Applied Survival Analysis

Chapter 10 - Competing/Semi-Competing Risks

Lu Mao

lmao@biostat.wisc.edu

Department of Biostatistics & Medical Informatics

University of Wisconsin-Madison

Outline

Cause-specific hazard and cumulative incidence
Non- and semi-parametric methods
Analysis of bone marrow transplantation study
Semi-competing risks and examples

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

Basic Quantities

Definition and Examples

Competing risks
- Definition: multiple types of failures, occurrence of one precludes others
  - Multiple latent risks competing with each other for first occurrence
- Examples: different causes of death
Defining feature
- One failure per subject
- Less info than multivariate failure times
- Implications for analysis and interpretation

Outcome Data & Identifiability

Target of inference: $(T, \Delta)$
- $T$: time to failure
- $\Delta \in \{1, \ldots, K\}$: type/cause of failure (categorical)
Multivariate perspective
- A conceptual framework \[\begin{equation}\label{eq:cmpr:mult} T=\min(T_1,\ldots, T_K) \hspace{1mm}\mbox{ and } \hspace{1mm}\Delta=\arg\min_{k=1,\ldots, K} T_k \end{equation}\]
  - $T_k$: time to $k$th latent risk $(k=1,\ldots, K)$ in absence of other risks
  - Competing risks $=$ partially observed multivariate failure times
- Identifiability: net distribution of latent $T_k$ not recognizable from $(T, \Delta)$
  - unless under unrealistic assumption of mutual independence of the $T_k$

Two Approaches

Two ways to characterize $(T, \Delta)$
- Cause-specific hazard
- Cumulative incidence (Sub-distribution)
Cause-specific hazard (CSH)
- Definition: instantaneous incidence of $k$th risk given overall “survival” \[\begin{equation}\label{eq:cmpr:cs_hazard} \dd\Lambda_k^\cc(t)=\pr(t\leq T<t+\dd t, \Delta=k\mid T\geq t) \end{equation}\]
- Example: $\dd\Lambda_1^\cc(t)=$ incidence rate for CV death among survivors at $t$
  - $k =1$: CV death; $k =2$: other causes of death
- Overall hazard: partitioned into CSH \[\begin{equation}\label{eq:cmpr:overall_hazard} \dd\Lambda(t)=\pr(t\leq T<t+\dd t \mid T\geq t) = \sum_{k=1}^K \dd\Lambda_k^\cc(t) \end{equation}\]

Cause-Specific Hazard

Limitations
- The cumulative CSH $\Lambda_k^\cc(t)$ not a meaningful quantity
- $\exp\{-\Lambda_k^\cc(t)\}$ not a survival function of any kind
Correspondence with “net hazard”
- Under (unrealistic) mutual independence of the latent $T_k$
- Hazard of $T_k$ identifiable and equal to $\Lambda_k^\cc(t)$
- Not recommended

Cumulative Incidence

Cumulative incidence function (CIF)
- Definition: probability of failure from $k$th risk in presence of other risks \[ F_k(t)=\pr(T\leq t, \Delta=k) \]
- Marginal quantity, easy to interpret (a real-world probability)
- a.k.a. Sub-distribution function
- $F(t)=\pr(T\leq t) = \sum_{k=1}^K F_k(t)$
Relationship
- CSH in terms of CIF (for a particular risk, no one-to-one correspondence) \[\begin{equation}\label{eq:cmpr:correspond} \dd\Lambda_k^\cc(t)=\frac{\dd F_k(t)}{1-\sum_{l=1}^K F_l(t-)} \end{equation}\]

CSH vs CIF

Example: effect on CSH may not align with CIF
- Treatment: $\Lambda_1^\cc(t)=\Lambda_2^\cc(t)=3t$
- Control: $\Lambda_1^\cc(t)=2t$ and $\Lambda_2^\cc(t)=t$

Comparison of Functions

CSH vs net hazard vs CIF

Methods for Competing Risks

Observed Data

A random $n$-sample \[(X_i, \delta_i, Z_i), \,\,\, i=1,\ldots, n\]
- $X=T\wedge C$
- $\delta=\Delta I(T\leq C)$ (0: censored; $k$: observed $k$th risk, $k=1,\ldots, K$)
- $C$: independent censoring time
- $Z$: covariates
- Observed counting process for $k$th risk \[ N_{ki}(t)=I(X_i\leq t, \delta_i=k) \]

Cause-Specific Hazard Models (I)

Proportional cause-specific hazards \[\begin{equation}\label{eq:cmpr:ph_csh} \pr(t\leq T<t+\dd t, \Delta=k\mid T\geq t, Z)=\exp(\beta_k^\T Z)\dd\Lambda_{k0}(t) \end{equation}\]
- A model on “survivors” (those who have not failed from any risk)
- $\beta_k$: log-hazard ratios for $k$th risk in survivors
- Estimation \[\begin{equation}\label{eq:cmpr:pls_csh} U_{nk}(\beta_k)=\Pn\int_0^\infty\left\{Z_i-\frac{\sum_{j=1}^n I(X_j\geq t)Z_j\exp(\beta_k^\T Z_j)}{\sum_{j=1}^n I(X_j\geq t)\exp(\beta_k^\T Z_j)}\right\}\dd N_{ki}(t) \end{equation}\]
  - Essentially treating non-$k$ risks as censoring
  - Easy to implement in survival::coxph()

Cause-Specific Hazard Models (II)

Under (unrealistic) independence of the latent $T_k$
- Proportional cause-specific hazards $=$ proportional net hazards
- Frailty to relax independence
- Not recommended (non-identifiability)
Log-rank test
- Risk-specific log-rank statisic with other risks treated as censoring
- “Kaplan-Meier” doesn’t work $\to$ estimand $\exp\{-\Lambda_k^\cc(t)\}$ is non-quantity
- Competing-risks equivalent of KM is Gray (1988) estimator of $F_k(t)$

CIF: Nonparametric Estimation

How to estimate $F_k(t)$
- Recall relationship \[\begin{equation*} \dd F_k(t)=S(t-)\dd\Lambda_k^\cc(t) \end{equation*}\]
- $S(t-)=\pr(T\geq t)$: By KM estimator $\hat S(t-)$ for overall failure
- $\dd\Lambda_k^\cc(t)$: By Nelsen-Aalen estimator (non-$k$ risks as censoring) \[ \dd\hat\Lambda_k^\cc(t)=\frac{\sum_{i=1}^n\dd N_{ki}(t)}{\sum_{i=1}^n I(X_i\geq t)} \]
- Gray estimator \[\hat F_k(t)=\int_0^t \hat S(u-)\dd\hat \Lambda_k^\cc(u)\]

CIF: Sub-Distribution Hazard

Sub-distribution hazard (SDH)
- Easier to work with hazard-like functions (unbounded)
- Time to $k$th risk in presence of other risks \[ T_k^*=TI(\Delta=k)+\infty I(\Delta\neq k) \]
  - $F_k(t)=\pr(T_k^*\leq t)$
- SDH: incidence of $k$th risk given not failing from it \[\begin{equation}\label{eq:cmpr:sdh} \dd\Lambda_k(t)=\pr(t\leq T_k^*<t+\dd t\mid T_k^*\geq t). \end{equation}\]
- Correspondence with CIF \[\begin{equation}\label{eq:cmpr:subdist} F_k(t)=1-\exp\{-\Lambda_k(t)\}. \end{equation}\]
- Model/test on SDH $\Leftrightarrow$ Model/test on CIF

CIF: Gray’s Test

Discrete version ($k$th risk) \[ \dd\Lambda_k(t)=\frac{\dd F_k(t)}{1-F_k(t-)} \]
Weighted log-rank-type test
- Testing $H_0:$ equality of CIF between two groups \[\begin{equation}\label{eq:cmpr:gray_tests} \int_0^\infty W(t)\big\{\dd\hat\Lambda_{k1}(t)-\dd\hat\Lambda_{k0}(t)\big\} \end{equation}\]
- $\hat\Lambda_{ka}(t)$: Gray’s CIF plug-in estimator for SDH in group $a$ $(a=1, 0)$
- $W(t)$: Weight function
- Extensions: multi-group, stratification, etc.

CIF: Fine-Gray Regression

Proportional sub-distribution hazards (Fine and Gray, 1999) \[\begin{equation}\label{eq:cmpr:fg} \pr(t\leq T_k^*<t+\dd t\mid T_k^*\geq t, Z)=\exp(\beta_k^\T Z)\dd\Lambda_{k0}(t) \end{equation}\]
- $\beta_k$: log-hazard ratios for $k$th risk in entire population (under other risks)
- Target population
  - Cause-specific hazard: survivors
  - Sub-distribution hazard: all, including those who have failed from other causes
- Estimation and inference: partial-likelihood score with IPCW to address dependent censoring by other risks

Software: `cmprk::cuminc()`

Basic syntax for Gray’s estimator & test

obj <- cuminc(ftime, fstatus, group, strata, rho = 0)

Input
- (ftime, fstatus): $(X, \delta)$
- group: group variable (optional); strata: strata variable (optional)
- rho: $\rho$ in weight $W(t)=\hat S(t-)^\rho$ (HF $G^\rho$ family)
Output: a list
- obj$Tests: tests results on each risk
- obj$"a k": CIF estimates for $k$th risk in group $a$
  - time: $t$; est: $\hat F_k(t)$; var: $\hat\var\{\hat F_k(t)\}$

Software: `cmprk::crr()` (I)

Basic syntax for Fine-Gray model

obj <- crr(ftime, fstatus, cov1, failcode = k)

Input
- (ftime, fstatus): $(X, \delta)$; cov1: $Z$
- failcode = k: models $k$th risk
Output: a list of class crr
- obj$coef: $\hat\beta_k$; obj$var: $\hat\var(\hat\beta_k)$
- obj$uftime: $t$; obj$bfitj: $\dd\hat\Lambda_{k0}(t)$

Software: `cmprk::crr()` (II)

Prediction of CIF by Fine-Gray model
- obj: a crr object for fit model
- z: new covariate data

# --- Method 1: use predict.crr() 
obj_pred <- predict(obj, z)

# --- Method 2: manual calculation
beta <- obj$coef
Lambda <- cumsum(obj$bfitj)
time <- obj$uftime
## calculate CIF based on FG model
cif <- 1- exp(- exp(sum(beta * z)) * Lambda)
## Same as obj_pred from Method 1
obj_pred <- cbind(time, cif)

Bone Marrow Transplant Study

Study Background

Study information
- Population: 864 multiple-myeloma leukemia patients undergoing allo-HCT cell transplantation
- Endpoints: Time from surgery to treatment-related mortality (TRM; death in remission) or relapse of leukemia
- Risk factors
  - Cohort indicator (years 1995–2000 or 2001–2005)
  - Type of donor (unrelated or HLA-identical sibling)
  - History of a prior transplant
  - Time from diagnosis to transplantation (<24 months, or ≥ 24 months)

Study Data

Data format

head(cibmtr)
#    time status cohort donor hist wait
# 1 0.010      2      1     1    1    1
# 2 0.066      2      1     1    1    0
# 3 0.099      1      1     1    1    1
#...

CIF by Donor Type

Gray’s nonparametric estimates/tests of CIFs by donor type

obj <- cuminc(cibmtr$time, cibmtr$status, cibmtr$donor, rho = 0)

Regression Analysis

Semiparametric regression
- Proportional cause-specific hazard
- Proportional sub-distribution hazard (FG)

#change k
k <- 1
#--- proportional cause-specific hazards -----------------------------
obj.cs <- coxph(Surv(time, status == k) ~ cohort + donor + hist + wait,
                data = cibmtr)
#--- Fine and Gray --------------------------------------------------
obj.fg <- crr(cibmtr$time, cibmtr$status, cibmtr[, 3:6], failcode = k)

Regression Results

Differential effects of prior surgery
- Relapse: increases risk by $42\%$
- TRM: decreases risk by $1-0.63 = 37\%$

Semi-Competing Risks

Definition and Examples

Semi-competing risks
- Terminal event (competing): death
- Non-terminal events (non-competing): hospitalization, relapse, etc.
Examples
- Death + relapse of cancer (German breast cancer study)
- Death terminates nonfatal event but not vice versa
Methods
- Marginal: cumulative incidence/frequency
- Frailty (Ch. 8); Multistate (Ch. 12); Composite (Ch. 13)

Data and Identifiability

Target of inference: $(T, D)$
- $T$: time to nonfatal event
- $D$: time to death
- Joint distribution \[H(s, t)=\pr(D>s, T>t)\]
- Identifiable region: $\{(s, t):0\leq t\leq s<\infty\}$
  - Possibly nonfatal $\to$ death, not vice versa
- Marginal models
  - Death: univariate event (KM, log-rank, Cox)
  - Nonfatal event: cumulative incidence (Gray, FG)

Recurrent Events

Outcome data: $\{N^*(\cdot), D\}$
- $N^*(t)$: number of recurrent events in presence of death
  - $\dd N^*(t)\equiv 0$ for $t>D$
  - Repeated tumor occurrences/hospitalizations before death
- Treating $D$ as censoring (AG/LWYY) $\to$ cause-specific event rate \[E\{\dd N^*(t)\mid D\geq t\}\]
- Incidence rate in survivors
Cumulative frequency
- Mean function in overall population, dead or alive \[ \mu(t)=E\{N^*(t)\} \]
- Extension of CIF

Marginal Approaches

Methods for cumulative frequency
- Gray-type nonparametric estimator/test (Ghosh and Lin, 2000)
- Proportional CF model (Fine-Gray-type) (Ghosh and Lin, 2002; 2003) \[\begin{equation}\label{eq:cmpr:pcf} E\{N^*(t)\mid Z\}=\exp(\beta^\T Z)\mu_0(t) \end{equation}\]
- Higher death rate can reduce CF
Joint analysis with mortality
- Ghosh-Lin for recurrent events $+$ Cox/log-rank for death
- $\chi^2$ test with 2 d.f.

Example: Bladder Tumor Study

Bladder cancer trial: thiotepa ($n=38$) vs placebo ($n=48$)
- Endpoints: tumor recurrences and death ($\chi_2^2$ test $p$-value 0.184)

Conclusion

Notes

Gray and FG
- Recommended by the European Group for Blood and Marrow Transplantation (EBMT) Statistical Committee
- Extensions
  - Proportional sub-distribution odds model (Eriksson et al., 2015)
  - Sub-distribution transformation models (Mao and Lin, 2017)
  - Regression of interval-censored competing risks data (Mao et al., 2018)
Semi-competing risks

Fine, J. P., Jiang, H., & Chappell, R. (2001). On semi-competing risks data. Biometrika, 88(4), 907-919.

Summary

Competing risks
- Cause-specific hazard: conditional failure rate on survivors
  - coxph(Surv(time, status == k) ~ covariates)
- Cumulative incidence: marginal failure probability under other risks
  - Gray’s estimator/test cmprk::cuminc(ftime, fstatus, group, strata, rho = 0)
  - Fine-Gray model cmprk::crr(ftime, fstatus, cov1, failcode = k)
Recurrent events in presence of death
- Ghosh-Lin methods (R packages mets, reReg, etc.)

Applied Survival Analysis

Outline

Basic Quantities

Definition and Examples

Outcome Data & Identifiability

Two Approaches

Cause-Specific Hazard

Cumulative Incidence

CSH vs CIF

Comparison of Functions

Methods for Competing Risks

Observed Data

Cause-Specific Hazard Models (I)

Cause-Specific Hazard Models (II)

CIF: Nonparametric Estimation

CIF: Sub-Distribution Hazard

CIF: Gray’s Test

CIF: Fine-Gray Regression

Software: cmprk::cuminc()

Software: cmprk::crr() (I)

Software: cmprk::crr() (II)

Bone Marrow Transplant Study

Study Background

Study Data

CIF by Donor Type

Regression Analysis

Regression Results

Semi-Competing Risks

Definition and Examples

Data and Identifiability

Recurrent Events

Marginal Approaches

Example: Bladder Tumor Study

Conclusion

Notes

Summary

Software: `cmprk::cuminc()`

Software: `cmprk::crr()` (I)

Software: `cmprk::crr()` (II)