A constrained polynomial regression procedure for estimating the local False Discovery Rate

<p>Abstract</p> <p>Background</p> <p>In the context of genomic association studies, for which a large number of statistical tests are performed simultaneously, the local False Discovery Rate (<it>lFDR</it>), which quantifies the evidence of a specific gene association with a clinical or biological variable of interest, is a relevant criterion for taking into account the multiple testing problem. The <it>lFDR </it>not only allows an inference to be made for each gene through its specific value, but also an estimate of Benjamini-Hochberg's False Discovery Rate (<it>FDR</it>) for subsets of genes.</p> <p>Results</p> <p>In the framework of estimating procedures without any distributional assumption under the alternative hypothesis, a new and efficient procedure for estimating the <it>lFDR </it>is described. The results of a simulation study indicated good performances for the proposed estimator in comparison to four published ones. The five different procedures were applied to real datasets.</p> <p>Conclusion</p> <p>A novel and efficient procedure for estimating <it>lFDR </it>was developed and evaluated.</p

The edge of discovery: Controlling the local false discovery rate at the margin

Despite the popularity of the false discovery rate (FDR) as an error control metric for large-scale multiple testing, its close Bayesian counterpart the local false discovery rate (lfdr), defined as the posterior probability that a particular null hypothesis is false, is a more directly relevant standard for justifying and interpreting individual rejections. However, the lfdr is difficult to work with in small samples, as the prior distribution is typically unknown. We propose a simple multiple testing procedure and prove that it controls the expectation of the maximum lfdr across all rejections; equivalently, it controls the probability that the rejection with the largest p-value is a false discovery. Our method operates without knowledge of the prior, assuming only that the p-value density is uniform under the null and decreasing under the alternative. We also show that our method asymptotically implements the oracle Bayes procedure for a weighted classification risk, optimally trading off between false positives and false negatives. We derive the limiting distribution of the attained maximum lfdr over the rejections, and the limiting empirical Bayes regret relative to the oracle procedure

Validation of differential gene expression algorithms: Application comparing fold-change estimation to hypothesis testing

<p>Abstract</p> <p>Background</p> <p>Sustained research on the problem of determining which genes are differentially expressed on the basis of microarray data has yielded a plethora of statistical algorithms, each justified by theory, simulation, or ad hoc validation and yet differing in practical results from equally justified algorithms. Recently, a concordance method that measures agreement among gene lists have been introduced to assess various aspects of differential gene expression detection. This method has the advantage of basing its assessment solely on the results of real data analyses, but as it requires examining gene lists of given sizes, it may be unstable.</p> <p>Results</p> <p>Two methodologies for assessing predictive error are described: a cross-validation method and a posterior predictive method. As a nonparametric method of estimating prediction error from observed expression levels, cross validation provides an empirical approach to assessing algorithms for detecting differential gene expression that is fully justified for large numbers of biological replicates. Because it leverages the knowledge that only a small portion of genes are differentially expressed, the posterior predictive method is expected to provide more reliable estimates of algorithm performance, allaying concerns about limited biological replication. In practice, the posterior predictive method can assess when its approximations are valid and when they are inaccurate. Under conditions in which its approximations are valid, it corroborates the results of cross validation. Both comparison methodologies are applicable to both single-channel and dual-channel microarrays. For the data sets considered, estimating prediction error by cross validation demonstrates that empirical Bayes methods based on hierarchical models tend to outperform algorithms based on selecting genes by their fold changes or by non-hierarchical model-selection criteria. (The latter two approaches have comparable performance.) The posterior predictive assessment corroborates these findings.</p> <p>Conclusions</p> <p>Algorithms for detecting differential gene expression may be compared by estimating each algorithm's error in predicting expression ratios, whether such ratios are defined across microarray channels or between two independent groups.</p> <p>According to two distinct estimators of prediction error, algorithms using hierarchical models outperform the other algorithms of the study. The fact that fold-change shrinkage performed as well as conventional model selection criteria calls for investigating algorithms that combine the strengths of significance testing and fold-change estimation.</p

A comparative review of estimates of the proportion unchanged genes and the false discovery rate

BACKGROUND: In the analysis of microarray data one generally produces a vector of p-values that for each gene give the likelihood of obtaining equally strong evidence of change by pure chance. The distribution of these p-values is a mixture of two components corresponding to the changed genes and the unchanged ones. The focus of this article is how to estimate the proportion unchanged and the false discovery rate (FDR) and how to make inferences based on these concepts. Six published methods for estimating the proportion unchanged genes are reviewed, two alternatives are presented, and all are tested on both simulated and real data. All estimates but one make do without any parametric assumptions concerning the distributions of the p-values. Furthermore, the estimation and use of the FDR and the closely related q-value is illustrated with examples. Five published estimates of the FDR and one new are presented and tested. Implementations in R code are available. RESULTS: A simulation model based on the distribution of real microarray data plus two real data sets were used to assess the methods. The proposed alternative methods for estimating the proportion unchanged fared very well, and gave evidence of low bias and very low variance. Different methods perform well depending upon whether there are few or many regulated genes. Furthermore, the methods for estimating FDR showed a varying performance, and were sometimes misleading. The new method had a very low error. CONCLUSION: The concept of the q-value or false discovery rate is useful in practical research, despite some theoretical and practical shortcomings. However, it seems possible to challenge the performance of the published methods, and there is likely scope for further developing the estimates of the FDR. The new methods provide the scientist with more options to choose a suitable method for any particular experiment. The article advocates the use of the conjoint information regarding false positive and negative rates as well as the proportion unchanged when identifying changed genes

Development of a calibration pipeline for a monocular-view structured illumination 3D sensor utilizing an array projector

Commercial off-the-shelf digital projection systems are commonly used in active structured illumination photogrammetry of macro-scale surfaces due to their relatively low cost, accessibility, and ease of use. They can be described as inverse pinhole modelled. The calibration pipeline of a 3D sensor utilizing pinhole devices in a projector-camera setup configuration is already well-established. Recently, there have been advances in creating projection systems offering projection speeds greater than that available from conventional off-the-shelf digital projectors. However, they cannot be calibrated using well established techniques based on the pinole assumption. They are chip-less and without projection lens. This work is based on the utilization of unconventional projection systems known as array projectors which contain not one but multiple projection channels that project a temporal sequence of illumination patterns. None of the channels implement a digital projection chip or a projection lens. To workaround the calibration problem, previous realizations of a 3D sensor based on an array projector required a stereo-camera setup. Triangulation took place between the two pinhole modelled cameras instead. However, a monocular setup is desired as a single camera configuration results in decreased cost, weight, and form-factor. This study presents a novel calibration pipeline that realizes a single camera setup. A generalized intrinsic calibration process without model assumptions was developed that directly samples the illumination frustum of each array projection channel. An extrinsic calibration process was then created that determines the pose of the single camera through a downhill simplex optimization initialized by particle swarm. Lastly, a method to store the intrinsic calibration with the aid of an easily realizable calibration jig was developed for re-use in arbitrary measurement camera positions so that intrinsic calibration does not have to be repeated