Skip to main content

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

From: An improved method for the effect estimation of the intermediate event on the outcome 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 βe 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 + γTx))]−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(te| s = 1, x) = d[1 − S(te| s = 1, x)]/dte.
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 ui 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 βz.
For the extended Cox regression method,
h(to| x, z(to)) = h0(to) exp(βoTx + βzz(to)).
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 to > tLM.