chap1.qmd

---
title: "Statistical Methods for Composite Endpoints: Win Ratio and Beyond"
subtitle: "Chapter 1. Introduction"
css: style.css
csl: apa.csl
author:
  name: Lu Mao
  affiliations: 
    - name: Department of Biostatistics & Medical Informatics
    - University of Wisconsin-Madison
    - May 31, 2025
  email: lmao@biostat.wisc.edu
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---

## Outline

-   Examples and regulatory guidelines
-   Traditional methods
    -   Time to first event
    -   Weighted total events ([`Wcompo`](https://cran.r-project.org/package=Wcompo){target="_blank"} package)
-   Win ratio and hierarchical endpoints
    -   The estimand issue

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# Example and Guidelines

## Motivating Example: Colon Cancer

::: fragment
-   **Landmark colon cancer trial**
    -   **Population**: 619 patients with stage C disease [@Moertel1990]
    -   **Arms**: Levamisole + fluorouracil ($n=304$) vs control ($n=315$)
    -   **Endpoint**: relapse-free survival (log-rank test p\<0.001)
        -   Death = Relapse
        -   258 deaths (89%) after relapse ignored

![](images/intro_colon.png){fig-align="center" width="60%"}
:::

## Motivating Example: HF-ACTION

::: fragment
-   **A cardiovascular trial (HF-ACTION)**
    -   **Subpopulation**: 426 heart failure patients [@OConnor2009]
    -   **Arms**: Exercise training + usual care ($n=205$) vs usual care ($n=221$)
    -   **Endpoint**: hospitalization-free survival (log-rank test p=0.100)
        -   Death = Hospitalization
        -   82 (88%) deaths + 707 (69%) recurrent hospitalizations ignored

![](images/intro_hfaction.png){fig-align="center" width="65%"}
:::

## Composite Endpoints

::: fragment
-   **Traditional composite endpoint (TCE)**
    -   **Time to first event**
        -   Relapse/Progression-free survival
        -   First major adverse cardiac event (MACE): death, heart failure, myocardio-infarction, stroke (event-free survival)
    -   **Limitations**
        -   Lack of clinical priority
        -   Statistical inefficiency (waste of data)
:::

::: fragment
-   **Hierarchical composite endpoint (HCE)**
    -   **Example**: Death \> nonfatal MACE \> six-minute walk test (6MWT)/NYHA class
:::

## Why Composite

::: fragment
-   **Advantages**

    -   More events $\to$ higher power $\to$ smaller sample size/lower costs

    -   No need for multiplicity adjustment

    -   A unified measure of treatment effect

    ::: callout-note
    ## ICH-E9 “Statistical Principles for Clinical Trials” [@ich1998]

    -   “There should generally be only one primary variable”
    -   “If a single primary variable cannot be selected …, another useful strategy is to integrate or combine the multiple measurements into a single or composite variable …”
    -   “\[composite endpoint\] addresses the multiplicity problem without adjustment to the type I error”
    :::
:::

## Regulatory Guidelines: FDA

::: fragment
-   **Main points**

    -   Typically first event but can do total events
    -   Component-wise analysis important for interpretation

    ::: callout-note
    ## FDA Guidance for Industry: “Multiple Endpoints in Clinical Trials” [@fda2022]

    -   “Composite endpoints are often assessed as the time to first occurrence of any one of the components, …, it also may be possible to analyze total endpoint events”
    -   “The treatment effect on the composite rate can be interpreted as characterizing the overall clinical effect when the individual events all have reasonably similar clinical importance”
    -   “…analyses of the components of the composite endpoint are important and can influence interpretation of the overall study results”
    :::
:::

## Regulatory Guidelines: Europe

::: fragment
-   **Main points**

    -   Combine events of similar importance
    -   Include mortality as a component

    ::: callout-note
    ## European Network for Health Technology Assessment “Endpoints used for Relative Effectiveness Assessment – Composite Endpoints” [@EUnetHTA2015]

    -   “All components of a composite endpoint should be separately defined as secondary endpoints and reported with the results of the primary analysis”
    -   “Components of similar clinical importance and sensitivity to intervention should preferably be combined”
    -   “If adequate, mortality should however be included if it is likely to have a censoring effect on the observation of other components”
    :::
:::

## A Tricky Example

::: fragment
-   **The EMPA-REG Trial** (NCT01131676)
    -   **Population**: 7,020 patients with type 2 diabetes [@Zinman2015]
    -   **Treatment arms**: Empagliflozin vs control
    -   **Endpoint**: Time to first CV death, nonfatal MI, nonfatal stroke

![](images/intro_empa.png){fig-align="center" width="50%"}
:::

# Traditional Composites

## Data and Notation

::: fragment
-   **Full data** $\mathcal H^*(\infty)$
    -   $D$: survival time; $N^*_D(t)=I(D\leq t)$
    -   $N^*_1(t), \ldots, N^*_K(t)$: counting processes for $K$ nonfatal event types
    -   Cumulative data: $\mathcal H^*(t)=\{N^*_D(u), N^*_1(u), \ldots, N^*_K(u):0\leq u\leq t\}$
:::

::: fragment
-   **Observed (censored) data** $\{\mathcal H^*(X), X\}$
    -   $\mathcal H^*(X)$: outcomes up to time $X$
    -   $X=D\wedge C$: length of follow-up ($a\wedge b = \min(a, b)$)
    -   $C$: independent censoring time
    -   **Goal**: estimate/test features of $\mathcal H^*(\infty)$ using $\{\mathcal H^*(X), X\}$
:::

## First Event

::: fragment
-   **Univariate endpoint**
    -   $N^*_{\rm TFE}(t) = I\{N^*_D(t)+\sum_{k=1}^KN^*_k(t)\geq 1\}$

        -   $I(\cdot)$: 0-1 indicator

    -   $\tilde T$: time to first event

        -   Kaplan--Meier curve, log-rank test, Cox model
:::

::: fragment
-   **Component-wise weighting**
    -   Upweight death over nonfatal events
        -   E.g., Death = 2 $\times$ hospitalization ![](images/intro_tfe.png){fig-align="center" width="50%"}
:::

## Total Events

::: fragment
-   **Weighted composite event process**
    -   $N^*_{\rm R}(t)=w_DN^*_D(t)+\sum_{k=1}^Kw_kN^*_k(t)$
        -   $w_D, w_1, \ldots, w_K$: weights to death and nonfatal events ![](images/intro_wcompo.png){fig-align="right" width="75%"}
    -   **Proportional means model** [@Mao2016] $$
        E\{N^*_{\rm R}(t)\mid Z\} = \exp(\beta^\T Z)\mu_0(t)
        $$
        -   $\exp(\beta)$: mean ratio of weighted total events comparing treatment $(Z=1)$ vs control $(Z=0)$
    -   **R-package**: [`Wcompo`](https://cran.r-project.org/package=Wcompo){target="_blank"}
:::

## Software: `Wcompo::CompoML()`

::: fragment
-   **Basic syntax**
    -   `id`: unique patient identifier; `time`: event times; `status`: event types (`1`: death; `2,...,K` nonfatal event types; `Z`: covariate matrix)
    -   `w`: $K$-vector of weights to event types `1,...K`; default is unweighted

::: big-code
```{r}
library(Wcompo)
obj <- CompoML(id, time, status, Z, w = c(2, 1))
```
:::
:::

::: fragment
-   **Output**: a list of class `CompoML`
    -   `obj$beta`: $\hat\beta$; `obj$var`: $\hat\var(\hat\beta)$
    -   `plot(obj, z)`: plot mean function $\exp(\hat\beta^{\rm T} z)\hat\mu_0(t)$
:::

## HF-ACTION: An Example

::: fragment
-   **High-risk subgroup (n=426)**
    -   Baseline cardiopulmonary exercise (CPX) test $\leq$ 9 min
:::

::: fragment
```{r}
#| eval: true
#| echo: false
#| label: tbl-desc
#| tbl-cap: Summary statistics for a high-risk subgroup (n=426)  in HF-ACTION trial.

library(tidyverse)
library(knitr)

descs <- readRDS("hf_tab1.rds")

hf_desc <- tibble(
  " " = c("Age", "", "Follow-up", "Death", "Hospitalizations", rep("", 3)),
  "  " = c("\u2264 60 years", "> 60 years", "(months)", 
           "", "0", "1-3", "4-10", ">10"),
  "Usual care (N = 221)" = descs[, 1],
  "Exercise training (N = 205)" = descs[, 2]
)

kable(hf_desc, align = c("lccc"))

```
:::

## HF-ACTION: Preparation

::: fragment
-   **Load packages and data**

```{r}
#| code-line-numbers: 1-4|6-16
library(survival) # for standard survival analysis
library(Wcompo) # for weighted total events
library(rmt) # for hfaction data
library(tidyverse) # for data wrangling

# Load data
data(hfaction)
head(hfaction) # trt_ab=1: training; 0: usual care
#>        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
```
:::

## HF-ACTION: Data

::: fragment
-   **Data processing**

```{r}
#| code-line-numbers: 1-9|11-16
# For weighted total analysis by compoML()
# Convert status=1 for death, 2=hospitalization
hfaction <- hfaction |> 
  mutate(
    status = case_when(
      status == 1 ~ 2,
      status == 2 ~ 1,
      status == 0 ~ 0)
  )

# TFE: take the first event per patient id
hfaction_TFE <- hfaction |> 
  arrange(patid, time) |> # sort by patid and time
  group_by(patid) |> 
  slice_head() |> # take first row
  ungroup()
```
:::

## HF-ACTION: Mortality

::: fragment
-   **Cox model for death**
    -   **HR**: $\exp(-0.3973) = 67.2\%$ ($32.8\%$ reduction in risk)
    -   $P$-value: 0.0621 (borderline significant)

```{r}
#| code-line-numbers: 1-3|5-10
## Get mortality data
hfaction_D <- hfaction |> 
  filter(status != 2) # remove hospitalization records

## Cox model for death against trt_ab
obj_D <- coxph(Surv(time, status) ~ trt_ab, data = hfaction_D)
summary(obj_D)
#> n= 426, number of events= 93 
#>           coef exp(coef) se(coef)      z      p
#> trt_ab -0.3973    0.6721   0.2129 -1.866 0.0621
```
:::

## HF-ACTION: TFE

::: fragment
-   **Cox model for hospitalization-free survival**
    -   **HR**: $\exp(-0.1770) = 83.8\%$ ($16.2\%$ reduction in risk)
    -   $P$-value: 0.111 (less significant than death)

```{r}
# Cox model for TFE against trt_ab
obj_TFE <- coxph(Surv(time, status > 0) ~ trt_ab, data = hfaction_TFE)
summary(obj_TFE)
#>   n= 426, number of events= 326 
#>           coef exp(coef) se(coef)      z Pr(>|z|)
#> trt_ab -0.1770    0.8378   0.1112 -1.592    0.111
```
:::

## HF-ACTION: Death vs TFE

::: fragment
-   Hospitalizations dilute effect on death ...
    -   An *EMPA-REG*-like situation ![](images/intro_hfaction_unis.png){fig-align="center" width="80%"}
:::

## HF-ACTION: Weighted Total

::: fragment
-   **Proportional means model** (death = $2\times$ hosp)

    -   **MR**: $\exp(-0.15398) = 85.7\%$ ($14.3\%$ reduction in total number of composite events)
    -   $P$-value: 0.170 (less significant than TFE)
    -   **Limitation**: Survival $\uparrow$ $\to$ cumulative total $\uparrow$ $\to$ attenuated effect

    ```{r}
    # Total events (proportional mean) -------------------------------
    obj_ML <- CompoML(hfaction$patid, hfaction$time, hfaction$status, 
                      hfaction$trt_ab, w = c(2, 1))
    obj_ML
    #>         Event 1 (Death) Event 2
    #> Weight               2       1
    #>         Estimate      se z.value p.value
    #> trt_ab -0.15398  0.11215 -1.3729  0.1698
    ```
:::

## HF-ACTION: Cumulative Means

::: fragment
-   **Model-based mean functions**

```{r}
plot(obj_ML, 0, ylim= c(0, 5), xlab="Time (years)", col= "red", lwd = 2)
plot(obj_ML, 1, add = TRUE, col = "blue", lwd = 2)
legend(0, 5, col=c("red","blue"), c("Usual care", "Training"), lwd = 2)
```

![](images/intro_hfaction_mean.png){fig-align="center" width="65%"}
:::

## Lessons Learned

::: fragment
-   **Adding nonfatal events** $\neq$ higher power
    -   Component may be less discriminating [@freemantle2003a]
    -   Length of exposure (death as competing risk) [@schmidli2023]
:::

::: fragment
-   **Solutions**
    -   Hierarchically prioritize death
        -   Evaluate nonfatal components only on survivors
    -   Quantitative weighting $\to$ adjust for survival time
        -   Loss rate = cumulative total / length of exposure ([Ch 3](chap3.html){target="_blank"})
:::

# Hierarchical Composites

## Win Ratio: Basics

::: fragment
-   **A common approach to HCE**
    -   **Proposed and popularized** by @pocock2012
    -   **Treatment vs control**: generalized pairwise comparisons
    -   **Win-loss**: sequential comparison on components
        -   Longer survival \> fewer/later nonfatal MACE \> better 6MWT/NYHA score
    -   **Effect size**: WR $=$ wins / losses
:::

::: fragment
-   **Alternative metrics**
    -   **Proportion in favor** (net benefit): PIF $=$ wins $-$ losses [@buyse2010]
    -   **Win odds**: WO $=$ (wins $+$ $2^{-1}$ties) / (losses $+$ $2^{-1}$ties) [@dong2019; @brunner2021]
:::

## Win Ratio: Gaining Popularity

::: fragment
-   More trials are using it...
:::

::: fragment
![](images/intro_Ntrial_title.png){fig-align="center" width="100%"}
:::

## An Important Caveat

::: fragment
-   **WR's estimand depends on censoring ...**

    -   @Luo2015, @Bebu2016, @oakes2016, @Mao2019, @Dong2020a, @li2024, etc.
:::

::: fragment
-   **What is an estimand?**

    -   Population-level quantity to be estimated
        -   Population-mean difference, (true) risk ratio, etc.
    -   Specifies how treatment effect is measured
    -   **ICH E9 (R1) addendum**: estimand construction one of the "*central questions for drug development and licensing*" [@ich2020]
:::

## Win-Loss Changes with Time

::: fragment
-   **Illustration**
    -   Win-loss status, and deciding component, changes with time ![](images/exmp.png){fig-align="center" width="85%"}
    -   Longer follow-up ...
        -   **Parameters**: win/loss proportions $\uparrow$ (WR uncertain); tie proportion $\downarrow$
        -   **Component contributions**: prioritized $\uparrow$; deprioritized $\downarrow$
:::

## Trial-Dependent Estimand

::: fragment
-   **Actual estimand**
    -   Average WR mixing shorter-term with longer-term comparisons
    -   Weight set (haphazardly) by censoring distribution
        -   Staggered entry, random withdrawal $\to$ non-scientific
:::

::: fragment
-   **Testing vs estimation**
    -   **Testing (qualitative)**: okay
        -   Valid under $H_0$, powerful if treatment *consistently* outperforms control over time
    -   **Estimation (quantitative)**: not okay
        -   Pre-define restriction time $\to$ use censoring weight for unbiased estimation ([Ch 3](chap3.html){target="_blank"})
        -   Specify a time-constant WR model ([Ch 4](chap4.html){target="_blank"})
:::

# Conclusion

## Notes

-   **More on**
    -   Regulatory guidelines for composite endpoints [@mao2021]
    -   ICH E9 (R1) implementation [@akacha2017; @ratitch2020; @qu2021; @ionan2022]
    -   Practical guidance [@redfors2020; @pocock:2024]
    -   Defining estimand for win ratio [@mao2024]
    -   Generalized pairwise comparisons [@péron2016; @deltuvaite-thomas2022; @dong2022; @verbeeck2023]
-   **Cumulative total events**
    -   Based on cumulative incidence/frequency under competing risks [@gray1988; @fine1999; @ghosh2000]

## Summary

-   **Composite endpoints**
    -   Death + hospitalization/progression/relapse
    -   Regulatory recommendation
-   **Traditional**
    -   **Time to first**: death = nonfatal (`survival::coxph()`)
    -   **Weighted total**: death = $w_D\times$ nonfatal (`Wcompo::compoML()`)
-   **Hierarchical**
    -   Win ratio, net benifit, win odds: death \> nonfatal
    -   Estimand issue - ICH E9 (R1)

## References