Table 1.

A summary of the PH, PM, and PW models for composite endpoints of N_D(·) and N(·).

Rule of univariatization
PH	$\mathcal{F}(Y)(t)=I\left\{N_D(t)+N(t)\ge 1\right\}$
PM	$\mathcal{R}(Y)(t)=N_D(t)+N(t)$
PW	$\mathcal{W}\left(Y_i,Y_j\right)(t)=I\left(D_j<D_i\wedge t\right)+I\left(D_i\wedge D_j>t,T_j<T_i\wedge t\right)$.
Modeling target (k = 1, 0)
PH	Hazard function: ${\lambda }_k(t)=E\{\mathcal{F}(Y)(\text{d}t)\mid \mathcal{F}(Y)(t-)=0,Z=k\}/\text{d}t$
PM	Mean function: ${\mu }_k(t)=E\{\mathcal{R}(Y)(t)\mid Z=k\}$
PW	Win-loss function: ${\omega }_k(t)=E\left\{\mathcal{W}\left(Y_i,Y_j\right)(t)\mid Z_i=k,Z_j=1-k\right\}$
Model assumption
PH	λ1(t) / λ0 (t) ≡ exp(β)
PM	μ1 (t) / μ0 (t) ≡ exp(β)
PW	ω0 (t) / ω1 (t) ≡ exp(β)
Estimand & Interpretation
PH	Hazard ratio: Treatment exp(β) times as likely to experience first event
PM	Mean ratio: Treatment experiencing exp(β) times as many events
PW	Loss ratio: Treatment exp(β) times as likely to produce worse outcome
	adjusting for priority of death over non-fatal events
Accounting for semi-competing risks
PH	Not needed
PM	Inverse probability censoring weighting
PW	Not needed if death is hierarchically prioritized
R-function
PH	survival::coxph
PM	Wcompo::CompoML
PW	WR::pwreg