Comparing Confidence Intervals for Multivariate Process Capability Indices

Multivariate process capability indices (MPCIs) are needed for process capability analysis when the quality of a process is determined by several univariate quality characteristics that are correlated. There are several different MPCIs described in the literature, but confidence intervals have been derived for only a handful of these. In practice, the conclusion about process capability must be drawn from a random sample. Hence, confidence intervals or tests for MPCIs are important. With a case study as a start and under the assumption of multivariate normality, we review and compare four different available methods for calculating confidence intervals of MPCIs that generalize the univariate index C_p. Two of the methods are based on the ratio of a tolerance region to a process region, and two are based on the principal component analysis. For two of the methods, we derive approximate confidence intervals, which are easy to calculate and can be used for moderate sample sizes. We discuss issues that need to be solved before the studied methods can be applied more generally in practice. For instance, three of the methods have approximate confidence levels only, but no investigation has been carried out on how good these approximations are. Furthermore, we highlight the problem with the correspondence between the index value and the probability of nonconformance. We also elucidate a major drawback with the existing MPCIs on the basis of the principal component analysis. Our investigation shows the need for more research to obtain an MPCI with confidence interval such that conclusions about the process capability can be drawn at a known confidence level and that a stated value of the MPCI limits the probability of nonconformance in a known way. Copyright © 2011 John Wiley & Sons, Ltd.     

基于广义置信区间的测量系统能力比较

研究了测量系统分析中对测量同一产品质量特性的两个测量系统能力的比较方法。基于广义推断法,提出了一种比较两个测量系统能力的方法。通过重复性与再现性（R&R）试验研究,利用广义推断法对两个测量系统能力的比较指数建立相应的广义置信区间,并进行计算机仿真。结果表明,基于广义置信区间的方法,可以有效地比较两个测量系统的能力。     

Confidence Intervals for Process Capability Indices for the Unbalanced One‐Way Random Effect ANOVA Model

Confidence intervals for process capability index C_pk are developed for the unbalanced one‐way random effect model using Bissell's approximation method. The proposed limit is compared with the generalized lower confidence limit obtained using the generalized pivotal quantity method. To assess the accuracy of the method, a simulation study is presented. The results are illustrated with an industrial example. Copyright © 2011 John Wiley & Sons, Ltd.     

Comparing Two Process Capability Indices under Balanced One‐Way Random Effect Model

Process capability indices such as C_p, C_pk, C_pmk and C_pm are widely used in manufacturing industries to provide a quantitative measurement of the performance of the products. In this article, we derived generalized confidence intervals for the difference between process capability indices for two processes under one‐way random effect model. Our study provides coverage probability close to the nominal value in almost all cases as shown via simulation. An example from industrial contexts is given to illustrate the results. Copyright © 2012 John Wiley & Sons, Ltd.     

A New Method for Constructing Confidence Intervals for the Index Cpm

In the statistical literature on the study of the capability of processes through the use of indices, C_pm appears to have been one of the most widely used capability indices and its estimation has attracted much interest. In this article, a new method for constructing approximate confidence intervals or lower confidence limits for this index is suggested. The method is based on an approximation of the non‐central chi‐square distribution, which was proposed by Pearson. Its coverage appears to be more satisfactory compared with that achieved by any of the two most widely used methods that were proposed by Boyles, in situations where one is interested in assessing a lower confidence limit for C_pm. This is supported by the results of an extensive simulation study. Copyright © 2004 John Wiley & Sons, Ltd.     

Bootstrap Confidence Limits for Short-Run Capability Indices

The capability indices are widely used by quality professionals as an estimate of process capability. Recently, techniques and tables were developed to construct confidence limits for each index. These techniques are based on the assumption that the underlying process is normally distributed. In industries it may not be possible to get large samples, and, hence, the normality assumption is often violated. For the short-run production processes where sample size is small, appropriate indices can be developed. In this article, the capability indices such as C_p, C_pk, and C_pm are modified, and appropriate capability indices are constructed. A further bootstrap technique is used to define the confidence intervals. A simulation using two distributions (one normal and the other nonnormal) is conducted, and a comparison is made to show the performances of the three nonparametric confidence intervals.     

基于Bootstrap方法的过程能力指数区间估计

针对传统统计过程控制方法的局限性,利用Bootstrap方法构建了过程能力指数的置信区间,分别对正态分布、指数分布进行仿真计算,对3种Bootstrap置信区间进行了比较。通过实证分析验证了在小样本条件下Bootstrap置信区间的有效性。     

Evaluation of Nonnormal Process Capability Indices using Generalized Lambda Distribution

Process capability indices (PCIs) are used to describe a manufacturing process expressing its ability to produce items within the specified limits. These indices are developed under the assumption that the underlying process distribution is normal. In industries, there are many manufacturing processes where process distribution can not be described by a normal distribution. In such cases, those PCIs will give misleading results about the process. The most commonly used approach for analysing a nonnormal process data is to fit a standard nonnormal distribution (e.g., weibull, gamma) or a family of distribution curves (e.g., Pearson, Johnson) to the process data and then to estimate the percentile points from the fitted distribution that can be used to compute generalized PCIs. In this article, we outline the procedure using the generalized lambda distribution (GLD) curve for modeling a set of process data and for estimating percentile points in order to compute generalized PCIs. The four-parameter GLD can assume a wide variety of curve shapes and hence it is very useful for the representation of data when the underlying model is unknown. Compared to the Pearson and Johnson family of distributions, the GLD is computationally simpler and more flexible. The article provides all necessary formulas for fitting a GLD curve, estimating its parameters, performing goodness-of-fit tests, and computing generalized PCIs. An example is used to illustrate the calculations that can be easily performed using spreadsheets.     

Yield‐Related Process Capability Indices for Processes of Multiple Quality Characteristics

Process capability indices (PCIs) have been widely used in industries for assessing the capability of manufacturing processes. Castagliola and Castellanos (Quality Technology and Quantitative Management 2005, 2(2):201–220), viewing that there were no clear links between the definition of the existing multivariate PCIs and theoretical proportion of nonconforming product items, defined a bivariate C_pk and C_p (denoted by BC_pk and BC_p, respectively) based on the proportions of nonconforming product items over four convex polygons for bivariate normal processes with a rectangular specification region. In this paper, we extend their definitions to MC_pk and MC_p for multivariate normal processes with flexible specification regions. To link the index to the yield, we establish a ‘reachable’ lower bound for the process yield as a function of MC_pk. An algorithm suitable for such processes is developed to compute the natural estimate of MC_pk from process data. Furthermore, we construct via the bootstrap approach the lower confidence bound of MC_pk, a measure often used by producers for quality assurance to consumers. As for BC_p, we first modify the original definition with a simple preprocessing step to make BC_p scale‐invariant. A very efficient algorithm is developed for computing a natural estimator of BC_p. This new approach of BC_p can be easily extended to MC_p for multivariate processes. For BC_p, we further derive an approximate normal distribution for , which enables us to construct procedures for making statistical inferences about process capability based on data, including the hypothesis testing, confidence interval, and lower confidence bound. Finally, the proposed procedures are demonstrated with three real data sets. Copyright © 2012 John Wiley & Sons, Ltd.     

Incorporating Confidence Intervals on the Decision Threshold in Logistic Regression

This paper discusses an application of confidence intervals to the threshold decision value used in logistic regres]sion and discusses its effect on changing the quantification of false positive and false negative errors. In doing this a grey area, in which observations are not classified as success (1) or failure (0), but rather ‘uncertain’ is developed. The size of this grey area is related to the level of confidence chosen to create the interval around the threshold as well as the quality of logistic regression model fit. This method shows that potential errors may be mitigated. Monte Carlo simulation and an experimental design approach are used to study the relationship between a number of responses relating to classification of observations and the following factors: threshold level, confidence level, noise in the data, and number of observations collected. Copyright © 2015 John Wiley & Sons, Ltd.     

基于工序成本的多元过程能力指数分析

单一的多元过程能力指数在反映企业制造多元质量特性产品能力时存在缺陷,无法反映由于各工序成本的差异而造成的工序能力重要性的差别.基于此,从企业工序质量控制和工序成本关系的角度来分析,构建了基于工序成本修正的多元过程能力指数,反映企业的成本控制能力和产品竞争能力.最后,通过修正得到的多元过程能力指数,一方面优选企业的质量改进或控制方案,另一方面判定企业在同类产品制造中的实际质量控制能力和过程成本损失差异.     

Approximate Confidence Intervals for the Log‐Normal Standard Deviation

In this work, approximate confidence intervals are derived for the standard deviation of a log‐normal distribution. Simulations are conducted to evaluate the coverage probability, average length, and coverage bias of the derived approximate confidence intervals. The simulation results indicate that the proposed approximate confidence intervals perform reasonably well when the standard deviation of the log‐transformed variable is small. The approximate confidence intervals are applied to a phase I pharmacokinetic study and a real data set concerning a lab testing of reference engine oil. Copyright © 2015 John Wiley & Sons, Ltd.     

一种基于产出合格率的过程能力指数及其置信下限的估计

介绍了一种基于加工过程产出合格率的过程能力指数Cpc,并推导了Cpc指数与过程合格率之间的弹性系数。指出：相对于传统过程能力指数来说,Cpc计算简单,含义明确,不受过程分布特性的影响,可适用于双边或单边规格限情况,尤其是Cpc指数相对产出合格率非常富于弹性,可以有效反映过程质量的波动情况。为了获得Cpc,指数的概率分布及相关统计推断,利用Bootstrap抽样技术,对Cpc样本估计量的分布情况进行仿真处理,得到Cpc的经验分布及其置信下限,为更有效的使用该指数提供了概率依据。最后对Cpc指数的具体应用给出了案例分析。     

Process Capability Indices for Logistic Regression Profile

A profile is a relationship between a response variable and one or more independent variables that can describe the quality of a process or product. On the other side, for an in‐control process, capability indices are the criteria for process quality improvement that allows meeting customer expectations. Recently, evaluating the process capability of profiles has been investigated by some researchers. In all of these efforts, the response variable in the profile follows a normal distribution. However, sometimes, this assumption is violated, and the response variable may follow a binary or binomial distribution. In this paper, we propose two methods to measure the process capability when the quality of a process is characterized by a logistic regression profile. The performance of the proposed indices is evaluated through simulation studies. Finally, the application of the proposed methods is illustrated through a real case. Copyright © 2015 John Wiley & Sons, Ltd.     

Functional Process Capability Indices for Circular Profile

Profile is a relation between one response variable and one or more explanatory variables that represent quality of a product or performance of a process. On the other hand, process capability indices are measures to help practitioners in improving the processes to satisfy the customer's expectations. Few researches are done to account for the process capability index in the areas of profile monitoring. All of these researches are focused on process capability index in simple linear profile. In all of these methods, response variables in different levels of explanatory variable are considered, and the relationship in all range of explanatory variable is neglected. In this paper, a functional method is proposed to measure process capability index of circular profiles in all range of explanatory variable. The proposed method follows the traditional definition of process capability indices. The functional method uses reference profile, functional specification limits and functional natural tolerance limits to present a functional form of process capability indices. This functional form results in measuring the process capability in each level of explanatory variable in circular profile as well as a unique value of process capability index for circular profile. The application of the proposed method is illustrated through a real case in automotive industry. Copyright © 2013 John Wiley & Sons, Ltd.     

Confidence limits for compliance testing using mixed acceptance criteria

For manufactured items sold by weight or volume, this article considers mixed acceptance criteria that put limits on the sample mean or on an upper confidence limit based on the sample mean and on the number of individual sample units that are nonconforming. For a normally distributed quality characteristic of interest, this article develops lower confidence limits for the mixed acceptance criteria applying the concept of a generalized pivotal quantity and applying a bias-corrected and accelerated parametric bootstrap. The accuracy of the confidence limits is assessed using estimated coverage probabilities, and the results are illustrated with an example.     

A Sampler On Sampling

The paper outlines a detailed study of three sets of large sample simultaneous confidence intervals for the probabilities of a multinomial distribution, all three making use of the Bonferroni inequality. One of them, originally proposed by Goodman, is based on the assumption that each cell frequency n_i is, marginally, normally distributed, while the other two require the normality of transformations of n_i —an angular transformation in one case and a square root in the other. It is shown that all three sets of intervals should be used with a correction for continuity; their coverage probabilities are investigated, and it is seen that the two sets based on transformations of n_i produce shorter intervals than Goodman's when n_i is small. There is little to choose between these two except that one of them is a little simpler to use than the other.