state j into state k at some point during the time for which they are in the system, conditional on the individual's death time, yi and covariate values, xi Nijk E-Step: N[ u] ijk = Pr(Xu = j, Xu+ u = k | yi 2 dy, xi) = Pr(Xu = j, Xu+ u = k, yi 2 dy | xi) Pr(yi 2 dy | xi) = Pr(Xu = j | xi) Pr(Xu+ u = k, | Xu = j, yi 2 dy, xi) Pr(yi 2 dy | Xu+ u = k, xi) Pr(yi 2 dy | xi) N[ u] ijk = Pr(Xu = j, Xu+ u = k | yi 2 dy, xi) = Pr(Xu = j, Xu+ u = k, yi 2 dy | xi) Pr(yi 2 dy | xi) = Pr(Xu = j | xi) Pr(Xu+ u = k, | Xu = j, yi 2 dy, xi) Pr(yi 2 dy | Xu+ u = k, xi) Pr(yi 2 dy | xi) = R yi 0 p exp n T exp( xi↵ ) uo ejtjkek 0 exp n T exp( xi↵ )( yi u ) o t exp( xi↵ ) du p exp n T exp( xi↵ ) yi o t exp( xi↵ ) Nijk EM approach to fitting Coxian phase-type regression models • The probability density function of the complete Markov process was amended to include covariate effects according to the phase-type regression model: f ( y ; p , T , ↵ ) = M Y i=1 ( n Y j=1 pBij j n Y j=1 exp ntjj exp( xi↵ ) Eij o n Y j=1 n Y k=0 k6=j ⇣tjk exp( xi↵ ) ⌘Nijk )