Confidence intervals and sample size determination for Cpm

The capability index, C_pm, sometimes called the Taguchi index, has the desirable characteristic of being sensitive to both dispersion and deviation of the process average from the engineering target. As a result, the proposed estimators of C_pm have a sampling distribution that is dependent on the non‐central chi‐square distribution. Hence, constructing confidence intervals, performing hypothesis testing or estimating sample size requirements necessitates manipulation of a rather complex functional expression, typically beyond the capabilities of practitioners who need readily available tools. Here, a simple graphical procedure is proposed and illustrated for obtaining exact confidence intervals for C_pm. The graphical procedure allows the user to simply enter the graph with an estimate of the index and a value of the non‐centrality parameter for a given sample size to arrive at end‐points of 90%, 95% or 99% one‐sided or two‐sided confidence intervals. Detailed tables are also provided to assist the user for a wider range of sample values and sample sizes. In addition, a procedure is also presented for determining the minimum sample size required for attaining a pre‐specified level of accuracy of the C_pm. Extensive tables are provided for the user with a simple example illustrating the facility of the technique. Copyright © 2001 John Wiley & Sons, Ltd.     

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.     

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。     

Obtaining confidence intervals for Cpk using percentiles of the distribution of Ĉp

?_pk is used as an estimate of process capability and can reflect performance degradation due to both shifts in the process mean and variability. Exact upper and lower confidence limits for the actual parameter value are elusive. Objections to existing estimators focus on two areas: the difficulty of executing the needed computations and the excessive widths of those confidence bounds. The CPE estimator (so called since it relies on ?_p ) which we derive in this paper is formed by simply multiplying ?_pk by a value from the inverse chi‐square distribution. A PASCAL computer program is supplied which generates those percentiles. Our CPE is compared via simulation to other estimators and is shown to be the preferred selection in most cases by providing the requisite probability coverage and narrow interval widths. Copyright © 2001 John Wiley & Sons, Ltd.     

Exact two‐sided statistical tolerance limits for sample variances

Statistical tolerance intervals are widely used in the industry and in various areas of sciences, especially in conformity assessment and acceptance of products or processes in terms of quality. When the interest is in precision, a tolerance interval for the variance is useful. In this paper, we consider two‐sided tolerance intervals for the population of sample variances for data that arise from a normal distribution. These intervals are useful in applications where one needs information about process deterioration as well as process improvement, to properly assess product quality. In this paper, the theory for these tolerance intervals is developed and tables for the tolerance factors, required to calculate the proposed tolerance limits, are provided for various settings. Construction and implementation of the proposed tolerance intervals are illustrated using a dataset from a real application. Summary and conclusions are offered.     

Estimated incapability index: reliability and decision making with sample information

The process incapability index , which provides an uncontaminated separation between information concerning the process precision and process accuracy, has been proposed to measure process performance for industry applications. In this paper, we investigate the reliability of the natural estimator computationally, based on the ‐level confidence relative error for various sample sizes. We also develop a decision‐making procedure for judging if the process satisfies the preset quality requirement. The investigation is useful to the practitioners in determining the sample sizes required in their applications for the decisions reliable to the desired level. Copyright © 2002 John Wiley & Sons, Ltd.     

A Bayesian Approach to Obtain a Lower Bound for the Cpm Capability Index

The Taguchi capability index C_pm, which incorporates the departure of the process mean from the target value, has been proposed to the manufacturing industry for measuring manufacturing capability. A Bayesian procedure has been considered for testing process performance assuming , which was generalized without assuming . Statistical properties of the natural estimator of the index C_pm for normal processes have been investigated extensively. However, the investigation was restricted to processes with symmetric tolerances. Recently, a generalized C_pm, referred to as , was proposed to cover processes with asymmetric tolerances. Under the normality assumption, the statistical properties of the estimated including the exact sampling distribution, the rth moment, expected value, variance, and the mean‐squared error were obtained. In this paper, we use a Bayesian approach to obtain the interval estimation for the generalized Taguchi capability index . Consequently, the manufacturing capability testing can be performed for quality assurance. Copyright © 2005 John Wiley & Sons, Ltd.     

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.     

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.     

A Note on “Capability Assessment for Processes with Multiple Characteristics: A Generalization of the Popular Index Cpk”

The generalized yield index establishes the relationship between the manufacturing specifications and the actual process performance, which provides a lower bound on process yield for two‐sided processes with multiple characteristics. The results attended are very practical for industrial application. In this article, we extended the results in cases with one‐sided specification and multiple characteristics. The generalized index was considered, and the asymptotic distribution of the natural estimator was developed. Then, we derived the lower confidence bounds as well as the critical values of index . We not only provided some tables but also presented an application example. Copyright © 2012 John Wiley & Sons, Ltd.     

Process Capability Indices Based on the Highest Density Interval

For process capability indices (PCIs) of non‐normal processes, the natural tolerance is defined as the difference between the 99.865 percentile and the 0.135 percentile of the process characteristic. However, some regions with relatively low probability density may still be included in this natural tolerance, while some regions with relatively high probability density may be excluded for asymmetric distributions. To take into account the asymmetry of process distributions and the asymmetry of tolerances from the viewpoint of probability density, the highest density interval is utilized to define the natural tolerance, and a family of new PCIs based on the highest density interval is proposed to ensure that regions with high probability density are included in the natural tolerance. Some properties of the proposed PCIs and two algorithms to compute the highest density interval are given. A real example is given to show the application of the proposed method. Copyright © 2014 John Wiley & Sons, Ltd.     

A Method of Estimating the Process Capability Index from the First Four Moments of Non‐normal Data

A method is presented to estimate the process capability index (PCI) for a set of non‐normal data from its first four moments. It is assumed that these four moments, i.e. mean, standard deviation, skewness, and kurtosis, are suitable to approximately characterize the data distribution properties. The probability density function of non‐normal data is expressed in Chebyshev–Hermite polynomials up to tenth order from the first four moments. An effective range, defined as the value for which a pre‐determined percentage of data falls within the range, is solved numerically from the derived cumulative distribution function. The PCI with a specified limit is hence obtained from the effective range. Compared with some other existing methods, the present method gives a more accurate PCI estimation and shows less sensitivity to sample size. A simple algebraic equation for the effective range, derived from the least‐square fitting to the numerically solved results, is also proposed for PCI estimation. Copyright © 2004 John Wiley & Sons, Ltd.     

An adaptive chart for monitoring the process mean and variance

Traditionally, an chart is used to control the process mean and an R chart is used to control the process variance. However, these charts are not sensitive to small changes in the process parameters. The adaptive and R charts might be considered if the aim is to detect small disturbances. Due to the statistical character of the joint and R charts with fixed or adaptive parameters, they are not reliable in identifying the nature of the disturbance, whether it is one that shifts the process mean, increases the process variance, or leads to a combination of both effects. In practice, the speed with which the control charts detect process changes may be more important than their ability in identifying the nature of the change. Under these circumstances, it seems to be advantageous to consider a single chart, based on only one statistic, to simultaneously monitor the process mean and variance. In this paper, we propose the adaptive non‐central chi‐square statistic chart. This new chart is more effective than the adaptive and R charts in detecting disturbances that shift the process mean, increase the process variance, or lead to a combination of both effects. Copyright © 2006 John Wiley & Sons, Ltd.     

Computing process capability indices for non‐normal data: a review and comparative study

When the distribution of a process characteristic is non‐normal, C_p and C_pk calculated using conventional methods often lead to erroneous interpretation of the process's capability. Though various methods have been proposed for computing surrogate process capability indices (PCIs) under non‐normality, there is a lack of literature that covers a comprehensive evaluation and comparison of these methods. In particular, under mild and severe departures from normality, do these surrogate PCIs adequately capture process capability, and which is the best method(s) in reflecting the true capability under each of these circumstances? In this paper we review seven methods that are chosen for performance comparison in their ability to handle non‐normality in PCIs. For illustration purposes the comparison is done through simulating Weibull and lognormal data, and the results are presented using box plots. Simulation results show that the performance of a method is dependent on its capability to capture the tail behaviour of the underlying distributions. Finally we give a practitioner's guide that suggests applicable methods for each defined range of skewness and kurtosis under mild and severe departures from normality. Copyright © 1999 John Wiley & Sons, Ltd.