While-alive estimands for recurrent events in presence of death

Summarizing patient experience under differential length of exposure

Lu Mao

lmao@biostat.wisc.edu

Department of Biostatistics & Medical Informatics

University of Wisconsin-Madison

Motivating example

A cardiovascular trial (HF-ACTION) (O’Connor et al. 2009)

• Subpopulation: 741 heart failure patients

  • Median and max follow-up 2.5 and 3.9 years, respectively

• Treatment arms

  • Exercise training + usual care \((n_1=364)\)
  • Usual care alone \((n_0=377)\)

• Endpoints: Death and repeated hospitalizations

	Exercise training	Usual care
Death rate	13.5%	19.9%
Avg # hospitalization (SD)	1.8 (2.1)	2.0 (2.1)

A fundamental question

How do we measure treatment effect on recurrent hospitalizations when patients survive different lengths in different arms?

Longer survivors tend to experience more events…

 

Mathematical notation (\(a=1\): Treatment; \(a=0\): Control)

• \(D^{(a)}\): Survival time
• \(N_D^{(a)}(t) = I\left(D^{(a)}\leq t\right)\): Counting process for death
• \(N^{(a)}(t)\): Counting process for recurrent events (e.g., hospitalization)
  • No event after death
  • \({\rm d}N^{(a)}(t)=0\) for \(t>D^{(a)}\)

Standard methods

Two broad-based approaches…

• Conditional event rate

  • Standard analysis treating death as censoring, e.g., proportional rates model (LWYY) (Lin et al. 2000)
  • Estimand: \(E\left\{{\rm d}N^{(a)}(t)\mid D^{(a)}\geq t\right\}\)
  • Lacks causal interpretation (condition on post-treatment status) (ICH 2020)

• Cumulative frequency

  • Mean number of events in presence of death as a competing risk (Ghosh and Lin 2000; Ghosh and Lin 2002; Mao and Lin 2016)
  • Estimand: \(E\left\{N^{(a)}(t)\right\}\)
  • Ignores length of exposure…

A new proposal

While-alive event rate

• Estimand \[\ell^{(a)}(\tau) = \frac{E\left\{N^{(a)}(\tau)\right\}}{E\left(D^{(a)}\wedge\tau\right)} =\frac{\mbox{Mean # of events by $\tau$}}{\mbox{Mean survival time by $\tau$}}\]

  • \(\tau\): prespecified time horizon; \(b\wedge c :=\min(b, c)\)
  • Average event rate in \([0, \tau]\) per person-time alive
  • Proposed as a clinically interpretable measure to Committee for Medicinal Products for Human Use (CHMP) of European Medicines Agency (Akacha et al. 2018; CHMP 2020)
  • Also called “exposure-weighted” event rate

Early work

A series of follow-up papers (Schmidli, Roger, and Akacha 2023a, 2023b; Wei et al. 2023; Fritsch et al. 2023)…

• Gamma shared-frailty models: analytic expression of \(\ell^{(a)}(\tau)\)

• Inference under minimal mortality: Poisson/negative-binomial/LWYY regressions

• Method-of-moment estimator under a fixed censoring point

A similar idea is (independently) considered for mortality (Uno and Horiguchi 2023) \[E\{N_D^{(a)}(\tau)\}/E(D^{(a)}\wedge\tau)\]

Gaps

• Generalization of estimand

• General nonparametric inference procedure

General estimands - definition

While-alive loss rate (Mao 2023) \[\ell^{(a)}(\tau) = \frac{E\left\{\mathcal L\left(\mathcal H^{(a)}\right)(\tau)\right\}}{E\left(D^{(a)}\wedge\tau\right)}\]

• \(\mathcal H^{(a)}(t)=\left\{N_D^{(a)}(u), N^{(a)}(u): 0\leq u\leq t\right\}\): total outcomes by \(t\)
• \(\mathcal L\left(\mathcal H^{(a)}\right)(t)\): user-specified loss function satisfying two conditions
  • A function only of \(\mathcal H^{(a)}(t)\)
  • \(\mathcal L\left(\mathcal H^{(a)}\right)({\rm d}t)\equiv 0\) for \(t>D^{(a)}\) (what does this mean?)
• Average loss rate in \([0, \tau]\) per person-time alive

General estimands - examples

• Original while-alive event rate (Schmidli, Roger, and Akacha 2023a) \[\mathcal L\left(\mathcal H^{(a)}\right)(t) = N^{(a)}(t)\]
• Per-person-time mortality rate (Uno and Horiguchi 2023) \[\mathcal L\left(\mathcal H^{(a)}\right)(t) = N_D^{(a)}(t)\]
• More generally… \[\mathcal L\left(\mathcal H^{(a)}\right)(t) = \int_0^t \left\{w^D_{N^{(a)}(u-)}(u){\rm d}N_D^{(a)}(u)+w_{N^{(a)}(u-)}(u){\rm d}N^{(a)}(u)\right\}\]
  • \(w^D_m(u), w_m(u)\): weights for incident death or nonfatal event at \(u\) if patient has experienced \(m\) nonfatal events by then

General estimands - effect size

Survival-completed (SC) cumulative loss

\[L^{(a)}(\tau)=\ell^{(a)}(\tau)\tau\]

• Better graphics: \(\ell^{(a)}(\tau)\approx 0/0\) unstable for \(\tau\approx 0\)
• Properties: \(L^{(a)}(0)=0\), \(L^{(a)}(t)\uparrow\) with \(t\), and \(L^{(a)}(t)\geq E\{\mathcal L(\mathcal H^{(a)})(t)\}\)

Measuring the treatment effect…

• Risk (loss rate) ratio (RR): \(r(\tau)=\ell^{(1)}(\tau)/\ell^{(0)}(\tau)\)
  • Treatment reduces average loss rate by \(100\{1-r(\tau)\}\%\)
• Absolute risk reduction (ARR): \(d(\tau)=\ell^{(1)}(\tau)-\ell^{(0)}(\tau)\)
  • Treatment reduces average loss rate by \(-d(\tau)\) (per person-time alive)

Nonparametric estimation

Observed data \(\left\{\mathcal H\left(X^{(a)}\right), X^{(a)}\right\}\)

• \(X^{(a)}=D^{(a)}\wedge C^{(a)}\); \(C^{(a)}\): independent censoring time
• Two samples: \(\left\{\mathcal H_i\left(X_i^{(a)}\right), X_i^{(a)}\right\}\) \((i=1,\ldots, n_a; a = 1, 0)\)

Estimating \(\ell^{(a)}(\tau)=E\{\mathcal L(\mathcal H^{(a)})(\tau)\} /E(D^{(a)}\wedge\tau)\)…

• Demonstrator (easy) \(=\int_0^\tau S^{(a)}(t){\rm d}t\) (restricted mean survival time; RMST) (Royston and Parmar 2011)
  • \(S^{(a)}(t)={\rm P}(D^{(a)} > t)\): plug in Kaplan–Meier estimator
• Numerator (moderate) \(=\int_0^\tau S^{(a)}(t-)E\{\mathcal L(\mathcal H^{(a)})({\rm d}t)\mid D^{(a)}\geq t\}\)
  • Integrator estimated by \(\sum_{i=1}^{n_a}I(X_i^{(a)}\geq t)\mathcal L(\mathcal H_i^{(a)})({\rm d}t)/ \sum_{i=1}^{n_a}I(X_i^{(a)}\geq t)\)
• Robust variance estimator by delta method

Nonparametric testing

\(J\)-sample testing \((a=0, 1,\ldots, J-1)\)

\[H_0: \ell^{(0)}(\tau)=\cdots=\ell^{(J-1)}(\tau)\]

• \(\chi_{J-1}^2\) test on the \(\log\hat r^{(a)}(\tau)=\log\hat\ell^{(a)}(\tau)-\log\hat\ell^{(0)}(\tau)\) \((a= 1,\ldots, J-1)\)

Joint test of morbidity & mortality

\[H_0: \ell^{(0)}(\tau)=\cdots=\ell^{(J-1)}(\tau),\hspace{5mm} \mu^{(0)}(\tau)=\cdots=\mu^{(J-1)}(\tau)\]

• \(\mu^{(a)}(\tau)=E(D^{(a)}\wedge\tau)\): \(\tau\)-RMST
• \(\chi_{2(J-1)}^2\) test on the \(\log\hat r^{(a)}(\tau)\) and \(\log\hat\mu^{(a)}(\tau)-\log\hat\mu^{(0)}(\tau)\) \((a=1,\ldots, J-1)\)

R-package WA - usage

CRAN: https://cran.r-project.org/web/packages/WA (Mao 2021)

Main function

LRfit(id, time, status, trt, Dweight = 0, wH = NULL, wD = NULL)

• id: vector of patient IDs
• time: vector of times
• status: vector of event types (1= recurrent event; 2= death; 0= censoring)
• trt: vector of (binary or multiclass) treatment groups
• Dweight: weight for death relative to each recurrent event
• wH and wD: user-supplied R-functions of (m, t) implementing \(w_m(t)\) and \(w^D_m(t)\) (override Dweight)

Summarize and plot results by summary() and plot()

R-package WA - code example (i)

First install the package if it hasn’t been installed…

# install package from CRAN
install.packages("WA")

Load the package and the HF-ACTION dataset…

# load the package
library(WA)
# load the HF-ACTION dataset
dat <- hfaction_cpx12
dat[1:16,] # what the data look like

           id       time status trt
1  HFACT00001 0.60506502      1   0
2  HFACT00001 1.04859685      0   0
3  HFACT00002 0.06297057      1   0
4  HFACT00002 0.35865845      1   0
5  HFACT00002 0.39698836      1   0
6  HFACT00002 3.83299110      0   0
7  HFACT00007 0.29021218      1   1
8  HFACT00007 1.80424367      1   1
9  HFACT00007 2.42573580      1   1
10 HFACT00007 2.68583162      1   1
11 HFACT00007 2.91307324      2   1
12 HFACT00008 0.01916496      1   0
13 HFACT00008 0.02737851      2   0
14 HFACT00011 0.06570842      0   0
15 HFACT00019 3.66598220      1   0
16 HFACT00019 4.23271732      0   0

R-package WA - code example (ii)

Unweighted (while-alive) hospitalization rate

obj <- LRfit(dat$id, dat$time, dat$status, dat$trt)
obj

Call:
LRfit(id = dat$id, time = dat$time, status = dat$status, trt = dat$trt)
    N Rec. event Death Med. Follow-up
0 377        747    75       2.496920
1 364        644    49       2.536619

Summarize inferential results for restriction time \(\tau=3.5\) years…

summary(obj, tau = 3.5)

Call:
LRfit(id = dat$id, time = dat$time, status = dat$status, trt = dat$trt)

Analysis of log loss rate (LR) by tau = 3.5:
               Estimate   Std.Err Z value  Pr(>|z|)    
Ref (Group 0) -0.223940  0.054308 -4.1235 3.731e-05 ***
Group 1 vs 0  -0.186019  0.084590 -2.1991   0.02787 *  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Test of group difference in while-alive LR
X-squared = 4.835905, df = 1, p = 0.02787301

Point and interval estimates for the LR ratio:
              LR ratio 95% lower CL 95% higher CL
Group 1 vs 0 0.8302577     0.703412     0.9799773

R-package WA - code example (iii)

If you want joint test \((\chi_2^2)\) with mortality …

summary(obj, tau = 3.5, joint.test = TRUE)

Call:
LRfit(id = dat$id, time = dat$time, status = dat$status, trt = dat$trt)

Analysis of log loss rate (LR) by tau = 3.5:
               Estimate   Std.Err Z value  Pr(>|z|)    
Ref (Group 0) -0.223940  0.054308 -4.1235 3.731e-05 ***
Group 1 vs 0  -0.186019  0.084590 -2.1991   0.02787 *  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Test of group difference in while-alive LR
X-squared = 4.835905, df = 1, p = 0.02787301

Point and interval estimates for the LR ratio:
              LR ratio 95% lower CL 95% higher CL
Group 1 vs 0 0.8302577     0.703412     0.9799773


Analysis of log RMST (restricted mean survival time) by tau = 3.5:
              Estimate  Std.Err Z value  Pr(>|z|)    
Ref (Group 0) 1.107544 0.016064 68.9443 < 2.2e-16 ***
Group 1 vs 0  0.056689 0.020349  2.7858  0.005339 ** 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1


Test of group difference in while-alive LR and RMST
X-squared = 9.913726, df = 2, p = 0.007034964

R-package WA - code example (iv)

Graphics …

# plot the estimated survival-completed cumulative loss
# by group, with 95% confidence intervals
plot(obj, conf = TRUE, xlab = "Time (years)", xlim = c(0, 3.5), 
     ylab = "SC cumulative frequency")

Data example - HF-ACTION (i)

	Exercise training	Usual care
Death rate	13.5%	19.9%
Avg # hospitalization (SD)	1.8 (2.1)	2.0 (2.1)

Blue: exercise training; Red: usual care

Data example - HF-ACTION (ii)

Blue: exercise training; Red: usual care

• Raw (left) shrinks treatment difference

  • Trained surviving longer \(\to\) hospitalized more

• SC (right) corrects this by using event rate while alive

Data example - HF-ACTION (iii)

For \(\tau=3.5\) years…

• In the first 3.5 years, exercise training on average reduces hospitalizations per person-year alive by 1 - 0.83 =17% (2%–30%; p-value 0.03)

• Joint test with RMST: \(\chi_2^2\)=9.88, p-value 0.007

Data example - HF-ACTION (iv)

Weighted composites: \(w_m(t)\equiv 1\) and \(w_m^D(t)\equiv 1, 2, 3\)…

• In the first 3.5 years, exercise training on average reduces death and hospitalizations per person-year alive by 18%, 19%, and 20% under weights 1:1, 2:1, and 3:1, respectively

Summary

General while-alive loss rate: \(\ell^{(a)}(\tau)=E\{\mathcal L(\mathcal H^{(a)})(\tau)\} /E(D^{(a)}\wedge\tau)\)

• Summarizes loss profile while adjusting for length of exposure

Risk (loss rate) ratio: \(r(\tau)=\ell^{(1)}(\tau)/\ell^{(0)}(\tau)\)

• Treated experience \(100r(\tau)\%\) as much loss as control does

R-package WA: https://cran.r-project.org/web/packages/WA

Open question: What if treatment delays nonfatal events without necessarily reducing their number by time \(\tau\)

• Use a decreasing \(w_m(t)\) to reward late occurrence?

Acknowledgments

This research is supported by NIH-NHLBI grant R01HL149875

Novel Statistical Methods for Complex Time-to-Event Data in Cardiovascular Clinical Trials

HF-ACTION study data are provided by BioLINCC depository of NHLBI

References

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CHMP. 2020. “Qualification Opinion of Clinically Interpretable Treatment Effect Measures Based on Recurrent Event Endpoints That Allow for Efficient Statistical Analyses.” https://www.ema.europa.eu/en/documents/other/qualification-opinion-clinically-interpretable-treatment-effect-measures-based-recurrent-event_en.pdf.

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———. 2023b. “Rejoinder to Commentaries on “Estimands for Recurrent Event Endpoints in the Presence of a Terminal Event”.” Statistics in Biopharmaceutical Research 15 (2): 255–56. https://doi.org/10.1080/19466315.2023.2166098.

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