Test

## [1] "Effloglog"

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), y_i is the efficacy response for participant i, treated at dose d_i and ε_i is an error term. The εs are iid N(0, ν^-1).

The posterior distribution of ν is currently Γ( 3.00, 0.98).

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.

The data observed to date are given in the following table:

ID	Cohort	Dose	Tox	Placebo	Response	Backfilled	W
1	1	25	FALSE	FALSE	NA	FALSE	0.31
2	2	50	FALSE	FALSE	NA	FALSE	0.42
3	2	50	FALSE	FALSE	NA	FALSE	0.59
4	3	75	FALSE	FALSE	NA	FALSE	0.45
5	4	100	TRUE	FALSE	NA	FALSE	0.60
6	4	100	TRUE	FALSE	NA	FALSE	0.70
7	5	225	TRUE	FALSE	NA	FALSE	0.60
8	6	300	TRUE	FALSE	NA	FALSE	0.52

Together, the pseudo and observed data give rise to the following posterior for the model parameters:

\[ \boldsymbol\theta = \begin{bmatrix}\theta_1 \\ \theta_2\end{bmatrix}\sim N \left(\begin{bmatrix}-2.82 \\ 2.71\end{bmatrix} , \begin{bmatrix} 6.00 & 8.27 \\ 8.27 & 11.63\end{bmatrix} \right) \]