Summarizing patient experience under differential length of exposure
Department of Biostatistics & Medical Informatics
University of Wisconsin-Madison
A cardiovascular trial (HF-ACTION) (O’Connor et al. 2009)
Subpopulation: 741 heart failure patients
Treatment arms
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) |
Longer survivors tend to experience more events…
Mathematical notation (\(a=1\): Treatment; \(a=0\): Control)
Two broad-based approaches…
Conditional event rate
Cumulative frequency
While-alive event rate
Estimand \[\ell^{(a)}(\tau) = \frac{E\left\{N^{(a)}(\tau)\right\}}{E\left(D^{(a)}\wedge\tau\right)} =\frac{\mbox{Mean # of events by $\tau$}}{\mbox{Mean survival time by $\tau$}}\]
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
While-alive loss rate (Mao 2023) \[\ell^{(a)}(\tau) = \frac{E\left\{\mathcal L\left(\mathcal H^{(a)}\right)(\tau)\right\}}{E\left(D^{(a)}\wedge\tau\right)}\]
Survival-completed (SC) cumulative loss
\[L^{(a)}(\tau)=\ell^{(a)}(\tau)\tau\]
Measuring the treatment effect…
Observed data \(\left\{\mathcal H\left(X^{(a)}\right), X^{(a)}\right\}\)
Estimating \(\ell^{(a)}(\tau)=E\{\mathcal L(\mathcal H^{(a)})(\tau)\} /E(D^{(a)}\wedge\tau)\)…
\(J\)-sample testing \((a=0, 1,\ldots, J-1)\)
\[H_0: \ell^{(0)}(\tau)=\cdots=\ell^{(J-1)}(\tau)\]
Joint test of morbidity & mortality
\[H_0: \ell^{(0)}(\tau)=\cdots=\ell^{(J-1)}(\tau),\hspace{5mm} \mu^{(0)}(\tau)=\cdots=\mu^{(J-1)}(\tau)\]
WA
- usageCRAN: 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 IDstime
: vector of timesstatus
: vector of event types (1
= recurrent event; 2
= death; 0
= censoring)trt
: vector of (binary or multiclass) treatment groupsDweight
: weight for death relative to each recurrent eventwH
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()
WA
- code example (i)First install the package if it hasn’t been installed…
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
WA
- code example (ii)Unweighted (while-alive) hospitalization rate
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…
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
WA
- code example (iii)If you want joint test \((\chi_2^2)\) with mortality …
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
WA
- code example (iv)Graphics …
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
Raw (left) shrinks treatment difference
SC (right) corrects this by using event rate while alive
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
Weighted composites: \(w_m(t)\equiv 1\) and \(w_m^D(t)\equiv 1, 2, 3\)…
General while-alive loss rate: \(\ell^{(a)}(\tau)=E\{\mathcal L(\mathcal H^{(a)})(\tau)\} /E(D^{(a)}\wedge\tau)\)
Risk (loss rate) ratio: \(r(\tau)=\ell^{(1)}(\tau)/\ell^{(0)}(\tau)\)
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 \(\tau\)
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