Table 1 Three steps of the proposed effect estimation method based on the susceptible pre-identification

Step 1: Fit the incidence and time of the intermediate event with the mixture cure model.

Based on the intermediate event data, maximize the following likelihood function to obtain the estimates of β_e and γ,

\( L\left({\boldsymbol{\upbeta}}_{\mathbf{e}},\boldsymbol{\upgamma} \right)=\prod \limits_{i=1}^N\left\{{\left[\pi \left({\mathbf{x}}_i\right)f\left({t}_{ei}|s=1,{\mathbf{x}}_i\right)\right]}^{\delta_{ei}}\times {\left[1-\pi \left({\mathbf{x}}_i\right)+\pi \left({\mathbf{x}}_i\right)S\left({t}_{ei}|s=1,{\mathbf{x}}_i\right)\right]}^{1-{\delta}_{ei}}\right\} \)

where π(x) = [1 + exp(−(γ₀ + γ^Tx))]⁻¹, \( S\left({t}_e|s=1,\mathbf{x}\right)=\exp \left(-{\lambda}_e{t_e}^{v_e}\exp \left({{\boldsymbol{\upbeta}}_{\mathbf{e}}}^T\mathbf{x}\right)\right) \) and f(t_e| s = 1, x) = d[1 − S(t_e| s = 1, x)]/dt_e.

Step 2: Pre-identification of the susceptible subpopulation.

(1) For patients that have experienced the intermediate event the susceptibility is s = 1, i.e., being susceptible to the intermediate event.

(2) For patients with censored intermediate event time the susceptibility is s = 1 when \( {u}_i>\frac{1-\pi \left({\mathbf{x}}_i\right)}{1-\pi \left({\mathbf{x}}_i\right)+\pi \left({\mathbf{x}}_i\right)S\left({C}_{ei}|s=1,{\mathbf{x}}_i\right)} \)and s = 0, i.e., being insusceptible, when \( {u}_i\le \frac{1-\pi \left({\mathbf{x}}_i\right)}{1-\pi \left({\mathbf{x}}_i\right)+\pi \left({\mathbf{x}}_i\right)S\left({C}_{ei}|s=1,{\mathbf{x}}_i\right)} \) where u_i is a random number from the uniform distribution U(0, 1).

Step 3: Effect estimation based on the identified susceptible subpopulation.

Based on the identified susceptible subpopulation, estimate the effect of the time-varying intermediate event which is quantified by β_z.

For the extended Cox regression method,

h(t_o| x, z(t_o)) = h₀(t_o) exp(β_o^Tx + β_zz(t_o)).

For the landmark method,

\( h\left({t}_o|\mathbf{x},{z}_{t_{LM}}\right)={h}_0\left({t}_o\right)\exp \left({{\boldsymbol{\upbeta}}_{\mathbf{o}}}^T\mathbf{x}+{\beta}_z{z}_{t_{LM}}\right), \) for patients with t_o > t_LM.