U-statistic Method Research Articles

Delineation of geochemical anomalies for mineral exploration using combining U-statistic method and fractal technique: U-N and U-A models

ABSTRACT In this study, by using the algorithm of the U-statistic and fractal methods and combining them with each other, a new combined method as U values fractal model (U-N and U-A) is introduced. Then, the proposed method is employed to determine the boundaries of background and anomalous populations. Results show that in U-N and U-A fractal models, the first fracture boundary is much clearer and more accurate than previous fractal models (C-N and C-A) in the same condition. In U-N model, due to the nature of the U method algorithm, there is a discontinuity as exact threshold between background and anomaly that in U-A model, this does not exist due to the homogenization of U values. In this method, the exact threshold between background and anomaly is determined by U-statistic method and by its combination with the fractal method, in each population, sub-populations are identified more accurately and simply than concentration fractal model.

Integrating principal component analysis and U-statistics for mapping polluted areas in mining districts

Detecting metal contaminations and mapping the areas affected by pollutants surrounding the mining activities are essential issues in environmental geochemistry. Toxic metals can be broadly distributed around the mines owing to the effects of dust, water, and diverse anthropogenic activities. In this study, the effects of metal pollution from mining activities in Irankuh Pb-Zn mine, located in central Iran, on the surrounding environment have been investigated. Toxic metals, especially lead and zinc, dispersed in the agricultural area in the south of the Irankuh mine have been deemed environmental stressors. This study introduces a new combined methodology based on principal component analysis (PCA) and U-statistics modeling to recognize the zones afflicted by toxic metal pollution. The stepwise principal component analysis (SPCA) is a multi-stage dimension reduction technique applied to extract significant pollution sources of toxic metals around agricultural areas. After five stages of SPCA, the Pollution Principal Factor (PPF) was determined based on the rotated component matrix, and the toxic metals related to the pollution area were identified. For mapping the multivariate metal contamination, the U-statistics method was applied to the PPF for identification of toxic metal anomalies. The concentration-area fractal method was also utilized for separating and mapping different populations of Pb, Zn, As, Mn, Ba, and Cd. To apply this method, ordinary Kriging was used to grid and interpolate raw data. The U-values of PPF in the proposed method were delineated and compared to the results of the fractal method. The proposed approach displays more accurate results for pollution mapping. The methodology adopted here can be applied to other case studies for the source apportionment of toxic metals.

The efficiency and accuracy of probability diagram, spatial statistic and fractal methods in the identification of shear zone gold mineralization: a case study of the Saqqez gold ore district, NW Iran

In this study, geochemical anomaly separation was carried out with methods based on the distribution model, which includes probability diagram (MPD), fractal (concentration-area technique), and U-statistic methods. The main objective is to evaluate the efficiency and accuracy of the methods in separation of anomalies on the shear zone gold mineralization. For this purpose, samples were taken from the secondary lithogeochemical environment (stream sediment samples) on the gold mineralization in Saqqez, NW of Iran. Interpretation of the histograms and diagrams showed that the MPD is capable of identifying two phases of mineralization. The fractal method could separate only one phase of change based on the fractal dimension with high concentration areas of the Au element. The spatial analysis showed two mixed subpopulations after U = 0 and another subpopulation with very high U values. The MPD analysis followed spatial analysis, which shows the detail of the variations. Six mineralized zones detected from local geochemical exploration results were used for validating the methods mentioned above. The MPD method was able to identify the anomalous areas higher than 90%, whereas the two other methods identified 60% (maximum) of the anomalous areas. The raw data without any estimation for the concentration was used by the MPD method using aminimum of calculations to determine the threshold values. Therefore, the MPD method is more robust than the other methods. The spatial analysis identified the detail soft hegeological and mineralization events that were affected in the study area. MPD is recommended as the best, and the spatial U-analysis is the next reliable method to be used. The fractal method could show more detail of the events and variations in the area with asymmetrical grid net and a higher density of sampling or at the detailed exploration stage.

An optimal kernel-based U-statistic method for quantitative gene-set association analysis.

Single-variant-based genome-wide association studies have successfully detected many genetic variants that are associated with a number of complex traits. However, their power is limited due to weak marginal signals and ignoring potential complex interactions among genetic variants. The set-based strategy was proposed to provide a remedy where multiple genetic variants in a given set (e.g., gene or pathway) are jointly evaluated, so that the systematic effect of the set is considered. Among many, the kernel-based testing (KBT) framework is one of the most popular and powerful methods in set-based association studies. Given a set of candidate kernels, the method has been proposed to choose the one with the smallest p-value. Such a method, however, can yield inflated Type 1 error, especially when the number of variants in a set is large. Alternatively one can get p values by permutations which, however, could be very time-consuming. In this study, we proposed an efficient testing procedure that cannot only control Type 1 error rate but also have power close to the one obtained under the optimal kernel in the candidate kernel set, for quantitative trait association studies. Our method, a maximum kernel-based U-statistic method, is built upon the KBT framework and is based on asymptotic results under a high-dimensional setting. Hence it can efficiently deal with the case where the number of variants in a set is much larger than the sample size. Both simulation and real data analysis demonstrate the advantages of the method compared with its counterparts.

Applying 3D U-statistic method for modeling the iron mineralization in Baghak mine, central section of Sangan iron mines

The U-statistic method is one of the most important structural methods to separate the anomaly from background. It considers the location of samples and carries out the statistical analysis of the data without judging from a geochemical point of view and tries to separate subpopulations and determine anomalous areas. In the present study, 3D U-statistic method has been applied for the first time through the three-dimensional (3D) modeling of an ore deposit. In order to achieve this purpose, 3D U-statistic is applied on the data (Fe grade) resulted from the drilling network in Baghak mine, central part of the Sangan iron mines (in Khorassan Razavi Province, Iran). Afterward, results from applying 3D U-statistic method are used for 3D modeling of the iron mineralization. Results show that the anomalous values are well separated from background so that the determined samples as anomalous are not dispersed and according to their positioning, denser areas of anomalous samples could be considered as anomaly areas. And also, final results (3D model of iron mineralization) show that output model using this method is compatible with designed model for mining operation. Moreover, seen that U-statistic method in addition for separating anomaly from background, could be very efficient for the 3D modeling of different ore type.

A comparative study of fractal models and U-statistic method to identify geochemical anomalies; case study of Avanj porphyry system, Central Iran

The most significant aspect of a geochemical exploration program is to define and separate the anomalous values from the background. In the past decades, geochemical anomalies have been identified by means of various methods. Most of the conventional statistical methods aiming at defining the geochemical concentration thresholds for separating anomalies from the background have limited the efficiency in the areas with complex geological settings. In this work, three methods including the Concentration-Area (C-A) and Spectrum-Area (S-A) fractal models, and the U-statistic method are applied to identify the geochemical anomalies in Avanj porphyry system due to a complex geological and tectonic setting. The results obtained show that the S-A and U-statistic methods present more acceptable outputs than the C-A method. The C-A model acts well to identify the geochemical anomalies within a region including a simple geochemical background; however, the model has limitations within a region including a complex geological setting, where each sub-area is characterized by different geochemical fields. The U-statistic method, by considering the location of sampling points, their spatial relation, and radius of influence for each point in the estimation of anomaly location, overcomes the limitations of the C-A model. The S-A model is a powerful tool to decompose mixed geochemical patterns into a geochemical anomaly map and a varied geochemical background map. The output of this method shows the analysis of geochemical data in the frequency domain, which can provide new exploratory information that may not be revealed in the spatial domain. Eventually, it can be pointed out that the accuracy of the S-A fractal model for determining the thresholds is higher than the other two methods mentioned.

Applying a Structural Multivariate Method Using the Combination of Statistical Methods for the Delineation of Geochemical Anomalies

Quantitative descriptions of geochemical patterns and providing geochemical anomaly map are important in applied geochemistry. Several statistical methodologies (nonstructural and structural) are presented in order to identify and separate geochemical anomalies. Structural methods take the sampling locations and their spatial relation into account for estimating the anomalous areas. In the present study, a nonstructural method (Mahalanobis distance method as a multivariate method) is used and U-statistic is considered as a structural method to assess prospective areas of Susanvar district as a gold mineralization index in the Torud-Chah Shirin mountain range of Semnan Province, northern Iran. Results show that the U-statistic method is an efficient method according to spatial distribution of the anomalous samples. In the present study, according to the ability of U-statistic method in combining with other methods, the goal is to use and develop Mahalanobis distance method in structural mode. For this purpose, Mahalanobis distance should be combined with a structural method which devotes a new value to each sample based on its surrounding samples. For this reason, the combination of efficient U-statistic method and multivariate Mahalanobis distance method has been used to separate geochemical anomalies from background. Combination results show that the performance of these two methods is more accurate than using just one of them. Because samples indicated by the combination of these methods as anomalous are less dispersed and closer to each other than in the case of using just the U-statistic and other nonstructural methods such as Mahalanobis distance. Also it was observed that the combination results are closely associated with the defined Au ore indication within the study area. Finally, univariate and bivariate geochemical anomaly maps are provided for Au and As, which have been, respectively, prepared using U-statistic and its combination with Mahalanobis distance method.

A method for extracting anomaly map of Au and As using combination of U-statistic and Euclidean distance methods in Susanvar district, Iran

Quantitative descriptions of geochemical patterns and providing geochemical anomaly map are important in applied geochemistry. Several statistical methodologies are presented in order to identify and separate geochemical anomalies. The U-statistic method is one of the most important structural methods and is a kind of weighted mean that surrounding points of samples are considered in U value determination. However, it is able to separate the different anomalies based on only one variable. The main aim of the presented study is development of this method in a multivariate mode. For this purpose, U-statistic method should be combined with a multivariate method which devotes a new value to each sample based on several variables. Therefore, at the first step, the optimum p is calculated in p-norm distance and then U-statistic method is applied on p-norm distance values of the samples because p-norm distance is calculated based on several variables. This method is a combination of efficient U-statistic method and p-norm distance and is used for the first time in this research. Results show that p-norm distance of p=2 (Euclidean distance) in the case of a fact that Au and As can be considered optimized p-norm distance with the lowest error. The samples indicated by the combination of these methods as anomalous are more regular, less dispersed and more accurate than using just the U-statistic or other nonstructural methods such as Mahalanobis distance. Also it was observed that the combination results are closely associated with the defined Au ore indication within the studied area. Finally, univariate and bivariate geochemical anomaly maps are provided for Au and As, which have been respectively prepared using U-statistic and its combination with Euclidean distance method.

A functional U-statistic method for association analysis of sequencing data.

Although sequencing studies hold great promise for uncovering novel variants predisposing to human diseases, the high dimensionality of the sequencing data brings tremendous challenges to data analysis. Moreover, for many complex diseases (e.g., psychiatric disorders) multiple related phenotypes are collected. These phenotypes can be different measurements of an underlying disease, or measurements characterizing multiple related diseases for studying common genetic mechanism. Although jointly analyzing these phenotypes could potentially increase the power of identifying disease-associated genes, the different types of phenotypes pose challenges for association analysis. To address these challenges, we propose a nonparametric method, functional U-statistic method (FU), for multivariate analysis of sequencing data. It first constructs smooth functions from individuals' sequencing data, and then tests the association of these functions with multiple phenotypes by using a U-statistic. The method provides a general framework for analyzing various types of phenotypes (e.g., binary and continuous phenotypes) with unknown distributions. Fitting the genetic variants within a gene using a smoothing function also allows us to capture complexities of gene structure (e.g., linkage disequilibrium, LD), which could potentially increase the power of association analysis. Through simulations, we compared our method to the multivariate outcome score test (MOST), and found that our test attained better performance than MOST. In a real data application, we apply our method to the sequencing data from Minnesota Twin Study (MTS) and found potential associations of several nicotine receptor subunit (CHRN) genes, including CHRNB3, associated with nicotine dependence and/or alcohol dependence.

Applying the combination of U-statistic and Mahalanobis distance as a multivariate structural method for the delineation of geochemical anomalies

ABSTRACTThere are several statistical methods for separating anomalous values from background leading to determination of anomalous areas. These methods range from simple approaches to complicated ones and include nonstructural and structural methods. Structural methods take the sampling locations and their spatial relation into account for estimating the anomalous areas. The U-statistic method is one of the most important structural methods. It considers the location of samples and carries out the statistical analysis of the data without judging from a geochemical point of view and tries to separate subpopulations and also to determine anomalous areas. In the present study, two nonstructural methods including gap statistic and Mahalanobis distance method (as a multivariate method) are used and U-statistic is considered as a structural method to assess prospective areas of Parkam district. Results show that the samples indicated by U-statistic and gap statistic methods as anomalous are more regular and less dispersed compared to those indicated by other nonstructural methods. Both gap statistic and U-statistic methods have shown almost similar results for estimating threshold and separation, but U-statistic method is even better than gap statistic because the U-statistic method devotes a new value to each sample which can be combined with other methods. For this reason, the combination of U-statistic and Mahalanobis distance method has been employed to separate geochemical anomaly from background. Combination results show that the performance of these two methods is more accurate than using just one of them. Finally, bivariate lithogeochemical map of the study area is provided for Cu and Mo which has been prepared using combination of the U-statistic and Mahalanobis distance method. In this map, the delineated Cu-Mo mineralization is closely associated with the defined zone of potassic alteration, which is also consistent with the field and microscopic observation of the Cu mineralization in this alteration zone. Moreover, it is associated with the phyllic alteration and is spatially conformable with the zone defined for it.

The Parkam exploration district, Kerman, Iran: Geology, alterations, and delineation of Cu- and Mo-mineralized zones using U-spatial statistic with associated software development

The Parkam exploration district represents an area of approximately 4 km2 located 50 km north of Shahr-E-Babak (Kerman Province, Iran) , and has several traces of old copper mining and smelting activities. This area lies in the Kerman Copper Belt which is part of the larger Sahand-Bazman igneous and metallogenic zone hosting numerous known porphyry copper deposits and systems. The geology of the Parkam exploration district demonstrates that the area contains a diorite-type porphyry copper system hosted by volcanic and pyroclastic rocks of predominantly andesitic composition. Based on field and microscopic investigation, it was determined that the dominant types of alteration were propylitic, phyllic, argillic, and potassic, and the alteration map of the study area was produced. Expect for the propylitic alteration which was observed mainly in the host rocks, the other types of alteration are associated mainly with the dioritic subvolcanic body. Accompanied by subordinate amounts of primary sulfides, fracture-filling malachite is widespread in the potassic and phyllic zones and comprises the dominant style of mineralization at the surface of the porphyry system. Lithogeoc hemical data resulting from 377 samples were analyzed, and the results of background and anomaly separation by means of conventional and the U-spatial statistic method were compared. The Cu and Mo mineralizations were subsequently delineated using the U-spatial statistic. The delineated Cu mineralization is closely associated with the defined zone of potassic alteration, which is also consistent with the field and microscopic observation of the Cu mineralization in this alteration zone. The Mo mineralization delineated by the U-statistic method is mostly associated with the phyllic alteration and is spatially conformable with the zone defined for it. The source code for a software program, which was developed in the MATLAB programming language in order to perform the calculations of the U-spatial statistic method, is additionally provided. This software is compatible with geochemical variates other than Cu and Mo and can be used in similar exploration projects.

Providing the bivariate anomaly map of Cu–Mo and Pb–Zn using combination of statistic methods in Parkam district, Iran

The U-statistic method is one of the most important univariate structural methods which considers spatial situation of samples. The U-statistic method could be combined with other methods because it devotes a new value to each sample. However, this method separates anomaly based on one variable. The goal in present study is to use and extend this method in multivariate mode. For this purpose, the U-statistic method should be combined with a multivariate method which devotes a new value to each sample based on several variables. Therefore, the U-statistic was applied on Mahalanobis distance values of samples because Mahalanobis distance is calculated based on several variables. This method is a combination of efficient U-statistic and Mahalanobis distance and is used for the first time. Combination results for Cu, Mo, Pb and Zn elements in Parkam district, Kerman, Iran, led to better performance of these two methods. Results show that samples indicated by the combination of these methods as anomalous are more regular; less dispersed and are more accurate than using just one of them. Also it was observed that combination results (especially for Cu and Mo) are closely associated with the defined zone of potassic alteration in the study area. Finally, bivariate lithogeochemical maps of the study area are provided for Cu–Mo and Pb–Zn which have been prepared using combination of the U-statistic and the Mahalanobis distance methods.

Separation of a geochemical anomaly from background by fractal and U-statistic methods, a case study: Khooni district, Central Iran

Separation of geochemical anomalies from background are one of the important steps in mineral exploration. The Khooni mineral district (Central Iran) has complex geochemical surface expression due to a complex geological background. This region was chosen as a study area for recognition of the spatial distribution of geochemical elements and separating anomalies from background using stream sediment geochemical data. In the past decades, geochemical anomalies have been identified by means of various methods. Some of these separation methods include: statistical analysis methods, spatial statistical methods and fractal and multi-fractal methods. In this article, two efficient methods, i.e. U-statistics and the fractal concentration-area for separation and detection of anomalous areas of the background were used. The U spatial statistic method is a weighted mean, which considers sampling point positions and their spatial relation in the estimation of anomaly location. Also, fractal and multi-fractal models have also been applied to separate anomalies from background values. In this paper, the concentration–area model (C–A) was suggested to separate the anomaly of background. For this purpose, about 256 stream sediment samples were collected and analyzed. Then anomaly maps of elements were generated based on U spatial statistics and the C-A fractal methods for Au, As and Sb elements. According to obtained results, the U-statistics method performed better than C-A method. Because the comparisons of the known deposits and occurrences against the anomalous area created using thresholds from U-statistics and C-A method show that the spatial U-statistics method hits all of 3 known deposits and occurrences, the C-A fractal method hits 1 and fails 2. In addition, the results showed that these methods with regard to spatial distribution and variability within neighboring samples, in addition to concentration value frequency distributions and correlation coefficients, have more accurate results than the traditional approaches.

Introducing 3D U-statistic method for separating anomaly from background in exploration geochemical data with associated software development

The U-statistic method is one of the most important structural methods to separate the anomaly from the background. It considers the location of samples and carries out the statistical analysis of the data without judging from a geochemical point of view and tries to separate subpopulations and determine anomalous areas. In the present study, to use U-statistic method in three-dimensional (3D) condition, U-statistic is applied on the grade of two ideal test examples, by considering sample Z values (elevation). So far, this is the first time that this method has been applied on a 3D condition. To evaluate the performance of 3D U-statistic method and in order to compare U-statistic with one non-structural method, the method of threshold assessment based on median and standard deviation (MSD method) is applied on the two example tests. Results show that the samples indicated by U-statistic method as anomalous are more regular and involve less dispersion than those indicated by the MSD method. So that, according to the location of anomalous samples, denser areas of them can be determined as promising zones. Moreover, results show that at a threshold of U = 0, the total error of misclassification for U-statistic method is much smaller than the total error of criteria of \(\bar {x}+n\times s\). Finally, 3D model of two test examples for separating anomaly from background using 3D U-statistic method is provided. The source code for a software program, which was developed in the MATLAB programming language in order to perform the calculations of the 3D U-spatial statistic method, is additionally provided. This software is compatible with all the geochemical varieties and can be used in similar exploration projects.

Comparing U-statistic and nonstructural methods for separating anomaly and generating geochemical anomaly maps of Cu and Mo in Parkam district, Kerman, Iran

In applied geochemistry, obtaining quantitative descriptions of geochemical patterns and identifying geochemical anomalies are important. To identify and separate geochemical anomalies, several statistical methodologies (nonstructural and structural) are presented by researchers. In this study, four nonstructural methods including threshold assessment method based on median and standard deviation, median absolute deviation (MAD), P.N product and Sinclair’s method are selected first and then U-statistic is considered as a structural method to compare their performance. Subsequently, the best method is used to assess prospective areas of Parkam district. Results show that P.N and Sinclair’s methods are not always efficient. MAD method reduced the background well and roughly increased the correlation factor of points. However, U-statistic method includes both mentioned advantages meaning in addition to reducing outlier data effect, it regularizes anomalous values and also their dispersion is reduced significantly. It is possible to determine anomaly areas according to anomalous samples positioning so that denser areas are more important. Finally, lithogeochemical map of study area is generated for copper and molybdenum. In this map, the Cu mineralization which is delineated by this method is closely associated with the defined potassic alteration zone (according to alteration map of the study area), and also, the delineated Mo mineralization is exactly associated with the phyllic alteration and is spatially conformable with the zone defined for it.

Assessment of prospective areas for providing the geochemical anomaly maps of lead and zinc in Parkam district, Kerman, Iran

There are several statistical methodologies presented for separating anomalous values from background leading to determination of anomalous areas. These methods range from simple approaches to complicated ones and include nonstructural and structural methods, subtraction separation method and so on. Structural methods take the sampling locations and their spatial relation into account for estimating the anomalous areas. The U-statistic method is one of the most important structural methods. It considers the location of samples and carries out the statistical analysis of the data without judging from a geochemical point of view and tries to separate subpopulations and also to determine anomalous areas. In the present study, several nonstructural methods including assessment of threshold based on median and standard deviation, median absolute deviation (MAD) and P.N product are used and U-statistic is considered as structural method to assess prospective areas of Parkam district. Results show that MAD method reduced background well and P.N method increased correlation of points. However, U-statistic method plays the role of both mentioned advantages meaning in addition to reducing outlier data effect, it regularizes anomalous values and also their dispersion is reduced significantly. It is possible to determine anomaly areas according to anomalous samples positioning so that denser areas are more important. Finally, lithogeochemical map of study area is provided for lead and zinc.

A Composite Model via Proportional Intensity Function and Additive Hazard Function

This paper mainly discusses a composite model which the end time is an additive hazard function and the recurrent event process is a proportional intensity function, the covariate is time-independent, and censoring is dependent on recurrent events process and end times. Based on the likelihood method, Delta method, U-statistic method and the idea of general estimation equation, the estimation of unknown parameters and unknown functions in this composite model is proposed.

Estimation Methods of a Joint Model Based on Proportional Intensity Function and Proportional Hazard Function

Recurrent events data refers to the observation of individuals, which contains the recurrent event time of interest. This paper mainly discusses a joint model when the end time is a multiplicable hazard function and the recurrent event process is a multiplicable intensity function. Based on the likelihood method, Delta method, U-statistic method and the idea of general estimation equation, the estimation of unknown parameters and unknown functions in the model is provided. It provides a new method of parameter estimation for the statistic analysis of recurrent events data.

Identifying Genetic Variants for Addiction via Propensity Score Adjusted Generalized Kendall’s Tau

Identifying replicable genetic variants for addiction has been extremely challenging. Besides the common difficulties with genome-wide association studies (GWAS), environmental factors are known to be critical to addiction, and comorbidity is widely observed. Despite the importance of environmental factors and comorbidity for addiction study, few GWAS analyses adequately considered them due to the limitations of the existing statistical methods. Although parametric methods have been developed to adjust for covariates in association analysis, difficulties arise when the traits are multivariate because there is no ready-to-use model for them. Recent nonparametric development includes U-statistics to measure the phenotype-genotype association weighted by a similarity score of covariates. However, it is not clear how to optimize the similarity score. Therefore, we propose a semiparametric method to measure the association adjusted by covariates. In our approach, the nonparametric U-statistic is adjusted by parametric estimates of propensity scores using the idea of inverse probability weighting. The new measurement is shown to be asymptotically unbiased under our null hypothesis while the previous nonweighted and weighted ones are not. Simulation results show that our test improves power as opposed to the nonweighted and two other weighted U-statistic methods, and it is particularly powerful for detecting gene-environment interactions. Finally, we apply our proposed test to the Study of Addiction: Genetics and Environment (SAGE) to identify genetic variants for addiction. Novel genetic variants are found from our analysis, which warrant further investigation in the future.

Multi-reader ROC Studies with Split-plot Designs: A Comparison of Statistical Methods

Multireader imaging trials often use a factorial design, in which study patients undergo testing with all imaging modalities and readers interpret the results of all tests for all patients. A drawback of this design is the large number of interpretations required of each reader. Split-plot designs have been proposed as an alternative, in which one or a subset of readers interprets all images of a sample of patients, while other readers interpret the images of other samples of patients. In this paper, the authors compare three methods of analysis for the split-plot design. Three statistical methods are presented: the Obuchowski-Rockette method modified for the split-plot design, a newly proposed marginal-mean analysis-of-variance approach, and an extension of the three-sample U-statistic method. A simulation study using the Roe-Metz model was performed to compare the type I error rate, power, and confidence interval coverage of the three test statistics. The type I error rates for all three methods are close to the nominal level but tend to be slightly conservative. The statistical power is nearly identical for the three methods. The coverage of 95% confidence intervals falls close to the nominal coverage for small and large sample sizes. The split-plot multireader, multicase study design can be statistically efficient compared to the factorial design, reducing the number of interpretations required per reader. Three methods of analysis, shown to have nominal type I error rates, similar power, and nominal confidence interval coverage, are available for this study design.