Test

## [1] "LogisticNormalFixedMixture"

A mixture of 2 logistic log normal models with fixed weights will describe the relationship between dose and toxicity: \[ p(Tox | d) = f(X = 1 | \theta, d) = \frac{e^{\alpha + \beta \cdot log(d/d^*)}}{1 + e^{\alpha + \beta \cdot log(d/d^*)}} \]where d* denotes a reference dose.

The prior for θ is given by\[ \theta = \begin{bmatrix} \alpha \\ \beta \end{bmatrix} \sim \sum_{i=1}^{2}w_i \cdot N \left( \mathbf{\mu}_i , \mathbf{\Sigma}_i \right) \] with \[ \sum_{i=1}^{2} w_i = 1 \] The individual components of the mixture are \[ \boldsymbol\theta_1 = \begin{bmatrix}\alpha \\ \beta\end{bmatrix}\sim N \left(\begin{bmatrix}-0.85 \\ 1.00\end{bmatrix} , \begin{bmatrix} 1.00 & -0.50 \\ -0.50 & 1.00\end{bmatrix} \right) \]

with weight 0.3 and \[ \boldsymbol\theta_2 = \begin{bmatrix}\alpha \\ \beta\end{bmatrix}\sim N \left(\begin{bmatrix} 1.00 \\ 1.50\end{bmatrix} , \begin{bmatrix} 1.20 & -0.45 \\ -0.45 & 0.60\end{bmatrix} \right) \]

with weight 0.7 The reference dose will be 50.00.