## Modified acceptance range, confirmatory vs. exploratory [Power / Sample Size]

Hi Laura,

❝ […] power a DP study for a 505b2:

❝ […] The PK of the drug is linear over the entire range however because of non-proportionality composition between T strengths we will run also a DP trial (for T only).

❝ Power model at alpha = 0.05 will be used, so the 90% CI of the slope will be compared to the [0.80, 1.25] bounds.

Let’s consider the power model:$$\small{\mu_j=\alpha\cdot D_{j}^{\;\beta}}\tag{1},$$where $$\small{\mu}$$ is the respective PK metric and $$\small{D}$$ the administered dose; both at level $$\small{j}$$. For convenience generally the linearized model is used:$$\small{\log_{e}(\mu_j)=\alpha+\beta\cdot\log_{e}(D_j)},\tag{2}$$Whether only the extent of absorption (AUC) or additionally the rate (Cmax) should be assessed is the topic of heated debates in the PK community…
Whilst you start with $$\small{\left\{\theta_1,\theta_2\right\}}$$ (e.g., $$\small{\left\{0.80,1.25\right\}}$$), you have to modify the acceptance range.1,2
When $$\small{r}$$ is the ratio of highest and lowest dose levels, the parameter of interest is $$\small{r^{\,\beta-1}}$$ or the ratio of dose-normalized means $$\small{r_\textrm{dnm}}$$. Dose proportionality is defined if $$\small{r^{\,\beta-1}}$$ is within a predefined acceptance range $$\small{\left\{\theta_\textrm{L},\theta_\textrm{U}\right\}}$$. Since $$\small{r_\textrm{dnm}}$$ is a function of $$\small{\beta}$$, evaluation of dose proportionality can be performed through a $$\small{100(1-2\alpha)}$$ confidence interval of $$\small{\beta}$$ with the following modified acceptance range:$$\small{\left\{\theta_\textrm{L},\theta_\textrm{U}\right\}=\left\{1+\frac{\log_{e}(\theta_1)}{\log_{e}(r)}, 1+\frac{\log_{e}(\theta_2)}{\log_{e}(r)}\right\}}\tag{3}$$Example for $$\small{\left\{\theta_1,\theta_2\right\}=\left\{0.80,1.25\right\}}$$:

$$\small{\begin{matrix} \:r & \theta_\textrm{L} & \theta_\textrm{U}\\ \; \: 3 & 0.7969 & 1.2031\\ \; \: 4 & 0.8390 & 1.1610\\ \; \: 5 & 0.8614 & 1.1386\\ 10 & 0.9031 & 1.0969 \end{matrix}}$$

Hence, the wider the dose range, the more restrictive the modified acceptance range gets (note that the limits are symmetric around 1). Science is cruel mistress.
In my European hybrids acc. to 2001/83/EC, Article 10(3) I used a mixed-effects model3 (fixed effect $$\small{D}$$ and random effect $$\small{subject}$$) with restricted maximum likelihood estimation and Satterthwaite’s degrees of freedom. This allows to use incomplete data (subjects with missing periods). I guess that’s fine for the FDA as well.

❝ Couldn't find any reference/guidance on the need to base sample size to achieve 80% or 90% power (in our case this will result in a very large study),…

Don’t know any reference but in my hybrids I used 80%.

❝ … and from review of precedents this sort of trials are many times non-powered…

If that’s accepted by the agency, fine. ❝ … and only descriptive.

Mine were confirmatory (luckily never beyond $$\small{r=8}$$). In a purely exploratory setting you might consider more liberal $$\small{\left\{\theta_1,\theta_2\right\}}$$. Hummel et al.4 proposed even $$\small{\left\{0.50,2.0\right\}}$$…
See also the vignette of the package PowerTOST.

1. Smith BP, Vandenhende FR, DeSante KA, Farid NA, Welch PA, Callaghan JT, Forgue S. Confidence Interval Criteria for Assessment of Dose Proportionality. Pharm Res. 2000; 17(10): 1278-1283. doi:10.1023/A:1026451721686.
2. Wolfsegger MJ, Bauer A, Labes D, Schütz H, Vonk R, Lang B, Lehr S, Jaki TF, Engl W, Hale MD. Assessing goodness-of-fit for evaluation of dose-proportionality. Pharm. Stat. Early View 15 Oct 2020. doi:10.1002/pst.2074.
3. Interesting, since in the context of BE the EMA prefers a model with all effects fixed (ANOVA).
4. Hummel J, McKendrick S, Brindley C, French R. Exploratory assessment of dose proportionality: review of current approaches and proposal for a practical criterion. Pharm. Stat. 2009; 8(1): 38–49. doi:10.1002/pst.326.

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