## [1] "DualResponsesSamplesDesign"A linear log-log model with a pseudo data prior will describe the relationship between dose and efficacy. The model is given by \[ y_i = \theta_1 + \theta_2 \cdot \log(\log(d_i + k)) + \epsilon_i \]where k is a constant (equal to 0), yi is the efficacy response for participant i, treated at dose di and εi is an error term. The εs are iid N(0, ν-1).
The prior distribution of ν is currently Γ( 1.00, 0.02).
The joint distribution of θ1 and θ2 is given by
\[ \boldsymbol\theta = \begin{bmatrix}\theta_1 \\ \theta_2\end{bmatrix} \sim N\left(\mu, \nu \boldsymbol{Q}^\intercal \right) \] where \(\boldsymbol{Q} = \boldsymbol{X_0}^\intercal\boldsymbol{X_0} + \boldsymbol{X}^\intercal\boldsymbol{X}\) and X0 is a design matrix based on the dose levels in the pseudo data and X is a design matrix based on the dose levels of participants’ no-DLAE efficacy responses in the observed data, if any.
No observed data has yet been recorded.
Together, the pseudo and observed data give rise to the following posterior for the model parameters:
\[ \boldsymbol\theta = \begin{bmatrix}\theta_1 \\ \theta_2\end{bmatrix}\sim N \left(\begin{bmatrix}-1.41 \\ 2.25\end{bmatrix} , \begin{bmatrix} 2.00 & 2.91 \\ 2.91 & 4.40\end{bmatrix} \right) \]
No participants are yet evaluable.
The dose grid is 25, 50, 75, 100, 125, 150, 175, 200, 225, 250, 275 and 300.
A logistic log normal model will describe the relationship between dose and DLAE: \[ p(Tox | d) = f(X = 1 | \theta, d) = \frac{e^{\phi_1 + \phi_2 \cdot log(d)}}{1 + e^{\phi_1 + \phi_2 \cdot log(d)}} \]The prior is expressed in terms of pseudo data and, consequently, the number of cases and of DLAEs need not be whole numbers.
The pseudo data are defined in the following table:
| Dose | N | DLAEs |
|---|---|---|
| 25 | 3 | 1.05 |
| 300 | 3 | 1.80 |
No observed data has yet been recorded.
Together, the pseudo and observed data give rise to the following posterior for the model parameters:
\[ \boldsymbol\phi = \begin{bmatrix}\phi_1 \\ \phi_2\end{bmatrix}\sim N \left(\begin{bmatrix}-1.95 \\ 0.41\end{bmatrix} , \begin{bmatrix} 10.05 & -2.08 \\ -2.08 & 0.46\end{bmatrix} \right) \]
≥ 36 patients dosed: If 36 or more participants have been treated.
| Min | Max | Increment |
|---|---|---|
| 25 | 300 | 2 |
| 300 | Inf | 2 |
Placebo will not be administered in the trial.
The dose recommended for the next cohort will be the one which is closest to Gstar, the dose for which the full Bayes posterior estimate of the probability of toxicity maximises the gain relative to the target toxicity rate, either during [0.35] or at the end of the trial [0.3].
A constant size of 3 participants.
The starting dose is 25.