Table 2.

Overview of weighted treatment effect measures using the parametric model assuming constant hazards.

Corresponding Measure a	Mathematical Formulation
Constant hazards
Death hazard w/o treatment	λ 02
Discharge hazard w/o treatment	λ 03
Hazard w/o treatment	${\lambda }_0={\lambda }_{02}+{\lambda }_{03}$
Death hazard with treatment	λ 12
Discharge hazard with treatment	λ 13
Hazard with treatment	${\lambda }_1={\lambda }_{12}+{\lambda }_{13}$
Mortality
Mortality risk w/o treatment at the end of follow-up	$M{R}_0=\frac{{\lambda }_{02}}{{\lambda }_{02}+{\lambda }_{03}}$
Mortality risk with treatment at the end of follow-up	$M{R}_1=\frac{{\lambda }_{12}}{{\lambda }_{12}+{\lambda }_{13}}$
Mortality risk ratio at the end of follow-up	$\frac{{\lambda }_{12}/{\lambda }_{02}}{({\lambda }_{13}+{\lambda }_{12})/({\lambda }_{02}+{\lambda }_{03})}$
Difference in mortality at the end of follow-up	$M{R}_1-M{R}_0$
Hazards and cumulative incidence functions
Hazard ratio of death (treatment vs. w/o treatment) at the end of follow-up	$H{R}_2=\frac{{\lambda }_{12}}{{\lambda }_{02}}$
Hazard ratio of discharge (treatment vs. w/o treatment) at the end of follow-up	$H{R}_3=\frac{{\lambda }_{13}}{{\lambda }_{03}}$
Cumulative risk of death w/o treatment at time t	$CI{F}_{02}\left(t\right)=\frac{{\lambda }_{02}}{{\lambda }_0}\times \left(1-\exp \left(-{\lambda }_{0}t\right)\right)$
Cumulative risk of discharge w/o treatment at time t	$CI{F}_{03}\left(t\right)=\frac{{\lambda }_{03}}{{\lambda }_0}\times \left(1-\exp \left(-{\lambda }_{0}t\right)\right)$
Cumulative risk of death with treatment at time t	$CI{F}_{12}\left(t\right)=\frac{{\lambda }_{12}}{{\lambda }_1}\times \left(1-\exp \left(-{\lambda }_{1}t\right)\right)$
Cumulative risk of discharge with treatment at time t	$CI{F}_{12}\left(t\right)=\frac{{\lambda }_{13}}{{\lambda }_1}\times \left(1-\exp \left(-{\lambda }_{1}t\right)\right)$
Risk differences and ratios
Risk difference functions for death at time t	$R{D}_2\left(t\right)=CI{F}_{12}\left(t\right)-CI{F}_{02}\left(t\right)$
Risk difference functions for discharge at time t	$R{D}_3\left(t\right)=CI{F}_{13}\left(t\right)-CI{F}_{03}\left(t\right)$
Risk ratios for death at time t	$R{R}_2\left(t\right)=\frac{CI{F}_{12}\left(t\right)}{CI{F}_{02}\left(t\right)}$
Risk ratios for discharge at time t	$R{R}_3\left(t\right)=\frac{CI{F}_{13}\left(t\right)}{CI{F}_{03}\left(t\right)}$
Length of stay
Length of stay w/o treatment	$LO{S}_0=\frac{1}{{\lambda }_{02}+{\lambda }_{03}}$
Length of stay with treatment	$LO{S}_1=\frac{1}{{\lambda }_{12}+{\lambda }_{13}}$
Difference in length of stay	$LO{S}_1-LO{S}_0$

^a Inverse probability censoring weighted. Abbreviations: CIF, cumulative incidence function; HR, hazard ratio; LOS, length of stay; MR, mortality risk, RD, risk difference; RR, risk ratio; w/o, without. Notes: ${\lambda }_0$ = non-X-treated; ${\lambda }_1$ = X-treated.