Application of the Modified Chi-Square Ratio Statistic in a Stepwise Procedure for Cascade Impactor Equivalence Testing

Abstract

Equivalence testing of aerodynamic particle size distribution (APSD) through multi-stage cascade impactors (CIs) is important for establishing bioequivalence of orally inhaled drug products. Recent work demonstrated that the median of the modified chi-square ratio statistic (MmCSRS) is a promising metric for APSD equivalence testing of test (T) and reference (R) products as it can be applied to a reduced number of CI sites that are more relevant for lung deposition. This metric is also less sensitive to the increased variability often observed for low-deposition sites. A method to establish critical values for the MmCSRS is described here. This method considers the variability of the R product by employing a reference variance scaling approach that allows definition of critical values as a function of the observed variability of the R product. A stepwise CI equivalence test is proposed that integrates the MmCSRS as a method for comparing the relative shapes of CI profiles and incorporates statistical tests for assessing equivalence of single actuation content and impactor sized mass. This stepwise CI equivalence test was applied to 55 published CI profile scenarios, which were classified as equivalent or inequivalent by members of the Product Quality Research Institute working group (PQRI WG). The results of the stepwise CI equivalence test using a 25% difference in MmCSRS as an acceptance criterion provided the best matching with those of the PQRI WG as decisions of both methods agreed in 75% of the 55 CI profile scenarios.

Appendix

Construction of 90% BCA Confidence Intervals for the MmCSRS

A sample size of 30 T and 30 R products will be assumed for simplicity. However, this method could be applied to any other sample size of T and R products. Moreover, this example assumes a bootstrapping sample size of 2000. For details and statistical background on the calculations, please refer to the original publication (14).

1. For both the T and R products obtain a bootstrapping sample with size equal to 30 by resampling with replacement.

2. Repeat (1) 2000 times.

3. Obtain the MmCSRS for each of the 2000 bootstrapping samples of 30 T and 30 R products.

4. For the distribution of 2000 MmCSRS that is obtained in (3), the Z _L and Z _U percentile of the 2000 MmCSRS are the lower and upper bounds of the BCA confidence interval, respectively. The computation of Z _L and Z _U is shown below.

$$ {Z}_{\mathrm{L}}=\varPhi \left(\frac{z_0-{z}_{0.95}}{1-a\left({z}_0-{z}_{0.95}\right)}+{z}_0\right) $$

(3)

$$ {Z}_{\mathrm{U}}=\varPhi \left(\frac{z_0-{z}_{0.95}}{1-a\left({z}_0-{z}_{0.95}\right)}+{z}_0\right) $$

(4)

where Φ is the cumulative distribution function of the standard normal distribution and ${z}_{0.95}$ is the 95th percentile of the standard normal distribution

$$ a=\frac{{\displaystyle {\sum}_{i=1}^{900}{\left({\widehat{\theta}}_{-i}-\overline{\theta}\right)}^3}}{6{\left[{\displaystyle {\sum}_{i=1}^{900}{\left({\widehat{\theta}}_{-i}-\overline{\theta}\right)}^2}\right]}^{1.5}} $$

(5)

where

${\widehat{\theta}}_{-i}$ is the MmCSRS obtained for the original sample of 30 T and 30 R products by leaving a single T and R sample out (i.e., 900 MmCSRS based upon either 30 T and 29 R or 29 T and 30 R are obtained)

and

$$ \overline{\theta}=\frac{1}{900}{\displaystyle \sum_{i=1}^{900}{\widehat{\theta}}_{-i}} $$

(6)

and

$$ {z}_0={\varPhi}^{-1}\left[\frac{1}{2000+1}\right]{\displaystyle \sum_{b=1}^{2000}I\left({\widehat{\theta}}_b\le \widehat{\theta}\right)} $$

(7)

where I is an indicator function (i.e., equal to one if the condition is true and zero otherwise), ${\widehat{\theta}}_b$ is the MmCSRS for bth bootstrapping sample, and $\widehat{\theta}$ is the MmCSRS of the original sample.