Analysis of multistate hierarchical outcomes by the restricted mean time in favor of treatment

Data type

To recap the methodology, we formulate patient experience with multiple events as a multistate process \(Y(t)\) taking values in the ordered set \(\{0,1,\ldots, K, K+1\}\), with a larger integer representing a more series condition. For example, \(Y(t)=0\) if a cancer patient is in a state of remission, \(Y(t)=1\) if the cancer has relapsed, \(Y(t)=2\) if it has metastasized, and \(Y(t)=3\) if the patient has died. The methodology in rmt is sufficiently general to be applicable to a wide range of settings, as long as the following assumptions are met:

1. The outcome process is progressive (can transition only from a less serious to a more serious state), i.e., \(Y(t)\leq Y(s)\) for \(t\leq s\);
2. Death is the only absorbing state, i.e., no competing risks to death (hence “cure models” are not allowed).

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 intermediate stages in disease progression. In fact, it can be shown that \(\mu(\tau)\) reduces to the net restricted mean survival time (Royston & Parmar, 2011) under the two-state life-death model. For details of the methodology, refer to Mao (2021).

The overall effect size can be decomposed into stage-wise components: \[\mu(\tau)=\sum_{k=1}^{K+1}\mu_k(\tau)\] with \[\begin{equation}\label{eq:comp}\tag{*} \mu_k(\tau)=E\int_0^\tau I\{Y^{(1)}(t)<k, Y^{(0)}(t)=k\}{\rm d}t - E\int_0^\tau I\{Y^{(0)}(t)<k, Y^{(1)}(t)=k\}{\rm d}t. \end{equation}\] The component \(\mu_k(\tau)\) can be interpreted as the average ``pre-state \(k\)’’ time gained by the treatment among those who have not transitioned to state \((k+1)\) or higher. It is clear that the last component \(\mu_{K+1}(t)\) always stands for the usual net restricted mean survival time.

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 times of the transitioning events, a status variable labeling the event type (status=k if transitioning to state \(k\) and status=0 if censored; note that death is represented by status=K+1), 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(ms(id,time,status)~trt,data)

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

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

The (default) type option specifies the input as multistate data rather than recurrent event data (type=="recurrent").

The returned object obj contains basically all the information about the overall and stage-wise RMTs. To extract relevant information for a particular \(\tau=\)tau, use

summary(obj,tau)

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 stage-wise restricted mean win/loss times, i.e., the first and second terms on the right hand side of \((*)\), 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)

Other graphical parameters can be specified and, if so, will be passed to the underlying generic plot method.

Data description

A landmark colon cancer trial on the efficacy of levamisole and fluorouracil was reported by Moertel et al. (1990). The trial recruited 929 patients with stage C disease and randomly assigned them to levamisole treatment alone, levamisole combined with fluorouracil, and the control. We focus on the comparison between the combined treatment and control groups, consisting of 304 and 314 patients, respectively. The endpoints of interest are cancer relapse and death. The death rates in the treatment and control groups are about 40% and 53%, relapse rates about 39% and 56%, and median follow-up times about 5.7 and 5.1 years, respectively.

The dataset colon_lev (a subset of the colon dataset in the survival package) is contained in the rmt package and can be loaded by

library(rmt)
head(colon_lev)
#>   id      time status      rx sex age
#> 1  1 2.6502396      1 Lev+5FU   1  43
#> 2  1 4.1642710      2 Lev+5FU   1  43
#> 3  2 8.4517454      0 Lev+5FU   1  63
#> 4  3 1.4839151      1 Control   0  71
#> 5  3 2.6365503      2 Control   0  71
#> 6  4 0.6707734      1 Lev+5FU   0  66

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

Estimation and inference

We first fit the data by

obj=rmtfit(ms(id,time,status)~rx,data=colon_lev)
## print the event numbers by group
obj
#> Call:
#> rmtfit.formula(formula = ms(id, time, status) ~ rx, data = colon_lev)
#> 
#>           N State 1 Death Med follow-up time
#> Control 315     177   168           5.081451
#> Lev+5FU 304     119   123           5.749487

# summarize the inference results for tau=7.5 years
summary(obj,tau=7.5)
#> Call:
#> rmtfit.formula(formula = ms(id, time, status) ~ rx, data = colon_lev)
#> 
#> Restricted mean winning time by tau = 7.5:
#>           State 1 Survival  Overall
#> Control 0.2659633 1.130967 1.396930
#> Lev+5FU 0.6120146 1.750924 2.362938
#> 
#> Restricted mean time in favor of group "Lev+5FU" by time tau = 7.5:
#>          Estimate  Std.Err Z value  Pr(>|z|)    
#> State 1  0.346051 0.072066  4.8018 1.572e-06 ***
#> Survival 0.619957 0.213610  2.9023  0.003704 ** 
#> Overall  0.966008 0.252585  3.8245  0.000131 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

From the above output, we conclude that, at 7.5 years, the combined treatment on average gains the patient 0.97 extra year in a more favorable state compared to the control. This total effect size comprises an additional 0.62 year of survival time and 0.35 year of relapse-free time among the survivors. The matrix containing the inferential results can be obtained from summary(obj,tau=7.5)$tab.

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,main="Bouquet plot",cex.group=0.8, xlab="Restricted mean win/loss time (years)",
        ylab="Follow-up time (years)") #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="Overall")

par(oldpar)

From the left panel, both the restricted mean survival (dark gray) and pre-relapse times (light gray) are clearly in favor of the treatment, both increasingly so as follow-up continues. From the right panel, judging from the lower 95% confidence limit, the treatment effect becomes significant at the 0.05 level soon after randomization and stays so till the end of the study. To plot the component-specific curves , run

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

# Plot of component-wise RMT in favor of treatment over time
plot(obj,k=2,conf=TRUE,col='red',conf.col='blue',conf.lty=2, xlab="Follow-up time (years)",
        ylab="RMT in favor of treatment (years)",main="Survival")
plot(obj,k=1,conf=TRUE,col='red',conf.col='blue',conf.lty=2, xlab="Follow-up time (years)",
        ylab="RMT in favor of treatment (years)",main="Pre-relapse")

par(oldpar)