Analysis of while-alive event rate for recurrent hospitalization

Data

Let \(D\) denote the survival time and write \(N_D(t)=I(D\leq t)\). Let \(N(t)\) count the number of recurrent nonfatal event by time \(t\). Since death is a terminal event, we must have that \(N(t)=N(t\wedge D)\), where \(b\wedge c=\min(b,c)\). Let \(\mathcal H(t)=\{N_D(u), N(u):0\leq u\leq t\}\) denote the event history up to time \(t\).

Estimands

Let \(\mathcal L(\mathcal H)({\rm d}t)\) be a user-specified loss function that measures the instantaneous loss incurred at time \(t\). We focus on loss functions of the form \[\begin{equation}\tag{1} \mathcal L(\mathcal H)({\rm d} t)=w_{N(t-)}{\rm d}N(t)+w^D_{N(t-)}{\rm d}N_D(t), \end{equation}\] where \(w_m(t)\) and \(w^D_m(t)\) are weights attached to an incident (i.e., new) nonfatal event or death, respectively, when the patient has already experienced \(m\) nonfatal events. This scheme allows the user to assign weights according not only to the type and timing of the incident event but also to past history. It can be useful, for example, if we want to make the loss metric more robust to patients who experience unusually large numbers of event. In that case, we can specify a \(w_m(t)\) that decays with larger \(m\).

The cumulative loss is then defined by \(\mathcal L(\mathcal H)(t)=\int_0^t L(\mathcal H)({\rm d}u)\). With \(w_m(t)\equiv w\) and \(w_m^D(t)\equiv w^D\), for example, the cumulative loss becomes \(\mathcal L(\mathcal H)(t)=wN(t)+w^DN_D(t)\), same as the weighted composite event process considered by Mao and Lin (2016).

Let \(\mathcal H^{(a)}\) denote the outcome data from group \(a\) (\(a=1, 2,\ldots, J-1\) for different treatments and \(a=0\) for the control). To characterize the loss profile over \([0,\tau]\) in the presence of death, define the while-alive loss rate by \[\begin{equation}\tag{2} l^{(a)}(\tau)=\frac{E\{\mathcal L(\mathcal H^{(a)})(\tau)\}}{E(D^{(a)}\wedge\tau)}. \end{equation}\] The numerator on the right hand side of (2) is the mean cumulative loss, akin to the mean cumulative frequency studied by Ghosh and Lin (2000), and the denominator is the restricted mean survival time (RMST; Royston & Parmar, 2011), the latter of which accounts for the length of exposure. We can interpret \(l^{(a)}(\tau)\) as the average loss experienced per unit time alive in group \(a\) over \([0, \tau]\). Using the loss function defined in (1) with \(w_m(t)\equiv 1\) and \(w_m^D(t)\equiv 0\) reduces (2) to \(l^{(a)}(\tau)=E\{N(\tau)\}/E(D^{(a)}\wedge t)\), the while-alive event rate for the recurrent event considered by Schmidli et al. (2021) and Wei et al. (2021). The survival-completed cumulative loss is defined by \(L^{(a)}(\tau)=l^{(a)}(\tau)\tau\), which can be considered roughly as the total loss over \([0,\tau]\) without being curtailed by death.

We can measure the effect size of treatment \(a\) as compared to the control by the loss rate ratio: \[\begin{equation}\tag{3} r^{(a)}(\tau)=l^{(a)}(\tau)/l^{(0)}(\tau). \end{equation}\] That is, patients under treatment \(a\) on average experience \(r^{(a)}(\tau)\) times as much loss (or as many events) per unit time alive as compared to the control by time \(\tau\).

Tests

Testing the equality of loss rate across the \(J\) groups, i.e., \[\begin{equation}\tag{4} H_0: l^{(0)}(\tau)=l^{(1)}(\tau)=\cdots=l^{(J-1)}(\tau), \end{equation}\] is equivalent to testing \(\{\log r^{(1)}(\tau),\ldots,\log r^{(J-1)}(\tau)\}^{\rm T}=\bf 0\), which can be done via a Wald-type chi-square test with \((J-1)\) degrees of freedom (df). Similarly, we can test the loss rate jointly with the RMST, i.e., \[\begin{equation}\tag{5} H_0: l^{(0)}(\tau)=l^{(1)}(\tau)=\cdots=l^{(J-1)}(\tau) \mbox{ and }\mu_D^{(0)}(\tau)=\mu_D^{(1)}(\tau)=\cdots=\mu_D^{(J-1)}(\tau), \end{equation}\] where \(\mu_D^{(a)}(\tau)=E(D^{(a)}\wedge\tau)\), by a Wald-type chi-square test with \(2(J-1)\) df. The \(2(J-1)\) df test will be useful if \(\mathcal L\) captures only recurrent event (e.g., when \(w_m^D(t)\equiv 0\) in (1)) so that (5) becomes a joint test of recurrent event and death.

Data fitting and summarization

The main function to fit the data is LRfit(). To use the function, the input data must be 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 recurrent non-fatal event, =2 for death, and =0 for censoring), and, finally, a categorical (either binary or multiclass) trt variable for the treatment arm. To analyze the (unweighted) recurrent event rate, i.e., \(w_m(t)\equiv 1\) and \(w_m^D(t)\equiv 0\) in (1), simply run the function in the default mode

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

Alternatively, we can add the Dweight argument to specify a non-zero (constant) \(w^D\) for death. Even more generally, we can supply self-defined \(w_m(t)\) and \(w_m^D(t)\) through the functional arguments wH and wD, respectively. For example, the above code for the unweighted event rate is equivalent to

obj<-LRfit(id,time,status,trt,Dweight=0)

and to

# define w_m(t)
wH<-function(t,m) return (1)
# define w_m^D(t)
wD<-function(t,m) return (0)
obj<-LRfit(id,time,status,trt,wH=wH,wD=wD)

The returned object obj contains all information about the group-specific loss rate and their standard errors. To extract relevant information for a particular \(\tau=\)tau, use

summary(obj,tau)

This will print out the inference results on the loss rate ratio defined in (3) comparing each group to the reference group (which can be specified through the optional ref= argument if you want to override the default) as well as the \((J-1)\)-df overall test of (4). To request the \(2(J-1)\)-df loss-rate/RMST joint test of (5), add the option joint.test=T to the summary() function.

Plot of \(L(\cdot)\)

To plot the estimated survival-completed cumulative loss \(L^{(a)}(\tau)\) as a function of \(\tau\), use

plot(obj,conf=TRUE)

This by default plots \(L^{(a)}(\tau)\) for all groups. The option conf=T requests the 95% confidence limits for each \(L^{(a)}(\tau)\) to be overlaid. To restrict to specific groups, use the group= option. To override the default color of the group-specific curves, use the group.col argument to supply a \(J\)-vector of colors. Other graphical parameters can be added and, if so, will be passed to the underlying generic plot method.

Data description

The Heart Failure: A Controlled Trial Investigating Outcomes of Exercise Training (HF-ACTION) study was conducted between 2003–2007 to investigate whether adding exercise training to the usual care of heart failure patients improves their cardiovascular outcomes (O’Conner et al., 2009). We focus on a high-risk subgroup consisting of 741 nonischemic patients with baseline cardiopulmonary test duration less than or equal to 12 minutes and analyze recurrent hospitalizations as well as overall survival.

Among the 741 patients, 364 were assigned to participate in exercise training sessions in addition to receiving usual care, and the remaining 377 received usual care alone as control. Over a median follow-up of 2.5 years, 49 (13.5%) patients died in the exercise training arm, with an average of 1.8 hospitalizations per patient; 75 (19.9%) patients died in the usual care arm, with an average of 2.0 hospitalizations per patient. While patients undergoing exercise training experience only moderately fewer hospitalizations in the course of follow-up, they are also much less likely to die than those in the control group. The differential survivorship needs to be taken into account when assessing the treatment effect on hospitalization.

The associated dataset hfaction_cpx12 is contained in the WA package and can be loaded by

library(WA)
head(hfaction_cpx12)
#>           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

The dataset is already in a format suitable for LRfit() (status= 1 for hospitalization and = 2 for death). The time variable is in units of years and trt=0 for usual care (control) and 1 for exercise training.

Graphical analysis

First, consider the unweighted while-alive hospitalization rate.

dat<-hfaction_cpx12
obj<-LRfit(dat$id,dat$time,dat$status,dat$trt)
## print some descriptive information 
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

Let’s plot the estimated group-specific survival-completed cumulative frequency.

plot(obj,conf=T,xlab="Time (years)",xlim=c(0, 3.5),ylim=c(0,3))

This pretty much reproduces the second panel in Figure 3 of Mao (2022). We can see that the curve in the exercise training arm is substantially lower than that in the usual care arm, with only slightly overlapping 95% confidence intervals at \(\tau=3.5\) years. This bodes well for the statistical significance of treatment effect on the while-alive hospitalization rate (recall that \(l^{(a)}(\tau)=L^{(a)}(\tau)/\tau\)).

Estimation and inference

Formal analysis results at \(\tau=3.5\) years can be obtained through

## summarize the inference results at tau=3.5 years
summary(obj,tau=3.5,joint.test=T)
#> 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

Besides the function call, the first part of the output is about the log-loss rate: \(\log l^{(0)}(\tau)\) (Ref) and \(\log r^{(1)}(\tau)=\log l^{(1)}(\tau)-\log l^{(0)}(\tau)\) (Group 1 vs 0). We can see that the two-sided \(z\)-test on the log-loss rate ratio generates a \(p\)-value of 0.028, which, of course, agrees with the \(p\)-value of the 1-df chi-square test of (4) since there are only two groups. Further below is a table for the point and interval estimates for the loss rate ratio: 0.830 with 95% confidence interval (0.703, 0.980) — exercise training on average reduces hospitalization by 1-0.830=17% per year alive as compared to usual care by year 3.5. As a result of specifying joint.test=T, we get the inference results for the RMST and the joint test of (5) at the end. The 2-df joint test gives a \(p\)-value of 0.007, suggesting that the (beneficial) treatment effect is highly significant on survival and hospitalization jointly.

As supplemental analysis, we consider a weighted composite loss in (1) with \(w_m(t)\equiv 1\) and \(w_m^D(t)\equiv 2\). This is implemented by

## fit the data using the weighted composite loss
obj2<-LRfit(dat$id,dat$time,dat$status,dat$trt,Dweight=2)
## summarize the results at tau=3.5 years
summary(obj2,tau=3.5)
#> Call:
#> LRfit(id = dat$id, time = dat$time, status = dat$status, trt = dat$trt, 
#>     Dweight = 2)
#> 
#> Analysis of log loss rate (LR) by tau = 3.5:
#>                Estimate   Std.Err Z value Pr(>|z|)  
#> Ref (Group 0) -0.038361  0.055785 -0.6877  0.49167  
#> Group 1 vs 0  -0.216208  0.086746 -2.4924  0.01269 *
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Test of group difference in while-alive LR
#> X-squared = 6.212179, df = 1, p = 0.01268744
#> 
#> Point and interval estimates for the LR ratio:
#>               LR ratio 95% lower CL 95% higher CL
#> Group 1 vs 0 0.8055679    0.6796163     0.9548617

This shows a significant treatment effect on the weighted composite event rate (\(p\)-value 0.013), consistent with the conclusion drawn from the earlier analysis.