Rectangular Confidence Regions for the Means of Multivariate Normal Distributions
01 Jun 1967-Journal of the American Statistical Association (Taylor & Francis Group)-Vol. 62, Iss: 318, pp 626-633
TL;DR: For rectangular confidence regions for the mean values of multivariate normal distributions, this paper proved that a confidence region constructed for independent coordinates is, at the same time, a conservative confidence region for any case of dependent coordinates.
Abstract: For rectangular confidence regions for the mean values of multivariate normal distributions the following conjecture of 0. J. Dunn [3], [4] is proved: Such a confidence region constructed for the case of independent coordinates is, at the same time, a conservative confidence region for any case of dependent coordinates. This result is based on an inequality for the probabilities of rectangles in normal distributions, which permits one to factor out the probability for any single coordinate.
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TL;DR: In this paper, a new general class of local indicators of spatial association (LISA) is proposed, which allow for the decomposition of global indicators, such as Moran's I, into the contribution of each observation.
Abstract: The capabilities for visualization, rapid data retrieval, and manipulation in geographic information systems (GIS) have created the need for new techniques of exploratory data analysis that focus on the “spatial” aspects of the data. The identification of local patterns of spatial association is an important concern in this respect. In this paper, I outline a new general class of local indicators of spatial association (LISA) and show how they allow for the decomposition of global indicators, such as Moran's I, into the contribution of each observation. The LISA statistics serve two purposes. On one hand, they may be interpreted as indicators of local pockets of nonstationarity, or hot spots, similar to the Gi and G*i statistics of Getis and Ord (1992). On the other hand, they may be used to assess the influence of individual locations on the magnitude of the global statistic and to identify “outliers,” as in Anselin's Moran scatterplot (1993a). An initial evaluation of the properties of a LISA statistic is carried out for the local Moran, which is applied in a study of the spatial pattern of conflict for African countries and in a number of Monte Carlo simulations.
8,933 citations
TL;DR: The results provide a novel molecular stratification of the breast cancer population, derived from the impact of somatic CNAs on the transcriptome, and identify novel subgroups with distinct clinical outcomes, which reproduced in the validation cohort.
Abstract: The elucidation of breast cancer subgroups and their molecular drivers requires integrated views of the genome and transcriptome from representative numbers of patients. We present an integrated analysis of copy number and gene expression in a discovery and validation set of 997 and 995 primary breast tumours, respectively, with long-term clinical follow-up. Inherited variants (copy number variants and single nucleotide polymorphisms) and acquired somatic copy number aberrations (CNAs) were associated with expression in 40% of genes, with the landscape dominated by cisand trans-acting CNAs. By delineating expression outlier genes driven in cis by CNAs, we identified putative cancer genes, including deletions in PPP2R2A, MTAP and MAP2K4. Unsupervised analysis of paired DNA–RNA profiles revealed novel subgroups with distinct clinical outcomes, which reproduced in the validation cohort. These include a high-risk, oestrogen-receptor-positive 11q13/14 cis-acting subgroup and a favourable prognosis subgroup devoid of CNAs. Trans-acting aberration hotspots were found to modulate subgroup-specific gene networks, including a TCR deletion-mediated adaptive immune response in the ‘CNA-devoid’ subgroup and a basal-specific chromosome 5 deletion-associated mitotic network. Our results provide a novel molecular stratification of the breast cancer population, derived from the impact of somatic CNAs on the transcriptome.
4,722 citations
Cites methods from "Rectangular Confidence Regions for ..."
...The Šidák correction [36] adjusts each P-value using a transformation of the form: p→ 1-(1-p)Nr, where Nr represents the effective number of tests: N is the number of predictors, while r is a ratio between 0 and 1....
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TL;DR: In this paper, the statistics Gi(d) and Gi*(d), introduced in Getis and Ord (1992) for the study of local pattern in spatial data, are extended and their properties further explored.
Abstract: The statistics Gi(d) and Gi*(d), introduced in Getis and Ord (1992) for the study of local pattern in spatial data, are extended and their properties further explored. In particular, nonbinary weights are allowed and the statistics are related to Moran's autocorrelation statistic, I. The correlations between nearby values of the statistics are derived and verified by simulation. A Bonferroni criterion is used to approximate significance levels when testing extreme values from the set of statistics. An example of the use of the statistics is given using spatial-temporal data on the AIDS epidemic centering on San Francisco. Results indicate that in recent years the disease is intensifying in the counties surrounding the city.
2,638 citations
TL;DR: Valsartan is as effective as captopril in patients who are at high risk for cardiovascular events after myocardial infarction and the combination of the two on mortality in this population of patients.
Abstract: background Angiotensin-converting–enzyme (ACE) inhibitors such as captopril reduce mortality and cardiovascular morbidity among patients with myocardial infarction complicated by left ventricular systolic dysfunction, heart failure, or both. In a double-blind trial, we compared the effect of the angiotensin-receptor blocker valsartan, the ACE inhibitor captopril, and the combination of the two on mortality in this population of patients. methods Patients receiving conventional therapy were randomly assigned, 0.5 to 10 days after acute myocardial infarction, to additional therapy with valsartan (4909 patients), valsartan plus captopril (4885 patients), or captopril (4909 patients). The primary end point was death from any cause. results During a median follow-up of 24.7 months, 979 patients in the valsartan group died, as did 941 patients in the valsartan-and-captopril group and 958 patients in the captopril group (hazard ratio in the valsartan group as compared with the captopril group, 1.00; 97.5 percent confidence interval, 0.90 to 1.11; P=0.98; hazard ratio in the valsartanand-captopril group as compared with the captopril group, 0.98; 97.5 percent confidence interval, 0.89 to 1.09; P=0.73). The upper limit of the one-sided 97.5 percent confidence interval for the comparison of the valsartan group with the captopril group was within the prespecified margin for noninferiority with regard to mortality (P=0.004) and with regard to the composite end point of fatal and nonfatal cardiovascular events (P<0.001). The valsartan-and-captopril group had the most drug-related adverse events. With monotherapy, hypotension and renal dysfunction were more common in the valsartan group, and cough, rash, and taste disturbance were more common in the captopril group. conclusions Valsartan is as effective as captopril in patients who are at high risk for cardiovascular events after myocardial infarction. Combining valsartan with captopril increased the rate of adverse events without improving survival.
2,305 citations
TL;DR: This paper proposes a new way of using forward selection of explanatory variables in regression or canonical redundancy analysis, and proposes a two-step procedure to prevent overestimation of the amount of explained variance.
Abstract: This paper proposes a new way of using forward selection of explanatory variables in regression or canonical redundancy analysis. The classical forward selection method presents two problems: a highly inflated Type I error and an overestimation of the amount of explained variance. Correcting these problems will greatly improve the performance of this very useful method in ecological modeling. To prevent the first problem, we propose a two-step procedure. First, a global test using all explanatory variables is carried out. If, and only if, the global test is significant, one can proceed with forward selection. To prevent overestimation of the explained variance, the forward selection has to be carried out with two stopping criteria: (1) the usual alpha significance level and (2) the adjusted coefficient of multiple determination (Ra(2)) calculated using all explanatory variables. When forward selection identifies a variable that brings one or the other criterion over the fixed threshold, that variable is rejected, and the procedure is stopped. This improved method is validated by simulations involving univariate and multivariate response data. An ecological example is presented using data from the Bryce Canyon National Park, Utah, U.S.A.
1,720 citations
Additional excerpts
...Two corrections can be applied when there are two tests (k¼ 2), the corrections of Sidak (1967), where PS¼ 1 (1 P)k, and Bonferroni (Bonferroni 1935), where PB ¼ kP and P is the P value....
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References
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Book•
14 Sep 1984
TL;DR: In this article, the distribution of the Mean Vector and the Covariance Matrix and the Generalized T2-Statistic is analyzed. But the distribution is not shown to be independent of sets of Variates.
Abstract: Preface to the Third Edition.Preface to the Second Edition.Preface to the First Edition.1. Introduction.2. The Multivariate Normal Distribution.3. Estimation of the Mean Vector and the Covariance Matrix.4. The Distributions and Uses of Sample Correlation Coefficients.5. The Generalized T2-Statistic.6. Classification of Observations.7. The Distribution of the Sample Covariance Matrix and the Sample Generalized Variance.8. Testing the General Linear Hypothesis: Multivariate Analysis of Variance9. Testing Independence of Sets of Variates.10. Testing Hypotheses of Equality of Covariance Matrices and Equality of Mean Vectors and Covariance Matrices.11. Principal Components.12. Cononical Correlations and Cononical Variables.13. The Distributions of Characteristic Roots and Vectors.14. Factor Analysis.15. Pattern of Dependence Graphical Models.Appendix A: Matrix Theory.Appendix B: Tables.References.Index.
9,693 citations
7,408 citations
TL;DR: In this paper, the authors considered the probability that a stationary Gaussian process with mean zero and covariance function r(τ) be nonnegative throughout a given interval of duration T. Several strict upper and lower bounds for P were given, along with some comparison theorems that relate P's for different covariance functions.
Abstract: This paper is concerned with the probability, P[T, r(τ)], that a stationary Gaussian process with mean zero and covariance function r(τ) be nonnegative throughout a given interval of duration T. Several strict upper and lower bounds for P are given, along with some comparison theorems that relate P's for different covariance functions. Similar results are given for F[T, r(τ)], the probability distribution for the interval between two successive zeros of the process.
786 citations
"Rectangular Confidence Regions for ..." refers background or methods in this paper
...For the variances unknown and unequal we may apply the procedure based on confidence ellipsoids, or, better, the procedure based on the Bonferroni inequality (for details see [3], [4]; also [8] is related to this topic); the latter procedure is in most cases very close to the "best" one....
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...Dunn [3], [4] is proved: Such a confidence region constructed for the case of independent coordinates is, at the same time, a conservative confidence region for any case of dependent coordinates....
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01 Feb 1955
552 citations
"Rectangular Confidence Regions for ..." refers background in this paper
...Andersoni's Corollary 2 in [1] which asserts the following: If X is a random vector with density g(x) such that g(x) = g( -x) and the set {x; g(x) > u } is convex for every non-negative u, and if E is a convex set, symmetric about the origin, y is a vector and k a number, 0< kI< I, then P{X+kyEEE} ?P{X+y&E}....
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395 citations