Two-sided Tolerance Research Articles

Numerical experiment to evaluating two-sided tolerance limit for safety analysis

Uncertainty analysis is required to quantify uncertainties in safety evaluation for industrial applications when the best-estimate methodology is employed. The nonparametric order statics method suggested by the GRS (Gesellschaftfür Anlagen-und Reaktorsicherheit) is one of the uncertainty evaluation methodologies to obtain the figure of merits with a probability of 95 % and confidence of 95 %, namely 95/95 value. In this method, the number of repeated calculations with perturbation to acquire the 95/95 value is decided by a formula suggested by Wilks and is dependent on the number of uncertainty parameters. Thus, the method is effective when the reference system has a large number of parameters which bring uncertainty in the analysis. Previous studies indicate that the method can estimate the 95/95 value successfully when the figure of merit has one-sided tolerance limit where either the upper or lower limit exits. However, when it is necessary to cut off the tails of 2.5% evenly in both ends, namely the centered two-sided tolerance limit, the suggested formula results in a lower confidence level. Thus, a modified formula is suggested in this study to account for such characteristics, and, as a result, the number of repeated calculations required to obtain the 95/95 value is calculated. The validity of the formula and the number of repeated calculations are examined using numerical experiments for 21 different distributions. The numerical experiment has been conducted with one to ten million sample sets to estimate the confidence level. The results of the numerical experiments indicate that the 95/95 value is predicted successfully by the repeated calculations decided by the modified formula when the figure of merit has characteristics of the centered two-sided distribution, while the existing formula results in a confidence level of 80%.

Echocardiographic heart ageing patterns predict cardiovascular and non-cardiovascular events and reflect biological age: the SardiNIA study.

Age is a crucial risk factor for cardiovascular (CV) and non-CV diseases. As people age at different rates, the concept of biological age has been introduced as a personalized measure of functional deterioration. Associations of age with echocardiographic quantitative traits were analysed to assess different heart ageing rates and their ability to predict outcomes and reflect biological age. Associations of age with left ventricular mass, geometry, diastolic function, left atrial volume, and aortic root size were measured in 2614 healthy subjects. Based on the 95% two-sided tolerance intervals of each correlation, three discrete ageing trajectories were identified and categorized as 'slow', 'normal', and 'accelerated' heart ageing patterns. The primary endpoint included fatal and non-fatal CV events, and the secondary endpoint was a composite of CV and non-CV events and all-cause death. The phenotypic age of the heart (HeartPhAge) was estimated as a proxy of biological age. The slow ageing pattern was found in 8.7% of healthy participants, the normal pattern in 76.9%, and the accelerated pattern in 14.4%. Kaplan-Meier curves of the heart ageing patterns diverged significantly (P = 0.0001) for both primary and secondary endpoints, with the event rate being lowest in the slow, intermediate in the normal, and highest in the accelerated pattern. In the Cox proportional hazards model, heart ageing patterns predicted both primary (P = 0.01) and secondary (P = 0.03 to <0.0001) endpoints, independent of chronological age and risk factors. Compared with chronological age, HeartPhAge was 9 years younger in slow, 4 years older in accelerated (both P < 0.0001), and overlapping in normal ageing patterns. Standard Doppler echocardiography detects slow, normal, and accelerated heart ageing patterns. They predict CV and non-CV events, reflect biological age, and provide a new tool to calibrate prevention timing and intensity.

A New Approach to Production Process Capability Assessment for Non-Normal Data

The process quality capability indicators Cp and Cpk are widely used to measure process capability. Traditional metric estimation methods require process data to be explicit and normally distributed. Often, the actual data obtained from the production process regarding the measurements of quality features are incomplete and do not have a normal distribution. This means that the use of traditional methods of estimating Cp and Cpk indicators may lead to erroneous results. Moreover, in the case of qualitative characteristics where a two-sided tolerance limit is specified, it should not be very difficult. The problem arises when the data do not meet the postulate of normality distribution and/or a one-sided tolerance limit has been defined for the process. Therefore, the purpose of this article was to present the possibility of using the Six Sigma method in relation to numerical data that do not meet the postulate of normality of distribution. The paper proposes a power transformation method using multiple-criteria decision analysis (MCDA) for the asymmetry coefficient and kurtosis coefficient. The task was to minimize the Jarque–Bera statistic, which we used to test the normality of the distribution. An appropriate methodology was developed for this purpose and presented on an empirical example. In addition, for the variable after transformation, for which the one-sided tolerance limit was determined, selected process quality evaluation indices were calculated.

A double integration method for generating exact tolerance limit factors for normal populations

This article introduces a new method for generating the exact one-sided and two-sided tolerance limit factors for normal populations. This method does not need to handle the noncentral t-distribution at all, but only needs to do a double integration of a joint probability density function with respect to the two independent variables “s” (standard deviation) and “x” (sample mean). The factors generated by this method are investigated through Monte Carlo simulations and compared with the existing factors. As a result, it is identified that the two-sided tolerance limit factors being currently used in practical applications are inaccurate. For the right understanding, some factors generated by this method are presented in Tables along with a guidance for correct use of them. The AQL (Acceptable Quality Level) is a good, common measure about quality of a product lot which was already produced or will be produced. Therefore, when performing sampling inspection on a given lot using a tolerance limit factor, there is a necessity to know the AQL assigned to the factor. This new double integration method even makes it possible to generate the AQLs corresponding to the one-sided and two-sided tolerance limit factors.

Empirical weighted Bayesian tolerance intervals

ABSTRACT Bayesian statistics has been widely utilized as an approach that can incorporate prior knowledge into statistical inference. Tolerance intervals (TI) are the most commonly used statistical methods for product quality assurance. There are two main Bayesian approaches for calculating statistical tolerance intervals: Hamada and Wolfinger. A simulation-based approach was implemented to compare two-sided Wolfinger, Hamada, and frequentist tolerance intervals which control the probability content at a specified level of confidence. As sample sizes increase, compared to frequentist, Hamada TI become more conservative while Wolfinger TI are more liberal. To address this issue, we propose an empirical weighted Bayesian TI approach that is a compromise between Hamada and Wolfinger approaches. The proposed Bayesian TI result in narrower limits in certain scenarios while ensuring the confidence content coverage remains comparable to frequentist.

A method for computing tolerance intervals for a location-scale family of distributions

The problems of computing two-sided tolerance intervals (TIs) and equal-tailed TIs for a location-scale family of distributions are considered. The TIs are constructed using one-sided tolerance limits with the Bonferroni adjustments and then adjusting the confidence levels so that the coverage probabilities of the TIs are equal to the specified nominal confidence level. The methods are simple, exact and can be used to find TIs for all location-scale families of distributions including log-location-scale families. The computational methods are illustrated for the normal, Weibull, two-parameter Rayleigh and two-parameter exponential distributions. The computational method is applicable to find TIs based on a type II censored sample. Factors for computing two-sided TIs and equal-tailed TIs are tabulated and R functions to find tolerance factors are provided in a supplementary file. The methods are illustrated using a few practical examples.

Use of a two-sided tolerance interval in the design and evaluation of biosimilarity in clinical studies.

In assessing biosimilarity between two products, the question to ask is always "How similar is similar?" Traditionally, the equivalence of the means between products is the primary consideration in a clinical trial. This study suggests an alternative assessment for testing a certain percentage of the population of differences lying within a prespecified interval. In doing so, the accuracy and precision are assessed simultaneously by judging whether a two-sided tolerance interval falls within a prespecified acceptance range. We further derive an asymptotic distribution of the tolerance limits to determine the sample size for achieving a targeted level of power. Our numerical study shows that the proposed two-sided tolerance interval test controls the type I error rate and provides sufficient power. A real example is presented to illustrate our proposed approach.

Computing exact factors for two-sided tolerance limits in a normal distribution with unknown parameters in MATLAB

MATLAB procedure is presented for calculating the exact factors used to find of two-sided tolerance limits in a normal distribution with unknown parameter values. The proposed procedure allows using arbitrary combinations of the sample size and the degrees of freedom of estimated variance. The procedure can be used in systems for processing the results of statistical quality control created on the basis of MATLAB.

Approximate two-sided tolerance interval for sample variances

Tolerance limits for variances are useful in quality assessments when the focus is on the precision of a quality characteristic. Two-sided tolerance intervals (limits) provide insight into a process degradation as well as improvement, in terms of process variability. Sarmiento, Chakraborti, and Epprecht constructed the exact two-sided tolerance intervals for the population of sample variances, assuming normality of the data. The required tolerance factors cannot be expressed in a closed-form and their computation is complex, depending on the numerical solutions of a system of three nonlinear equations. Motivated by this, from a practical point of view, we consider a simpler, approximate tolerance interval based on the approximate tolerance interval for the gamma distribution, which uses the Wilson–Hilferty approximation. The required tolerance factors for the proposed interval are readily obtained using existing tables and software and therefore can be implemented more easily in practice. The performance of the proposed tolerance interval is compared with that of the exact interval in terms of accuracy and robustness in simulation studies. In addition, the tolerance intervals are illustrated with a dataset from a real application. A summary and some conclusions are offered. It is seen that the proposed approximate tolerance intervals are fairly accurate, reasonably robust and being much simpler to calculate, can be useful in practical applications.

Tolerance limits and tolerance intervals for ratios of normal random variables using a bootstrap calibration

This paper addresses the problem of deriving one-sided tolerance limits and two-sided tolerance intervals for a ratio of two random variables that follow a bivariate normal distribution, or a lognormal/normal distribution. The methodology that is developed uses nonparametric tolerance limits based on a parametric bootstrap sample, coupled with a bootstrap calibration in order to improve accuracy. The methodology is also adopted for computing confidence limits for the median of the ratio random variable. Numerical results are reported to demonstrate the accuracy of the proposed approach. The methodology is illustrated using examples where ratio random variables are of interest: an example on the radioactivity count in reverse transcriptase assays and an example from the area of cost-effectiveness analysis in health economics.

Two-sided tolerance intervals for members of the (log)-location-scale family of distributions

In this paper, we propose methods to calculate exact factors for two-sided control-the-centre and control-both-tails tolerance intervals for the (log)-location-scale family of distributions, based on complete or Type II censored data. With Type I censored data, exact factors do not exist. For this case, we developed an algorithm to compute approximate factors. Our approaches are based on Monte Carlo simulations. We also provide algorithms for computing TIs that control the probability in both tails of a distribution. A simulation study for Type I censored data shows that the estimated coverage probability is close to the nominal confidence level when the expected number of uncensored observations is moderate to large. We illustrate the methods with applications using different combinations of distributions and types of censoring.

Approximate tolerance intervals for the discrete Poisson–Lindley distribution

The Poisson–Lindley distribution is a compound discrete distribution that can be used as an alternative to other discrete distributions, like the negative binomial. This paper develops approximate one-sided and equal-tailed two-sided tolerance intervals for the Poisson–Lindley distribution. Practical applications of the Poisson–Lindley distribution frequently involve large samples, thus we utilize large-sample Wald confidence intervals in the construction of our tolerance intervals. A coverage study is presented to demonstrate the efficacy of the proposed tolerance intervals. The tolerance intervals are also demonstrated using two real data sets. The R code developed for our discussion is briefly highlighted and included in the tolerance package.

Exact parametric confidence intervals for Bland-Altman limits of agreement.

The previous literature on Bland-Altman analysis only describes approximate methods for calculating confidence intervals for 95% limits of agreement (LoAs). This article describes exact methods for calculating such confidence intervals based on the assumption that differences in measurement pairs are normally distributed. Two basic situations are considered for calculating LoA confidence intervals: the first, where LoAs are considered individually (i.e., using one-sided tolerance factors for a normal distribution); and the second, where LoAs are considered as a pair (i.e., using two-sided tolerance factors for a normal distribution). Equations underlying the calculation of exact confidence limits are briefly outlined. To assist in determining confidence intervals for LoAs (considered individually and as a pair), tables of coefficients have been included for degrees of freedom between 1 and 1000. Numerical examples, showing the use of the tables for calculating confidence limits for Bland-Altman LoAs, have been provided. Exact confidence intervals for LoAs can differ considerably from the Bland and Altman approximate method, especially for sample sizes that are not large. There are better, more precise methods for calculating confidence intervals for LoAs than the Bland and Altman approximate method, although even an approximate calculation of confidence intervals for LoAs is likely to be better than none at all. Reporting confidence limits for LoAs considered as a pair is appropriate for most situations; however, there may be circumstances where it is appropriate to report confidence limits for LoAs considered individually.

Sample Size Determination for a Two-Sided Multiple Components Tolerance Interval

In the conventional concept, the variance of a tolerance interval from the measurements is a single component, and the sample size for quality control process was estimated by the variance of a single component. However, we can find examples in recent about several components that could vary in their measurements, so an approximate method must be found to modify the conventional tolerance interval. In our paper, we develop an approach to calculate the sample size for a two-sided tolerance interval including several components in the variance from the measurements. An example is presented to illustrate our proposed method.

Equivalence Assessment for Interchangeability Based on Two-Sided Tests

Interchangeability was originally developed in order to assess drug bioequivalence beyond average bioequivalence. In 2003, the Food and Drug Administration (FDA) published a Guidance documenting the procedures on using in vivo bioequivalence crossover trial to assess interchangeability between test and reference products. In general, this FDA Guidance describes interchangeability in terms of population and individual bioequivalence. The Guidance procedures were criticized for their lack of sampling distribution of the test statistics. As a result, the critical points were generated from simulation studies without adjusting for sample size. Further more, they lack consistency with average bioequivalence required in the 1992 FDA Guidance. Alternative interchangeability or interchangeability procedures were proposed to measure the probability of individual response difference under two treatments within prespecified lower and upper limits. Interchangeability is claimed if this probability is greater than a prespecified threshold. Tse et al. (2006) proposed an approximate distribution of the estimated probability based on the second-order Taylor expansion. For the same interchangeability probability hypothesis, Liu and Chow (1997) and Tsong and Shen (2007) also proposed a tolerance interval-based approach that can be extended to clinical trials with parallel arm design under the normality assumption. In this article, we first generalized the two-sided tolerance interval based interchangeability without equal sample size and variance assumption. We also derived a power function for the proposed method, and performed simulation studies to compare the type I error rate, power, and sample size between the Tse approximated test and the generalized tolerance interval approach for interchangeability assessment.

Using Tolerance Intervals for Assessment of Pharmaceutical Quality

In quality control of drug products, tolerance intervals are commonly used methods to assure a certain proportion of the products covered within a pre-specified acceptance interval. Depending on the nature of the quality attributes, the corresponding acceptance interval could be one-sided or two-sided. Thus, the tolerance intervals can also be one-sided or two-sided. To better utilize tolerance intervals for quality assurance, we reviewed the computation method and studied their statistical properties in terms of batch acceptance probability in this article. We also illustrate the application of one-sided and two-sided tolerance, as well as two one-sided tests through the examples of dose content uniformity test, delivered dose uniformity test, and dissolution test.

Tolerance Intervals for Hypergeometric and Negative Hypergeometric Variables

Tolerance intervals for discrete variables are widely used, especially in industrial applications. However, there is no thorough treatment of tolerance intervals when sampling without replacement. This paper proposes methods for constructing one-sided tolerance limits and two-sided tolerance intervals for hypergeometric and negative hypergeometric variables. Equal-tailed tolerance intervals (i.e., tolerance intervals that control the percentages in both tails) are studied followed by a small adjustment to the nominal coverage level to obtain tolerance intervals that control a specified inner percentage of the sampled distribution. The tolerance interval calculations implicitly use confidence bounds for M, the unknown number of elements possessing a certain attribute in the finite population of size N. Three different methods for obtaining such confidence bounds are suggested: a large sample approach, an approach with a continuity correction, and an exact method based on nonrandomization. The intervals are examined for desirable coverage probabilities and expected widths. The methods are also illustrated using some examples.

Statistical Properties of Large Sample Tests for Dose Content Uniformity.

The European Union (EU) test for uniformity of dosage units using large sample sizes was published in European Pharmacopoeia 7.7 in 2012. There are 2 alternative tests. Option 1 is a parametric two-sided tolerance interval-based method modified with an indifference zone and counting units outside of (0.75 M, 1.25 M) (here, M is defined by sample mean, [Formula: see text], as M = 98.5% if [Formula: see text] < 98.5%, M = 101.5% if [Formula: see text] > 101.5%, and M = [Formula: see text] otherwise). Option 2 is a nonparametric counting method with an additional indifference-zone concept. The authors extended the parametric two one-sided tolerance interval-based method that was proposed for dose content uniformity testing based on 30 tablets to large sample sizes with the restriction that all operating characteristic curves of the two one-sided tolerance intervals for any given sample size intersect with the operating characteristic curve of the US Pharmacopoeia harmonized method for a sample size of 30 at the acceptance probability of 90% when the individual tablets with on-target mean are assumed to be normally distributed. This paper studies the acceptance probabilities in relation to the batch mean and batch standard deviation among the 2 EU options and the authors' proposed method. The acceptance probabilities of EU options 1 and 2 and the proposed method were compared using simulation; results revealed that both EU options 1 and 2 produce larger acceptance probabilities when the batch mean is off-target. Furthermore, for a given standard deviation, the acceptance probability of EU option 2 at a mean 102% of the label claim is larger than that at a mean of 100% of the label claim under the normality assumption.

Sample size determination for quality control process using a multiple components tolerance interval

ABSTRACTIn the past, a tolerance interval was used for the statistical quality control process on raw materials and/or the final product. In the traditional concept of the tolerance interval, the variance from the measurements is a single component. However, we can find examples about several components that could vary in their measurements, so an approximate method must be found to modify the traditional tolerance interval. Now we employ a tolerance interval considering multiple components in the variance from the measurements to deal with quality control process. In our paper, the proposed method is used to solve the sample size determination for a two-sided tolerance interval approach considering multiple components on the variance of measurements.

On the Exact Two-Sided Tolerance Intervals for Univariate Normal Distribution and Linear Regression

Statistical tolerance intervals are another tool for making statistical inference on anunknown population. The tolerance interval is an interval estimator based on the resultsof a calibration experiment, which can be asserted with stated confidence level 1 ? ,for example 0.95, to contain at least a specified proportion 1 ? , for example 0.99, ofthe items in the population under consideration. Typically, the limits of the toleranceintervals functionally depend on the tolerance factors. In contrast to other statisticalintervals commonly used for statistical inference, the tolerance intervals are used relativelyrarely. One reason is that the theoretical concept and computational complexity of thetolerance intervals is significantly more difficult than that of the standard confidence andprediction intervals.In this paper we present a brief overview of the theoretical background and approachesfor computing the tolerance factors based on samples from one or several univariate normal(Gaussian) populations, as well as the tolerance factors for the non-simultaneousand simultaneous two-sided tolerance intervals for univariate linear regression. Such toleranceintervals are well motivated by their applicability in the multiple-use calibrationproblem and in construction of the calibration confidence intervals. For illustration, wepresent examples of computing selected tolerance factors by the implemented algorithmin MATLAB.