Calculations, including statistical tolerance intervals, can assist in the development and revision of specification acceptance criteria. Manufacturing results for attributes of a biopharmaceutical product can be positively autocorrelated. The sample standard deviation — calculated from limited, positively autocorrelated data — tends to underestimate the long-term process standard deviation (**1**). In this article, simulated data are used to assess the relative performance of statistical tolerance intervals, intervals calculated using the minimum process performance index P_{pk} approach, and the sample range.

**Prevalence of Positively Autocorrelated Data**

The estimate of process standard deviation (σ) calculated from a limited number of batches (*n*) can be markedly underestimated when data are positively autocorrelated. The presence of positive autocorrelation (values that are closely related to each other in sequential order) for a biopharmaceutical manufacturing process can be caused by a number of factors, including lack of sensitivity of an analytical method and homogeneity of input factors (e.g., multiple drug product batches sourced from a common drug substance lot). In a recent assessment, over 75% of quality attributes in continued process verification (CPV) activities for one biopharmaceutical product had lag-one sample autocorrelation function estimates between 0.0 and 0.5 (**1**). Therefore, analysts recommended using the sample standard deviation as the estimate of σ when at least *n* = 30 results are available for biopharmaceutical CPV activities (e.g., control charting).

**Statistical Tolerance Intervals and Acceptance Criteria Determination**

A statistical calculation such as a tolerance interval (**2**) can assist in the decision-making exercise for developing specification acceptance criteria (SAC). One aim is to find reasonable limits to represent expected results from a current manufacturing process. That information is used with nonstatistical considerations, including clinical experience, to determine SAC. Before a drug product is commercialized, SAC-related calculations may involve fewer than 30 batches.

Consider a situation in which *n* = 10 drug product lots have been manufactured. To obtain an interval in which a large proportion of individual values would fall, a two-sided 95% confidence/99% coverage tolerance interval (95/99 TI) is calculated as the sample mean ± 4.44 sample standard deviation when assuming a normal distribution (**3**). That interval covers at least 99% of individual values with 95% confidence (95% of the time).

Monte Carlo simulations as described in a previous study (**1**) were performed using 108 samples for each value of n to assess the long-run relative frequency (confidence level) for which at least 99% of individual results were contained inside the interval. For example, generating independent data (autocorrelation function, ACF(1) = 0) and calculating two-sided 95/99 TIs using the TI multipliers for 95% confidence as described by Hahn and Meeker (**3**), the confidence level for 99% coverage is, as expected, 95.0% for each sample size (Table 1). However, if data are positively autocorrelated with ACF(1) = 0.5, then the achieved confidence level with* n* = 10 is not 95.0% but about 86.9%. Therefore, TIs have lower confidence levels than expected when limited data are positively autocorrelated, and this includes the situation in which *n* = 30.

**Minimum P _{pk} and Sample Range Intervals**

Another calculation strategy to assist in determining specification acceptance criteria the minimum P

_{pk}approach (

**4**). This approach uses the sample size, sample mean, and sample standard deviation to generate an interval such that the lower bound of an approximate two-sided 95% confidence interval for P

_{pk}is ≥1. This approach tends to generate intervals closer to a 95/99 TI when encountering positively autocorrelated data (Table 1). For example, with

*n*= 10, when results are positively autocorrelated with ACF(1) = 0.5, this interval would correspond to a 96.5/99 TI.

Simulations also were performed to assess sample range, namely the difference between the largest and smallest results of a sample of size n. The conclusion from that analysis is that practitioners would be ill-advised to use sample range to assist in setting SAC because of the low confidence levels for capturing a high percentage of the distribution. For example, when n = 10, the achieved confidence level is only 0.4% with independent results and 0.3% when results are positively autocorrelated with ACF(1) = 0.5 for 99% coverage. That should not be surprising, because it can be shown that the expected percentage of the distribution to be covered by the sample range using independent results equals (*n* – 1)/(*n* + 1) (5). Thus, with *n* = 30 independent results, the average percentage covered by the sample range is 100 (29/31)% ≈ 93.5%. The sample range calculated using *n* = 30 is equivalent to only a 3.6/99 TI (Table 1) and illustrated using simulated results in Figure 1.

**Practical Implications for Specification Acceptance Criteria Determination**

Positively autocorrelated data are likely to be used in calculations to assist in developing SAC for a biopharmaceutical manufacturing system. A TI calculated using limited amounts of such data provides intervals with a lower confidence level than expected. The minimum P_{pk} approach guarantees that the calculated interval is such that the lower two-sided 95% confidence bound of P_{pk} is at least one, and it has further been illustrated to provide intervals closer to the expected performance of 95/99 TIs with positively autocorrelated data. Use of a sample range to assist in SAC calculations has been shown to lack adequate confidence for high coverage levels and thus is not recommended.

**References**

**1** Bower KM. Determining Control Chart Limits for Continued Process Verification with Autocorrelated Data. *BioProcess Int.* 17(4) 2019: 14–16.

**2** Dong X, Tsong Y, Shen M. Statistical Considerations in Setting Product Specifications. *J. Biopharm. Statistics* 25, 2015: 280–294.

**3** Hahn GJ, Meeker WQ. *Statistical Intervals: A Guide for Practitioners.* John Wiley & Sons: Hoboken, NJ, 1991; 291.

**4** Coffey T, Bower KM. A Statistical Approach to Assess and Justify Potential Product Specifications. *BioProcess Int.* 15(2) 2017: 38–39.

**5** Mood AF.* Introduction to the Theory of Statistics.* McGraw-Hill: New York, NY, 1950; 387.

**Keith M. Bower,** MS, is senior principal CMC statistician at Seattle Genetics, Inc. and an affiliate assistant professor in the Department of Pharmacy at the University of Washington; kbower@seagen.com; www.seattlegenetics.com.

*SAS Enterprise Guide 7.15 was used to calculate the results for Table 1 and to generate Figure 1.*