The futility boundary \(\tilde{b}\) is computed by solving (under the alternative)
mHP.btilde(beta, cov.J)the type II error
the 3 x 3 covariance matrix
$$ P(\tilde{Z}_J^1\le\tilde{b} or \tilde{Z}_J^2\le\tilde{b}) = \epsilon\beta $$
where the superscripts denote the stage and \(\epsilon\) is the fraction of the type I error (\(\alpha\)) spent and \(\beta\) is the type II error. We make use of the joint normal density of \(Z_{J}\) (the overall group) at each of the three stages and the fact that the \(\tilde{Z_J}\) is merely a translation of \(Z_J\). So here the calculation is based on a mean of zero and has to be translated during use!