Optimal temperature and thermal tolerance of postlarvae of the freshwater prawn Cryphiops (Cryphiops) caementarius acclimated to different temperatures.

In this study, the optimum temperature and thermal tolerance of postlarvae of the commercially important freshwater prawn Cryphiops (Cryphiops) caementarius were determined after acclimation to six different rearing temperatures (19 °C, 22 °C, 24 °C, 26 °C, 28 °C, and 30 °C) during a 45 day-culture period. Best growth parameter values were obtained within the temperature range of 24 °C to 28 °C, where the optimum temperature for growth was found to be at 26 °C (weight gain 81.70%; specific growth rate 1.33 %/day) but had not significant effect (p > 0.05) on survival (64%-71%) of postlarvae. Increasing the acclimation temperature significantly (p < 0.05) increased both the critical thermal maximum (CTMax: from 33.82 °C to 38.48 °C) and minimum (CTMin: from 9.27 °C to 14.58 °C). The thermal tolerance interval increased (p < 0.05) from 24.55 °C to 25.48 °C in postlarvae acclimated at 28 °C but decreased (p < 0.05) to 23.90 °C in those acclimated at 30 °C. The acclimation response rate was lower for CTMax and higher for CTMin. The current (12.48 °C) and future (9.48 °C) thermal safety margins were like those reported for other tropical crustaceans. A thermal tolerance polygon over the range of 19-30 °C resulted in a calculated area of 242.25 °C2. The presented results can be used for aquaculture activities and also to help to protect this species against expected climate warming impacts.

Distribution-free hyperrectangular tolerance regions for setting multivariate reference regions in laboratory medicine.

Reference regions are important in laboratory medicine to interpret the test results of patients, and usually given by tolerance regions. Tolerance regions of p ( ≥ 2 ) $$ p\;\left(\ge 2\right) $$ dimensions are highly desirable when the test results contains p $$ p $$ outcome measures. Nonparametric hyperrectangular tolerance regions are attractive in real problems due to their robustness with respect to the underlying distribution of the measurements and ease of intepretation, and methods to construct them have been recently provided by Young and Mathew [Stat Methods Med Res. 2020;29:3569-3585]. However, their validity is supported by a simulation study only. In this paper, nonparametric hyperrectangular tolerance regions are constructed by using Tukey's [Ann Math Stat. 1947;18:529-539; Ann Math Stat. 1948;19:30-39] elegant results of equivalence blocks. The validity of these new tolerance regions is proven mathematically in [Ann Math Stat. 1947;18:529-539; Ann Math Stat. 1948;19:30-39] under the only assumption that the underlying distribution of the measurements is continuous. The methodology is applied to analyze the kidney function problem considered in Young and Mathew [Stat Methods Med Res. 2020;29:3569-3585].

Frequentist and Bayesian tolerance intervals for setting specification limits for left-censored gamma distributed drug quality attributes.

Tolerance intervals from quality attribute measurements are used to establish specification limits for drug products. Some attribute measurements may be below the reporting limits, that is, left-censored data. When data has a long, right-skew tail, a gamma distribution may be applicable. This paper compares maximum likelihood estimation (MLE) and Bayesian methods to estimate shape and scale parameters of censored gamma distributions and to calculate tolerance intervals under varying sample sizes and extents of censoring. The noninformative reference prior and the maximal data information prior (MDIP) are used to compare the impact of prior choice. Metrics used are bias and root mean square error for the parameter estimation and average length and confidence coefficient for the tolerance interval evaluation. It will be shown that Bayesian method using a reference prior overall performs better than MLE for the scenarios evaluated. When sample size is small, the Bayesian method using MDIP yields conservatively too wide tolerance intervals that are unsuitable basis for specification setting. The metrics for all methods worsened with increasing extent of censoring but improved with increasing sample size, as expected. This study demonstrates that although MLE is relatively simple and available in user-friendly statistical software, it falls short in accurately and precisely producing tolerance limits that maintain the stated confidence depending on the scenario. The Bayesian method using noninformative prior, even though computationally intensive and requires considerable statistical programming, produces tolerance limits which are practically useful for specification setting. Real-world examples are provided to illustrate the findings from the simulation study.

The calculation of historical control limits in toxicology: Do's, don'ts and open issues from a statistical perspective.

For reporting toxicology studies, the presentation of historical control data and the validation of the concurrent control group with respect to historical control limits have become requirements. However, many regulatory guidelines fail to define how such limits should be calculated and what kind of target value(s) they should cover. Hence, this manuscript is aimed to give a brief review on the methods for the calculation of historical control limits that are in use as well as on their theoretical background. Furthermore, this manuscript is aimed to identify open issues for the use of historical control limits that need to be discussed by the community. It seems that, even after 40 years of discussion, more issues remain open than solved, both, with regard to the available methodology as well as its implementation in user-friendly software. Since several of these topics equally apply to several research fields, this manuscript is addressed to all relevant stakeholders who deal with historical control data obtained from toxicological studies, regardless of their background or field of research.

Quantifying conditioned place preference: a review of current analyses and a proposal for a novel approach.

Conditioned place preference (CPP) is used to measure the conditioned rewarding effects of a stimulus, including food, drugs, and social interaction. Because various analytic approaches can be used to quantify CPP, this can make direct comparisons across studies difficult. Common methods for analyzing CPP involve comparing the time spent in the CS+ compartment (e.g., compartment paired with drug) at posttest to the time spent in the CS+ compartment at pretest or to the CS- compartment (e.g., compartment paired with saline) at posttest. Researchers can analyze the time spent in the compartment(s), or they can calculate a difference score [(CS+post - CS+pre) or (CS+post - CS-post)] or a preference ratio (e.g., CS+post/(CS+post + CS-post)). While each analysis yields results that are, overall, highly correlated, there are situations in which different analyses can lead to discrepant interpretations. The current paper discusses some of the limitations associated with current analytic approaches and proposes a novel method for quantifying CPP, the adjusted CPP score, which can help resolve the limitations associated with current approaches. The adjusted CPP score is applied to both hypothetical and previously published data. Another major topic covered in this paper is methodologies for determining if individual subjects have met criteria for CPP. The paper concludes by highlighting ways in which researchers can increase transparency and replicability in CPP studies.

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.

Estimating Shelf Life Through Tolerance Intervals Extended to Nonlinear Response Trends.

Methods for estimating pharmaceutical shelf life based on tolerance intervals are proposed by Schwenke, et al. AAPS PharmSciTech. 2020;21:290, [1] where a critical quality attribute that follows a simple linear (straight line) response trend across storage time is presented as the traditional example. A random coefficient mixed linear regression model is used to characterize the between batch and within batch variation. These methods are further discussed for various stability study scenarios, number of stability batches, and levels of assumed risk in Schwenke, et al. AAPS PharmSciTech. 2021;22:273, [4] through a simulation study, again based on a critical quality attribute assuming a simple linear response. However, in practice, not all stability response profiles conveniently follow straight line or linear trends. The purpose of this paper is to extend the proposed tolerance interval and random coefficient mixed regression methods for estimating pharmaceutical shelf life to critical quality attributes that follow more complex stability response profiles. As an example, a nonlinear response is typically characterized by either an increasing or decreasing response, starting from an initial concentration, trending with storage time towards some limiting response or asymptote. Nonlinear responses cannot be statistically analyzed with linear model methods. Practical information supported by simulation results based on a pharmaceutical stability study are discussed to allow for appropriate statistical analyses and shelf life estimates through random coefficient mixed nonlinear regression and tolerance interval methods.

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.

A Practical Discussion on Estimating Shelf Life Through Tolerance Intervals.

This paper is a companion article to the research originally presented in "Estimating Shelf Life through Tolerance Intervals" (Schwenke et al., 21:290, 2020) published in AAPS PharmSciTech where tolerance intervals are introduced as an alternative methodology for estimating pharmaceutical shelf life. An industry stability shelf life example data set was used to demonstrate the proposed methods. Although using industry data does give relevance to examples demonstrating shelf life estimation, measures of how well the proposed methods accurately and effectively estimate shelf life cannot be obtained because the true shelf life values are not known for example data sets. In this current paper, the results of a computer simulation are reported where the tolerance interval estimates of shelf life are compared to theoretically known true shelf life values. Various factors that affect a tolerance interval estimate of pharmaceutical shelf life are investigated. A critical decision factor is the choice of the proportion of the stability distribution allowed out of specification at expiry to define the pharmaceutical risk. The number of stability batches available for shelf life estimation and the storage time at which the estimate is made are also considered in this simulation study. The industry example data are again used as the basis for the simulation study to give relevance to this research.

Capturing the underlying distribution in meta-analysis: Credibility and tolerance intervals.

Tolerance intervals provide a bracket intended to contain a percentage (e.g., 80%) of a population distribution given sample estimates of the mean and variance. In random-effects meta-analysis, tolerance intervals should contain researcher-specified proportions of underlying population effect sizes. Using Monte Carlo simulation, we investigated the coverage for five relevant tolerance interval estimators: the Schmidt-Hunter credibility intervals, a prediction interval, two content tolerance intervals adapted to meta-analysis, and a bootstrap tolerance interval. None of the intervals contained the desired percentage of coverage at the nominal rates in all conditions. However, the prediction worked well unless the number of primary studies was small (<30), and one of the content tolerance intervals approached nominal levels with small numbers (<20) of primary studies. The bootstrap tolerance interval achieved near nominal coverage if there were sufficient numbers of primary studies (30+) and large enough sample sizes (N ≅ 70) in the included primary studies, although it slightly exceeded nominal coverage with large numbers of large-sample primary studies. Next, we showed the results of applying the intervals to real data using a set of previously published analyses and provided suggestions for practice. Tolerance intervals incorporate error of estimation into the construction of proper brackets for fractions of population true effects. In many contexts, such intervals approach the desired nominal levels of coverage.

Reference range: Which statistical intervals to use?

Reference ranges, which are data-based intervals aiming to contain a pre-specified large proportion of the population values, are powerful tools to analyse observations in clinical laboratories. Their main point is to classify any future observations from the population which fall outside them as atypical and thus may warrant further investigation. As a reference range is constructed from a random sample from the population, the event 'a reference range contains (100 P)% of the population' is also random. Hence, all we can hope for is that such event has a large occurrence probability. In this paper we argue that some intervals, including the P prediction interval, are not suitable as reference ranges since there is a substantial probability that these intervals contain less than (100 P)% of the population, especially when the sample size is large. In contrast, a (P,γ) tolerance interval is designed to contain (100 P)% of the population with a pre-specified large confidence γ so it is eminently adequate as a reference range. An example based on real data illustrates the paper's key points.

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.

Empirical weighted Bayesian tolerance intervals.

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.

Coverage Intervals.

Since coverage intervals are widely used expressions of measurement uncertainty, this contribution reviews coverage intervals as defined in the Guide to the Expression of Uncertainty in Measurement (GUM), and compares them against the principal types of probabilistic intervals that are commonly used in applied statistics and in measurement science. Although formally identical to conventional confidence intervals for means, the GUM interprets coverage intervals more as if they were Bayesian credible intervals, or tolerance intervals. We focus, in particular, on a common misunderstanding about the intervals derived from the results of the Monte Carlo method of the GUM Supplement 1 (GUM-S1), and offer a novel interpretation for these intervals that we believe will foster realistic expectations about what they can deliver, and how and when they can be useful in practice.

Estimating Shelf Life Through Tolerance Intervals.

This paper is a continuation of the research published by the Stability Shelf Life Working Group as chartered under the Product Quality Research Institute. The Working Group was formed in 2006 and disbanded in late 2019. Following the philosophy presented by the Working Group on how to characterize the stability shelf life paradigm (Capen et al., 2012), shelf life is estimated here in terms of defining risk as a specified proportion of the pharmaceutical stability distribution of interest being out of specification. Shelf life can be defined for the batch mean distribution for regulatory issues, as well as for the product distributions for patient interests. Estimates of shelf life are proposed corresponding to each stability distribution through the use of statistical tolerance intervals. Appropriate estimates of the between-batch and within-batch variance components are obtained through a random coefficient mixed regression model analysis based on the best fit to batch stability response data. Tolerance interval estimates are computed as part of the mixed model analysis and computed directly using the statistical definition of the stability distributions. A proposed rationale is offered on how to select an appropriate proportion allowed out of specification to define a meaningful shelf life. Examples of the proposed shelf life estimates are presented using industry stability batch data. For each example, the traditional ICH shelf life estimate is given for comparison.

To tolerate or to agree: A tutorial on tolerance intervals in method comparison studies with BivRegBLS R Package.

The well-known agreement interval by Bland and Altman is extensively applied in method comparison studies. Two clinical measurement methods are considered interchangeable if their differences are not clinically significant. The agreement interval is commonly applied to assess the spread of the differences. However, this interval is approximate (too narrow) and several authors propose calculating a confidence interval around each bound. This article demonstrates that this approach is misleading, awkward, and confusing. On the other hand, tolerance intervals are exact and can include a confidence level if needed. Tolerance intervals are also easier to calculate and to interpret. Real data sets are used to illustrate the tolerance intervals with the R package BivRegBLS under normal or log-normal assumptions. Furthermore, it is also explained how to assess the coverage probabilities of the tolerance intervals with simulations.

Use of tolerance intervals for assessing biosimilarity.

A biosimilar is a biological product that is highly similar to an existing approved reference drug and has no clinically meaningful difference from it. Biosimilars are composed of or derived from living cells or organisms. Therefore, they are often sensitive to slight variations in the manufacturing process. Consequently, in demonstrating biosimilarity, it might be inappropriate to focus solely on the mean difference, or ratio of means, while ignoring the variabilities associated with the test and reference products. It is important to account for the entire population of clinical outcomes. Thus, we propose using the concept of tolerance intervals and related hypothesis testing for assessing biosimilarity. Our approach has the advantage of considering entire populations associated with both groups. A real example is used to illustrate our proposed method, and our approach is more stringent than those that employ confidence intervals. This is specifically the case when the mean difference of two drugs is not sufficiently large, but the biosimilar has a higher variability than that in the reference drug.

Controlling the Reproducibility of AC50 Estimation during Compound Profiling through Bayesian ß-Expectation Tolerance Intervals.

During drug discovery, compounds/biologics are screened against biological targets of interest to find drug candidates with the most desirable activity profile. The compounds are tested at multiple concentrations to understand the dose-response relationship, often summarized as AC50 values and used directly in ranking compounds. Differences between compound repeats are inevitable because of experimental noise and/or systematic error; however, it is often desired to detect the latter when it occurs. To address this, the ß-expectation tolerance interval is proposed in this article. Besides the classical acceptance criteria on assay performance, based on control compounds (e.g., quality control samples), this metric permits us to compare new estimates against historical estimates of the same study compound. It provides a measure that detects whether observed differences are likely due to systematic error. The challenge here is that limited information is available to build such compound-specific acceptance limits. To this end, we propose the use of Bayesian ß-expectation tolerance intervals to validate agreement between replicate potency estimates for individual study compounds. This approach allows the variability of the compound-testing process to be estimated from reference compounds within the assay and used as prior knowledge in the computation of compound-specific intervals as from the first repeat of the compound and then continuously updated as more information is acquired with subsequent repeats. A repeat is then flagged when it is not within limits. Unlike a fixed threshold such as 0.5log, which is often used in practice, this approach identifies unexpected deviations on each compound repeat given the observed variability of the assay.

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

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 [Formula: see text]-content, [Formula: see text]-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.

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.