Parametric Tolerance Interval Research Articles

Methodology for Reliable Assessment of Water Quality. V. Statistical Methods for Water Quality Research: Parametric Approach

The possibility of using a parametric tolerance interval to improve the accuracy of studying water quality indicators was analyzed. It was demonstrated that the proposed approach provides advantages over non-parametric interval rank, but not in all cases. The advantages of the new method due to the use of more complete experimental information were substantiated, and the reasons for the decrease in the accuracy of water quality assessment at its application were analysed.

Improving coverage probabilities for parametric tolerance intervals via bootstrap calibration.

Statistical tolerance intervals are commonly employed in biomedical and pharmaceutical research, such as in lifetime analysis, the assessment of biosimilarity of branded and generic versions of biopharmaceutical drugs, and in quality control of drug products to ensure that a specified proportion of the products are covered within established acceptance limits. Exact two-sided parametric tolerance intervals are only available for the normal distribution, while exact one-sided parametric tolerance limits are available for a limited number of distributions. Approximations to two-sided parametric tolerance intervals often use the Bonferroni correction on the one-sided tolerance interval calculation; however, this often incurs a higher coverage probability than the nominal level. Recently, the usage of a bootstrap calibration has been demonstrated as a way to improve coverage probabilities of tolerance intervals for very specific and complex distributional settings. We present a focused treatment on using a single-layer bootstrap calibration to improve the coverage probabilities of two-sided parametric tolerance intervals. Simulation results clearly demonstrate the improved coverage probabilities towards the nominal level over the uncalibrated setting. Applications to medical data for various parametric distributions also highlight the utility of constructing these calibrated tolerance intervals.

Efficient Computing for One and Two Variance Components Parametric Tolerance Interval Testing

ABSTRACTIn biopharmaceutical settings, testing of content uniformity measures the amount of active ingredient in solid dosage units of a manufactured batch. Though content uniformity has historically been evaluated via USP⟨905⟩, the two one-sided parametric tolerance interval test (PTI-TOST) has emerged as a viable alternative. In this work, sufficient statistics for the PTI-TOST are developed for two-tiered testing and are shown to dramatically speed up the computer time spent generating operating characteristic curves. While the original PTI-TOST assumes that the data are independent and identically distributed Gaussian random variables, a two-variance components PTI-TOST has been established for the situation in which solid dosage units are sampled in strata (location). Sufficient statistics were also developed for the two-tier, two-variance components PTI-TOST. Computer simulations evaluated the computational time savings from using the efficient algorithms suggested and discussed in this work.

The rat bone marrow micronucleus test: Statistical considerations on historical negative control data

Recent updates of the OECD Guidelines for the Testing of Chemicals (Section 4: Health Effects) on genotoxicity testing emphasize the use of appropriate statistical methods for data analysis and proficiency proof. Updates also concern the mammalian erythrocyte micronucleus test (OECD 474), as the currently most often performed regulatory in vivo test. As the updated guideline gives high importance to adequate statistical assessment of historical negative control data to estimate validity of experiments and judge results, the present study evaluated statistical methodologies for handling of historical negative control data sets, and comes forward with respective proposals and reference data. Therefore, the working group “Statistics” within the German-speaking “Gesellschaft für Umwelt-Mutationsforschung e.V.” (GUM) compiled a data set of 891 negative control rats from valid OECD 474-studies of four laboratories. Based on these data, Analysis-of-Variance (ANOVA) identified “laboratory” and “strain”, but not “gender” as relevant stratification parameters, and argued for approximately normally distributed micronucleus frequencies in polychromatic erythrocytes per animal. This assumption provided the basis for further specifying one-sided parametric tolerance intervals for determination of corresponding upper historical negative control limits. Finally, the stability of such limits was investigated as a function of the number of experiments performed, using a simulation-based statistical strategy.

Real time release testing of tablet content and content uniformity

A comprehensive commercial control strategy for tablet content and content uniformity focussed on the unit operation of compression is presented and is proposed to enable real time release for these critical quality attributes. The control strategy is based on process understanding, process control through compaction force weight control on the tablet press, periodic checks of mean and individual tablet weight combined with at-line testing of tablet content by near infrared (NIR). The application of the at-line NIR tablet content method is discussed and an acceptance criteria based on a parametric tolerance interval test (PTIT) is proposed. Sample handling limitations and spectral acquisition time for the NIR content method limit the sample size, however the chosen PTIT assures an appropriate level of batch coverage. Data are presented for ten commercial-scale batches that demonstrates the control strategy delivered the quality standard for content and content uniformity.

Content uniformity testing for stratified samples via parametric tolerance interval testing

ABSTRACTHistorically in the biopharmaceutical setting, USP<905> has been used to establish that a batch of drug product has acceptable content uniformity. More recently, alternative approaches such as the two one-sided parametric tolerance interval test (PTI-TOST) have been proposed to establish content uniformity. Traditionally, the PTI-TOST is implemented as a sequential, two-tiered test, under the generally accepted assumption that the data are independently and identically distributed. Since the material is sequenced through the manufacturing process over a period of time, there are conceptually arguable locations within each batch, for instance: beginning, middle, and end. In such a situation, a practitioner may wish to evaluate potential effects of these batch locations, for example, during process validation. If location (stratified) differences exist within the batch and if multiple samples are taken from each location, significant within-location correlations may be induced in the data. In such a case, the traditional PTI-TOST underestimates the total variability, thereby improperly boosting the power of the test method. When there is reason to believe that location variances exist, the batch may be evaluated using stratified sampling, and the location effect may be modeled. In this paper, a two-tiered PTI-TOST that accounts for both between-location and within-location variance components is introduced. Operating characteristic curves and practical advice are given to aid the practitioner’s uptake of the proposed method.

Sample size determination strategies for normal tolerance intervals using historical data

ABSTRACTStatistical tolerance intervals are often used during design verification or process validation in diverse applications, such as the manufacturing of medical devices, the construction of nuclear reactors, and the development of protective armor for the military. Like other statistical problems, the determination of a minimum required sample size when using tolerance intervals commonly arises. Under the Faulkenberry-Weeks approach for sample size determination of parametric tolerance intervals, the user must specify two quantities—typically set to rule-of-thumb values—that characterize the desired precision of the tolerance interval. Practical applications of sample size determination for tolerance intervals often have historical data that one expects to closely follow the distribution of the future data to be collected. Moreover, such data are typically required to meet specification limits. We provide a strategy for specifying the precision quantities in the Faulkenberry-Weeks approach that utilizes both historical data and the required specification limits. Our strategy is motivated by a sampling plan problem for the manufacturing of a certain medical device that requires calculation of normal tolerance intervals. Both classical and Bayesian normal tolerance intervals are considered. Additional numerical studies are provided to demonstrate the general applicability of our strategy for setting the precision quantities.

Choice of Measurement Scale for Evaluating Compliance of the Vertical Landing Velocity of Aircraft with Established Specifications

A scheme is proposed for correct analysis of the results of measurements in the concept of acceptable risk. The three basic methods for estimating the probability of finding a measured characteristic (nonparametric and parametric tolerance intervals and separate confirmation of specifications for the mathematical expectation and dispersion) within tolerance for different scales are compared for the example of measuring the accuracy characteristics of automatic aircraft landing systems. Recommendations are given for the use of the proposed methods and for the required number of measurements.

Model-Based Tolerance Intervals Derived from Cumulative Historical Composition Data: Application for Substantial Equivalence Assessment of a Genetically Modified Crop

Compositional analysis is a requisite component of the substantial equivalence framework utilized to assess genetically modified (GM) crop safety. Statistical differences in composition data between GM and non-GM crops require a context in which to determine biological relevance. This context is provided by surveying the natural variation of key nutrient and antinutrient levels within the crop population with a history of safe use. Data accumulated from various genotypes with a history of safe use cultivated in relevant commercial crop-growing environments over multiple seasons are discussed as the appropriate data representative of this natural variation. A model-based parametric tolerance interval approach, which accounts for the correlated and unbalanced data structure of cumulative historical data collected from multisite field studies conducted over multiple seasons, is presented. This paper promotes the application of this tolerance interval approach to generate reference ranges for evaluation of the biological relevance of statistical differences identified during substantial equivalence assessment of a GM crop.

A Generalized Pivotal Quantity Approach to the Parametric Tolerance Interval Test for Dose Content Uniformity Batch Testing

The Food and Drug Administration (FDA) has proposed a parametric tolerance interval test (PTIT) for batch-release testing of inhalation devices. The proposed test examines dose uniformity based on several inhalation units from a batch, with two observations per unit. An underlying assumption is that the observations are a random sample from a univariate normal distribution. Because there are two observations per unit, it may be more appropriate to model the data as stemming from a bivariate normal distribution. We take a bivariate approach and use generalized confidence interval methodology to derive a parametric tolerance interval for the distribution of doses within a batch. We then use Monte Carlo simulation to compare results based on this bivariate approach with those based on the FDA-proposed PTIT.

Challenges and Opportunities in Implementing the FDA Default Parametric Tolerance Interval Two One-Sided Test for Delivered Dose Uniformity of Orally Inhaled Products

The goal of this article is to discuss considerations regarding implementation of the parametric tolerance interval two one-sided test (PTI-TOST) for delivered dose uniformity (DDU) of orally inhaled products (OIPs). That test was proposed by FDA in 2005 as an alternative to the counting test described in the 1998 draft FDA guidance for metered dose inhalers and dry powder inhalers. The 2005 PTI-TOST, however, still has not found much use in practice despite the general desirability of parametric approaches in modern pharmaceutical quality control. A key reason for its slow uptake is that it rejects, with high probability, batches whose quality is considered acceptable by all other published regulatory and pharmacopeial standards as well as by the DDU specifications for many approved OIPs. Manufacturers therefore continue using nonparametric counting tests for control of DDU. A simulated case study presented here compares the consequences of the PTI-TOST compared to the counting test. The article discusses three possibilities that would help increase the uptake of the PTI-TOST approach, namely: product-specific quality standards, a different default standard suitable for the majority of OIPs, and integration of the PTI-TOST with a continuous verification control strategy rather than using it as an isolated-batch (transactional) end-product testing. In any of these efforts, if a parametric test is used, it is critical not to set the target quality close to, or at the boundary of the process/product capabilities, because PTI tests are designed to reject with high probability the identified target quality.

A Two One-Sided Parametric Tolerance Interval Test for Control of Delivered Dose Uniformity. Part 1—Characterization of FDA Proposed Test

The FDA proposed a parametric tolerance interval (PTI) test at the October 2005 Advisory Committee meeting as a replacement of the attribute (counting) test for delivered dose uniformity (DDU), published in the 1998 draft guidance for metered dose inhalers (MDIs) and dry powder inhalers (DPIs) and the 2002 final guidance for inhalation sprays and intranasal products. This article (first in a series of three) focuses on the test named by the FDA "87.5% coverage." Unlike a typical two-sided PTI test, which controls the proportion of the DDU distribution within a target interval (coverage), this test is comprised of two one-sided tests (TOST) designed to control the maximum amount of DDU values in either tail of the distribution above and below the target interval. Through simulations, this article characterizes the properties and performance of the proposed PTI-TOST under different scenarios. The results show that coverages of 99% or greater are needed for a batch to have acceptance probability 98% or greater with the test named by the FDA "87.5% coverage" (95% confidence level), while batches with 87.5% coverage have less than 1% probability of being accepted. The results also illustrate that with this PTI-TOST, the coverage requirement for a given acceptance probability increases as the batch mean deviates from target. The accompanying articles study the effects of changing test parameters and the test robustness to deviations from normality.

A Two One-Sided Parametric Tolerance Interval Test for Control of Delivered Dose Uniformity—Part 3—Investigation of Robustness to Deviations from Normality

The robustness of the parametric tolerance interval test, which was proposed by the Food and Drug Administration for control of delivered dose uniformity in orally inhaled and nasal drug products, is investigated in this article using different scenarios for deviations from a univariate normal distribution. The studied scenarios span a wide range of conditions, the purpose of which is to provide an understanding of how the test performs depending on the nature and degree of the deviation from normality. Operating characteristic curves were generated to compare the performance of the test for different types of distributions (normal and non-normal) having the same proportion of doses in the tails (on one or both sides) outside the target interval. The results show that, in most cases, non-normality does not increase the probability of accepting a batch of unacceptable quality (i.e., the test is robust) except in extreme situations, which do not necessarily represent commercially viable products. The results also demonstrate that, in the case of bimodal distributions where the life-stage means differ from each other by up to 24% label claim, the test's criterion on life-stage means does not affect pass rates because the tolerance interval portion of the test reacts to shifting means as well.

Parametric Two-Stage Sequential Quality Assurance Test of Dose Content Uniformity

The United States Pharmacopeia (USP) content uniformity sampling acceptance plan consisting of a two-stage sampling plan with criteria on sample mean and number of out-of-range tablets is the standard for compendium. It is, however, often used mistakenly for lot quality assurance. In comparison to the Japan Phamacopeia (JP) procedure, USP procedure is less discriminative between lots with on-target mean and small variance and lots with off-target mean and large variance. The new European Pharmacopeia (EP) and USP harmonized test adopted a tolerance interval approach. But the “no-difference zone” criteria modification for off-target products make the approaches biased in favor of off-target products. We propose a parametric tolerance interval procedure to test a two-sided specification that is equivalent to the test of two one-sided hypotheses. Testing against a lower specification is to assure that the drug product is not under-dosed for the sake of efficacy. On the other hand, testing against an upper specification is to assure that the drug product is not over-dosed for the sake of safety. The operating curves of the proposed procedure are compared with those of the USP test to illustrate the difference in acceptance probability against the mean and variance of the lot.

Oral Dosage Form Performance Tests: New Dissolution Approaches

The performance test is one of a series of tests that compose the specification in a United States Pharmacopeia (USP) dosage form monograph. For an orally administered, nonsolution dosage form, it is usually satisfied by either a dissolution or disintegration procedure. Dissolution acceptance criteria are usually set in private negotiations between an applicant and a regulatory agency. With information about this private agreement and other information provided in a sponsor's Request for Revision to USP, the USP's Council of Experts elaborates a public dosage form monograph. Based on the relationship between the regulatory decisions and the Request for Revision, the USP dissolution procedure links to a regulatory judgment about bioavailability and bioequivalence and, ultimately, to a judgment about safety and efficacy. The current dissolution procedure and acceptance criteria are perceived as having worked well over the years and are generally accepted. This article discusses new approaches that merit consideration. These approaches focus on a) explicit use of hypothesis testing, b) use of parametric tolerance intervals, c) improved ways to set dissolution acceptance criteria, and d) a more flexible protocol to assess conformity. Application of the proposed approaches may better assess, manage, and communicate both manufacturer and consumer risk for dissolution testing.

Extension to the use of tolerance intervals for the assessment of individual bioequivalence

For the determination of bioequivalence, researchers have recently shifted their emphasis from average bioequivalence alone to average and individual bioequivalence. Existing methods for assessing average bioequivalence were first developed for the standard 2 x 2 crossover design, but these methods are easily generalized to the two-treatment, p-period crossover designs (e.g., TRR, RTT, and TTRR, RRTT, TRRT, RTTR). With respect to individual bioequivalence, Westlake (1,2) implemented the use of parametric and distribution-free tolerance intervals for assessing individual bioequivalence. Anderson and Hauck (3) described what they call the test of individual equivalence ratios (TIER) for the same purpose. Note that these methods have been applied and/or developed only for the standard 2 x 2 crossover design. The present work extends the method of using parametric tolerance intervals for assessing individual bioequivalence.