LogisticKadaneBetaGamma-class: `LogisticKadaneBetaGamma`

Description

LogisticKadaneBetaGamma is the class for the logistic model in the parametrization of KadaneDickeyWinklerSmithPeters1980;textualcrmPack, using a beta and a gamma distribution as the model priors.

Usage

LogisticKadaneBetaGamma(theta, xmin, xmax, alpha, beta, shape, rate)
.DefaultLogisticKadaneBetaGamma()

Arguments

theta: (proportion)
the target toxicity probability.
xmin: (number)
the minimum of the dose range.
xmax: (number)
the maximum of the dose range.
alpha: (number)
the first shape parameter of the Beta prior distribution rho0 = p(xmin) the probability of a DLT at the minimum dose xmin.
beta: (number)
the second shape parameter of the Beta prior distribution rho0 = p(xmin) the probability of a DLT at the minimum dose xmin.
shape: (number)
the shape parameter of the Gamma prior distribution gamma the dose with target toxicity probability theta.
rate: (number)
the rate parameter of the Gamma prior distribution gamma the dose with target toxicity probability theta.

Slots

theta: (proportion)
the target toxicity probability.

xmin

(number)
the minimum of the dose range.

xmax

(number)
the maximum of the dose range.

alpha

(number)
the first shape parameter of the Beta prior distribution of rho0 = p(xmin) the probability of a DLT at the minimum dose xmin.

beta

(number)
the second shape parameter of the Beta prior distribution of rho0 = p(xmin) the probability of a DLT at the minimum dose xmin.

shape

(number)
the shape parameter of the Gamma prior distribution of gamma the dose with target toxicity probability theta.

rate

(number)
the rate parameter of the Gamma prior distribution of gamma the dose with target toxicity probability theta.

Details

Let rho0 = p(xmin) be the probability of a DLT at the minimum dose xmin, and let gamma be the dose with target toxicity probability theta, i.e. $p(gamma) = theta$. Then it can easily be shown that the logistic regression model has intercept $$[gamma * logit(rho0) - xmin * logit(theta)] / [gamma - xmin]$$ and slope $$[logit(theta) - logit(rho0)] / [gamma - xmin].$$

The prior for gamma, is $$gamma ~ Gamma(shape, rate).$$. The prior for rho0 = p(xmin), is $$rho0 ~ Beta(alpha, beta).$$

References

Examples

Run this code

my_model <- LogisticKadaneBetaGamma(
  theta = 0.3,
  xmin = 0,
  xmax = 7,
  alpha = 1,
  beta = 19,
  shape = 0.5625,
  rate = 0.125
)

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