A Two-Level Toeplitz Model for Large-Scale Simultaneous Hypothesis - PowerPoint PPT Presentation

Empirical Bayes Methods for Simultaneous Hypothesis Testing Estimating Level II Covariance Matrix Data Results Conclusion A Two-Level Toeplitz Model for Large-Scale Simultaneous Hypothesis Testing Dan Cervone Advisor: Carl Morris December 10, 2012

Empirical Bayes Methods for Simultaneous Hypothesis Testing Estimating Level II Covariance Matrix Data Results Conclusion Empirical Bayes Methods for Simultaneous Hypothesis Testing 1 Estimating Level II Covariance Matrix 2 Data Results 3 Conclusion 4

Empirical Bayes Methods for Simultaneous Hypothesis Testing Estimating Level II Covariance Matrix Data Results Conclusion Efron’s fdr[3] Suppose we have M test statistics (assumed to be z scores): ind z i | µ i ∼ N ( µ i , 1) for i = 1 , ..., M . iid µ i ∼ p 0 δ 0 + (1 − p 0 ) g ( µ i ) iid z i ∼ f ( z i ) (marginally) Define fdr( z ) = P ( µ i = 0 | z i = z ) = p 0 φ ( z i ) f ( z i ) fdr( z ) = p 0 φ ( z i ) ˆ ˆ f ( z i ) where ˆ f is an estimate of the density function f . Declare the i th test statistic nonnull if: ˆ fdr( z i ) ≤ q . Bayesian posterior probability interpretation relies on independence!