On the Confidence Intervals in Bioequivalence Studies

In the section, we proceed to review the TOST, or two one-sided tests. The TOST involves performing two one-sided tests, one to test if the difference between the treatments or populations is greater than a predetermined non-inferiority margin and another to test if the difference is less than a predetermined equivalence margin. If the results of both tests are significant, it suggests that the treatments or populations are equivalent within the defined margins. Before proceeding, we first review the union-interaction test and intersection-union test, which provide the theoretical foundation for the TOST. Suppose the parameter space of a hypothesis testing is $\Theta=\Theta_{0}\bigcup\Theta_{0}^{c}$, where $\Theta_{0},\Theta_{0}^{c}$ are the parameter space of null and alternative hypothesis, respectively. Furthermore, let $R$ be the rejection region of the test, then the power function of the test, as a function of $\theta\in\Theta$, is defined as $\beta(\theta)=\mathbb{P}_{\theta\in\Theta}(T\in R)$, where $T$ is the test statistics. Before moving on, we first clarify two definitions that are commonly confused with one another.: the size and the level of a test.

These definitions help distinguish between the size and the level of a test, which are related but distinct concepts. The size of a test is the maximum probability of making a Type I error (i.e., rejecting the null hypothesis when it is true) under the null hypothesis parameter space $\Theta_{0}$. In contrast, the level of a test refers to the upper bound on the probability of making a Type I error. When the size of a test equals $\alpha$, it is considered a size-$\alpha$ test, whereas when the size of a test is less than or equal to $\alpha$, it is considered a level-$\alpha$ test. For a level-$\alpha$ test, the size of the test may be much less then $\alpha$, and in such case, if we still use $\alpha$ as the significance level, then the test will be less powerful.

Obviously, the null hypothesis in (2) can be viewed as the union of two simple null hypothesis $H_{01}:\mu_{T}-\mu_{R}\leq\theta_{L}$ and $H_{02}:\mu_{T}-\mu_{R}\geq\theta_{U}$. Not only in bioequivalence study, but also in some other applications, such complicated hypotheses can be developed from tests for simpler hypotheses. These hypothesis can be generalized from the so called intersection-union tests (IUT) and union-intersection test (UIT), which play an important role in bioequivalence testing and multiple comparison procedures.

It is straightforward to show that the rejection region of IUT is $\bigcap_{i=1}^{k}R_{i}$. The logic behind this is simple. $H_{0}$ is false if and only all of the $H_{0i},i=1,\ldots,k$ are false. So rejecting $H_{0}:\theta\in\bigcup_{i=1}^{k}\Theta_{i}$ is equivalent as rejecting each individual $H_{0i}:\theta\in\Theta_{i}$. Similarly, in terms of UIT, using the same logic, if any one of the hypotheses $H_{0i}$ is rejected, then $H_{0}$ must also be rejected, and only if each of $H_{0i}$ is accepted will the intersection $H_{0}$ be accepted. Thus, the rejection region of UIT is $\bigcup_{i=1}^{k}R_{i}$. Clearly, the hypothesis testing in 2 belongs to IUT. Now, a natural question rises: given the significance of each individual $H_{0i}$, what is the significance level of IUT? The following theorem gives an upper bound for the size of the IUT.

It should be remarked that Theorem 2 only shows that the level of an IUT is $\alpha=\sup_{i=1,\ldots,k}\alpha_{i}$, that is, an upper bound for the size of the IUT. In fact, the size of the IUT could be much less than $\alpha$. The following theorem provides the conditions under which the size of the IUT is exactly $\alpha$, ensuring that the IUT is not too conservative.

We are now in a position to formally introduce the method for bioequivalence testing in (2). Albeit good theoretical guarantees for IUT, Westlake, (1981); Schuirmann, (1981, 1987) proposed the following so-called two one-sided tests (TOST), which has now been one of the standard procedures of (2). As its name implies, TOST consists two one-sided hypotheses

$\displaystyle H_{01}:\mu_{T}-\mu_{R}\leq\theta_{L}\textrm{ versus }H_{a1}:\mu_{T}-\mu_{R}>\theta_{L}$

(5)

and

$\displaystyle H_{02}:\mu_{T}-\mu_{R}\geq\theta_{U}\textrm{ versus }H_{a2}:\mu_{T}-\mu_{R}<\theta_{U}.$

(6)

The $H_{0}:\mu_{T}-\mu_{R}\leq\theta_{L}\textrm{ or }\mu_{T}-\mu_{R}\geq\theta_{U}$ in (2) can be expressed as the union of $H_{01}$ and $H_{02}$. This procedure establishes bioequivalence at significance level $\alpha$ if both $H_{01}$ and $H_{02}$ are rejected at level-$\alpha$. The rationale underlying is simple. If one may conclude that $\mu_{T}-\mu_{R}>\theta_{L}$ and also $\mu_{T}-\mu_{R}<\theta_{U}$, then it has in effect been concluded that $\theta_{L}<\mu_{T}-\mu_{R}<\theta_{U}$. In practice, size-$\alpha$ Student’s $t$-test is used for (5) and (6) and $H_{0}$ is rejected if

$$T_{L}=\frac{(\bar{X}_{T}-\bar{X}_{R})-\theta_{L}}{\hat{\sigma}_{\Delta\bar{X}}}>t_{1-\alpha,r},\quad T_{U}=\frac{(\bar{X}_{T}-\bar{X}_{R})-\theta_{U}}{\hat{\sigma}_{\Delta\bar{X}}}<-t_{1-\alpha,r},$$

(7)

where $\bar{X}_{T},\bar{X}_{R}$ are the average of log-transformed PK data for treatment/reference product, and $\hat{\sigma}_{\Delta\bar{X}}=S_{p}\sqrt{(\frac{1}{n_{T}}+\frac{1}{n_{R}})}$ is the standard error of $\bar{X}_{T}-\bar{X}_{R}$ and $S_{p}^{2}=\frac{1}{n_{T}+n_{R}-2}[(n_{T}-1)S_{T}^{2}+(n_{R}-1)S_{R}^{2}]$, with $S_{T},S_{R}$ being the standard deviation of two groups, and $t_{1-\alpha,r}=\mathbb{P}(X\leq t_{1-\alpha,r})=1-\alpha$ is the critical value from $t$ distribution with degree of freedom $r=n_{T}+n_{R}-2$. In addition, it should be mentioned that the size for each individual $t$-test $H_{01},H_{02}$ is $\alpha$, not $\alpha/2$. There is no need for multiplicity adjustment in testing each the two one-sided null hypotheses for a univariate PK parameter. Suppose the size for each $t$-test is $\alpha$, since TOST belongs to IUT, from Theorem 2, we know the level of TOST is $\alpha$, that is, the size is at most $\alpha$. To prove the size of TOST is exactly $\alpha$, we need to check the conditions of Theorem 3. First, consider a parameter point ${\theta_{l}}=\mu_{T}-\mu_{R}=\theta_{U}$, then we have

$$\lim_{l\rightarrow\infty}\mathbb{P}_{\theta_{l}}(\bm{X}\in R_{2})=\mathbb{P}_{\theta_{U}}(\bm{X}\in R_{2})=\mathbb{P}_{\theta_{U}}(T_{U}<-t_{1-\alpha,r})=\alpha,$$

thus the first condition in Theorem 3 holds true. Furthermore,

$$\lim_{l\rightarrow\infty}\mathbb{P}_{\theta_{l}}(\bm{X}\in R_{1})=\mathbb{P}_{\theta_{U}}(\bm{X}\in R_{1})=\mathbb{P}_{\theta_{U}}(T_{L}>t_{1-\alpha,r})\rightarrow 1,\textrm{ as }\sigma^{2}\rightarrow 0,$$

therefore, the second condition in Theorem 3 also remains valid, from where, we summarize the above analysis to the following theorem

To compute the power of TOST, Phillips, (1990) derived the explicit form of the power, as a function of $(\mu_{T},\mu_{R},n_{T},n_{R},\sigma^{2},\theta_{L},\theta_{U})$ given by

	$\displaystyle\textrm{Power}\left(\mu_{T},\mu_{R},n_{T},n_{R},\sigma^{2},\theta_{L},\theta_{U}\right)$	$\displaystyle=Q_{r}\left(-t_{1-\alpha,r},\frac{\mu_{T}-\mu_{R}-\theta_{U}}{\sigma\sqrt{\frac{1}{n_{T}}+\frac{1}{n_{T}}}};0\frac{\left(\theta_{U}-\theta_{L}\right)\sqrt{r}}{2\sigma\sqrt{\frac{1}{n_{T}}+\frac{1}{n_{T}}}t_{1-\alpha,r}}\right)$
		$\displaystyle\quad-Q_{r}\left(t_{1-\alpha,r},\frac{\mu_{T}-\mu_{R}-\theta_{L}}{\sigma\sqrt{\frac{1}{n_{T}}+\frac{1}{n_{T}}}};0\frac{\left(\theta_{U}-\theta_{L}\right)\sqrt{r}}{2\sigma\sqrt{\frac{1}{n_{T}}+\frac{1}{n_{T}}}t_{1-\alpha,r}}\right)$

where

$$Q_{v}(t,\delta;a,b)=\frac{\sqrt{2\pi}}{\Gamma\left(\frac{v}{2}\right)\cdot 2^{\frac{v-2}{2}}}\int_{a}^{b}\Phi\left(\frac{tx}{\sqrt{v}}-\delta\right)x^{v-1}\phi(x)dx,r=n_{1}+n_{2}-2$$

is referred as the Owen’s $Q$ function and, $\Phi(\cdot),\phi(\cdot)$ are the cdf and pdf of standard normal distribution, respectively.

In the previous sections, the theoretical foundations of the TOST method for bioequivalence testing have been established. However, the relationship between the TOST and $100(1-\alpha)\%$ and $100(1-2\alpha)\%$ confidence intervals is not yet clear. In the next section, we will explore the connections between the TOST and these two types of confidence intervals, shedding light on their similarities, differences, and implications for bioequivalence testing. Understanding the connections between the TOST and different confidence intervals will provide insights into how these statistical methods relate to each other, and how they can be effectively applied in bioequivalence testing for univariate pharmacokinetic parameters. This will enhance the interpretation and application of these methods in practice, contributing to more accurate and reliable conclusions about bioequivalence.

Intuitively, a hypothesis is associated with an equivalent confidence interval approach. Traditionally, bioequivalence is claimed if the $100(1-2\alpha)\%$ two-sided confidence interval of the geometric mean ratio, or equivalently, the arithmetic mean difference of the log-transformed data, falls within the BE limits. The use of a $100(1-2\alpha)\%$ confidence interval for bioequivalence testing, rather than a $100(1-\alpha)\%$ confidence interval, has been a subject of debate among statisticians. The relationship between the size-$\alpha$ TOST procedure and the $100(1-2\alpha)\%$ confidence interval approach has been questioned by some researchers, who argue that the similarity is based more on an algebraic coincidence than a true statistical equivalence. Brown et al., (1995) suggested that the association between the TOST procedure and the $100(1-2\alpha)\%$ confidence interval approach may be somewhat of a fiction, while Berger and Hsu, (1996) pointed out that using a $100(1-2\alpha)\%$ confidence interval for bioequivalence testing can be conservative and may only work in specific cases. Other discussions on the use of the $100(1-2\alpha)\%$ confidence interval procedure in bioequivalence testing can be found in works by Westlake, (1976),Westlake, (1981), Rani et al., (2004), Choi et al., (2008). These discussions highlight the ongoing debate about the appropriateness of using a $100(1-2\alpha)\%$ confidence interval for bioequivalence testing, and the importance of understanding the underlying statistical concepts and assumptions in order to make informed decisions about the most suitable methods for a given study.

From the previous analysis, we know that there are various procedures for testing (2), and a natural question rises, which one is better? Unfortunately, even though FDA suggests using $100(1-2\alpha)\%$ to test bioequivalence, the procedure is not the best. In fact, it should be remarked that TOST has been criticized by many authors. On the one hand, TOST yields a biased test, on the other hand, the power of TOST is quite low. In practice, we would like a hypothesis test to be more likely to reject $H_{0}$ if $\theta\in\Theta_{0}^{c}$ than if $\theta\in\Theta_{0}$, that is, $\beta(\theta_{2})\geq\beta(\theta_{1})$ for every $\theta_{1}\in\Theta_{0}$ and $\theta_{2}\in\Theta_{0}^{c}$. The following definition summarize the tests summarizing this property.

Unfortunately, even using a $100(1-2\alpha)\%$ confidence interval approach, TOST is still a biased test. Furthermore, it has also been shown that TOST is conservative and inefficient under asymptotic setup. Much attention has been paid to equivalence testing problem in (2) for decades, to mention but a few, Anderson and Hauck, (1983), Brown et al., (1997), Mart´ın Andrés and Herranz Tejedor, (2004), Choi et al., (2008), Fogarty and Small, (2014), Pesarin et al., (2016). See Wellek, (2002), Meyners, (2012) for a comprehensive review. Even though many methodologies have been proposed to test bioequivalence, the theoretical results are limited. Except some special models, like normal distribution with known variance, no finite sample optimality theory is available for tests of equivalence. To the best of our knowledge, the work by Romano, (2005) is the first theoretical result where the asymptotic optimality theory is established. Since the optimality theory is beyond the scope of the paper, we only show two examples to give the author some intuitions about optimal testing of equivalence.

However, the uniformly most powerful test does not exist is $\sigma$ is unknown. The next example generalize the results to the exponential family.