Optimal Discovery Procedure Research Articles

Optimal Sufficient Statistics for Parametric and Non-Parametric Multiple Simultaneous Hypothesis Testing

In multiple simultaneous hypothesis testing (MSHT), a significance thresholding function as a scalar statistic can be designed in an adaptive manner by sharing information among many tests performed simultaneously. By using such an adapted statistic, MSHT has greater detection power than tests using simple individual statistics. To systematically obtain an optimal thresholding function that maximizes the detection power in MSHT, Storey (2007) proposed a theoretical framework called the optimal discovery procedure (ODP). He also proposed an empirical estimation of the ODP thresholding function for a parametric MSHT that presupposes parametric forms of the null and alternative likelihood functions. Empirical Bayesian testing (Efron et al. 2001), which is based on a non-parametric treatment of arbitrary test statistics, has sometimes exhibited comparable power to the ODP. These two MSHT frameworks appear to be closely related but, because of differences in their approach (frequentist vs. Bayesian), the relationship is not well understood.We present the new concept of an optimal sufficient statistic that links the ODP and empirical Bayesian frameworks, and we show that the local false discovery rate based on the empirical Bayes can be an optimal thresholding function if a certain condition holds. We lay out exhaustive sets of presumptions to achieve optimal thresholding functions and show that, if an optimal thresholding function is derived for a parametric MSHT problem, it is still optimal for a more general and broader range of MSHT problems defined in a non- or semi-parametric way. A guide to designing optimal thresholding functions for general MSHT problems is thus provided by our study.

Bayesian optimal discovery procedure for simultaneous significance testing

BackgroundIn high throughput screening, such as differential gene expression screening, drug sensitivity screening, and genome-wide RNAi screening, tens of thousands of tests need to be conducted simultaneously. However, the number of replicate measurements per test is extremely small, rarely exceeding 3. Several current approaches demonstrate that test statistics with shrinking variance estimates have more power over the traditional t statistic.ResultsWe propose a Bayesian hierarchical model to incorporate the shrinkage concept by introducing a mixture structure on variance components. The estimates from the Bayesian model are utilized in the optimal discovery procedure (ODP) proposed by Storey in 2007, which was shown to have optimal performance in multiple significance tests. We compared the performance of the Bayesian ODP with several competing test statistics.ConclusionWe have conducted simulation studies with 2 to 6 replicates per gene. We have also included test results from two real datasets. The Bayesian ODP outperforms the other methods in our study, including the original ODP. The advantage of the Bayesian ODP becomes more significant when there are few replicates per test. The improvement over the original ODP is based on the fact that Bayesian model borrows strength across genes in estimating unknown parameters. The proposed approach is efficient in computation due to the conjugate structure of the Bayesian model. The R code (see Additional file 1) to calculate the Bayesian ODP is provided.

The Optimal Discovery Procedure: A New Approach to Simultaneous Significance Testing

SummaryThe Neyman–Pearson lemma provides a simple procedure for optimally testing a single hypothesis when the null and alternative distributions are known. This result has played a major role in the development of significance testing strategies that are used in practice. Most of the work extending single-testing strategies to multiple tests has focused on formulating and estimating new types of significance measures, such as the false discovery rate. These methods tend to be based on p-values that are calculated from each test individually, ignoring information from the other tests. I show here that one can improve the overall performance of multiple significance tests by borrowing information across all the tests when assessing the relative significance of each one, rather than calculating p-values for each test individually. The ‘optimal discovery procedure’ is introduced, which shows how to maximize the number of expected true positive results for each fixed number of expected false positive results. The optimality that is achieved by this procedure is shown to be closely related to optimality in terms of the false discovery rate. The optimal discovery procedure motivates a new approach to testing multiple hypotheses, especially when the tests are related. As a simple example, a new simultaneous procedure for testing several normal means is defined; this is surprisingly demonstrated to outperform the optimal single-test procedure, showing that a method which is optimal for single tests may no longer be optimal for multiple tests. Connections to other concepts in statistics are discussed, including Stein's paradox, shrinkage estimation and the Bayesian approach to hypothesis testing.

Detecting differential expression in microarray data: comparison of optimal procedures.

BackgroundMany procedures for finding differentially expressed genes in microarray data are based on classical or modified t-statistics. Due to multiple testing considerations, the false discovery rate (FDR) is the key tool for assessing the significance of these test statistics. Two recent papers have generalized two aspects: Storey et al. (2005) have introduced a likelihood ratio test statistic for two-sample situations that has desirable theoretical properties (optimal discovery procedure, ODP), but uses standard FDR assessment; Ploner et al. (2006) have introduced a multivariate local FDR that allows incorporation of standard error information, but uses the standard t-statistic (fdr2d). The relationship and relative performance of these methods in two-sample comparisons is currently unknown.MethodsUsing simulated and real datasets, we compare the ODP and fdr2d procedures. We also introduce a new procedure called S2d that combines the ODP test statistic with the extended FDR assessment of fdr2d.ResultsFor both simulated and real datasets, fdr2d performs better than ODP. As expected, both methods perform better than a standard t-statistic with standard local FDR. The new procedure S2d performs as well as fdr2d on simulated data, but performs better on the real data sets.ConclusionThe ODP can be improved by including the standard error information as in fdr2d. This means that the optimality enjoyed in theory by ODP does not hold for the estimated version that has to be used in practice. The new procedure S2d has a slight advantage over fdr2d, which has to be balanced against a significantly higher computational effort and a less intuititive test statistic.

Semi-supervised discovery of differential genes

BackgroundVarious statistical scores have been proposed for evaluating the significance of genes that may exhibit differential expression between two or more controlled conditions. However, in many clinical studies to detect clinical marker genes for example, the conditions have not necessarily been controlled well, thus condition labels are sometimes hard to obtain due to physical, financial, and time costs. In such a situation, we can consider an unsupervised case where labels are not available or a semi-supervised case where labels are available for a part of the whole sample set, rather than a well-studied supervised case where all samples have their labels.ResultsWe assume a latent variable model for the expression of active genes and apply the optimal discovery procedure (ODP) proposed by Storey (2005) to the model. Our latent variable model allows gene significance scores to be applied to unsupervised and semi-supervised cases. The ODP framework improves detectability by sharing the estimated parameters of null and alternative models of multiple tests over multiple genes. A theoretical consideration leads to two different interpretations of the latent variable, i.e., it only implicitly affects the alternative model through the model parameters, or it is explicitly included in the alternative model, so that the interpretations correspond to two different implementations of ODP. By comparing the two implementations through experiments with simulation data, we have found that sharing the latent variable estimation is effective for increasing the detectability of truly active genes. We also show that the unsupervised and semi-supervised rating of genes, which takes into account the samples without condition labels, can improve detection of active genes in real gene discovery problems.ConclusionThe experimental results indicate that the ODP framework is effective for hypotheses including latent variables and is further improved by sharing the estimations of hidden variables over multiple tests.

The optimal discovery procedure for large-scale significance testing, with applications to comparative microarray experiments

As much of the focus of genetics and molecular biology has shifted toward the systems level, it has become increasingly important to accurately extract biologically relevant signal from thousands of related measurements. The common property among these high-dimensional biological studies is that the measured features have a rich and largely unknown underlying structure. One example of much recent interest is identifying differentially expressed genes in comparative microarray experiments. We propose a new approach aimed at optimally performing many hypothesis tests in a high-dimensional study. This approach estimates the optimal discovery procedure (ODP), which has recently been introduced and theoretically shown to optimally perform multiple significance tests. Whereas existing procedures essentially use data from only one feature at a time, the ODP approach uses the relevant information from the entire data set when testing each feature. In particular, we propose a generally applicable estimate of the ODP for identifying differentially expressed genes in microarray experiments. This microarray method consistently shows favorable performance over five highly used existing methods. For example, in testing for differential expression between two breast cancer tumor types, the ODP provides increases from 72% to 185% in the number of genes called significant at a false discovery rate of 3%. Our proposed microarray method is freely available to academic users in the open-source, point-and-click EDGE software package.