An Improved Boxplot for Univariate Data

The boxplot is an effective data-visualization tool useful in diverse applications and disciplines. Although more sophisticated graphical methods exist, the boxplot remains relevant due to its simplicity, interpretability, and usefulness, even in the age of big data. This article highlights the origins and developments of the boxplot that is now widely viewed as an industry standard as well as its inherent limitations when dealing with data from skewed distributions, particularly when detecting outliers. The proposed Ratio-Skewed boxplot is shown to be practical and suitable for outlier labeling across several parametric distributions.     

Robustness of the EWMA control chart for individual observations

The traditional exponentially weighted moving average (EWMA) chart is one of the most popular control charts used in practice today. The in-control robustness is the key to the proper design and implementation of any control chart, lack of which can render its out-of-control shift detection capability almost meaningless. To this end, Borror et al. [5 Borror, C. M., Montgomery, D. C. and Runger, G. C. 1999. Robustness of the EWMA control chart to non-normality. J. Qual. Technol., 31(3): 309–316. [Taylor & Francis Online], [Web of Science ®] [Google Scholar]] studied the performance of the traditional EWMA chart for the mean for i.i.d. data. We use a more extensive simulation study to further investigate the in-control robustness (to non-normality) of the three different EWMA designs studied by Borror et al. [5 Borror, C. M., Montgomery, D. C. and Runger, G. C. 1999. Robustness of the EWMA control chart to non-normality. J. Qual. Technol., 31(3): 309–316. [Taylor & Francis Online], [Web of Science ®] [Google Scholar]]. Our study includes a much wider collection of non-normal distributions including light- and heavy-tailed and symmetric and asymmetric bi-modal as well as the contaminated normal, which is particularly useful to study the effects of outliers. Also, we consider two separate cases: (i) when the process mean and standard deviation are both known and (ii) when they are both unknown and estimated from an in-control Phase I sample. In addition, unlike in the study done by Borror et al. [5 Borror, C. M., Montgomery, D. C. and Runger, G. C. 1999. Robustness of the EWMA control chart to non-normality. J. Qual. Technol., 31(3): 309–316. [Taylor & Francis Online], [Web of Science ®] [Google Scholar]], the average run-length (ARL) is not used as the sole performance measure in our study, we consider the standard deviation of the run-length (SDRL), the median run-length (MDRL), and the first and the third quartiles as well as the first and the 99th percentiles of the in-control run-length distribution for a better overall assessment of the traditional EWMA chart's in-control performance. Our findings sound a cautionary note to the (over) use of the EWMA chart in practice, at least with some types of non-normal data. A summary and recommendations are provided.     

Attitude Toward Menopause and Aging: A Study on Postmenopausal Women of West Bengal

We examined the attitude of postmenopausal women toward menopause and aging with respect to sociodemographic variables and postmenopausal years. Four hundred and eighty postmenopausal women representing Bengali-speaking Hindu ethnic group of West Bengal, India were interviewed about their attitude toward menopause and aging. Information on sociodemographic and reproductive characteristics and menopausal symptoms were also collected. The participants were categorized into four groups based on postmenopausal years (Group 1: ≤2; Group 2: >2 to ≤5; Group 3: >5 to ≤8; and Group 4: >8). The attitude did not differ significantly among different groups, but it differed significantly when compared for residential status and per capita monthly household expenditure (pooled groups), for residential and educational status (Groups 3 and 4), and per capita monthly household expenditure (Group 3). Hierarchical linear regression (stepwise) shows per capita monthly expenditure, age at menopause, years after menopause, and menopausal symptoms (irritability and inability to hold urine) significantly predict attitude.     

Precedence tests and Lehmann alternatives

G = F ^k (k > 1); G = 1 − (1−F) ^k (k < 1); G = F ^k (k < 1); and G = 1 − (1−F) ^k (k > 1), where F and G are two continuous cumulative distribution functions. If an optimal precedence test (one with the maximal power) is determined for one of these four classes, the optimal tests for the other classes of alternatives can be derived. Application of this is given using the results of Lin and Sukhatme (1992) who derived the best precedence test for testing the null hypothesis that the lifetimes of two types of items on test have the same distibution. The test has maximum power for fixed κ in the class of alternatives G = 1 − (1−F) ^k , with k < 1. Best precedence tests for the other three classes of Lehmann-type alternatives are derived using their results. Finally, a comparison of precedence tests with Wilcoxon's two-sample test is presented. Received: February 22, 1999; revised version: June 7, 2000     

Parameter estimation and design considerations in prospective applications of the � chart

The effects of parameter estimation on the in-control performance of the Shewhart X¯ chart are studied in prospective (phase 2 or stage 2) applications via a thorough examination of the attained false alarm rate (AFAR), the conditional false alarm rate (CFAR), the conditional and the unconditional run-length distributions, some run-length characteristics such as the ARL, the conditional ARL (CARL), some selected percentiles including the median, and cumulative run-length probabilities. The examination involves both numerical evaluations and graphical displays. The effects of parameter estimation need to be accounted for in designing the chart. To this end, as an application of the exact formulations, chart constants are provided for a specified in-control average run-length of 370 and 500 for a number of subgroups and subgroup sizes. These will be useful in the implementation of the X¯ chart in practice.     

Nonparametric Shewhart-Type Sign Control Charts Based on Runs

Nonparametric control charts are considered for the median and other percentiles based on runs of sign statistics above and below the control limits. It is noted that the sign charts are advantageous in certain practical situations. Expressions for the run-length distributions are derived using Markov chain theory; several examples are given. The in-control (IC) and the out-of-control (OOC) performance of these charts are studied and compared to the existing nonparametric Wilcoxon signed-ranked charts of Chakraborti and Eryilmaz (2007 Chakraborti , S. , Eryilmaz , S. ( 2007 ). A nonparametric Shewhart-type signed-rank control chart based on runs . Commun. Statist. Simul. Computat. 36 ( 2 ): 335 – 356 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) under the normal, the double exponential and the Cauchy distributions, using the average run-length (ARL), the standard deviation of the run-length (SDRL), the false alarm rate (FAR) and some percentiles of the run-length, including the median run-length (MDRL). It is shown that the proposed “runs-rules enhanced” sign charts offer more practically desirable IC ARL (ARL ₀) and FAR values and perform better for some heavy-tailed distributions. Some concluding remarks are offered.     

Run length,average run length and false alarm rate of shewhart x-bar chart: exact derivations by conditioning

The effects of estimation of the control limits on the performance of the popular Shewhart X-bar chart are examined via the average run length and the probability of a false alarm, when one or both of the process mean and variance are unknown. Exact expressions for the run length, the average run length (ARL) and the false alarm rate are obtained, in each case, using expectation by conditioning. Applying Jensen's inequality, together with expectation by conditioning, a simple lower bound to the ARL is obtained. This could be useful in designing the charts. The expressions for the exact ARL and the exact probabilities of false alarm are evaluated, using simulations, for various numbers of subgroups and shift sizes. The calculations throw new light on the performance of the Shewhart X-bar chart. Some recommendations are given.     

Distribution-Free Exceedance CUSUM Control Charts for Location

Distribution-free (nonparametric) control charts can be useful to the quality practitioner when the underlying distribution is not known. A Phase II nonparametric cumulative sum (CUSUM) chart based on the exceedance statistics, called the exceedance CUSUM chart, is proposed here for detecting a shift in the unknown location parameter of a continuous distribution. The exceedance statistics can be more efficient than rank-based methods when the underlying distribution is heavy-tailed and/or right-skewed, which may be the case in some applications, particularly with certain lifetime data. Moreover, exceedance statistics can save testing time and resources as they can be applied as soon as a certain order statistic of the reference sample is available. Guidelines and recommendations are provided for the chart's design parameters along with an illustrative example. The in- and out-of-control performances of the chart are studied through extensive simulations on the basis of the average run-length (ARL), the standard deviation of run-length (SDRL), the median run-length (MDRL), and some percentiles of run-length. Further, a comparison with a number of existing control charts, including the parametric CUSUM chart and a recent nonparametric CUSUM chart based on the Wilcoxon rank-sum statistic, called the rank-sum CUSUM chart, is made. It is seen that the exceedance CUSUM chart performs well in many cases and thus can be a useful alternative chart in practice. A summary and some concluding remarks are given.     

Shortest Expected Length Confidence Interval for the Power of the t-Test

Tarasińska (2005 Tarasińska , J. ( 2005 ). Confidence intervals for the power of student's t-test . Statistics & Probability Letters 73 ( 2 ): 125 – 130 .[Crossref], [Web of Science ®] , [Google Scholar]) considered a method to construct the shortest length confidence interval on the power of the t-test using a confidence interval for the population standard deviation in the non centrality parameter. Gilliland and Li (2008 Gilliland , D. , Li , M. ( 2008 ). A note on confidence intervals for the power of t-test . Statistics & Probability Letters 78 ( 5 ): 488 – 489 .[Crossref], [Web of Science ®] , [Google Scholar]) used simulations to show that this confidence interval has less than the nominal coverage, particularly in small samples. We propose to find the shortest expected length confidence interval for the power of the t-test by accounting for the variation in the sample standard deviation, and provide the necessary constants for its implementation for some selected sample and shift sizes. It is seen that the proposed interval is reasonably robust to the specification of the population standard deviation and maintains the nominal coverage.