Regina Y. Liu

A Quality Index Based on Data Depth and Multivariate Rank Tests

Let F and G be the distribution functions of two given populations on Rp, p ≥ 1. We introduce and study a Parameter Q = Q(F, G), which measures the Overall “outlyingness” of population G relative to population F. The Parameter Q can be defined using any concept of data depth. Its value ranges from 0 to 1, and is .5 when F and G are identical. We show that within the dass of elliptical distributions when G departs from F in location or G has a larger spread, or both, the value of Q dwindles down from .5. Hence Q can be used to detect the loss of accuracy or precision of a manufacturing process, and thus it should serve as an important measure in quality assurance. This in fact is the reason why we refer to Q as a quality index in this article. In addition to studying the properties of Q, we provide an exact rank test for testing Q = .5 vs. Q < .5. This can be viewed as a multivariate analog of Wilcoxons rank sum test. The tests proposed here have power against location change and scale increase simultaneo...

Control charts for multivariate processes

Abstract This article uses the concept of data depth to introduce several new control charts for monitoring processes of multivariate quality measurements. For any dimension of the measurements, these charts are in the form of two-dimensional graphs that can be visualized and interpreted just as easily as the well-known univariate X, X, and CUSUM charts. Moreover, they have several significant advantages. First, they can detect simultaneously the location shift and scale increase of the process, unlike the existing methods, which can detect only the location shift. Second, their construction is completely nonparametric; in particular, it does not require the assumption of normality for the quality distribution, which is needed in standard approaches such as the χ2 and Hotellings T 2 charts. Thus these new charts generalize the principle of control charts to multivariate settings and apply to a much broader class of quality distributions.

Notions of Limiting P Values Based on Data Depth and Bootstrap

Abstract In this article we propose some new notions of limiting P values for hypothesis testing. The limiting P value (LP) here not only provides the usual attractive interpretation of a P value as the strength in support of the null hypothesis coming from the observed data, but also has several advantages. First, it allows us to resample directly from the empirical distribution (in the bootstrap implementations), rather than from the estimated population distribution satisfying the null constraints. Second, it serves as a test statistic and as a P value simultaneously, and thus enables us to obtain test results directly without having to construct an explicit test statistic and then establish or approximate its sampling distribution. These are the two steps generally required in a standard testing procedure. Using bootstrap and the concept of data depth, we have provided LPs for a broad class of testing problems where the parameters of interest can be either finite or infinite dimensional. Some compute...

Control Charts for Dependent and Independent Measurements Based on Bootstrap Methods

Abstract Shewhart charts are widely accepted as standard tools for monitoring manufacturing processes of univariate, independent “nearly” normal measurements. They are not as well developed beyond these types of data. We generalize the idea of Shewhart charts to cover other types of data commonly encountered in practice. More specifically, we develop some valid control charts for dependent data and for independent data that are not necessarily “nearly” normal. We derive the proposed charts from the moving blocks bootstrap and the standard bootstrap methods. Their constructions are completely nonparametric no distributional assumptions are required. Some simulated as well as real data examples are included they are very supportive of the proposed methods.

DD-Classifier: Nonparametric Classification Procedure Based on DD-Plot

Using the DD-plot (depth vs. depth plot), we introduce a new nonparametric classification algorithm and call it DD-classifier. The algorithm is completely nonparametric, and it requires no prior knowledge of the underlying distributions or the form of the separating curve. Thus, it can be applied to a wide range of classification problems. The algorithm is completely data driven and its classification outcome can be easily visualized in a two-dimensional plot regardless of the dimension of the data. Moreover, it has the advantage of bypassing the estimation of underlying parameters such as means and scales, which is often required by the existing classification procedures. We study the asymptotic properties of the DD-classifier and its misclassification rate. Specifically, we show that DD-classifier is asymptotically equivalent to the Bayes rule under suitable conditions, and it can achieve Bayes error for a family broader than elliptical distributions. The performance of the classifier is also examined using simulated and real datasets. Overall, the DD-classifier performs well across a broad range of settings, and compares favorably with existing classifiers. It can also be robust against outliers or contamination.

New Nonparametric Tests of Multivariate Locations and Scales Using Data Depth

Multivariate statistics plays a role of ever increasing importance in the modern era of information technology. Using the center-outward ranking induced by the notion of data depth, we describe several nonparametric tests of location and scale differences for multivariate distributions. The tests for location differences are derived from graphs in the so-called DD plots (depth vs. depth plots) and are implemented through the idea of permutation tests. The proposed test statistics are scale-standardized measures for the location difference and they can be carried out without estimating the scale or variance of the underlying distributions. The test for scale differences introduced in Liu and Singh (2003) is a natural multivariate rank test derived from the center-outward depth ranking and it extends the Wilcoxon rank-sum test to the testing of multivariate scale. We discuss the properties of these tests, and provide simulation results as well as a comparison study under normality. Finally, we apply the tests to compare airlines’ performances in the context of aviation safety evaluations.

Practical Engineering Statistics

Description: PRACTICAL ENGINEERING STATISTICS This lucidly written book offers engineers and advanced students all the essential statistical methods and techniques used in day-today engineering work. Without unnecessary digressions into formal proofs or derivations, Practical Engineering Statistics shows how to select the appropriate statistical method for a specific task and then how to apply it correctly and confidently. Clear explanations supported by real-world examples lead the reader step-by-step through each procedure. Topics covered include product design and development; estimations of the mean value and variability of measured data; comparison of processes or products; the relationships between variables; and more. With its emphasis on practical use and its full range of engineering applications, Practical Engineering Statistics serves as an indispensable, time-saving reference for all engineers working in design, reliability, assurance, scheduling, and manufacturing. PRACTICAL ENGINEERING STATISTICS While engineers are frequently involved in projects that require the application of statistical methods to analysis, prediction, and planning, their background in statistics is often insufficient to the task. In many cases the engineer has had little training in statistics beyond the concepts of the mean, the standard deviation, the median, and the quartile. Even those who have had one or more courses in statistics will, at times, encounter problems which are beyond their capacity to solve or understand. Practical Engineering Statistics is designed to give engineers the knowledge to select the statistical approach that is most appropriate to the problem at hand and the skills to confidently apply this approach to specific cases. It provides the engineer with the statistical tools needed to perform the job effectively, whether it is product design and development, estimation of the mean value and variability of measured data, comparison of processes or products, or the relationship between variables. Its authors bring two different areas of expertise to this unique book: statistics and engineering physics. In Practical Engineering Statistics their collaboration has produced a book that clearly leads engineers step-by-step through each procedure, without time-consuming and unnecessary discussions of proofs and derivations. Statistical procedures are discussed and explained in detail and demonstrated through real-world sample problems, with correct answers always provided. Readers learn how to determine which data represent true observations and which, through human error or flawed data, are false observations. Complex problems are presented with computer printouts of the database, intermediate steps, and results. Numerous illustrations and tables of all commonly used distributions enhance the usefulness of this …

Monitoring multivariate aviation safety data by data depth: control charts and threshold systems

Abstract Aviation safety analysis is increasingly needed in regulating air traffic and safety, in light of the rapid growth in air traffic density. With the recent advances in computer technology, large amounts of multivariate aviation safety data are now routinely collected in databases. Many existing analysis methods prescribed in those databases and corresponding safety indictors are based on classical statistical analysis, and their applicability are considerably restricted by the requirement of normality. An alternative nonparametric methodology based on data depthis pursued in this paper. For a given multivariate sample, a data depth can be used to measure their depth or outlyingness with respect to the underlying distribution. The measure of depth leads to a center-outward ordering of the sample points. Derived from this ordering, Liu (1995) introduced a simple, yet effective, control chart for monitoring multivariate observations. The control chart is combined here with properly chosen false alarm rates to develop meaningful threshold systems for multivariate aviation safety data for both regulating and monitoring purposes. The developed procedure is applied to the aviation inspection results collected by the Federal Aviation Administration (FAA) inspection system. The threshold system serves as a standard for evaluating the performance of aircraft operators, and provides clear guidelines for identifying unexpectedperformances and for assigning appropriate corrective actions.

Psychophysiologic treatment for patients with medically unexplained symptoms: a randomized controlled trial.

BACKGROUND Patients presenting with medically unexplained physical symptoms (MUPS) typically present with significant distress and marked impairment in functioning and pose a unique challenge to health care providers. The purpose of this study was to examine the efficacy of a psychophysiological treatment (PT) for MUPS. METHODS Thirty-eight participants meeting criteria for subthreshold somatization disorder (abridged somatization) were randomly assigned to one of two conditions: (1) standard medical care augmented by a psychiatric consultation intervention (wait-list) or (2) a 10-session, manualized, individually-administered PT added to the psychiatric consultation intervention. Assessments were conducted at baseline, at midpoint (after four sessions), and after completing the last session. The primary outcome measure was the severity scale of the Clinical Global Impression Scale anchored for Somatic Symptoms (CGI-SD). Secondary outcome measures were responder status as determined by clinical ratings, self-report measures of mental and physical functioning. RESULTS At the end of the trial, the severity (and frequency) of physical symptoms improved significantly more (p<0.05) in the intervention group. The average improvement in the CGI-SD was 0.80 points greater in the intervention group than in the wait-list group. PT was also associated with greater improvements in self-reported functioning and depressive symptomatology. The effect sizes at the final assessment point indicate that this intervention had a robust effect on complex somatic symptom presentations. CONCLUSION For patients with high levels of MUPS (abridged somatization), PT produces significant improvements in symptoms and functional status.

Multivariate Meta-Analysis of Heterogeneous Studies Using Only Summary Statistics: Efficiency and Robustness

Meta-analysis has been widely used to synthesize evidence from multiple studies for common hypotheses or parameters of interest. However, it has not yet been fully developed for incorporating heterogeneous studies, which arise often in applications due to different study designs, populations, or outcomes. For heterogeneous studies, the parameter of interest may not be estimable for certain studies, and in such a case, these studies are typically excluded from conventional meta-analysis. The exclusion of part of the studies can lead to a nonnegligible loss of information. This article introduces a meta-analysis for heterogeneous studies by combining the confidence density functions derived from the summary statistics of individual studies, hence referred to as the CD approach. It includes all the studies in the analysis and makes use of all information, direct as well as indirect. Under a general likelihood inference framework, this new approach is shown to have several desirable properties, including: (i) it is asymptotically as efficient as the maximum likelihood approach using individual participant data (IPD) from all studies; (ii) unlike the IPD analysis, it suffices to use summary statistics to carry out the CD approach. Individual-level data are not required; and (iii) it is robust against misspecification of the working covariance structure of parameter estimates. Besides its own theoretical significance, the last property also substantially broadens the applicability of the CD approach. All the properties of the CD approach are further confirmed by data simulated from a randomized clinical trials setting as well as by real data on aircraft landing performance. Overall, one obtains a unifying approach for combining summary statistics, subsuming many of the existing meta-analysis methods as special cases.