Motivating Examples

More Generally

• Complex patient outcomes
  • Multistate process \(Y(t)\in\{0, 1,\ldots, K,\infty\}\) (Cook & Lawless, 2018)
  • Traditional composite endpoint
    • Time to first event \(\tilde T =\inf\{t: Y(t) > 0\}\) \(\to\) event-free survival
  • Hierarchical composite endpoint
    • Death (> metastasis) > relapse
    • Death > number of hospitalizations

Model-free Effect Size

• Two-sample comparison
  • \(Y^\a(t)\): patient in group \(a\) (\(a=1\): treatment; \(0\): control)
• Estimand for traditional composite
  • Event-free survival rate: \(\tilde S^\a(t)=P(\tilde T^\a >t)\)
  • Restricted mean event-free survival time (RMEST) \[ \mu_E^\a(\tau) = E\{\min(\tilde T^\a ,\tau)\}=\int_0^\tau \tilde S^\a(t)\d t \]
  • Net event-free survival time attributable to treatment \[ \mu_E(\tau) = \mu_E^\t(\tau) - \mu_E^\c(\tau) \]

Estimand for Hierarchical Composite

• Win ratio (Pocock et al., 2012)
  • Standard estimand involves censoring, but fixable (Mao & Wang, 2024)
  • Win probability \[ w^\a(t) = {\mbox{probability group $a$ wins by time $t$}} \]
    • Estimable by IPCW (Dong et al., 2020b, 2021)
  • Restricted WR: \(w^\t(\tau)/w^\c(\tau)\)
  • Restricted proportion in favor (PIF): \(w^\t(\tau) - w^\c(\tau)\)
• Time difference?
  • Win-loss: proportion \(\to\) average time

Time on a Win or Loss

• Pairwise win-loss time (Buyse et al., 2025)
  • \(Y^\t(t)\) vs \(Y^\c(t)\) over \([0, \tau]\)
  • Win time \(=\) time residing in a lower-tiered state \[ W^{(a, 1-a)}(\tau)=\int_0^\tau I\{Y^\a(t)<Y^\b(t)\}\d t \]

Net Average Win Time

• Etimand of treatment effect
  • Restricted mean time in favor (RMT-IF) of treatment (Mao, 2023a) \[ \mu(\tau) = E\{W^{(1, 0)}(\tau) - W^{(0, 1)}(\tau)\} \]
    • Net average time in a more favorable state gained by treatment
  • Decomposition \[\begin{align} \mu(\tau) = \sum_{k=1}^{K,\infty} \mu_k(\tau) := \sum_{k=1}^{K,\infty} E\left\{W_k^{(1, 0)}(\tau) - W_k^{(0, 1)}(\tau)\right\} \end{align}\]
    • \(W_k^{(a, 1-a)}(\tau)=\int_0^\tau I\{Y^\a(t)<Y^\b(t) = k\}\d t\): time won on state \(k\) (being in a better state)
    • \(\mu_k(\tau)\): net average time won on state \(k\) (remission vs relapse; alive vs dead)
      • \(\mu_\infty(\tau)\): difference in RMST

Simplify for Progressive Processes

• Progressive process
  • Definition: \(Y^\a(t)\leq Y^\a(s)\) for all \(0\leq t\leq s\)
    • Only marching forward (all earlier examples)
  • Transition time \(T_k^\a\): time to transition to a state \(\geq k\)
    • \(T_1^\a\): time to relapse/metastasis/death
    • \(T_2^\a\): time to metastasis/death
    • \(T_\infty^\a=D^\a\): time to death
  • Reformulation: \(Y^\a(\cdot)\equiv \big\{0\leq T_1^\a\leq\cdots\leq T_K^\a\leq T_\infty^\a\big\}\)
    • A progressive process \(\Longleftrightarrow\) a sequence of transition marks

Delve into Estimand

• Average win time on state \(k\)
  • Re-expression with \(S_k^\a(t)=P\{T_k^\a > t\}\) \[\begin{align} E\{W_k^{(a, 1-a)}(\tau)\}&=E\left\{\int_0^\tau I\{Y^\a(t)<Y^\b(t) = k\}\d t\right\}\\ &=\int_0^\tau P\{Y^\a(t)< k\}P\{Y^\b(t) = k\}\d t\\ &=\int_0^\tau P\{T_k^\a > t\}P\{T_k^\b \leq t < T_{k+1}^\b \}\d t\\ &=\int_0^\tau S_k^\a(t)\left\{S_{k+1}^\b(t) - S_k^\b(t)\right\}\d t\\ \end{align}\]
  • Net average win time \[\mu_k(\tau)=E\{W_k^{(1, 0)}(\tau)\}-E\{W_k^{(0, 1)}(\tau)\}= \int_0^\tau \left\{S_k^\t(t)S_{k+1}^\c(t) - S_k^\c(t)S_{k+1}^\t(t)\right\}\d t\]

Observed Data & Estimation

• Censored observations \[ (X_k^\a, \delta_k^\a),\,\,\, k =1,\ldots, K, \infty \]
  • \(X_k^\a = \min(T_k^\a, C^\a)\); \(\delta_k^\a = I(T_k^\a \leq C^\a)\); \(C^\a=\)censoring time
  • Kaplan–Meier estimator \(\hat S_k^\a(t)\)
• Estimation
  • Plug-in KM estimator \[ \hat\mu_k(\tau)= \int_0^\tau \left\{\hat S_k^\t(t)\hat S_{k+1}^\c(t) - \hat S_k^\c(t)\hat S_{k+1}^\t(t)\right\}\d t \]
  • Robust variance estimator

Hypothesis Testing

• Test of overall effect
  • \(\chi_1^2\) test based on \(\hat\mu(\tau)=\sum_{k=1}^{K,\infty}\hat\mu_k(\tau)\) for \[ H_0: \mu(\tau)= 0 \]
• Joint test on components
  • \(\chi_{K+1}^2\) test based on \(\hat\mu_1(\tau),\ldots,\hat\mu_K(\tau),\hat\mu_\infty(\tau)\) \[ H_0: \mu_1(\tau)=\cdots=\mu_K(\tau)=\mu_\infty(\tau) \]
    • Or individual components for secondary analyses
  • Further decomposition: \(\mu_k(\tau) = \sum_{j=0}^{k-1}\mu_{jk}(\tau)\) (Mao & Wang, 2024)
    • \(\mu_2(\tau)=\mu_{02}(\tau) + \mu_{12}(\tau)\): net non-metastic time in remission or after relapse
    • \(\mu_\infty(\tau)=\mu_{0,\infty}(\tau), \mu_{1,\infty}(\tau), \mu_{2,\infty}(\tau)\): net survival time in different states

Sample Size Calculation

• Semi-competing risks (illness-death model)
  • Gumbel–Hougaard copula (Mao, 2023b)
  • Input parameters
    • Baseline: Hazards for death & relapse, association parameter
    • Effect sizes: Component-wise hazard ratios
• Recurrent events with death
  • Homogeneous Markov model (Mao, 2023c)
  • Input parameters
    • Baseline: Intensities for another hospitalization or death, having had \(k-1\) hospitalizations \((k=1,2,\ldots)\)
    • Effect sizes: Intensity (risk) ratios for all transitions

Software: rmt::rmtfit() (I)

• Input data format (long)
  • Standard multistate
    • status = k for entry into state \(k\), K+1 for death, 0 for censoring
  • Recurrent events with death
    • status = 1 for nonfatal event, 2 for death, 0 for censoring

head(hfaction)
#>       patid       time status trt_ab age60
#>  HFACT00001 0.60506502      1      0     1
#>  HFACT00001 1.04859685      0      0     1
#>  HFACT00002 0.06297057      1      0     1
#>  HFACT00002 0.35865845      1      0     1
#>  HFACT00002 0.39698836      1      0     1
#>  HFACT00002 3.83299110      0      0     1
#>  ...

Software: rmt::rmtfit() (II)

• Basic syntax

library(rmt)
# trt: binary treatment
obj <- rmtfit(id, time, status, trt, 
              type = c("multistate", "recurrent"))

• Output: a list of class rmtfit
  • obj$t: \(t\); obj$mu: a matrix of \((K+2)\) rows, \(\hat\mu_k(t)\) in \(k\)th row, \(\hat\mu(t)\) in last; obj$var: variances of point estimates in mu
  • summary(obj, tau) for summary results on \(\mu(\tau)\) (tau: \(\tau\))
  • plot(obj) to plot \(\hat\mu(t)\) against \(t\)

Example: HF-ACTION

• Exercise training vs usual care

Standard Analyses

• Traditional composite and overall survival

R-Code

library(rmt)
head(hfaction)
#>       patid       time status trt_ab age60
#>  HFACT00001 0.60506502      1      0     1
#>  HFACT00001 1.04859685      0      0     1
#> ...

# fit RMT-IF
obj <- rmtfit(hfaction$patid, hfaction$time, hfaction$status, hfaction$trt, 
              type = "recurrent")
summary(obj, Kmax=4, tau=3.97) ## combined recurrent events >= 4
# Restricted mean time in favor of group "1" by time tau = 3.97:
#   Estimate    Std.Err Z value Pr(>|z|)    
# Event 1   0.0140515  0.0498836  0.2817 0.778184    
# Event 2   0.0358028  0.0499618  0.7166 0.473619    
# Event 3   0.1385287  0.0409533  3.3826 0.000718 ***
# Event 4+ -0.0064731  0.0600813 -0.1077 0.914203    
# Survival  0.2384169  0.1143484  2.0850 0.037069 *  
# Overall   0.4203268  0.1777363  2.3649 0.018035 * 

Graphics

• \(\hat\mu(t)\) as a function of \(t\)

  • Overall RMT-IF becomes significant after 1 year (see lower CL)

  plot(obj, conf = TRUE, conf.col = "gray", lwd=2, xlab="t (years)",
       ylab = "expression(mu(t))", main = "")

Inference Results

• 4-year RMT-IF of exercise training

  • Training on average gains 5.1 months (\(P\)=0.018) in favorable state

    • 2.9 months net survival \(+\) 2.2 months net time with fewer hospitalizations (little effect on 1st)

    		Estimate	SE	P-value
    Hopitalization		2.18	1.22	0.073
    	1	0.17	0.60	0.778
    	2	0.43	0.60	0.474
    	3	1.66	0.49	<0.001
    	4+	-0.08	0.72	0.914
    Death		2.86	1.37	0.037
    Overall		5.04	2.13	0.018

Favorability Plot

• 4-year average win-loss times

Summary

• RMT-IF
  • A model-free estimand to measure treatment effect on hierarchical composite endpoints
  • Net average win time \(=\) Average win \(-\) loss time
    • \(\sum_k\) (Net average win time on \(k\)th component)
  • R-package: rmt
    • obj <- rmtfit(id, time, status, trt, type = c("multistate", "recurrent"))
    • summary(plot)
    • plot(obj)

Discussion

• Standard win-loss measures
  • Longer survivor wins categorically
• RMT-IF

  • Quantitative win-loss times

  • Loss on nonfatal event can (more than) offset win on survival

    • Disregard win/loss time on nonfatal event in presence of death?

Acknowledgments

• Funding

  • R01HL149875 (11/2019 – 7/2028)
  • DMS2015526 (7/2020 – 6/2024)

• Collaborators

  • KyungMann Kim, Tuo Wang, etc.