Theory Of U-statistics Research Articles

Asymptotic distribution of ∆AUC, NRIs, and IDI based on theory of U-statistics.

The change in area under the curve (∆AUC), the integrated discrimination improvement (IDI), and net reclassification index (NRI) are commonly used measures of risk prediction model performance. Some authors have reported good validity of associated methods of estimating their standard errors (SE) and construction of confidence intervals, whereas others have questioned their performance. To address these issues, we unite the ∆AUC, IDI, and three versions of the NRI under the umbrella of the U-statistics family. We rigorously show that the asymptotic behavior of ∆AUC, NRIs, and IDI fits the asymptotic distribution theory developed for U-statistics. We prove that the ∆AUC, NRIs, and IDI are asymptotically normal, unless they compare nested models under the null hypothesis. In the latter case, asymptotic normality and existing SE estimates cannot be applied to ∆AUC, NRIs, or IDI. In the former case, SE formulas proposed in the literature are equivalent to SE formulas obtained from U-statistics theory if we ignore adjustment for estimated parameters. We use Sukhatme-Randles-deWet condition to determine when adjustment for estimated parameters is necessary. We show that adjustment is not necessary for SEs of the ∆AUC and two versions of the NRI when added predictor variables are significant and normally distributed. The SEs of the IDI and three-category NRI should always be adjusted for estimated parameters. These results allow us to define when existing formulas for SE estimates can be used and when resampling methods such as the bootstrap should be used instead when comparing nested models. We also use the U-statistic theory to develop a new SE estimate of ∆AUC. Copyright © 2017 John Wiley & Sons, Ltd.

Inverse probability weighting estimation of the volume under the ROC surface in the presence of verification bias.

In diagnostic medicine, the volume under the receiver operating characteristic (ROC) surface (VUS) is a commonly used index to quantify the ability of a continuous diagnostic test to discriminate between three disease states. In practice, verification of the true disease status may be performed only for a subset of subjects under study since the verification procedure is invasive, risky, or expensive. The selection for disease examination might depend on the results of the diagnostic test and other clinical characteristics of the patients, which in turn can cause bias in estimates of the VUS. This bias is referred to as verification bias. Existing verification bias correction in three-way ROC analysis focuses on ordinal tests. We propose verification bias-correction methods to construct ROC surface and estimate the VUS for a continuous diagnostic test, based on inverse probability weighting. By applying U-statistics theory, we develop asymptotic properties for the estimator. A Jackknife estimator of variance is also derived. Extensive simulation studies are performed to evaluate the performance of the new estimators in terms of bias correction and variance. The proposed methods are used to assess the ability of a biomarker to accurately identify stages of Alzheimer's disease.

Testing homogeneity of several covariance matrices and multi-sample sphericity for high-dimensional data under non-normality

ABSTRACTA test for homogeneity of g ⩾ 2 covariance matrices is presented when the dimension, p, may exceed the sample size, ni, i = 1, …, g, and the populations may not be normal. Under some mild assumptions on covariance matrices, the asymptotic distribution of the test is shown to be normal when ni, p → ∞. Under the null hypothesis, the test is extended for common covariance matrix to be of a specified structure, including sphericity. Theory of U-statistics is employed in constructing the tests and deriving their limits. Simulations are used to show the accuracy of tests.

Rank-based kernel estimation of the area under the ROC curve

In medical diagnostics, the ROC curve is the graph of sensitivity against 1-specificity as the diagnostic threshold runs through all possible values. The ROC curve and its associated summary indices are very useful for the evaluation of the discriminatory ability of biomarkers/diagnostic tests with continuous measurements. Among all summary indices, the area under the ROC curve (AUC) is the most popular diagnostic accuracy index, which has been extensively used by researchers for biomarker evaluation and selection. Sometimes, taking the actual measurements of a biomarker is difficult and expensive, whereas ranking them without actual measurements can be easy. In such cases, ranked set sampling based on judgment order statistics would provide more representative samples yielding more accurate estimation. In this study, Gaussian kernel is utilized to obtain a nonparametric estimate of the AUC. Asymptotic properties of the AUC estimates are derived based on the theory of U-statistics. Intensive simulation is conducted to compare the estimates using ranked set samples versus simple random samples. The simulation and theoretical derivation indicate that ranked set sampling is generally preferred with smaller variances and mean squared errors (MSE). The proposed method is illustrated via a real data analysis.

Adjusting for unmeasured confounding due to either of two crossed factors with a logistic regression model.

Motivated by an investigation of the effect of surface water temperature on the presence of Vibrio cholerae in water samples collected from different fixed surface water monitoring sites in Haiti in different months, we investigated methods to adjust for unmeasured confounding due to either of the two crossed factors site and month. In the process, we extended previous methods that adjust for unmeasured confounding due to one nesting factor (such as site, which nests the water samples from different months) to the case of two crossed factors. First, we developed a conditional pseudolikelihood estimator that eliminates fixed effects for the levels of each of the crossed factors from the estimating equation. Using the theory of U-Statistics for independent but non-identically distributed vectors, we show that our estimator is consistent and asymptotically normal, but that its variance depends on the nuisance parameters and thus cannot be easily estimated. Consequently, we apply our estimator in conjunction with a permutation test, and we investigate use of the pigeonhole bootstrap and the jackknife for constructing confidence intervals. We also incorporate our estimator into a diagnostic test for a logistic mixed model with crossed random effects and no unmeasured confounding. For comparison, we investigate between-within models extended to two crossed factors. These generalized linear mixed models include covariate means for each level of each factor in order to adjust for the unmeasured confounding. We conduct simulation studies, and we apply the methods to the Haitian data. Copyright © 2016 John Wiley & Sons, Ltd.

An Accurate Kernelized Energy Detection in Gaussian and non-Gaussian/Impulsive Noises

Motivated by the simplicity of energy detector and capability of higher order and fractional lower order statistics in non-Gaussian signal processing, this paper proposes a new spectrum sensing method based on kernel theory, referred to as Kerenlized Energy Detector (KED), which exhibits a moderate complexity, it is easy to implement, and it compares favourably against competing solutions in the case of various Gaussian and non-Gaussian impulsive noises. The incorporation of the nonlinear kernel function in the KED test statistic allows for the development of a nonlinear algorithm capable of considering both higher order and fractional lower order moments (FLOMs) in the sensing task. We show that the proposed KED detector can serve as an optimal spectrum sensing method under both Gaussian and non-Gaussian noise scenarios. In addition, the detection performance of the proposed KED scheme is analyzed by employing U-statistics theory. The Kernel parameter selection for the KED method has been discussed in both theoretical and practical points of view. Potential of considering the KED scheme in either single user multi-antennas or cooperative spectrum sensing is investigated.

Tests for high-dimensional covariance matrices using the theory of U-statistics

Test statistics for sphericity and identity of the covariance matrix are presented, when the data are multivariate normal and the dimension, p, can exceed the sample size, n. Under certain mild conditions mainly on the traces of the unknown covariance matrix, and using the asymptotic theory of U-statistics, the test statistics are shown to follow an approximate normal distribution for large p, also when p≫n. The accuracy of the statistics is shown through simulation results, particularly emphasizing the case when p can be much larger than n. A real data set is used to illustrate the application of the proposed test statistics.

Comparison of relative density of two random geometric digraph families in testing spatial clustering

We compare the performance of relative densities of two parameterized random geometric digraph families called proximity catch digraphs (PCDs) in testing bivariate spatial patterns. These PCD families are proportional edge (PE) and central similarity (CS) PCDs and are defined with proximity regions based on relative positions of data points from two classes. The relative densities of these PCDs were previously used as statistics for testing segregation and association patterns against complete spatial randomness. The relative density of a digraph, D, with n vertices (i.e., with order n) represents the ratio of the number of arcs in D to the number of arcs in the complete symmetric digraph of the same order. When scaled properly, the relative density of a PCD is a U-statistic; hence, it has asymptotic normality by the standard central limit theory of U-statistics. The PE- and CS-PCDs are defined with an expansion parameter that determines the size or measure of the associated proximity regions. In this article, we extend the distribution of the relative density of CS-PCDs for expansion parameter being larger than one, and compare finite sample performance of the tests by Monte Carlo simulations and asymptotic performance by Pitman asymptotic efficiency. We find the optimal expansion parameters of the PCDs for testing each alternative in finite samples and in the limit as the sample size tending to infinity. As a result of our comparisons, we demonstrate that in terms of empirical power (i.e., for finite samples) relative density of CS-PCD has better performance (which occurs for expansion parameter values larger than one) for the segregation alternative, while relative density of PE-PCD has better performance for the association alternative. The methods are also illustrated in a real-life data set from plant ecology.

Tests of Covariance Matrices for High Dimensional Multivariate Data Under Non Normality

Ahmad and von Rosen (2014) presented test statistics for sphericity and identity of the covariance matrix of a multivariate normal distribution when the dimension, p, exceeds the sample size, n. In this note, we show that their statistics are robust to normality assumption, when normality is replaced with certain mild assumptions on the traces of the covariance matrix. Under such assumptions, the test statistics are shown to follow the same asymptotic normal distribution as under normality for large p, also when p > >n. The asymptotic normality is proved using the theory of U-statistics, and is based on very general conditions, particularly avoiding any relationship between n and p.

On asymptotic distribution of sample central moments in normal-uniform distribution

The central moments are very useful in topics identified distribution as they reveal the measure of skewness. In this article, we present the asymptotic distribution of estimator of third central moment as well as the distribution of estimator of skewed measure by employing of the theory of U-statistics in the family of skewed normal-uniform distribution. By changing the three parameters of this distribution, one related to normal distribution, the other related to uniform distribution, and the third to skew parameter, we characterize this kind of distribution.

Averaged shifted chi-square test

A simple procedure based on the average of shifted chi-square statistics (ASCS) is proposed to improve the classical chi-square procedure for testing whether a random sample has been drawn from a specified continuous distribution. We repeatedly partition the sample space, say, ℓ times to obtain ℓ respective chi-square statistics. The proposed test statistic is defined as the average value of the resultant ℓ shifted chi-square statistics. We prove that the ASCS is asymptotically distributed as a weighted sum of a finite number of chi-square variables by the theory of U-statistics. The proposed procedure is shown to be markedly less sensitive to the choice of the anchor position and Monte Carlo experiments demonstrate that it leads to noticeable gains in power.

Resolving statistical uncertainty in correlation dimension estimation

In this paper we propose a novel method for obtaining standard errors and confidence intervals for the correlation dimension estimated on an observed chaotic time series. This method is based on the U-Statistics theory and an ingenious combination of the moving block and parametric bootstrap procedures. We test the method on the basis of computer simulations for both clean and noisy series. We show that the distribution of the correlation dimension estimate obtained by our method agrees very well with the "true" distribution obtained by the Monte Carlo simulation. One of the main advantage of our method is the ability to estimate the distribution (and hence, the standard error) of the correlation dimension estimate using only one observed time series.

The distribution of the relative arc density of a family of interval catch digraph based on uniform data

We study a family of interval catch digraph called proportional-edge proximity catch digraph (PCD) which is also a special type of intersection digraphs parameterized with an expansion and a centrality parameter. PCDs are random catch digraphs that have been developed recently and have applications in classification and spatial pattern analysis. We investigate a graph invariant of the PCDs called relative arc density. We demonstrate that relative arc density of PCDs is a U-statistic and using the central limit theory of U-statistics, we derive the (asymptotic) distribution of the relative arc density of proportional-edge PCD for uniform data in one dimension. We also determine the parameters for which the rate of convergence to asymptotic normality is fastest.

Asymptotic Distribution of Coefficients of Skewness and Kurtosis

Problem statement: In literature, a classic method which has been used to recognize the distribution so far is the measurement of its skewedness and kurtosis. However, there remains a question: how would these measurements work for skewed normal distribution when the size of the sample is large? Approach: This research aimed to determine the asymptotic distribution of skewedness and kurtosis measures in skewed normal distribution. In conducting this research, two groups of inferential findings will help. First, skewed normal distribution which has already been studied by a lot of researchers and we apply its characteristics. Second, there is the U-statistics theory which guides us to the determining of asymptotic distribution measures for skewedness and kurtosis. The combination of these two will solve the problem of this study. Results: Asymptotic distribution of measures for skewdness and kurtosis falls in the normal families. With the size of large samples, the values of expectation of these measures are also determined. By letting zero for skewedness parameter, asymptotic distribution for normal distribution can also be obtained. Conclusion: The findings of this study show new characteristics for skew normal distribution and this results in a new way for skew normal distribution recognition.

Nonparametric statistical inference method for partial areas under receiver operating characteristic curves, with application to genomic studies

Recently ROC50 index-the area under the lower portion of the receiver operating characteristic (ROC) curve up to the first 50 false positives-has been increasingly widely used in genomic research. Unfortunately, statistical inferences on the ROC50 index are not commonly drawn due to a lack of handy statistical inference methods and/or software tools. In this paper, we reviewed developments in statistical methods for the partial areas under ROC curves and using nonparametric methods we derived a simple and direct variance calculation formula for the partial areas, different from existing methods in the literature. We have also verified our method through simulation studies and compared our method with existing bi-normal approaches. We have shown that the partial area has an asymptotic normal distribution using trimmed U-statistics theory. On the basis of this asymptotic normality, we have given formulas for the confidence interval and the test statistic and we reported on their application to a genomic study of sample size approximately 10,000.

Association testing by haplotype-sharing methods applicable to whole-genome analysis

We propose two new haplotype-sharing methods for identifying disease loci: the haplotype sharing statistic (HSS), which compares length of shared haplotypes between cases and controls, and the CROSS test, which tests whether a case and a control haplotype show less sharing than two random haplotypes. The significance of the HSS is determined using a variance estimate from the theory of U-statistics, whereas the significance of the CROSS test is estimated from a sequential randomization procedure. Both methods are fast and hence practical, even for whole-genome screens with high marker densities. We analyzed data sets of Problems 2 and 3 of Genetic Analysis Workshop 15 and compared HSS and CROSS to conventional association methods. Problem 2 provided a data set of 2300 single-nucleotide polymorphisms (SNPs) in a 10-Mb region of chromosome 18q, which had shown linkage evidence for rheumatoid arthritis. The CROSS test detected a significant association at approximately position 4407 kb. This was supported by single-marker association and HSS. The CROSS test outperformed them both with respect to significance level and signal-to-noise ratio. A 20-kb candidate region could be identified. Problem 3 provided a simulated 10 k SNP data set covering the whole genome. Three known candidate regions for rheumatoid arthritis were detected. Again, the CROSS test gave the most significant results. Furthermore, both the HSS and the CROSS showed better fine-mapping accuracy than straightforward haplotype association. In conclusion, haplotype sharing methods, particularly the CROSS test, show great promise for identifying disease gene loci.

ROC Graphs for Assessing the Ability of a Diagnostic Marker to Detect Three Disease Classes with an Umbrella Ordering

Receiver operating characteristic (ROC) curves and the area under these curves are commonly used to assess the ability of a continuous diagnostic marker (e.g., DNA methylation markers) to correctly classify subjects as having a particular disease or not (e.g., cancer). These approaches, however, are not applicable to settings where the gold standard yields more than two disease states or classes. ROC surfaces and the volume under the surfaces have been proposed for settings with more than two disease classes. These approaches, however, do not allow one to assess the ability of a marker to differentiate two disease classes from a third disease class without requiring a monotone order for the three disease classes under study. That is, existing approaches do not accommodate an umbrella ordering of disease classes. This article proposes the construction of an ROC graph that is applicable for an umbrella ordering. Furthermore, this article proposes that a summary measure for this umbrella ROC graph can be used to summarize the classification accuracy, and corresponding variance estimates can be obtained using U-statistics theory or bootstrap methods. The proposed methods are illustrated using data from a study assessing the ability of a DNA methylation marker to correctly classify lung specimens into three histologic classes: squamous cell carcinoma, large cell carcinoma, and nontumor lung.

Rank methods for the analysis of clustered data in diagnostic trials

Methods for comparing areas under receiver operating characteristic curves usually depend on the assumption of independence between diseased and nondiseased units in the trial. However, if several parts of the same subject have to be classified as diseased or nondiseased, such observations are no longer independent. This situation is referred to as clustered data. First ideas for the analysis of such data are based on the theory of U-statistics. The idea of the multivariate nonparametric Behrens–Fisher problem is extended to clustered data and to factorial designs. ANOVA-type statistics are suggested for both small and moderate sample sizes. They are evaluated in a simulation study and applied to a real-data example.

Concordance probability and discriminatory power in proportional hazards regression

SUMMARY The concordance probability is used to evaluate the discriminatory power and the predictive accuracy of nonlinear statistical models. We derive an analytical expression for the concordance probability in the Cox proportional hazards model. The proposed estimator is a function of the regression parameters and the covariate distribution only and does not use the observed event and censoring times. For this reason it is asymptotically unbiased, unlike Harrell's c-index based on informative pairs. The asymptotic distribution of the concordance probability estimate is derived using U-statistic theory and the methodology is applied to a predictive model in lung cancer.

Feasibility of real-time calculation of correlation integral derived statistics applied to EEG time series

Background: This study assessed the feasibility of online calculation of the correlation integral ( C( r)) aiming to apply C( r)-derived statistics. For real-time application it is important to reduce calculation time. It is shown how our method works for EEG time series. Methods: To achieve online calculation of C( r) a non-randomly subset of inter vector distances was chosen and computer code was optimized. The effect of distance exclusion was investigated for both non-randomly and randomly chosen subsets. A C( r)-derived statistic was computed: an effective correlation dimension (CD) following the Grassberger–Procaccia–Takens approach. Results: By taking a subset of the maximum possible number of distances, C( r)-computation time could be easily 100 times reduced with marginal changes and minor variability in the C( r)-derived statistic CD. Applied to the EEG CD gives a good indication of the depth of anesthesia. Conclusions: If applied to the EEG, apparently a large number of distances can be omitted in the calculation of C( r) with minimal consequences. This outcome confirms Hoeffding's theory of U-statistics and hence it is expected to occur generally, also for non-EEG time series.