crmPack (version 1.0.6)

LogisticLogNormal-class: Standard logistic model with bivariate (log) normal prior

Description

This is the usual logistic regression model with a bivariate normal prior on the intercept and log slope.

Slots

mean

the prior mean vector \(\mu\)

cov

the prior covariance matrix \(\Sigma\)

refDose

the reference dose \(x^{*}\)

Details

The covariate is the natural logarithm of the dose \(x\) divided by the reference dose \(x^{*}\):

$$logit[p(x)] = \alpha + \beta \cdot \log(x/x^{*})$$ where \(p(x)\) is the probability of observing a DLT for a given dose \(x\).

The prior is $$(\alpha, \log(\beta)) \sim Normal(\mu, \Sigma)$$

The slots of this class contain the mean vector and the covariance matrix of the bivariate normal distribution, as well as the reference dose.

Note that the parametrization inside the class uses alpha0 and alpha1. alpha0 is identical to the intercept \(\alpha\) above and is the log-odds for a DLT at the reference dose x*. Therefore, the prior mean for alpha0 is the expected log-odds at the reference dose x* before observing any data. Note that the expected odds is not just the exp of the prior mean of alpha0, because the non-linearity of the exp transformation. The log-normal distribution on Wikipedia gives the formula for computing the prior mean of exp(alpha0). alpha0 is the log(alpha) in the Neuenschwander et al. (2008) paper. alpha1 is identical to \(\beta\) above and equals the beta in the Neuenschwander et al paper. exp(alpha1) gives the odds-ratio for DLT between two doses that differ by the factor exp(1) ~ 2.7. alpha1 has a log-normal distribution in the LogisticLogNormal model in order to ensure positivity of alpha1 and thus exp(alpha1) > 1.

Examples

model <- LogisticLogNormal(mean = c(-0.85, 1),
                           cov = matrix(c(1, -0.5, -0.5, 1), nrow = 2),
                           refDose = 50)