## 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

Whilst you

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 \(\small{\left\{\theta_1,\theta_2\right\}=\left\{0.80,1.25\right\}}\):

In my European hybrids acc. to 2001/83/EC, Article 10(3) I used a mixed-effects model

» 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

See also the vignette of the package

» […] 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 (*C*_{max}) 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}}\)

In my European hybrids acc. to 2001/83/EC, Article 10(3) I used a mixed-effects model

^{3}(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`

.- 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.

- 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.

- Interesting, since in the context of BE the EMA prefers a model with all effects fixed (ANOVA).

- 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.

—

Helmut Schütz

The quality of responses received is directly proportional to the quality of the question asked. 🚮

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*Dif-tor heh smusma*🖖Helmut Schütz

The quality of responses received is directly proportional to the quality of the question asked. 🚮

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### Complete thread:

- Sample size dosage strength proportionality study Laura R 2021-02-01 09:19
- Modified acceptance range, confirmatory vs. exploratoryHelmut 2021-02-01 11:34
- Modified acceptance range, confirmatory vs. exploratory Laura R 2021-02-02 18:03

- Modified acceptance range, confirmatory vs. exploratoryHelmut 2021-02-01 11:34