A comparison of the Benjamini-Hochberg procedure with some Bayesian rules for multiple testing

In the spirit of modeling inference for microarrays as multiple testing for sparse mixtures, we present a similar approach to a simplified version of quantitative trait loci (QTL) mapping. Unlike in case of microarrays, where the number of tests usually reaches tens of thousands, the number of tests performed in scans for QTL usually does not exceed several hundreds. However, in typical cases, the sparsity $p$ of significant alternatives for QTL mapping is in the same range as for microarrays. For methodological interest, as well as some related applications, we also consider non-sparse mixtures. Using simulations as well as theoretical observations we study false discovery rate (FDR), power and misclassification probability for the Benjamini-Hochberg (BH) procedure and its modifications, as well as for various parametric and nonparametric Bayes and Parametric Empirical Bayes procedures. Our results confirm the observation of Genovese and Wasserman (2002) that for small p the misclassification error of BH is close to optimal in the sense of attaining the Bayes oracle. This property is shared by some of the considered Bayes testing rules, which in general perform better than BH for large or moderate $p$'s.Comment: Published in at http://dx.doi.org/10.1214/193940307000000158 the IMS Collections (http://www.imstat.org/publications/imscollections.htm) by the Institute of Mathematical Statistics (http://www.imstat.org

Asymptotic Bayes-optimality under sparsity of some multiple testing procedures

Within a Bayesian decision theoretic framework we investigate some asymptotic optimality properties of a large class of multiple testing rules. A parametric setup is considered, in which observations come from a normal scale mixture model and the total loss is assumed to be the sum of losses for individual tests. Our model can be used for testing point null hypotheses, as well as to distinguish large signals from a multitude of very small effects. A rule is defined to be asymptotically Bayes optimal under sparsity (ABOS), if within our chosen asymptotic framework the ratio of its Bayes risk and that of the Bayes oracle (a rule which minimizes the Bayes risk) converges to one. Our main interest is in the asymptotic scheme where the proportion p of "true" alternatives converges to zero. We fully characterize the class of fixed threshold multiple testing rules which are ABOS, and hence derive conditions for the asymptotic optimality of rules controlling the Bayesian False Discovery Rate (BFDR). We finally provide conditions under which the popular Benjamini-Hochberg (BH) and Bonferroni procedures are ABOS and show that for a wide class of sparsity levels, the threshold of the former can be approximated by a nonrandom threshold.Comment: Published in at http://dx.doi.org/10.1214/10-AOS869 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

On the asymptotic properties of SLOPE

Sorted L-One Penalized Estimator (SLOPE) is a relatively new convex optimization procedure for selecting predictors in large data bases. Contrary to LASSO, SLOPE has been proved to be asymptotically minimax in the context of sparse high-dimensional generalized linear models. Additionally, in case when the design matrix is orthogonal, SLOPE with the sequence of tuning parameters $\lambda^{BH}$, corresponding to the sequence of decaying thresholds for the Benjamini-Hochberg multiple testing correction, provably controls False Discovery Rate in the multiple regression model. In this article we provide new asymptotic results on the properties of SLOPE when the elements of the design matrix are iid random variables from the Gaussian distribution. Specifically, we provide the conditions, under which the asymptotic FDR of SLOPE based on the sequence $\lambda^{BH}$ converges to zero and the power converges to 1. We illustrate our theoretical asymptotic results with extensive simulation study. We also provide precise formulas describing FDR of SLOPE under different loss functions, which sets the stage for future results on the model selection properties of SLOPE and its extensions.Comment: 43 pages, 4 figure

Regionalisation of unutilised agricultural area in Poland

Until the marketization of economy in 1990, practically there was no unutilised agricultural land in Poland. After the political transformation, the use of marginal land and part of small agricultural parcels located in areas of better soil agricultural suitability became unprofitable for economic reasons. Despite the reform of Polish agriculture, the problem of not using a large part of agricultural land is constantly present. There are currently no detailed data available on the actual scale of regionalisation and the structure of the landuse abandonment in Poland. Due to the above, the objective of this study is to fill knowledge gaps on this phenomenon. Analyses were performed based on cadastral maps for the whole country. It is the first comprehensive and detailed study of this issue, giving the full review of the regionalisation of abandoned land. Unutilised land defined as: cadastral parcels located on rural land, which is not declared as production area by farmers.The study has shown that currently in Poland more than 2.7 million ha of agricultural land is not declared by the farmers as area for agricultural activity. This assessment includes 2.03 million ha of unutilized areas of effective production (parcels  > 0.3 ha), which constitutes 14.2% of the overall agricultural area. A significant proportion of the unutilised agricultural land constitute medium and high productivity soils: 59.7 thousand ha of class III, 73.87 thousand ha of class IIIa, 116.6 thousand ha of class IIIb, 240 thousand ha of class IV, 225 thousand ha of class IVa, 221 thousand ha of class IVb. Analyses showed clear regionalisation of the problem of unused potential in the agricultural production area. This situation is particularly visible in Małopolskie, Podkarpackie, Świętokrzyskie, Śląskie, and part of Mazowieckie voivodeships

SLOPE - Adaptive variable selection via convex optimization

We introduce a new estimator for the vector of coefficients $\beta$ in the linear model $y=X\beta+z$, where $X$ has dimensions $n\times p$ with $p$ possibly larger than $n$. SLOPE, short for Sorted L-One Penalized Estimation, is the solution to $$\min_{b\in\mathbb{R}^p}\frac{1}{2}\Vert y-Xb\Vert _{\ell_2}^2+\lambda_1\vert b\vert _{(1)}+\lambda_2\vert b\vert_{(2)}+\cdots+\lambda_p\vert b\vert_{(p)},$$ where $\lambda_1\ge\lambda_2\ge\cdots\ge\lambda_p\ge0$ and $\vert b\vert_{(1)}\ge\vert b\vert_{(2)}\ge\cdots\ge\vert b\vert_{(p)}$ are the decreasing absolute values of the entries of $b$. This is a convex program and we demonstrate a solution algorithm whose computational complexity is roughly comparable to that of classical $\ell_1$ procedures such as the Lasso. Here, the regularizer is a sorted $\ell_1$ norm, which penalizes the regression coefficients according to their rank: the higher the rank - that is, stronger the signal - the larger the penalty. This is similar to the Benjamini and Hochberg [J. Roy. Statist. Soc. Ser. B 57 (1995) 289-300] procedure (BH) which compares more significant $p$-values with more stringent thresholds. One notable choice of the sequence $\{\lambda_i\}$ is given by the BH critical values $\lambda_{\mathrm {BH}}(i)=z(1-i\cdot q/2p)$, where $q\in(0,1)$ and $z(\alpha)$ is the quantile of a standard normal distribution. SLOPE aims to provide finite sample guarantees on the selected model; of special interest is the false discovery rate (FDR), defined as the expected proportion of irrelevant regressors among all selected predictors. Under orthogonal designs, SLOPE with $\lambda_{\mathrm{BH}}$ provably controls FDR at level $q$. Moreover, it also appears to have appreciable inferential properties under more general designs $X$ while having substantial power, as demonstrated in a series of experiments running on both simulated and real data.Comment: Published at http://dx.doi.org/10.1214/15-AOAS842 in the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Late outcome of mitral valve replacement with the Cross-Jones prosthesis 36 years after initial surgery

A 60 year-old woman with rheumatic mitral stenosis underwent re-replacement of Cross- Jones caged lens mitral valve prosthesis, 36 years after valve implantation. In 1968, she underwent mitral commissurotomy. In 1992, she had a stroke, and in July 2009 echocardiography revealed the malfunction of the prosthesis with pannus and reduced mitral prosthetic area < 1.0 cm2 with the elevated transprosthetic gradient of 30 mm Hg. To begin with, she did not approve of the reoperation. Finally, she consented to this therapeutic option. In October 2009 Medtronic prosthesis Advantage 27 was re-implanted. We report the longest period of working Cross-Jones mitral valve in the literature. (Cardiol J 2011; 18, 6: 698&#8211;700

Grinding of titanium alloys on cylindrical grinding machine.

W procesie szlifowania stopów tytanu na parametry wynikowe obróbki ma wpływ szereg czynników związanych z zastosowaną ściernicą oraz wyborem właściwych parametrów kinematycznych szlifowania. Z powodu specyficznych właściwości stopów tytanu siła szlifowania może osiągać duże wartości, co może powodować powstawanie drgań samowzbudnych, czy też uszkodzenia cieplne obrabianego przedmiotu. W artykule przedstawiono wyniki prób szlifowania stopu tytanu Gr 19 na szlifierce kłowej do wałków dla różnych ściernic i warunków obróbki.In a grinding process of titanium alloys the grinding results depend on several factors related to the used grinding wheel and selection of appropriate grinding kinematic parameters. Because of the specific properties of titanium alloys grinding force components may reach high values which in turn may cause the development of self-excited vibrations or the thermal damage of the workpiece surface. In the paper experimental study was carried out on cylindrical grinding machine to investigate the influence of different grinding wheels and conditions on grinding results

Grinding of a nickel-based alloys on cylindrical grinding machine.

W artykule przedstawiono badania szlifowania stopu niklu Inconel 718 dla różnych parametrów kinematycznych szlifowania i siły różnych ściernic. Przeprowadzono analizę wpływu warunków obróbki na zmiany składowych szlifowania oraz emisji akustycznej. W końcowej części pracy przedstawiono wpływ zastosowanych ściernic i warunków obróbki na wybrane parametry wynikowe szlifowanych przedmiotów.In the paper grinding process has been studied in the machining of Inconel 718 alloy for different grinding kinematic parameters and grinding wheels. An analysis was carried out to investigate the influence of grinding conditions on the course of grinding force components and acoustic emission signal. In the end the influence of grinding wheels and conditions on grinding results is presented