LogisticKadane2-class: Reparametrized logistic model (version 2)
In 0liver0815/onc-crmpack-test: Object-Oriented Implementation of CRM Designs

This is the logistic model in the parametrization (version 2) of Kadane et al. (1980).

Let ρ_{0} = p(x_{0}) be the probability of a DLT of the placebo (no drug) dose x_{0}, and let MTD be the dose with target toxicity probability θ, i.e. p(MTD) = θ. Then it can easily be shown that the logistic regression model has intercept

\frac{MTD logit(ρ_{0})}{MTD}

and slope

\frac{logit(θ) - logit(ρ_{0})}{MTD}

The prior for MTD is a Gamma(shape,rate) distribution. The prior for ρ_{0} is a Beta(α,β) distribution.

The minimum d_{min} and maximum d_{max} planned dose, are used to set the initial value of the MTD arbitrarily as the average of those two. The initial value of ρ_{0} is set arbitrarily as

\frac{θ}{10}

The slots of this class, required for creating the model, are the target toxicity, the Beta and Gamma distribution parameters, as well as the minimum and maximum of the dose range. Note that these can be different from the minimum and maximum of the dose grid in the data later on.

theta: the target toxicity probability θ
dmin: the minimum of the dose range d_{min}
dmax: the maximum of the dose range d_{max}
alpha: the α shape parameter of the Beta(α,β) distribution
beta: the β shape parameter of the Beta(α,β) distribution
shape: the shape parameter of the Gamma(shape,rate) distribution
rate: the rate parameter of the Gamma(shape,rate) distribution.