Introduction
Setup
Load the package hce and check the version:
For citing the package run citation("hce") (Samvel B. Gasparyan 2025).
Definitions
Hierarchical composite endpoints (HCE) are a general class of endpoints combining different clinical outcomes of patients into a composite so as to preserve their different natures. A particular case of these endpoints is defined in a fixed follow-up period and accounts for the patient’s clinically most important outcome for the analysis. HCEs are analyzed using win odds and other win statistics (Samvel B. Gasparyan et al. 2023).
Examples
Here we provide examples of HCE using in clinical trials from different therapeutic areas. General considerations for creating HCEs can be found in Samvel B. Gasparyan et al. (2022).
COVID-19
The DARE-19 (M. Kosiborod et al. 2021; M. N. Kosiborod et al. 2021) trial used an HCE to assess outcomes in patients hospitalized for COVID-19 and treated for 30 days. The COVID-19 HCE is presented below. It combines death, in hospital organ dysfunction events with clinical status at Day 30 for patients alive, still hospitalized but without previous organ dysfunction events, and hospital discharge as the most favorable outcome for patients discharging without organ dysfunction events and being alive at Day 30.
Below a higher category signifies a better outcome. Patients are
ranked into one and only one category based on their clinically most
severe event. For example, patients experiencing an in-hospital new or
worsening organ dysfunction event then dying will be included in the
category I.
#> Order Category
#> 1 I Death
#> 2 II More than one new or worsened organ dysfunction events
#> 3 III One new or worsened organ dysfunction event
#> 4 IV Hospitalized at the end of follow-up (Day 30)
#> 5 V Discharged from hospital before Day 30Patients in the category I are compared using the timing
of the event, with an earlier event being a worse outcome (are assigned
a lower rank). Similarly, in the category III the timing of
the event is used for ranking patients within this category. In the
category II patients are compared using the number of
events with a higher number signifying a worse outcome. Patients in the
category IV - hospitalized at the end of follow-up without
previous worsening events - are further ranked according to oxygen
support requirements at the hospital (IV.1 on high flow
oxygen devices, IV.2 requiring supplemental oxygen,
IV.3 not requiring supplemental oxygen, with a higher rank
being a better outcome). Patients in the category V are
compared using the timing of the event, but, the hospital discharge
being a favorable outcome, here the earlier event signifies a better
outcome than the late event (reverse of the ranking in categories
I and III).
The simplest case of a COVID-19 HCE is an endpoint with ordinal scale
outcomes assessed at a given timepoint. The endpoint uses 1-8 categories
for assessing the physical limitations of hospitalized patients with
COVID-19 after 15 or 30 days of treatment. But unlike the DARE-19 HCE,
within each category it does not use the timing of events to reduce the
ties in a paiwrise comparison of patients in the active group with
patients in the control group. See, for example, COVID-19
and COVID-19b datasest ordinal scale outcomes (Beigel et al. 2020).
table(COVID19)
#> GROUP
#> TRTP 1 2 3 4 5 6 7 8
#> Active 34 95 28 58 38 14 117 157
#> Placebo 58 121 24 60 33 8 102 115The function hce::summaryWO() provides the number of
wins, losses, and ties by categories. We can calculate the probability
of ties from the provided numbers.
COVID19HCE <- hce(GROUP = COVID19$GROUP, TRTP = COVID19$TRTP)
SUM <- summaryWO(COVID19HCE, ref = "Placebo")$summary
SUM$Ptie <- round(SUM$TIE/SUM$TOTAL, 2)
SUM
#> TRTP WIN LOSS TIE TOTAL WR WO Ptie
#> 1 A 135744 97143 48974 281861 1.3973627 1.3173641 0.17
#> 2 P 97143 135744 48974 281861 0.7156338 0.7590916 0.17Kidney HCE
The kidney HCE defined in Heerspink et al. (2023) has the following ordinal outcomes (for the review of the topic see Little et al. (2023)).
#> Order Category
#> 1 I Death
#> 2 II Dialysis or kidney transplantation
#> 3 III Sustained GFR < 15 ml/min per 1.73 m2
#> 4 IV Sustained GFR decline from baseline of >= 57%
#> 5 V Sustained GFR decline from baseline of >= 50%
#> 6 VI Sustained GFR decline from baseline of >= 40%
#> 7 VII Individual GFR slopeThe dataset KHCE contains data on a kidney HCE
outcomes
dat <- KHCE
Order <- c("Death (adj)", "Chronic dialysis (adj) >=90 days",
"Sustained eGFR<15 (mL/min/1.73 m2)", "Sustained >=57% decline in eGFR",
"Sustained >=50% decline in eGFR", "Sustained >=40% decline in eGFR", "eGFR slope")
dat$GROUP <- factor(dat$GROUP, levels = Order)
table(dat$GROUP, dat$TRTP)
#>
#> A P
#> Death (adj) 40 50
#> Chronic dialysis (adj) >=90 days 17 29
#> Sustained eGFR<15 (mL/min/1.73 m2) 16 28
#> Sustained >=57% decline in eGFR 2 9
#> Sustained >=50% decline in eGFR 7 22
#> Sustained >=40% decline in eGFR 36 34
#> eGFR slope 632 578This dataset is derived from ADSL which contains
baseline characteristics, ADLB laboratory measurements of
kidney function, and ADET for the time-to-event outcomes
with their timing. For the detailed derivation see the Technical
Appendix in Heerspink et al.
(2023).
Heart Failure
In the Heart Failure population (see Kondo et al. (2023)) the following HCE was considered
#> Order Category
#> 1 I Cardiovascular death
#> 2 II Total (first and recurrent) HF hospitalizations
#> 3 III Total urgent HF visits
#> 4 IV Improvement/deterioration in KCCQ-TSSDependent outcomes
To model dependent outcomes, several methods are available:
Joint Distribution Modeling Using Copulas: This method employs copulas to model the joint distribution of outcomes, capturing their dependence.
Random Frailty Modeling: This approach captures patient-level dependence between outcomes using a random frailty model.
Conditional Distribution Specification Through Multi-State Modeling: This technique uses multi-state models to describe the conditional distribution of outcomes.
Joint Distribution Modeling Using Copulas
Sklar’s theorem (Sklar 1959) shows that multivariate distribution functions can be expressed using a copula and univariate distributions. For a random vector \(X^d=(X_1,\cdots,X_d)\) with a multivariate distribution function \(H(x_1,\cdots,x_d),\) Sklar’s theorem states that there is a copula \(C(\cdot)\) such that: \[H(x_1,\cdots,x_d)=C(F_1(x_1),\cdots,F_d(x_d)),\] where each component \(X_j\) has the univariate distribution \(F_j.\) A copula is essentially a multivariate distribution function where each univariate marginal distribution is uniform, describing the dependency structure of the multivariate distribution function \(H(\cdot)\). To construct the multivariate distribution function, one combines each variable’s univariate distributions \(F_j\) with the copula.
If \(X_j\) has distribution function \(F_j,\) then \(U_j=F_j(X_j)\) is uniformly distributed, allowing random variables \(X_j\sim F_j\) to be simulated by generating uniform random variables \(U_j\) and applying the inverse transformation:
\[F_j^{-1}(y)=\inf\{x\in {\mathbf R}: \ \ F_j(x)\geq y\}, \ \ \inf\varnothing=\infty.\] Thus, if one has simulated a uniform random vector \(U^d=(U_1,\cdots, U_d)\) from the copula \(C(\cdot)\), the random vector \(X^d=(X_1,\cdots,X_d)\) can be simulated as: \[(X_1,\cdots,X_d)=(F_1^{-1}(U_1),\cdots,F_d^{-1} (U_d)).\] The main challenge remains in simulating from the given copula.
An Archimedean copula (Nelsen 2006) is one where:
\[C(u^d;\varphi)=\varphi(\varphi^{-1}(u_1)+\cdots+\varphi^{-1}(u_d)).\] The function \(\varphi:[0,+\infty]\rightarrow [0,1]\) is a generator - continuous, decreasing, with \(\varphi(0)=1\) and \(\lim_{t\rightarrow+\infty}\varphi(t)=0.\) When \(\varphi(t)=e^{-t^\theta},\ \ \theta>1,\) the copula is called a Gumbel copula.
The Marshall-Olkin algorithm
By Bernstein’s theorem, completely monotone Archimedean generators coincide with Laplace-Stieltjes transforms of distribution functions \(F,\) determined by \(\varphi=LS[F].\) The Marshall-Olkin algorithm (Marshall and Olkin 1988) for sampling from an Archimedean copula involves:
- Sampling \(V\sim F=LS^{-1}[\varphi].\)
- Sampling \(R_j\sim Exp(1),\,j\in\{1,\cdots,d\}.\)
- Setting \(U_j=\varphi\left(\frac{R_j}{V}\right),\,j\in\{1,\cdots,d\}.\)
The vector \(U=(U_1,\cdots, U_d)\) is then a random vector from the Archimedean copula with generator \(\varphi\). For the Gumbel copula, one needs to use the inverse Laplace-Stieltjes transform of a stable distribution (Nolan 2020; Hofert and Mächler 2011): \[F\sim S(1/\theta, 1, \cos^\theta(\pi/(2\theta)), {\mathbf I}_{\{\theta=1\}},1)\] modifying the first step to sample from a stable distribution.
Chambers-Mallows-Stuck method for simulating stable random variables
Chambers-Mallows-Stuck method (Chambers, Mallows, and Stuck 1976) efficiently simulates stable variables with:
Generating independent uniform and exponential random variables \[\Theta\sim U\left[-\frac{\pi}{2},\frac{\pi}{2}\right] \text{ and } W\sim Exp(1).\]
Defining \(\alpha=1/\theta,\) setting \(b_{\tan}=\beta\tan\left(\frac{\alpha\pi}{2}\right),\) and \(\theta_0=\arctan(b_{\tan})/\alpha,\) with \[C_{\tan}=(1+b_{\tan}^2)^\frac{1}{2\alpha}.\]
Utilizing the transformations: \[Z(\theta) = \frac{\sin(a_0) C_{\tan}}{\cos(\Theta)^\frac{1}{\alpha}}\left(\frac{\cos(a_0-\Theta)}{W}\right)^\frac{1-\alpha}{\alpha}, \ \ a_0=\alpha(\Theta+\theta_0),\,\theta>1.\] \[Z(1)=\frac{2}{\pi}\left(\pi_\beta\tan(\Theta)-\beta\log\left(\frac{\pi}{2}W\frac{\cos(\Theta)}{\pi_\beta}\right)\right), \ \ \pi_\beta=\frac{\pi}{2}+\beta\Theta.\] Finally, \(\gamma Z + \delta\) has the desired distribution with \(\gamma =[\cos(\pi/(2\theta))]^\theta\) and \(\delta={\mathbf I}_{\{\theta=1\}}\) (and one needs to set \(\beta=1\)).
Note
In the original Chambers-Mallows-Stuck formula, the term \[\frac{1}{[\cos(\alpha\theta_0)\cos(\Theta)]^\frac{1}{\alpha}}\]
is replaced by \(\frac{C_{\tan}}{[\cos(\Theta)]^\frac{1}{\alpha}},\)
as suggested by the copula package (Hofert et al. 2025), which is based on the fact
that \(C_{\tan} =
1/(\cos(\alpha\theta_0))^{1/\alpha}.\) Indeed, one needs to show
that
\[(1+b_{\tan}^2)^\frac{1}{2\alpha}=1/(\cos(\alpha\theta_0))^{1/\alpha},\] which is equivalent to showing that \[1+\left[\beta\tan\left(\frac{\alpha\pi}{2}\right)\right]^2=\frac{1}{[\cos(\alpha\theta_0)]^2}.\] And this is true because of the trigonometric identity \[1+[\tan(y)]^2=\frac{1}{[\cos(y)]^2}.\] Here we have set \(y=\alpha\theta_0=\arctan(b_{\tan})=\arctan\left(\beta\tan\left(\frac{\alpha\pi}{2}\right)\right)\) and hence \(\tan(y)=\beta\tan\left(\frac{\alpha\pi}{2}\right).\)
Implementation
The function simHCE() provides the implementation above
with the argument theta specifying the dependence of
outcomes.
Rates_A <- c(10, 20)
Rates_P <- c(20, 20)
dat1 <- simHCE(n = 2500, TTE_A = Rates_A, TTE_P = Rates_P,
CM_A = -3, CM_P = -6, CSD_A = 15, fixedfy = 3, theta = 1, seed = 1)
dat2 <- simHCE(n = 2500, TTE_A = Rates_A, TTE_P = Rates_P,
CM_A = -3, CM_P = -6, CSD_A = 15, fixedfy = 3, theta = 1.0001, seed = 1)
dat3 <- simHCE(n = 2500, TTE_A = Rates_A, TTE_P = Rates_P,
CM_A = -3, CM_P = -6, CSD_A = 15, fixedfy = 3, theta = 10, seed = 1)
calcWO(dat1)
#> WO LCL UCL SE WOnull alpha Pvalue WP LCL_WP
#> 1 1.483251 1.38986 1.582918 0.03318096 1 0.05 0 0.5973021 0.5816594
#> UCL_WP SE_WP SD_WP N
#> 1 0.6129447 0.007981092 0.5643484 5000
calcWO(dat2)
#> WO LCL UCL SE WOnull alpha Pvalue WP LCL_WP
#> 1 1.52306 1.426936 1.625659 0.03326189 1 0.05 0 0.6036558 0.5880583
#> UCL_WP SE_WP SD_WP N
#> 1 0.6192534 0.00795809 0.5627219 5000
calcWO(dat3)
#> WO LCL UCL SE WOnull alpha Pvalue WP LCL_WP
#> 1 1.387231 1.299852 1.480482 0.03319378 1 0.05 0 0.5811046 0.5652679
#> UCL_WP SE_WP SD_WP N
#> 1 0.5969413 0.008080097 0.5713491 5000