Two-sided Tolerance Intervals Research Articles

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.

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.

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.

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.

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.

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.

Improved nonparametric tolerance intervals based on interpolated and extrapolated order statistics

The standard approach to construct nonparametric tolerance intervals is to use the appropriate order statistics, provided a minimum sample size requirement is met. However, it is well-known that this traditional approach is conservative with respect to the nominal level. One way to improve the coverage probabilities is to use interpolation. However, the extension to the case of two-sided tolerance intervals, as well as for the case when the minimum sample size requirement is not met, have not been studied. In this paper, an approach using linear interpolation is proposed for improving coverage probabilities for the two-sided setting. In the case when the minimum sample size requirement is not met, coverage probabilities are shown to improve by using linear extrapolation. A discussion about the effect on coverage probabilities and expected lengths when transforming the data is also presented. The applicability of this approach is demonstrated using three real data sets.

A Monte Carlo simulation study of two-sided tolerance intervals in balanced one-way random effects model for non-normal errors

Recently, Ong and Mukerjee [Probability matching priors for two-sided tolerance intervals in balanced one-way and two-way nested random effects models. Statistics. 2011;45:403–411] developed two-sided Bayesian tolerance intervals, with approximate frequentist validity, for a future observation in balanced one-way and two-way nested random effects models. These were obtained using probability matching priors (PMP). On the other hand, Krishnamoorthy and Lian [Closed-form approximate tolerance intervals for some general linear models and comparison studies. J Stat Comput Simul. 2012;82:547–563] studied closed-form approximate tolerance intervals by the modified large-sample (MLS) approach. We compare the performances of these two approaches for normal as well as non-normal error distributions. Monte Carlo simulation methods are used to evaluate the resulting tolerance intervals with regard to achieved confidence levels and expected widths. It turns out that PMP tolerance intervals are less conservative for data with large number of classes and small number of observations per class and the MLS procedure is preferable for smaller sample sizes.

Simultaneous Two-Sided Tolerance Intervals for a Univariate Linear Regression Model

In this article we deal with simultaneous two-sided tolerance intervals for a univariate linear regression model with independent normally distributed errors. We present a method for determining the intervals derived by the general confidence-set approach (GCSA), i.e. the intervals are constructed based on a specified confidence set for unknown parameters of the model. The confidence set used in the new method is formed based on a suggested hypothesis test about all parameters of the model. The simultaneous two-sided tolerance intervals determined by the presented method are found to be efficient and fast to compute based on a preliminary numerical comparison of all the existing methods based on GCSA.

Two-sided Bayesian and frequentist tolerance intervals: general asymptotic results with applications

It is well known that that the construction of two-sided tolerance intervals is far more challenging than that of their one-sided counterparts. In a general framework of parametric models, we derive asymptotic results leading to explicit formulae for two-sided Bayesian and frequentist tolerance intervals. In the process, probability matching priors for such intervals are characterized and their role in finding frequentist tolerance intervals via a Bayesian route is indicated. Furthermore, in situations where matching priors are hard to obtain, we develop purely frequentist tolerance intervals as well. The findings are applied to real data. Simulation studies are seen to lend support to the asymptotic results in finite samples.

One-Sided and Two-Sided Tolerance Intervals in General Mixed and Random Effects Models Using Small-Sample Asymptotics

The computation of tolerance intervals in mixed and random effects models has not been satisfactorily addressed in a general setting when the data are unbalanced and/or when covariates are present. This article derives satisfactory one-sided and two-sided tolerance intervals in such a general scenario, by applying small-sample asymptotic procedures. In the case of one-sided tolerance limits, the problem reduces to the interval estimation of a percentile, and accurate confidence limits are derived using small-sample asymptotics. In the case of a two-sided tolerance interval, the problem does not reduce to an interval estimation problem; however, it is possible to derive an approximate margin of error statistic that is an upper confidence limit for a linear combination of the variance components. For the latter problem, small-sample asymptotic procedures can once again be used to arrive at an accurate upper confidence limit. In the article, balanced and unbalanced data situations are treated separately, and computational issues are addressed in detail. Extensive numerical results show that the tolerance intervals derived based on small-sample asymptotics exhibit satisfactory performance regardless of the sample size. The results are illustrated using some examples. Some technical derivations, additional simulation results, and R codes are available online as supplementary materials.

Tolerance intervals for symmetric location-scale families based on uncensored or censored samples

Exact methods for constructing two-sided tolerance intervals (TIs) and tolerance intervals that control percentages in both tails for a location-scale family of distributions are proposed. The proposed methods are illustrated by constructing TIs for a normal, logistic, and Laplace (double exponential) distributions based on type II singly censored samples. Factors for constructing one-sided and two-sided TIs for a logistic distribution are tabulated for the case of uncensored samples. Factors for constructing TIs based on censored samples for all three distributions are also tabulated. The factors for all cases are estimated by Monte Carlo simulation. An adjustment to the tolerance factors based on type II censored samples is proposed so that they can be used to find approximate TIs based on type I censored samples. Coverage studies of the approximate TIs based on type I censored samples indicate that the approximation is satisfactory as long as the proportion of censored observations is no more than 0.70. The methods are illustrated using some practical examples.

Probability matching priors for two-sided tolerance intervals in balanced one-way and two-way nested random effects models

We consider two-sided Bayesian tolerance intervals, with approximate frequentist validity, for a future observation in balanced one-way and two-way nested random effects models. Probability matching conditions, specific to this problem, are derived in either case via a technique that involves inversion of approximate posterior characteristic functions. In addition to yielding probability matching priors for the present problem, these conditions are useful in evaluating certain other priors that have received attention in the literature.