Algorightm PenCoxFrail

(1)

Initialization Choose starting values ${\widehat{\beta }}^{(0)}$, ${\widehat{\alpha }}^{(0)}$, ${\widehat{\mathit{b}}}^{(0)}$, ${\widehat{\theta }}^{(0)}$ (see Section (A.3) of supplementary materials).

(2)

Iteration  For l = 1, 2,… until convergence:  (a) Computation of parameters for given ${\widehat{\theta }}^{(l-1)}$     Based on the penalized score function spen(δ) = ∂lpen/∂δ and the penalized information matrix Fpen(δ) (see Section (A.1) of supplementary materials) the general form of a single Newton-Raphson step is given by $${\widehat{\delta }}^{(\mathrm{l})}={\widehat{\delta }}^{(l-1)}+({\mathit{F}}^{\mathit{pen}}({\widehat{\delta }}^{(l-1)}){)}^{-1}{\mathit{s}}^{\mathit{pen}}({\widehat{\delta }}^{(l-1)}).$$     As the fit is within an iterative procedure it is sufficient to use just one single step.  (b) Computation of variance-covariance components     Estimates ${\widehat{\mathit{Q}}}^{(\mathrm{l})}$ are obtained as approximate EM-type estimates (see Section (A.2) of supplementary materials), yielding the update ${\widehat{\theta }}^{(\mathrm{l})}$.