\(0=\) event-free, \(k=1,\ldots, K\): cumulative number of hospitalizations (\(K + 1 =\) dead)
Life history \[\begin{equation}\label{eq:multi_state:lh1}
\mathcal H^*(t)=\{Y(u):0\leq u\leq t\}
\end{equation}\]
Past experience up to time \(t\)
Counting Process Notation
Counting transitions
\(N_{kj}^*(t)\): number of \(k\to j\) transitions by \(t\)\[
\dd N_{kj}^*(t)=I\{Y(t)=j, Y(t-)=k\}
\]
\(0\leq k\neq j\leq K\) (not all are possible)
Examples
Life-death model: \(N_{01}^*(t)\) counting process for death
Illness-death model: \(N_{01}^*(t)\) counting process for onset of illness, \(N_{02}^*(t)\) for death without illness, \(N_{12}^*(t)\) for death with illness
Life history re-formulated\[
\mathcal H^*(t)=\{Y(0), N_{kj}^*(u): 1\leq k\neq j\leq K; 0\leq u\leq t\}
\]
Intensity Function: Definition
Transition intensity
Instantaneous rate of transition given past \[\begin{align}\label{eq:multi_state:intensity}
\lambda_{kj}\{t\mid\mathcal H^*(t-)\}\dd t&=\pr\{\underbrace{Y(t+\dd t)=j}_{\text{Next state}}\mid \underbrace{Y(t-)=k}_{\text{Current state}}, \underbrace{\mathcal H^*(t-)}_{\text{Past experience}}\}\notag\\
&=E\{\dd N_{kj}^*(t)\mid Y(t-)=k,\mathcal H^*(t-)\}
\end{align}\]
Self-generating: completely determines distribution of \(Y(\cdot)\)
Semi-Markov: transition depends on past through time in current state \[
\lambda_{kj}\{t\mid\mathcal H^*(t-)\}=\lambda_{kj}\{t, B(t)\}
\]
\(B(t)\): time since last entry into current state
Example: risk of death depends on how long cancer has reappeared
Markov & Semi-Markov Examples
Illness-death model (semi-competing risks)
Transition intensities for \(0\to 1\) (onset of illness), \(0\to 2\) (death w.o. illness) \[
\lambda_{01}(t) \mbox{ and } \lambda_{02}(t): \,\, \mbox{always deterministic}
\]
No prior data to condition on
Transition intensity for \(1\to 2\) (death with illness) \[\begin{align}
\mbox{Markov model}:&\,\,\, \lambda_{12}\{t\mid\mathcal H^*(t-)\}=\lambda_{12}(t) \\
\mbox{Semi-Markov model}:&\,\,\, \lambda_{12}\{t\mid\mathcal H^*(t-)\}=\lambda_{12}(t, t - T)
\end{align}\]\(T\): time to onset of illness
\(\lambda_{01}(t), \lambda_{02}(t)\): cause-specific hazard for onset of illness or death w.o. illness \[
P_{00}(0, t)=\exp\left[-\int_0^t\{\lambda_{01}(u)+\lambda_{02}(u)\}\dd u\right]
\]
\(P_{01}(0, t)\) and \(P_{02}(0, t)\) more complicated and depends on \(\lambda_{12}\{t\mid\mathcal H^*(t-)\}\)
Example: probability patient will be ill but still alive in one year
Restricted mean sojourn time\[
\mu_k(\tau) = E\left[ \int_0^\tau I\{Y(t) = k\} \dd t \right] = \int_0^\tau P_k(t) \dd t
\]
Average time spent in state \(k\) (healthy, ill but alive, etc.) by time \(\tau\)
Cox-Type Models for Intensity
Observed Data
Censored life history
\(\mathcal H(t)=\{Y(u):0\leq u\leq t\wedge C\}\)
Alternate formulation
Focusing on (observed) transitions\(k\to j\)
\(u_k^{(r)}\): \(r\)th time subject enters state \(k\)
\(v_k^{(r)}\): exit or censoring time
\(\delta_{kj}^{(r)}\): indicator of observed transition to state \(j\) at \(v_k^{(r)}\) (0 if censored or going to other states)
Progressive process\(\to\)\(r\) redundant (why?)
Suitable for models on intensities \(\lambda_{kj}\{t\mid\mathcal H^*(t-)\}\)
Likelihood Function
Observed data\[\begin{equation}\label{eq:multi_state:obs}
(u_k^{(r)}, v_k^{(r)}, \delta_{kj}^{(r)}),\hspace{3mm}r=1,2,\ldots;\hspace{2mm} 1\leq k\neq j\leq K
\end{equation}\]
Collection of all observed transitions
Likelihood for a single subject\[\begin{equation}\label{eq:multi_state:lik}
\prod_{k\neq j}^K\prod_r\lambda_{kj}\left\{v_k^{(r)}\mid\mathcal H^*(v_k^{(r)}-)\right\}^{\delta_{kj}^{(r)}}
\exp\left[-\int_{u_k^{(r)}}^{v_k^{(r)}}\lambda_{kj}\left\{t\mid\mathcal H^*(t-)\right\}\dd t\right]
\end{equation}\]