Test

## [1] "DualEndpointBeta"

The relationships between dose and toxicity and between dose and PD biomarker will be modelled simultaneously.

A probit log normal model will describe the relationship between dose and toxicity: \[ \Phi^{-1}(Tox | d) = f(X = 1 | \theta, d) = \alpha + \beta \cdot log(d/d^*) \]where d* denotes a reference dose.

The prior for θ is given by\[ \boldsymbol\theta = \begin{bmatrix}\alpha \\ \beta\end{bmatrix}\sim N \left(\begin{bmatrix} 0.00 \\ 1.00\end{bmatrix} , \begin{bmatrix} 1.00 & 0.00 \\ 0.00 & 1.00\end{bmatrix} \right) \]

The reference dose will be 10.00.

The PD biomarker response w at dose d is modelled as \[ w(d) \sim N(f(d), \sigma_w^2) \]

where f(d) is a parametric rescaled beta density function such that

\[ f(d) = E_0 + (E_{max} - E_0) \times Beta(\delta_1, \delta_2) \times \left(\frac{d}{d_{max}}\right)^{\delta_1} \times \left(1 - \frac{d}{d_{max}}\right)^{\delta_2} \]

where d_max is the maximum dose in the dose grid, δ₁ and δ₂ are the parameters of the Beta function and E₀ and E_max are, respectively, the minimum and maximum levels of the biomarker. The mode can be written as

\[ \text{mode} = \frac{\delta_1}{\delta_1 + \delta_2} \]

and this is the parameterisation used to define the model.

In this case,

\[ E_0 \sim U(0, 100) \]

\[ E_{max} \sim U(0, 500) \]

\[ \delta_1 \sim U(0, 5) \]

\[ \text{mode} \sim U(1, 15) \]

and

\[ d_{max} = 1000 \]