Xiaoyu Dong

Using Tolerance Intervals for Assessment of Pharmaceutical Quality

In quality control of drug products, tolerance intervals are commonly used methods to assure a certain proportion of the products covered within a pre-specified acceptance interval. Depending on the nature of the quality attributes, the corresponding acceptance interval could be one-sided or two-sided. Thus, the tolerance intervals can also be one-sided or two-sided. To better utilize tolerance intervals for quality assurance, we reviewed the computation method and studied their statistical properties in terms of batch acceptance probability in this article. We also illustrate the application of one-sided and two-sided tolerance, as well as two one-sided tests through the examples of dose content uniformity test, delivered dose uniformity test, and dissolution test.

Statistical Considerations in Setting Product Specifications

According to ICH Q6A (1999), a specification is defined as a list of tests, references to analytical procedures, and appropriate acceptance criteria, which are numerical limits, ranges, or other criteria for the tests described. For drug products, specifications usually consist of test methods and acceptance criteria for assay, impurities, pH, dissolution, moisture, and microbial limits, depending on the dosage forms. They are usually proposed by the manufacturers and subject to the regulatory approval for use. When the acceptance criteria in product specifications cannot be pre-defined based on prior knowledge, the conventional approach is to use data from a limited number of clinical batches during the clinical development phases. Often in time, such acceptance criterion is set as an interval bounded by the sample mean plus and minus two to four standard deviations. This interval may be revised with the accumulated data collected from released batches after drug approval. In this article, we describe and discuss the statistical issues of commonly used approaches in setting or revising specifications (usually tighten the limits), including reference interval, (Min, Max) method, tolerance interval, and confidence limit of percentiles. We also compare their performance in terms of the interval width and the intended coverage. Based on our study results and review experiences, we make some recommendations on how to select the appropriate statistical methods in setting product specifications to better ensure the product quality.

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.

Equivalence Tests for Interchangeability Based on Two One-Sided Probabilities

A test treatment is considered to be interchangeable with its reference treatment if they are equivalent and expected to produce the same clinical result in any given patient. To assess interchangeability, FDA Draft Guidance (1999) and Guidance for Industry (2001, 2003) recommend using individual bioequivalence (IBE) and population bioequivalence (PBE) procedures. Chow (1999) and Chow and Liu (1999) gave a discussion on the limitation of the aggregate criteria of the IBE and PBE proposed therein. They mentioned that it is not clear whether IBE or PBE can imply average bioequivalence. Alternative approaches have been proposed to address the weakness of IBE and PBE. Dong et al. (2014) discuss the tolerance interval method and an approximate test for interchangeability defined by a two-sided probability. These tests may not be able to test for the two one-sided tests (TOST) with asymmetric margins around the true mean difference. In addition, the tests of two-sided probability provide no direction when failing the equivalence in interchangeability. Thus, we reexamine the statistical properties of the two one-sided tolerance interval approaches proposed by Tsong and Shen (2007, 2008). In this project, we extend their approach for parallel arms trials and paired/crossover data without the assumption of equal sample sizes and variances. We also develop the exact power function and assess the type I error rate of our proposed approach. In addition, we study the sample size determination based on the interchangeability testing utilizing the tolerance interval method.

Bayesian Approach to Assay Sensitivity Analysis of Thorough QT Trials

One of the analyses recommended in ICH E14 Guidance (International Conference on Harmonisation, 2005) after the test drug is shown to be negative in QT interval prolongation after subjects treated with the test drug is an assay sensitivity analysis of a positive control drug with known effect on QT prolongation. The assay sensitivity is validated if the response profile is shown to be expected and the QT interval after administration of the positive control drug is shown to be at least 5 ms more than placebo. The negative result of the test treatment is validated if the prolongation of the positive control is verified among the study subjects. One of the most frequently used positive control drugs in thorough QT (TQT) trials is moxifloxacin. In order to improve the efficiency of the study and to reduce the number of subjects exposed to moxifloxacin, we explore the potential sample size reduction with a Bayesian approach to the assay sensitivity utilizing the data of historical TQT trials. We derived the distribution of moxifloxacin-induced QT prolongation based on 14 crossover trials and six parallel trials. The estimated distribution is used as a prior distribution to assess the posterior probability that the moxifloxacin-induced QT prolongation is larger than 5 ms. Sample size based on such Bayesian approach will be compared with the conventional frequentist approach for efficiency assessment.

Statistical Evaluation of Several Methods for Cut-Point Determination of Immunogenicity Screening Assay

The cut point of the immunogenicity screening assay is the level of response of the immunogenicity screening assay at or above which a sample is defined to be positive and below which it is defined to be negative. The Food and Drug Administration Guidance for Industry on Assay Development for Immunogenicity Testing of Therapeutic recommends the cut point to be an upper 95 percentile of the negative control patients. In this article, we assume that the assay data are a random sample from a normal distribution. The sample normal percentile is a point estimate with a variability that decreases with the increase of sample size. Therefore, the sample percentile does not assure at least 5% false-positive rate (FPR) with a high confidence level (e.g., 90%) when the sample size is not sufficiently enough. With this concern, we propose to use a lower confidence limit for a percentile as the cut point instead. We have conducted an extensive literature review on the estimation of the statistical cut point and compare several selected methods for the immunogenicity screening assay cut-point determination in terms of bias, the coverage probability, and FPR. The selected methods evaluated for the immunogenicity screening assay cut-point determination are sample normal percentile, the exact lower confidence limit of a normal percentile (Chakraborti and Li, 2007) and the approximate lower confidence limit of a normal percentile. It is shown that the actual coverage probability for the lower confidence limit of a normal percentile using approximate normal method is much larger than the required confidence level with a small number of assays conducted in practice. We recommend using the exact lower confidence limit of a normal percentile for cut-point determination.

Sample Size Determination for a Three-Arm Equivalence Trial of Normally Distributed Responses

The equivalence assessment is often conducted through a three-arm clinical trial (namely, test, reference, and placebo) and it usually consists of three tests. The first two tests are to demonstrate the superiority of the test and the reference treatment to the placebo, and they are followed by an equivalence test between the test treatment and the reference treatment. When the response variable is continuous, equivalence is commonly defined in terms of mean difference, mean ratio, or ratio of mean differences, that is, the mean difference of the test and the placebo to the mean difference of the reference and the placebo. These equivalence tests can be performed with both a hypothesis-testing approach and a confidence-interval approach. The advantage of applying the equivalence test by ratio of mean differences is that it can test both superiority of the test treatment over placebo and equivalence between the test and the reference simultaneously through a single hypothesis. In this article, we derive the test statistics and the power function for the ratio of mean differences hypothesis and solve the required sample size for a three-arm clinical trial. Examples of required sample size are given in this article, and are compared with the required sample size by the traditional mean difference equivalence test. After a careful examination, we suggest increasing the power of the ratio of mean differences approach by appropriately adjusting the lower limit of the equivalence interval.

Statistical Properties of Large Sample Tests for Dose Content Uniformity

The European Union (EU) test for uniformity of dosage units using large sample sizes was published in European Pharmacopoeia 7.7 in 2012. There are 2 alternative tests. Option 1 is a parametric two-sided tolerance interval-based method modified with an indifference zone and counting units outside of (0.75 M, 1.25 M) (here, M is defined by sample mean, X ˉ , as M = 98.5% if X ˉ < 98.5%, M = 101.5% if X ˉ > 101.5%, and M = X ˉ otherwise). Option 2 is a nonparametric counting method with an additional indifference-zone concept. The authors extended the parametric two one-sided tolerance interval-based method that was proposed for dose content uniformity testing based on 30 tablets to large sample sizes with the restriction that all operating characteristic curves of the two one-sided tolerance intervals for any given sample size intersect with the operating characteristic curve of the US Pharmacopoeia harmonized method for a sample size of 30 at the acceptance probability of 90% when the individual tablets with on-target mean are assumed to be normally distributed. This paper studies the acceptance probabilities in relation to the batch mean and batch standard deviation among the 2 EU options and the authors’ proposed method. The acceptance probabilities of EU options 1 and 2 and the proposed method were compared using simulation; results revealed that both EU options 1 and 2 produce larger acceptance probabilities when the batch mean is off-target. Furthermore, for a given standard deviation, the acceptance probability of EU option 2 at a mean 102% of the label claim is larger than that at a mean of 100% of the label claim under the normality assumption.

Quality Assurance Test of Delivered Dose Uniformity of Multiple-Dose Inhaler and Dry Powder Inhaler Drug Products

The delivered dose uniformity is one of the most critical requirements for dry powder inhaler (DPI) and metered dose inhaler products. In 1999, the Food and Drug Administration (FDA) issued a Draft Guidance entitled Nasal Spray and Inhalation Solution, Suspension, and Spray Drug Products–Chemistry, Manufacturing and Controls Documentation and recommended a two-tier acceptance sampling plan that is a modification of the United States Pharmacopeia (USP) sampling plan of dose content uniformity (USP34<601>). This sampling acceptance plan is also applied to metered dose inhaler (MDI) and DPI drug products in general. The FDA Draft Guidance method is shown to have a near-zero probability of acceptance at the second tier. In 2000, under the request of The International Pharmaceutical Aerosol Consortium, the FDA developed a two-tier sampling acceptance plan based on two one-sided tolerance intervals (TOSTIs) for a small sample. The procedure was presented in the 2005 Advisory Committee Meeting of Pharmaceutical Science and later published in the Journal of Biopharmaceutical Statistics (Tsong et al., 2008). This proposed procedure controls the probability of the product delivering below a pre-specified effective dose and the probability of the product delivering over a pre-specified safety dose. In this article, we further propose an extension of the TOSTI procedure to single-tier procedure with any number of canisters.

Sample Size Determination for Equivalence Assessment with Multiple Endpoints

Equivalence assessment between a reference and test treatment is often conducted by two one-sided tests (TOST). The corresponding power function and sample size determination can be derived from a joint distribution of the sample mean and sample variance. When an equivalence trial is designed with multiple endpoints, it often involves several sets of two one-sided tests. A naive approach for sample size determination in this case would select the largest sample size required for each endpoint. However, such a method ignores the correlation among endpoints. With the objective to reject all endpoints and when the endpoints are uncorrelated, the power function is the production of all power functions for individual endpoints. With correlated endpoints, the sample size and power should be adjusted for such a correlation. In this article, we propose the exact power function for the equivalence test with multiple endpoints adjusted for correlation under both crossover and parallel designs. We further discuss the differences in sample size for the naive method without and with correlation adjusted methods and illustrate with an in vivo bioequivalence crossover study with area under the curve (AUC) and maximum concentration (Cmax) as the two endpoints.