When AUC and Cmax follow a log-normal distribution, the BE criterion for each parameter is defined as a ratio of the population geometric means of the test product and reference product, falling within the range of 0.80 to 1.25. BE is declared for a product if the 90% confidence interval of the difference in the average values of logarithmic parameters between the test and reference products is contained within the acceptable range of log(0.80) to log(1.25).
When the population geometric means before log-transformation of the parameters for evaluation of BE in the test product and products are described \({\mu }_{T}\) and \({\mu }_{R}\), the hypotheses of the BE test are :
$${H}_{0}: {\mu }_{T}/{\mu }_{R}\le {\theta }_{1} \text{o}\text{r} {\mu }_{T}/{\mu }_{R}\ge {\theta }_{2}$$
$${H}_{1}: {\theta }_{1}<{\mu }_{T}/{\mu }_{R}<{\theta }_{2}$$
The bioequivalence margins (\({\theta }_{1}\), \({\theta }_{2}\)) are \({\theta }_{1}=0.80\) and \({\theta }_{2}=1.25.\)The null hypothesis can be evaluated with two one-tailed t-tests at a 5% significance level as follows,
$$\frac{\text{log}\left({\theta }_{2}\right)-(\stackrel{-}{{X}_{T}}-\stackrel{-}{{X}_{R}})}{s\sqrt{2/n}}>t(1-\alpha ,n-2)$$
$$\frac{\left(\stackrel{-}{{X}_{T}}-\stackrel{-}{{X}_{R}}\right)-\text{l}\text{o}\text{g}\left({\theta }_{1}\right)}{s\sqrt{2/n}}>t\left(1-\alpha ,n-2\right)$$
1
\(\stackrel{-}{{X}_{T}}, \stackrel{-}{{X}_{R}}\) are means of the test and reference products, \(\alpha\) is significant level, \(n\) is total sample size and \(s\) is standard deviation of within-subject error.
In this manuscript we consider study designs that allow for the termination of the study if the BE criteria are met at the interim analysis. In the following section, we will explain the application of GSD and AD to the study design.
2.1 Group sequential design
Given the multiple hypothesis tests conducted in GSDs, it becomes necessary to adjust the significance level for each analysis to maintain the overall significance level 𝛼. These repeated analyses incorporate data from earlier interim analyses, resulting in correlated test statistics within GSDs. Various strategies exist for determining the interim-wise significance levels in GSDs. An early approach, as proposed by Pocock (1977), is to utilize the same level for all analyses. Another option is to select more stringent significance levels at earlier time points and less stringent levels at later stages, a concept put forth by O'Brien and Fleming (1979). Additionally, predefined alpha-spending functions for various fractions of the total sample size, as outlined by Lan and DeMets (1983), provide diverse methods for establishing appropriate local levels. The spending functions which approximate Pocock and O’Brien-Fleming designs have the following forms:
Pocock-type function: \(\alpha \times \text{log}\left((1+\left(e+1\right)t)\right)\) (2)
O’Brien-Fleming-type function: \(\bullet 2\left(1-{\Phi }\left(\frac{{{\Phi }}^{-1}\left(1-\frac{\alpha }{2}\right)}{\sqrt{t}}\right)\right)\) (3)
α is the overall significance level of the study, \(e\) is Napier’s constant and \({\Phi }\) is the cumulative standard normal distribution function. BE is evaluated at the time of an interim analysis using the appropriate significance level for the t-test in (1).
2.2 Adaptive design
Adaptive designs allow a flexible modification of design characteristics during an ongoing study while at the same time controlling the overall type I error rate. Adaptive designs offer the option of mid-course sample size recalculation based on interim results. There exist various approaches to construct adaptive designs that control the type I error rate also in case of sample size recalculation. One standard method is the inverse normal approach (Cui L et al., 1999; Lehmacher W and Wassmer G, 1999; Kieser and Rauch, 2015). In essence, it transforms p-values (\({p}_{1}\) and \({p}_{2}\)) from each stage of a two-stage design. When these p-values follow a uniform distribution under the null hypothesis, the inverse normal transformation converts them into standard normal random variables. For the inverse-normal combination test (Wassmer and Brannath, 2016), the test statistics \({Z}_{1}\) at the interim analysis and \({Z}_{2}\) at the final analysis are
$${T}_{1}^{*}: {Z}_{1}={{\Phi }}^{-1}(1-{p}_{1})$$
$${Z}_{2}={{\Phi }}^{-1}(1-{p}_{2})$$
Respectively, where \({T}_{1}^{*}\) at the interim analysis, \({p}_{1}\) and \({p}_{2}\) are the p-values at the interim and final analysis respectively. Then, the overall test statistics \(Z\) is:
$${T}_{2}^{*}: Z=\sqrt{w}{Z}_{1}+\sqrt{1-w}{Z}_{2}$$
4
In the manuscript, we set weight of \(w\) as \(w=\sqrt{{n}_{1}/({n}_{1}+{n}_{2})}\). \(Z\) follows a standard normal distribution under the null hypothesis and critical values used in the GSD can also be used in the AD. Consequently the probability of terminating the study at the interim analysis in the AD is same as that in the GSD.
Sample size re-estimation is a characteristic feature of AD, and involves adjusting the sample size during an ongoing clinical study based on the accumulating data to preserve or increase power. The conditional power that significance is achieved at the second stage conditioned on the test statistics at the first stage (at the interim analysis) is:
$$\text{Pr}\left({T}_{2}^{*}>{c}_{2}∣{T}_{1}^{*}\right)=\text{Pr}\left(\sqrt{w}{{\Phi }}^{-1}\left(1-{p}_{1}\right)+\sqrt{1-w}{{\Phi }}^{-1}\left(1-{p}_{2}\right)>{c}_{2}∣{p}_{1}\right) =\text{Pr}\left\{{p}_{2}<1-{\Phi }\left(\sqrt{\frac{{n}_{1}+{n}_{2}}{{n}_{2}}}{c}_{2}-\sqrt{\frac{{n}_{1}}{{n}_{2}}}{{\Phi }}^{-1}\left(1-{p}_{1}\right)\right)∣{p}_{1}\right\}$$
5
where \({c}_{2}\) is critical value at the final analysis. From (5), we can consider the significance level at stage 2 (\({\alpha }^{*}\)) as:
$$1-{\Phi }\left(\sqrt{\frac{{n}_{1}+{n}_{2}}{{n}_{2}}}{c}_{2}-\sqrt{\frac{{n}_{1}}{{n}_{2}}}{{\Phi }}^{-1}\left(1-{p}_{1}\right)\right).$$
In other words, conditional power can also be calculated using the following formula.
$${F}_{t}\left(\frac{\text{log}\left(1.25/\text{G}\text{M}\text{R}\right)}{{\sigma }_{W}\sqrt{2/{n}_{2}^{*}}}-{t}_{1-{\alpha }^{*},{n}_{2}^{*} },{n}_{2}^{*}-2\right)-{F}_{t}\left(\frac{-\text{log}\left(1.25\times \text{G}\text{M}\text{R}\right)}{{\sigma }_{W}\sqrt{2/{n}_{2}^{*}}}+{t}_{1-{\alpha }^{*},{n}_{2}^{*} },{n}_{2}^{*}-2\right)$$
6
GMR is calculated from the data at the interim analysis, and we search for \({n}_{2}^{*}\) where the following conditional powers exceeds \(1-\beta\) and is no larger than the maximum allowable sample size \({n}_{2}^{\text{m}\text{a}\text{x}}\) (\({n}_{2}^{\text{m}\text{i}\text{n}}\le {n}_{2}^{*}\le {n}_{2}^{\text{m}\text{a}\text{x}}\)).