Tolerance Intervals for the Distribution of the Difference Between Two Independent Normal Random Variables

Determining reference ranges and sample sizes in parallel-group studies

BACKGROUND AND OBJECTIVES

Reference ranges are widely used to locate the major range of the target probability distribution. When future measurements fall outside the reference range, they are classified as atypical and require further investigation. The fundamental principles and statistical properties of reference ranges are closely related to those of tolerance interval procedures. Existing investigations of reference ranges and tolerance intervals mainly devoted to the primitive cases of one- and paired-sample designs. Although reference ranges hold considerable promise for parallel group designs, the corresponding methodological and computational issues for determining reference limits and sample sizes have not been adequately addressed.

METHODS

This paper describes a complete collection of one- and two-sided reference ranges for assessing measurement differences in parallel-group studies that assume variance homogeneity.

RESULTS

The problem of sample size determination for precise reference ranges is also examined under the expected half-width and assurance probability considerations. Unlike the current methods, the suggested sample size criteria explicitly accommodate desired interval width in precise interval estimation.

CONCLUSIONS

Theoretical examinations and empirical assessments are presented to validate the usefulness of the proposed reference range and sample size procedures. To enhance the usages of the recommended techniques in practical applications, computer programs are developed for efficient calculation and exact analysis. A real data example regarding tablet absorption rate and extent is presented to illustrate the suggested assessments between two drug formulations.

Limitations of the quality range approach in analytical similarity assessment: Effect of mean shift and relative variability

Recently in 2019, the United States Food and Drug Administration (FDA) circulated a new draft guidance for comparative analytical assessment. They suggest the use of quality range (QR) methods. In this article, selection of the k value, and the effect of mean shifts and relative variability are evaluated. These are expressed as a ratio between the two standard deviations of the tested product and the reference product, σ_T/σ_R. In a second step, the two modified versions of the QR method proposed by Son et al. (2020) are also analysed under several scenarios, through simulation studies using real data from a biotechnology company and our own data for bevacizumab. Results indicate that k has a great impact on the probability of passing similarity tests. Pass rates higher than 90 % can be achieved for small relative variabilities (σ_T/σ_R ≤ 0.6) and large mean shifts (≈4%) by using k = 3. The situation is totally different for k = 2: the pass rate is higher than 90 % for scenarios with small (<0.5 %) or no differences between the means of the two products, but this percentage decreases by up to 50 % for σ_T/σ_R = 1. Effectiveness in detecting the various scenarios was quantified by calculating the probability curves of passing the similarity test, as a function of the two variables for each k value. Alternative methods present the same limitations but with different magnitude in comparison with QR, this being most pronounced in the plausibility-interval QR method.

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.

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.

Bayesian confidence intervals for the difference between variances of delta-lognormal distributions

Unnatural rainfall fluctuation can result in such severe natural phenomena as drought and floods. This variability not only occurs in areas with unusual natural features such as land formations and drainage but can also be due to human intervention. Since rainfall data often contain zero values, evaluating rainfall change is an important undertaking, which can be estimated via the confidence intervals for the difference between delta-lognormal variances using the highest posterior density-based reference (HPD-ref) and probability-matching (HPD-pm) priors. Simulation results indicate that HPD-pm performances were better than other methods in terms of coverage rates and relative average lengths for the difference in delta-lognormal variances, even with a large difference in variances. To illustrate the efficacy of our proposed methods, we applied them to daily rainfall data sets for the lower and upper regions of northern Thailand.

Some thoughts on the QR method for analytical similarity evaluation

As indicated in a recent published draft guidance on comparative analytical assessment, the United States (US) Food and Drug Administration (FDA) seems to suggest the use of quality range (QR) method for analytical similarity evaluation. It is a concern that the use of QR method for analytical similarity evaluation could potentially approve biological products which are not deemed biosimilar to the reference biological products. In this article, the limitations and potential risk for the use of the QR method for analytical similarity evaluation are discussed. Alternatively, two modified versions of the QR method, which are referred to as effect size (ES) mQR and plausibility interval (PI) mQR methods are suggested. The performance and statistical properties of the mQR methods are evaluated via extensive clinical trial simulation under various scenarios. The results indicate that the modified versions of the QR method not only overcome the limitations of the QR method for analytical similarity evaluation, but also can potentially help in detecting reference product changes during manufacturing process.

A Bayesian approach to construct confidence intervals for comparing the rainfall dispersion in Thailand

Natural disasters such as drought and flooding are the consequence of severe rainfall fluctuation, and rainfall amount data often contain both zero and positive observations, thus making them fit a delta-lognormal distribution. By way of comparison, rainfall dispersion may not be similar in enclosed regions if the topography and the drainage basin are different, so it can be evaluated by the ratio of variances. To estimate this, credible intervals using the highest posterior density based on the normal-gamma prior (HPD-NG) and the method of variance estimates recovery (MOVER) for the ratio of delta-lognormal variances are proposed. Monte Carlo simulation was used to assess the performance of the proposed methods in terms of coverage probability and relative average length. The results of the study reveal that HPD-NG performed very well and was able to meet the requirements in various situations, even with a large difference between the proportions of zeros. However, MOVER is the recommended method for equal small sample sizes. Natural rainfall datasets for the northern and northeastern regions of Thailand are used to illustrate the practical use of the proposed credible intervals.