Analysis of composite endpoints of recurrent event and death by the restricted mean time in favor of treatment

Data type

Let \(N(t)\) denote the counting process for the recurrent event, e.g., repeated hospitalizations, and let \(N_D(t)\) denote that for death. The composite outcome process is defined by \[Y(t)=N(t)+\infty N_D(t).\] That is, \(Y(t)\) counts the number of non-fatal event on the living patient and jumps to \(\infty\) when the patient dies. Traditional ways of combining the components include:

1. Time to the first event: \(Y^*(t)=I\{N(t)+N_D(t)>0\}\);
2. Weighted composite event process (Mao and Lin, 2016): \(Y^{**}(t)=N(t)+w_DN_D(t)\) for some \(w_D\geq 1\);

Compared to these approaches, \(Y(t)\) has the advantage of including all events while prioritizing death in a natural, hierarchical way.

Effect size estimand

Let \(Y^{(a)}\) denote the outcome process from group \(a\) (\(a=1\) for the treatment and \(a=0\) for the control). The estimand of interest is constructed under a generalized pairwise comparison framework (Buyse, 2010). With \(Y^{(1)}\perp Y^{(0)}\), let \[\mu(\tau)=E\int_0^\tau I\{Y^{(1)}(t)< Y^{(0)}(t)\}{\rm d}t - E\int_0^\tau I\{Y^{(1)}(t)> Y^{(0)}(t)\}{\rm d}t,\] for some pre-specified follow-up time \(\tau\). We call \(\mu(\tau)\) the restricted mean time (RMT) in favor of treatment and interpret it as the average time gained by the treatment in a more favorable condition. It generalizes the familiar restricted mean survival time to account for the non-fatal events. In fact, it can be shown that \(\mu(\tau)\) reduces to the net restricted mean survival time (Royston & Parmar, 2011) if \(N(t)\equiv 0\). For details of the methodology, refer to Mao (2021).

The overall effect size admits a component-wise decomposition: \[\mu(\tau)=\mu_D(\tau)+\mu_H(\tau),\] where \[\begin{equation}\tag{*} \mu_D(\tau)=E\int_0^\tau I\{Y^{(1)}(t)<\infty, Y^{(0)}(t)=\infty\}{\rm d} t- E\int_0^\tau I\{Y^{(0)}(t)<\infty, Y^{(0)}(t)=\infty\}{\rm d} t \end{equation}\] is equivalent to the standard net restricted mean survival time and \[\mu_H(\tau)=E\int_0^\tau I\{Y^{(1)}(t)< Y^{(0)}(t)<\infty\}{\rm d}t - E\int_0^\tau I\{Y^{(0)}(t)< Y^{(1)}(t)<\infty\}{\rm d}t\] is the average time gained by the treatment with fewer non-fatal events among the living patients. The second component can be further decomposed by \(\mu_H(\tau)=\sum_{k=1}^K\mu_k(\tau)\), where \[\begin{equation}\tag{**} \mu_k(\tau)=E\int_0^\tau I\{Y^{(1)}(t)<k, Y^{(0)}(t)=k\}{\rm d} t- E\int_0^\tau I\{Y^{(0)}(t)<k, Y^{(0)}(t)=k\}{\rm d} t, \end{equation}\] and \(K\) is the maximum number of non-fatal event. The quantity \(\mu_k(\tau)\) can be interpreted as the average time gained by the treatment before experiencing the \(k\)th non-fatal event among the living patients.

Data fitting and summarization

The main data-fitting function is rmtfit(). To use the function, the input data must be organized in the “long” format. Specifically, we need an id variable containing the unique patient identifiers, a time variable containing the event times, a status variable labeling the event type (status=1 for non-fatal event, =2 for death, and =0 for censoring), and, finally, a binary trt variable containing the subject-level treatment arm indicators. If id, time, status, and trt are all variables in a data frame data, we can then use the formula form of the function:

obj=rmtfit(rec(id,time,status)~trt,data)

Otherwise, we can feed the vector-valued variables directly into the function:

obj=rmtfit(id,time,status,trt,type="recurrent")

Note that the last type option must be specified; otherwise the input will be treated as multistate rather than recurrent event data. The returned object obj contains basically all the information about the overall and component-wise RMTs. To extract relevant information for a particular \(\tau=\)tau, use

summary(obj,tau,Kmax)

If the last option Kmax is specified, say, as \(l\), then the estimates for the \(\mu_k(\tau)\) over \(k=l,\ldots, K\) will be aggregated, i.e., \(\sum_{k=l}^K\mu_k(\tau)\). Therefore, to make inference on \(\mu_H(\tau)\), use

summary(obj,tau,Kmax=1)

Plot of \(\mu(\cdot)\)

To plot the estimated \(\mu(\tau)\) as a function of \(\tau\), use

plot(obj,conf=TRUE)

The option conf=T requests the 95% confidence limits to be overlaid. The color and line type of the confidence limits can be controlled by arguments conf.col and conf.lty, respectively. Other graphical parameters can be specified and, if so, will be passed to the underlying generic plot method.

Bouquet plot

The dynamic profile of treatment effects as follow-up progresses is captured by the bouquet plot, which puts \(\tau\) on the vertical axis and plots the component-wise restricted mean win/loss times, i.e., the first and second terms on the right hand side of \((*)\) and \((**)\), as functions of \(\tau\) on the two sides. The bouquet plot is useful because it visualizes the component-wise contributions to the overall effect. To plot it, use

bouquet(obj,Kmax)

The option Kmax performs a similar task to that in the summary() function. For better visualization, we should almost always specify a Kmax\(<K\), especially when \(K\) is large. Other graphical parameters can be specified and, if so, will be passed to the underlying generic plot method.

Data description

A total of over two thousand heart failure patients across the USA, Canada, and France participated in the Heart Failure: A Controlled Trial Investigating Outcomes of Exercise Training between 2003–2007 (O’Connor et al., 2009). The primary objective of the trial was to evaluate the effect of adding exercise training to the usual patient care on the composite endpoint of all-cause hospitalization and death. We consider a subgroup of 426 non-ischemic patients with baseline cardio-pulmonary exercise test less than or equal to nine minutes. In this subgroup, 205 patients were randomly assigned to receive exercise training in addition to usual care and 221 to receive usual care alone. With a median follow-up time about 28 months, the death rates in the exercise training and usual care groups are about 18% and 26%, and the average numbers of recurrent hospitalizations per patient about 2.2 and 2.6, respectively. The maximum number of hospitalizations per patient is \(K=26\).

The dataset hfaction is contained in the rmt package and can be loaded by

library(rmt)
head(hfaction)
#>        patid       time status trt_ab age60
#> 1 HFACT00001 0.60506502      1      0     1
#> 2 HFACT00001 1.04859685      0      0     1
#> 3 HFACT00002 0.06297057      1      0     1
#> 4 HFACT00002 0.35865845      1      0     1
#> 5 HFACT00002 0.39698836      1      0     1
#> 6 HFACT00002 3.83299110      0      0     1

The dataset is already in a format suitable for rmtfit() (status= 1 for hospitalization and = 2 for death).

Estimation and inference

We first fit the data by

obj=rmtfit(rec(patid,time,status)~trt_ab,data=hfaction)
## print the event numbers by group
obj
#> Call:
#> rmtfit.formula(formula = rec(patid, time, status) ~ trt_ab, data = hfaction)
#> 
#>     N Event 1 Event 2 Event 3 Event 4 Event 5 Event 6 Event 7 Event 8 Event 9
#> 0 221     170     117      86      56      33      23      15      13      13
#> 1 205     145      89      55      43      32      21      15      11       7
#>   Event 10 Event 11 Event 12 Event 13 Event 14 Event 15 Event 16 Event 17
#> 0       11        7        6        6        5        3        2        2
#> 1        5        4        3        2        2        2        2        2
#>   Event 18 Event 19 Event 20 Event 21 Event 22 Event 23 Event 24 Event 25
#> 0        2        1        0        0        0        0        0        0
#> 1        2        2        1        1        1        1        1        1
#>   Event 26 Death Med follow-up time
#> 0        0    57           2.390144
#> 1        1    36           2.302533

# summarize the inference results for tau=3.5 years
# 
summary(obj,tau=3.5,Kmax=4)
#> Call:
#> rmtfit.formula(formula = rec(patid, time, status) ~ trt_ab, data = hfaction)
#> 
#> Restricted mean winning time by tau = 3.5:
#>     Event 1   Event 2    Event 3    Event 4    Event 5    Event 6    Event 7
#> 0 0.2461459 0.1797341 0.07400189 0.05705096 0.06778913 0.03229824 0.02901336
#> 1 0.2606647 0.2246581 0.18911776 0.07928876 0.05043218 0.04036515 0.01434557
#>      Event 8    Event 9    Event 10    Event 11    Event 12     Event 13
#> 0 0.02467620 0.01351584 0.007900133 0.001056981 0.007932054 0.0006445581
#> 1 0.01169075 0.01232759 0.009301100 0.004563094 0.001627585 0.0024747931
#>       Event 14     Event 15     Event 16     Event 17     Event 18     Event 19
#> 0 0.0007787642 0.0003834044 0.0003137311 0.0001010352 0.0001123888 0.0004848228
#> 1 0.0077805017 0.0019915287 0.0007116988 0.0003040931 0.0003580146 0.0001079202
#>       Event 20     Event 21     Event 22     Event 23    Event 24     Event 25
#> 0 0.0001781611 0.0007311137 0.0003162343 0.0005731831 0.001192464 0.0002290646
#> 1 0.0000000000 0.0000000000 0.0000000000 0.0000000000 0.000000000 0.0000000000
#>      Event 26  Survival  Overall
#> 0 0.004352256 0.2967478 1.048254
#> 1 0.000000000 0.4958881 1.407999
#> 
#> Restricted mean time in favor of group "1" by time tau = 3.5:
#>           Estimate   Std.Err Z value Pr(>|z|)   
#> Event 1   0.014519  0.047535  0.3054 0.760034   
#> Event 2   0.044924  0.045661  0.9839 0.325185   
#> Event 3   0.115116  0.035992  3.1984 0.001382 **
#> Event 4+ -0.013954  0.049358 -0.2827 0.777403   
#> Survival  0.199140  0.093300  2.1344 0.032810 * 
#> Overall   0.359745  0.154062  2.3351 0.019540 * 
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

From the above output, we conclude that, at 3.5 years, the combined treatment on average gains the patient \(\mu(\tau)=0.36\) extra year in a more favorable state compared to the control. This total effect size comprises an additional \(\mu_D(\tau)=0.20\) year of survival time and \(\mu_H(\tau)=0.36-0.20=0.16\) year with fewer hospitalizations among the living. The latter component is mainly driven by a prolonging of time to the third hospitalization. The matrix containing the inferential results can be obtained from summary(obj,tau=3.5,Kmax=4)$tab.

To obtain the inferential result for \(\mu_H(\tau)\) as a whole, run

# summarize the inference results for hospitalization as a whole
obj_sum=summary(obj,tau=3.5,Kmax=1)
obj_sum$tab
#>           Estimate    Std.Err  Z value   Pr(>|z|)
#> Event 1+ 0.1606050 0.08986723 1.787136 0.07391549
#> Survival 0.1991404 0.09330039 2.134400 0.03281004
#> Overall  0.3597453 0.15406184 2.335071 0.01953971

Graphical analysis

Use the following code to construct the bouquet plot and the plot for the estimated \(\mu(\cdot)\):

# set-up plot parameters
oldpar <- par(mfrow = par("mfrow"))
par(mfrow=c(1,2))

# Bouquet plot
bouquet(obj,Kmax=4,main="Bouquet plot",cex.group=0.8, xlab="Restricted mean win/loss time (years)", 
        ylab="Follow-up time (years)", cex.main=0.8) #cex.group: font size of group labels#
# Plot of RMT in favor of treatment over time
plot(obj,conf=TRUE,col='red',conf.col='blue',conf.lty=2, xlab="Follow-up time (years)",
        ylab="RMT in favor of treatment (years)",main="Exercise training vs usual care",
     cex.main=0.8)

par(oldpar)

In the bouquet plot, the four bands with different shades of gray, from the darkest to the lightest, correspond to survival, 4+ hospitalizations, 3 hospitalization, 2 hospitalizations, and 1 hospitalization, respectively. We can see that the restricted mean survival time is clearly in favor of exercise training. The restricted mean win times on the second and third hospitalizations are also visibly greater in the treatment group. The 95% confidence limits for \(\mu(\tau)\) in the right panel suggests that the overall treatment effect becomes significant at the 0.05 level after approximately 1 year of follow-up and stays so till the end of the study.