---
title: "Analysis of multistate hierarchical outcomes by the restricted mean time in favor of treatment"
author: "Lu Mao"
#date: "8/19/2019"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Analysis of multistate hierarchical outcomes by the restricted mean time in favor of treatment}
  %\VignetteEngine{knitr::rmarkdown}
  \usepackage[utf8]{inputenc}
  \usepackage{amsmath}
---

```{r, echo = FALSE, message = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
```


## INTRODUCTION
This vignette demonstrates the use of the R-package `rmt` 
for the restricted-mean-time-in-favor-of-treatment approach to 
the analysis of multiple prioritized outcomes.

### 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 $t\leq s$;
1. 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.

## BASIC SYNTAX

### 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:
```{r,eval=F}
obj=rmtfit(ms(id,time,status)~trt,data)
```
Otherwise, we can feed the vector-valued variables directly into the function:
```{r,eval=F}
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
```{r,eval=F}
summary(obj,tau)
```

### Plot of $\mu(\cdot)$
To plot the estimated $\mu(\tau)$ as a function of $\tau$, use 
```{r,eval=F}
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
```{r,eval=F}
bouquet(obj)
```
Other graphical parameters 
can be specified and, if so, will be passed to the underlying generic `plot` method.



## AN EXAMPLE WITH A COLON CANCER TRIAL

### 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
```{r setup}
library(rmt)
head(colon_lev)
```
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
```{r}
obj=rmtfit(ms(id,time,status)~rx,data=colon_lev)
## print the event numbers by group
obj

# summarize the inference results for tau=7.5 years
summary(obj,tau=7.5)
```
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 $\mu(\cdot)$:

```{r,fig.align='center',fig.width=7.2,fig.height=4.5}
# 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 \eqn{\mu_k(\tau)},
run
```{r,fig.align='center',fig.width=7.2,fig.height=4.5}
# 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)
```






## References

* Buyse, M. (2010). Generalized pairwise comparisons of prioritized outcomes in the two‐sample problem. *Statistics in Medicine*, 29, 3245--3257.
* Mao, L. (2021). On restricted mean time in favour of treatment. *Submitted*.
* Moertel, C. G., Fleming, T. R., Macdonald, J. S., Haller, D. G., Laurie, J. A., Goodman, P. J., ... & Veeder, M. H. (1990). Levamisole and fluorouracil for adjuvant therapy of resected colon carcinoma. *New England Journal of Medicine*, 322, 352--358.
* Royston, P. & Parmar, M. K. (2011). The use of restricted mean survival time to estimate the treatment effect in randomized clinical trials when the proportional hazards assumption is in doubt. *Statistics in Medicine*, 30, 2409--2421.