25]  S. Sarkar.  False discovery and false nondiscovery rates in single-step multiple testing procedures , 2006, math/0605607.

[26]  J. Troendle,et al.  Stepwise normal theory multiple test procedures controlling the false discovery rate , 2000 .

[27]  E. George,et al.  Journal of the American Statistical Association is currently published by American Statistical Association. , 2007 .

[28]  Sanat K. Sarkar,et al.  FDR-CONTROLLING STEPWISE PROCEDURES AND THEIR FALSE NEGATIVES RATES , 2004 .