Tolerance Interval Approach Research Articles

Stability analysis using mixed models: A critique of tolerance interval methods and a probabilistic solution.

Recently, tolerance interval approaches to the calculation of a shelf life of a drug product have been proposed in the literature. These address the belief that shelf life should be related to control of a certain proportion of batches being out of specification. We question the appropriateness of the tolerance interval approach. Our concerns relate to the computational challenges and practical interpretations of the method. We provide an alternative Bayesian approach, which directly controls the desired proportion of batches falling out of specification assuming a controlled manufacturing process. The approach has an intuitive interpretation and posterior distributions are straightforward to compute. If prior information on the fixed and random parameters is available, a Bayesian approach can provide additional benefits both to the company and the consumer. It also avoids many of the computational challenges with the tolerance interval methodology.

Tolerance Interval Approach for the Determination of Overfill of Liquid Parenteral Drug Products.

Liquid parenteral products contain an overfill to ensure withdrawal of the declared volume. The overfill must be sufficiently high to compensate for the expected loss during product preparation and administration, but it should also be minimized to prevent accidental overdosing and unforeseen dose splitting of single-dose products. Finding the right balance between too much and too little overfill with an acceptable risk of product failure is challenging and requires consideration of the relevant sources of variability of the extractable volume. This article provides a novel approach for the calculation of the required overfill based on tolerance interval methodology. In a first step, a tolerance interval multiplier from the literature is proposed, and a simulation study is conducted to assess the appropriateness of its use for overfill determination. In a second step, this multiplier is adapted to cover operator-to-operator variability in the loss data and compared with other multipliers via a second simulation study. Use of a tolerance interval multiplier enables adaptation of the overfill such that the risk of not reaching the minimum extractable volume fulfills predefined acceptance criteria. By this, the scientific justification of the selected overfill is strengthened and control over a critical quality attribute is improved.

Use of a tolerance interval approach as a statistical quality control tool for traditional Chinese medicine

ABSTRACT Raw materials for traditional Chinese medicine (TCM) are often from different resources and its final product may also be made by different sites. Therefore, variabilities from different resources such as site-to-site or within site component-to-component may be expected. Consequently, test for consistency in raw materials, in-process materials, and/or final product has become an important issue in the quality control (QC) process in TCM development. In this paper, a statistical QC process for raw materials and/or the final product of TCM is proposed based on a two sided –content, –confidence tolerance interval. More specifically, we construct the tolerance interval for a random-effects model to assess the QC of TCM products from different regions and possibly different product batches. The products can be claimed to be consistency when the constructed tolerance interval is within the permitted range. Given the region and batch effects, sample sizes can also be calculated to ensure the desired measure of goodness. An example is presented to illustrate the proposed approach.

Process capability indices when two sources of variability present, a tolerance interval approach

AbstractThe sound tolerance interval–based method and two Pp–based approximations are compared on the proportion of nonconforming parts. As output distribution of the process, one possible model is examined here: two sources of variations are in a one‐way structure. It was found that the uncertainty of variance components estimates plays the major role in the goodness of the three calculation methods.SummaryStatistical indices – like process capability (CP) or process performance (PP) index – make the relationship between the width of the specification interval and the extent of the process variability illustrative. The latter is characterized by the tolerance interval, which contains the major part of the population with high confidence. In the original concept this tolerance interval is calculated using oversimplified models. In practice this model is not conform with reality. As improvement two sources of variation is assumed in a one‐way structure.The proportion of non‐conforming parts of the population is the quantity of interest. Non‐conforming means that the characteristic is beyond the specification limits. According to this, the quantile of the distribution shall be determined that is equal to the specification limits. Thus, the task is to calculate the tolerance interval for the distribution.In practical cases the variance components are unknown and are to be estimated. To estimate the ratio of non‐conforming parts, two approximate calculation methods which are coherent with the definition of PP are investigated, as well.The aim of this work is to compare the results of the two PP based approximate methods with the tolerance interval based (theoretically sound) calculation method. Two situations from the practice are considered – datasets from process validation and process monitoring environment are evaluated during which the effect of the number of experiments is given. Our study indicated that the main difference of the goodness of the approximations is from the uncertainty of parameter estimates.

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.

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.

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.

Flexibility and applicability of β‐expectation tolerance interval approach to assess the fitness of purpose of pharmaceutical analytical methods

An innovative versatile strategy using Total Error has been proposed to decide about the method's validity that controls the risk of accepting an unsuitable assay together with the ability to predict the reliability of future results. This strategy is based on the simultaneous combination of systematic (bias) and random (imprecision) error of analytical methods. Using validation standards, both types of error are combined through the use of a prediction interval or β-expectation tolerance interval. Finally, an accuracy profile is built by connecting, on one hand all the upper tolerance limits, and on the other hand all the lower tolerance limits. This profile combined with pre-specified acceptance limits allows the evaluation of the validity of any quantitative analytical method and thus their fitness for their intended purpose. In this work, the approach of accuracy profile was evaluated on several types of analytical methods encountered in the pharmaceutical industrial field and also covering different pharmaceutical matrices. The four studied examples depicted the flexibility and applicability of this approach for different matrices ranging from tablets to syrups, different techniques such as liquid chromatography, or UV spectrophotometry, and for different categories of assays commonly encountered in the pharmaceutical industry i.e. content assays, dissolution assays, and quantitative impurity assays. The accuracy profile approach assesses the fitness of purpose of these methods for their future routine application. It also allows the selection of the most suitable calibration curve, the adequate evaluation of a potential matrix effect and propose efficient solution and the correct definition of the limits of quantification of the studied analytical procedures.

Application of the β-expectation tolerance interval to method validation of the M30 and M65 ELISA cell death biomarker assays

Method validation should focus on demonstrating that an assay is fit for its intended purpose. We have applied the β-expectation tolerance interval – a statistical approach that predicts the accuracy of assay measurements in the future – to the validation of two different cell death biomarker assays, the M30 and M65 ELISAs. A meta-analysis was conducted on a total of 57 different M30 and M65 assays run over a 2 year period. All code utilised in calculations was developed using MATLAB. The optimal fit to the calibration curve for the M30 assay was shown to be a quartic curve which yielded a β-expectation tolerance interval of +20.5% and −23.6% at β = 95% over a wide range of QC standards (88–810 U/L). However, such a fit required at least 7 points to avoid problems with over fitting. A linear fit to the M65 calibration curve normally produced a tolerance interval of less than ±20%, however, marked inter-batch variations were evident. Amelioration of batch to batch variations was accomplished by fitting M65 calibration data preferably to a 4-parameter logistic function or a cubic spline. The minimum number of QC replicates and different assays required to produce reliable accuracy profiles was determined. The β-expectation tolerance interval approach has resulted in further optimisation of the M30 and M65 ELISAs as biomarker assays that should translate into greater accuracy in results generated from clinical trials samples.

Novel tolerance interval model for the estimation of the shelf life of pharmaceutical products

AbstractThe main objective of the stability study of pharmaceutical products is to estimate the shelf life indicated on the container label. For the determination of the label shelf life the International Conference on Harmonisation (ICH) Guideline recommends the confidence interval approach, but its adequacy is questionable. Instead of the current regulation, several proposals are known in the literature applying the tolerance interval approach. The aim of our paper is to extend the tolerance interval model assuming two distinct influence sources, namely the tablet‐to‐tablet variability and the analytical measurement error. Another main concept in the paper is that the tolerance interval is constructed for the true (and not for the measured) assay content. Investigating the influence of several factors on the width of the tolerance interval, a simulation is performed applying different number of samples and repeated analysis, as well as different variance components regarding the tablet‐to‐tablet variability and the analytical measurement error. An example is also given to illustrate the difference between the former and the proposed tolerance interval models and to compare them with the present regulation of the ICH. Copyright © 2010 John Wiley & Sons, Ltd.

Acceptance Criteria for Method Equivalency Assessments

Quality by design (ICH-Topic Q8) requires that process control strategy requirements are met and maintained. The challenging task of setting appropriate acceptance criteria for assessment of method equivalence is a critical component of satisfying these requirements. The use of these criteria will support changes made to methods across the product lifecycle. A method equivalence assessment is required when a change is made to a method which may pose a risk to its ability to monitor the quality of the process. Establishing appropriate acceptance criteria are a vital, but not clearly understood, prerequisite to deciding the appropriate design/sample size of the equivalency study. A number of approaches are proposed in the literature for setting acceptance criteria for equivalence which address different purposes. This perspective discusses those purposes and then provides more details on setting acceptance criteria based on patient and producer risk, e.g., tolerance interval approach and the consideration of method or process capability. Applying these to a drug substance assay method for batch release illustrates that, for the equivalence assessment to be meaningful, a clear understanding and appraisal of the control requirements of the method is needed. Rather than a single exact algorithm, the analyst's judgment on a number of aspects is required in deciding the appropriate acceptance criteria.

Statistical Assessment of Analytical Method Transfer*

Analytical method transfer is an important part of analytical method development and maintenance. The current common practice of analytical method transfer is based on the equivalence of the means between the original laboratory and the receiving laboratory. However, for some assays the most scientifically sound approach would be to show the equivalency of individual sample readings between the two laboratories. In this study, statistical approaches such as limit of agreement, total deviation index, and tolerance interval approach to address individual equivalence between laboratories are discussed. Using a tolerance interval approach that is equivalent to the two one-sided hypothesis testing is proposed. These approaches are compared with each other and also with the mean equivalence approach on their statistical properties. Examples are presented to illustrate each analysis approach and the comparisons.

Parametric Two-Tier Sequential Quality Assurance Test of Delivery Dose Uniformity of Multiple-Dose Inhaler and Dry Powder Inhaler Drug Products

The delivery dose uniformity is one of the most critical requirements of dry powder inhaler and metered dose inhaler products. In 1998, the U.S. Food and Drug Administration recommended a two-tier acceptance sampling plan in the Draft Guidance of Metered Dose Inhaler and Dry Powder Inhaler Drug Products Chemistry, Manufacturing and Controls. The two-tier procedure is a modification of the United States Pharmacopeia (USP) sampling plan of dose content uniformity. It employed a zero tolerance criterion. In addition, it has a near-zero probability acceptance at the second tier. In this article, a two-tier sequential tolerance interval approach is proposed that is equivalent to a two-tier two one-sided testing procedure. It controls the probability of the product delivering below a prespecified effective dose and the probability of the product delivering over a prespecified safety dose.

A Tolerance Interval Approach for Assessment of Agreement with Left Censored Data

We describe a tolerance interval approach for assessing agreement in method comparison data that may be left censored. We model the data using a mixed model and discuss a Bayesian and a frequentist methodology for inference. A simulation study suggests that the Bayesian approach with noninformative priors provides a good alternative to the frequentist one for moderate sample sizes as the latter tends to be liberal. Both may be used for sample sizes 100 or more, with the Bayesian one being slightly conservative. The proposed methods are illustrated with real data involving comparison of two assays for quantifying viral load in HIV patients.

A tolerance interval approach for assessment of agreement in method comparison studies with repeated measurements

This paper generalizes the tolerance interval approach for assessing agreement between two methods of continuous measurement for repeated measurement data—a common scenario in applications. The repeated measurements may be longitudinal or they may be replicates of the same underlying measurement. Our approach is to first model the data using a mixed model and then construct a relevant asymptotic tolerance interval (or band) for the distribution of appropriately defined differences. We present the methodology in the general context of a mixed model that can incorporate covariates, heteroscedasticity and serial correlation in the errors. Simulation for the no-covariate case shows good small-sample performance of the proposed methodology. For the longitudinal data, we also describe an extension for the case when the observed time profiles are modelled nonparametrically through penalized splines. Two real data applications are presented.

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.

Alternative approach for establishing acceptable thresholds on macroinvertebrate community metrics

We demonstrated the use of statistical tolerance intervals as a method for deriving acceptable thresholds for benthic macroinvertebrate community metrics. Tolerance intervals are simply confidence intervals based on percentiles, and they allow selection of acceptable limits (referred to as tolerance limits) and a desired level of statistical confidence for a metric distribution (e.g., of a reference population). We used benthic macroinvertebrate community data from several long-term monitoring projects for streams on the US Department of Energy’s Oak Ridge Reservation in eastern Tennessee, USA, for the demonstration. We focused on 3 benthic macroinvertebrate community metrics: density, total taxonomic richness, and taxonomic richness of Ephemeroptera, Plecoptera, and Trichoptera (EPT) taxa. Tolerance intervals yielded less restrictive thresholds than those produced by simple percentiles because the former approach includes variation of the reference data, whereas percentiles are distribution-free. The less restrictive thresholds produced by tolerance intervals will decrease the frequency with which metric values from test sites will be classified as unacceptable because data variation is included, but thresholds calculated using the tolerance interval approach may be better suited for studies that require a greater level of statistical rigor than routine monitoring or general screening surveys (e.g., biocriteria, environmental impact assessments). Conversely, approaches that use simple percentiles may be more appropriate for screening studies. Greater accuracy of tolerance limits can be achieved by increasing sample size, reducing variation (e.g., removing outliers, data transformation), and including data that incorporate both spatial and temporal variation. However, alternative approaches should be used if the data are not normally distributed. Tolerance limits can be adjusted easily to achieve the level of environmental protection desired or required, while providing a level of statistical confidence in derived thresholds. For this reason, a tolerance interval approach should be considered seriously, particularly if the study objectives require a known level of statistical certainty.