Applied Survival Analysis

Chapter 12 - Multistate Models

Lu Mao
lmao@biostat.wisc.edu

Department of Biostatistics & Medical Informatics

University of Wisconsin-Madison

Outline

  1. Basic concepts and examples
  2. Transition intensity and probability
  3. Cox-type Markov and semi-Markov models
  4. Analysis of the German breast cancer study

\[\newcommand{\d}{{\rm d}}\] \[\newcommand{\T}{{\rm T}}\] \[\newcommand{\dd}{{\rm d}}\] \[\newcommand{\cc}{{\rm c}}\] \[\newcommand{\pr}{{\rm pr}}\] \[\newcommand{\var}{{\rm var}}\] \[\newcommand{\se}{{\rm se}}\] \[\newcommand{\indep}{\perp \!\!\! \perp}\] \[\newcommand{\Pn}{n^{-1}\sum_{i=1}^n}\] \[ \newcommand\mymathop[1]{\mathop{\operatorname{#1}}} \]

Concepts & Examples

Definition and Concepts

  • Multistate processes
    • Stochastic process with discrete values indicating the subject’s current state
      • An alternative way of looking at (complex) time-to-event data
    • Example: healthy \(\to\) ill (hospitalized) \(\to\) dead
  • Terminology
    • Transition: change of state, e.g., relapse, hospitalization, death
    • Absorbing state: a terminal state (no transition after entering), e.g., dead
    • Transient state: a non-terminal state, e.g., hospitalized \(\to\) discharge/death

Life-Death & Competing Risks

  • Multistate model
    • A model specifying possible transitions
  • Examples
    • Life-death (two states: transient \(\to\) absorbing)
    • Competing risks (\(K+1\) states: transient \(\to\) \(K\) absorbing states)

Illness-Death Model

  • Examples
    • Illness-death model (three states: healthy, ill/hospitalized, dead)
    • Semi-competing risks
      • Progressive: healthy \(\to\) ill \(\to\) dead
      • Reversible: healthy \(\rightleftharpoons\) ill \(\to\) dead

Recurrent Events

  • Examples
    • Recurrent events (with death) (progressive transient states \(\to\) absorbing)
  • Two types of processes
    • Progressive: only forward, no turning back
    • Reversible: may go back and forth (hospital admission/discharge)

Transition Intensity & Probability

Outcome Data

  • Outcome: multistate process \[ Y(t)\in \{0, 1,\ldots, K\},\,\,\, t\geq 0 \]
    • \((K+1)\) distinct states
    • Example
      • \(0=\) remission, \(1=\) relapsed, \(2\) = dead
      • \(0=\) 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 \(Y(\cdot)\)
    • With (time-varying) covariates \[\begin{equation}\label{eq:multi_state:lh2} \mathcal H^*(t)=\{Y(u), Z:0\leq u\leq t\} \end{equation}\] \[\begin{equation}\label{eq:multi_state:lh3} \mathcal H^*(t)=\{Y(u), Z(u):0\leq u\leq t\} \end{equation}\]
      • Past data include baseline and previous trajectory of covariates

Intensity Function: Examples (I)

  • With \(Y(0)\equiv 0\)
    • Life-death model (\(T\): time to death) \[ \lambda_{01}\{t\mid\mathcal H^*(t-)\} = \lambda(t):\,\, \mbox{hazard of $T$} \]
    • Competing risks (\(T\): time to overall failure) \[ \lambda_{0k}\{t\mid\mathcal H^*(t-)\} = \lambda_k(t):\,\, \mbox{$k$th cause-specific hazard} \]

Intensity Function: Examples (II)

  • Recurrent events
    • Counting process: \(N^*(t)\)
    • Intensity for recurrent-event process (Chapter 9) \[\begin{equation}\label{eq:rec:intensity} \ell\{t\mid \overline{N}^*(t-)\}\dd t=E\{\dd N^*(t)\mid \overline{N}^*(t-)\} \end{equation}\]
    • Intensity for transition \[\begin{align} \lambda_{k-1,k}\{t\mid\mathcal H^*(t-)\}\dd t &= E\{\dd N^*(t)\mid N^*(t-) =k-1, \overline{N}^*(t-)\}\\ &= \mbox{A sub-function of }\ell\{t\mid \overline{N}^*(t-)\} \end{align}\]

Markov & Semi-Markov Processes

  • Markov: transition independent of past given current state \[ \lambda_{kj}\{t\mid\mathcal H^*(t-)\}=\lambda_{kj}(t)\,\,\, \mbox{fixed function} \]
    • Homogeneous Markov: \(\lambda_{kj}(t)\equiv \lambda_{kj}\)

  • 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

Transition Probability: Definition

  • Transition Probability
    • Definition \[\begin{equation}\label{eq:multi_state:trans_prob} P_{kj}\{s,t\mid\mathcal H^*(s-)\}=\pr\{\underbrace{Y(t)=j}_{\text{Future state}}\mid \underbrace{Y(s-)=k}_{\text{Current state}}, \underbrace{\mathcal H^*(s-)}_{\text{Past experience}}\} \end{equation}\]
      • Probability of being in state \(j\) at future time \(t\) given current state \(k\) at \(s\)
    • Relation with intensity \[\lambda_{kj}\{t\mid\mathcal H^*(t-)\}\dd t=P_{kj}\{t,t\mid\mathcal H^*(t-)\}\] \[\begin{align} \mbox{Transition probability } \mymathop{\rightleftharpoons}_{\mbox{cumulative}}^{\mbox{instantaneous}} \mbox{ intensity} \end{align}\]
      • Similarly to c.d.f. \(\leftrightarrows\) hazard

Transition Probability: Examples (I)

  • Life-death model
    • \(\lambda_{01}(t)\): hazard for death \[\begin{align} P_{00}(0, t)&=\exp\left\{-\int_0^t\lambda_{01}(u)\dd u\right\}\,\,\, \mbox{(Survival)}\\ P_{01}(0, t)&=1-\exp\left\{-\int_0^t\lambda_{01}(u)\dd u\right\}\,\,\, \mbox{(c.d.f. of death)} \end{align}\]
  • Competing risks
    • \(\lambda_{0k}(t)\): cause-specific hazard for \(k\)th risk \[\begin{align} P_{00}(0, t)&=\exp\left\{-\int_0^t\sum_{k=1}^K\lambda_{0k}(u)\dd u\right\}\,\,\, \mbox{(Overall survival)}\\ P_{0k}(0, t)&=\int_0^t P_{00}(0, u-)\lambda_{0k}(u)\dd u\,\,\, \mbox{(Cumulative incidence of $k$th risk)} \end{align}\]

Transition Probability: Examples (II)

  • Illness-death model (semi-competing risks)
    • \(\lambda_{01}(t), \lambda_{02}(t)\): 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 \(\lambda_{12}\{t\mid\mathcal H^*(t-)\}\)
    • Markov Process: \(\lambda_{12}\{t\mid\mathcal H^*(t-)\}=\lambda_{12}(t)\) \[\begin{equation}\label{eq:multi_state:ill_death} P_{12}(s, t)=\exp\left\{-\int_s^t\lambda_{12}(u)\dd u\right\} \end{equation}\]

Markov Process

  • Transition intensity matrix \[ \dd\Lambda_{kj}(u) \;=\; \begin{cases} \lambda_{kj}(u) \dd u, & \text{if } k \neq j, \\ -\sum_{l \neq k} \lambda_{kl}(u) \dd u, & \text{if } k = j \end{cases} \] i.e. \[ \dd\mathbf{\Lambda}(u) \;=\; \begin{bmatrix} -\sum_{k=1}^K \lambda_{0k}(u) & \lambda_{01}(u) & \cdots & \lambda_{0K}(u) \\ \lambda_{10}(u) & -\sum_{k \neq 1} \lambda_{1k}(u) & \cdots & \lambda_{1K}(u) \\ \vdots & \vdots & \ddots & \vdots \\ \lambda_{K0}(u) & \lambda_{K1}(u) & \cdots & -\sum_{k=0}^{K-1} \lambda_{Kk}(u) \end{bmatrix} \dd u \]

Transition Probability vs. Intensity

  • Transition probability matrix
    • Definition \[ \textbf{P}(s, t) = \begin{bmatrix} P_{00}(s, t) & P_{01}(s, t) & \cdots & P_{0K}(s, t) \\ P_{10}(s, t) & P_{11}(s, t) & \cdots & P_{1K}(s, t) \\ \vdots & \vdots & \ddots & \vdots \\ P_{K0}(s, t) & P_{K1}(s, t) & \cdots & P_{KK}(s, t) \end{bmatrix} \]
  • Product limit
    • \(s = u_0 < u_1 < \cdots < u_m = t\), with \(u_r - u_{r-1} = \dd u\) \[ \textbf{P}(s, t) \;=\; \lim_{\dd u \to 0} \prod_{r = 1}^m \bigl\{ \mathbf{I} \,+\, \dd\mathbf{\Lambda}(u_r) \bigr\} \]
    • \(\mathbf{I}\): Identity matrix

Estimating Discrete Intensities

  • Observed data
    • \(C\): censoring time
    • \(N_{kj}(t) = N^*_{kj}(t \wedge C)\) \[\begin{equation}\label{eq:multi_state:censoredY} \tilde Y(t) \;=\; \begin{cases} Y(t), & \text{if uncensored by } t, \\ -1, & \text{if censored by } t. \end{cases} \end{equation}\]
  • Nelsen–Aalen-type estimator \[\begin{align}\label{eq:multi_state:np_intensity} \dd\hat\Lambda_{kj}(t) &\;=\; \frac{\text{Number of observed }k \to j \text{ transitions}} {\text{Number of subjects at risk of a } k \to j \text{ transition}} \notag\\ &\;=\; \frac{\sum_{i=1}^n \dd N_{kji}(t)}{\sum_{i=1}^n I\left\{ \tilde Y_i(t) = k \right\}} \end{align}\]

Aalen–Johansen Estimator

  • Product-limit estimator \[ \hat{\textbf{P}}(s, t) \;=\; \lim_{\dd u \to 0} \prod_{r = 1}^m \bigl\{ \mathbf{I} \,+\, \dd\hat{\mathbf{\Lambda}}(u_r) \bigr\} \]
    • \(s = u_0 < u_1 < \cdots < u_m = t\), with \(u_r - u_{r-1} = \dd u\)
    • Reduces to KM, Gray’s estimator in life-death, competing risks models
  • Robustness property
    • Valid even without Markov assumption

Useful Functionals

  • State-occupancy probability \[ P_k(t) = \pr\{Y(t) = k\} = \sum_{l=0}^K \underbrace{\pr\{Y(0) = l\}}_{\text{Initial state}} \, \underbrace{P_{lk}(0, t)}_{\text{Transition probability}} \]
    • 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 \[\begin{equation}\label{eq:multi_state:lik} \prod_{k\neq j}^K\prod_r\lambda_{kj}\left\{v_k^{(r)}\mid\mathcal H^*(v_k^{(r)}-)\right\}^{\delta_{kj}^{(r)}} \exp\left[-\int_{u_k^{(r)}}^{v_k^{(r)}}\lambda_{kj}\left\{t\mid\mathcal H^*(t-)\right\}\dd t\right] \end{equation}\]
    • Factorize by transition intensities

Inference for Transition Intensity

  • Likelihood for \(k\to j\) transition
    • Progressive process \[ \lambda_{kj}\left\{v_k\mid\mathcal H^*(v_k-)\right\}^{\delta_{kj}} \exp\left[-\int_{u_k}^{v_k}\lambda_{kj}\left\{t\mid\mathcal H^*(t-)\right\}\dd t\right] \]
    • Left-truncated, right-censored \(N_{kj}^*(t)\) with “hazard” \(\lambda_{kj}\left\{t\mid\mathcal H^*(t-)\right\}\)
      • \(u_k\): left-truncation (late entry) time
      • \(v_k\): possibly censored event time
      • \(\delta_{kj}\): event indicator

Semiparametric Regression (I)

  • Modeling target
    • Covariate-specific transition intensity \[ \lambda_{kj}\{t\mid\mathcal H^*(t-), \overline Z(t)\} = E\{\dd N_{kj}^*(t)\mid Y(t-)=k,\mathcal H^*(t-), \overline Z(t)\} \]
      • \(Z(t)\): possibly time-dependent covariates
  • Multiplicative intensity models
    • Modulated (by covariates) Markov models \[\begin{equation}\label{eq:multi_state:markov} \lambda_{kj}\{t\mid\mathcal H^*(t-), \overline Z(t)\}=\exp\{\beta_{kj}^\T Z(t)\}\lambda_{kj0}(t) \end{equation}\]
      • Risk of transition independent of past given (current values of) covariates
      • \(\beta_{kj}\): log-intensity (risk) ratios of going to state \(j\) for those in state \(k\)

Semiparametric Regression (II)

  • Multiplicative intensity models
    • Modulated semi-Markov models \[\begin{equation}\label{eq:multi_state:semi_markov} \lambda_{kj}\{t\mid\mathcal H^*(t-)\}=\exp\{\beta_{kj}^\T Z(t)+\gamma_{kj} B(t)\}\lambda_{kj0}(t) \end{equation}\]
      • Risk of transition depend on past through time last spent in current state and (current values) of covariates
      • \(\gamma_{kj}\): log-intensity (risk) ratio with unit increase in the time spent in current state
    • Example: illness-death model
      • Semi-Markov for \(1\to 2\): risk of death in illness depends on how long the patient has been ill
      • \(\gamma_{kj}\): log-intensity (risk) ratio for death with unit increase in the time being ill

Software: survival::coxph() (I)

  • Input data format
    • Long format, one transition per row
    • (start, stop): \((u_k^{(r)}, v_k^{(r)})\); status: \(\delta^{(r)}_{kj}\)
    • (from, to): \((k, j)\)
# > gbc_ms
# id     start      stop status from to hormone age meno size grade nodes 
#  1  0.000000 43.836066      1    0  1       1  38    1   18     3     5  
#  1  0.000000 43.836066      0    0  2       1  38    1   18     3     5 
#  1 43.836066 74.819672      0    1  2       1  38    1   18     3     5 
#  2  0.000000 46.557377      1    0  1       1  52    1   20     1     1 
#  2  0.000000 46.557377      0    0  2       1  52    1   20     1     1 
#  2 46.557377 65.770492      0    1  2       1  52    1   20     1     1 
#   ...

Software: survival::coxph() (II)

  • Basic syntax for multiplicative intensity model
    • Fit each type of transition \((k\to j)\) separately
      • Factorization of likelihood
    • covariates: \(Z(t)\) and/or elements of \(\mathcal H^*(t-)\), e.g., \(B(t)\)
# Fit a model for transition k->j
obj <- coxph(Surv(start, stop, status) ~ covariates, 
       subset = (from == k) & (to == j))

GBC: A Multistate Perspective

What’s New

  • Study information
    • Population: 686 patients with primary node positive breast cancer
    • Endpoints: relapse of cancer and death (semi-competing risks)
      • Previously analyzed by time to earlier of relapse/death (RFS)
  • Multistate perspective
    • Does hormonal treatment (or other risk factors) have differential effects on
      1. Relapse
      2. Death in remission
      3. Death after relapse

Multistate Setup

  • Illness-death model
# > gbc_ms
# id     start      stop status from to hormone age meno size grade nodes 
#  1  0.000000 43.836066      1    0  1       1  38    1   18     3     5  
#  1  0.000000 43.836066      0    0  2       1  38    1   18     3     5 
#  1 43.836066 74.819672      0    1  2       1  38    1   18     3     5 
#  2  0.000000 46.557377      1    0  1       1  52    1   20     1     1 
#   ...

Multiplicative Intensity Models (I)

  • Three (semi-Markov) models
# 1. Transition 0 -> 1: remission to relapse
obj01 <- coxph(
  Surv(start, stop, status) ~ hormone + meno + agec + size + nodes + prog + 
    estrg + strata(grade),
  data   = gbc_ms,
  subset = (from == 0 & to == 1)
)

# 2. Transition 0 -> 2: remission to death
obj02 <- coxph(
  Surv(start, stop, status) ~ hormone + meno + size + nodes + prog + estrg +
    strata(grade),
  data   = gbc_ms,
  subset = (from == 0 & to == 2)
)

Multiplicative Intensity Models (II)

  • Three (semi-Markov) models
# 3. Transition 1 -> 2: relapse to death
# tt(start) is used to represent time spent in the current state
obj12 <- coxph(
  Surv(start, stop, status) ~ hormone + meno + agec + size + nodes + prog + 
    estrg + strata(grade) + tt(start),
  data   = gbc_ms,
  subset = (from == 1 & to == 2),
  tt     = function(x, t, ...) {
    t - x
  }
)

Regression Results (I)

  • Regression table
    • Hormonal therapy beneficial on relapse (32.2% risk reduction; \(p\)-value 0.005)

Regression Results (II)

  • Regression table
    • Hormonal therapy no significant effect on progression from relapse to death
    • One month away from relapse patient is at 1.7% higher risk of mortality (non-significant)

Conclusion

Notes

  • Methods and Software
    • Nonparametric estimation of transition probability for Markov process: Chapter IV.4 of Andersen et al. (1991)
    • Text: Cook and Lawless (2018)
    • R-packages
      • mstate: competing risks models and prediction of transition probabilities
      • msm: Hidden Markov Models (HMMs) & standard models

Summary

  • Multistate models
    • Multistate process \(Y(t)\in \{0, 1,\ldots, K\}\)
    • Transition intensity: risk of going to another state given current state and history
  • Cox-type multiplicative intensity models
    • Markov or semi-Markov models for intensity \(k\to j\)
      • coxph(Surv(start, stop, status) ~ covariates, subset = (from==k)&(to==j))

HW5 (Due April 15)

  • Problem 8.18
  • Choose one
    • Problem 10.6
    • Problem 10.15
  • Problem 11.18 (a) (“Data > Renal Transplants Study”)
  • (Extra credit) Problem 11.2