Inference of Transition Probabilities in Multi-State Models Using Adaptive Inverse Probability Censoring Weighting Technique

Abstract

Inverse probability censoring weighting (IPCW) technique is often used to adjust for right censoring or recover information for censored individuals in survival analysis and in multi-state modeling. A simple IPCW (SIPCW) technique which does not consider the intermediate states, has been proposed for analyzing multi-state data. However, our simulation studies show that the SIPCW technique may lead to biased estimates when being applied in complex multi-state models. We thereby propose a model-specific, state-dependent adaptive IPCW (AIPCW) technique for estimating transition probabilities in multi-state models. Intensive simulation results verified that the proposed AIPCW technique improves the accuracy of transition probability estimates compared to the SIPCW technique and leads to asymptotic unbiased estimates. We applied the proposed technique to a real-world hematopoietic stem cell transplant (HSCT) data to assess the acute and chronic graft-versus-host disease (GVHD) effects on disease relapse rates and mortality rates.

Appendices

Appendix 1

Large sample property of IPCW-based estimator has been studied extensively [8, 9, 16]. We present a brief derivation for the variance estimations. Let $T_{i,123}=(T_{i,123}^\ast \wedge C_i)$, $N_i^{C_{00}}(t)=I(T_{i,123} \le t, \varDelta _{C_{00},i}=1)$, $Y_i^{C_{00}}(t)=I(T_{i,123}\ge t)$, and $Y_{\bullet }^{C_{00}}(t)=\sum _i Y_i^{C_{00}}(t)$. Under regularity condition [9], we have that

$$\displaystyle \begin{aligned} \widehat G_{C_{00}}(T_{i,k}^\ast \wedge t) - G_{C_{00}}(T_{i,k}^\ast \wedge t) &\approx_p - \widehat G_{C_{00}}(T_{i,k}^\ast \wedge t) \sum_{j=1}^n \int_0^{T_{i,k}^\ast \wedge t} \frac{d\widehat{M}_j^{C_{00}}(u)}{Y_{\bullet}^{C_{00}}(u)}\\ d\widehat{M}_j^{C_{00}}(t)&=dN_j^{C_{00}}(t) - Y_j^{C_{00}}(t)d\widehat{\varLambda}_{C_{00}}(t)\\ d\widehat{\varLambda}_{C_{00}}(t)&=\sum_j \frac{dN_j^{C_{00}}(t)}{Y_{\bullet}^{C_{00}}(t)} \end{aligned} $$

Thus, variance of $\left \{\widehat P_{0k}^{\mathrm {AIPCW}}(0,t) - P_{0k}(0,t)\right \}$, for k = 0, 1, 2, can be estimated by

$$\displaystyle \begin{aligned} \widehat{\varSigma}_{P_{0k}}(t)=\frac{1}{n^2}\sum_{i=1}^n \left \{\widehat{W}_{i,0k}(t)\right \}^2, {} \end{aligned} $$

(4)

where

$$\displaystyle \begin{aligned} \widehat{W}_{i,00}(t)&=\left \{\frac{R_{i,0}(t)}{\widehat G_{C_{00}}(t)} - \widehat P_{00}^{\mathrm{AIPCW}}(0,t)\right \} +\left \{\sum_{j=1}^n \frac{R_{j,0}(t)}{\widehat G_{C_{00}}(t)}\right \} \int_0^t \frac{d\widehat M_i^{C_{00}}(u)}{Y_{\bullet}^{C_{00}}(u)}\\ \widehat{W}_{i,0k}(t) &=\left \{\frac{R_{i,k}(t)}{\widehat G_{C_{00}}(T_{i,k}^\ast)} - \widehat P_{0k}^{\mathrm{AIPCW}}(0,t)\right \}\\ &\quad +\int_0^t \left \{\sum_{j=1}^n \frac{R_{j,k}(t)}{\widehat G_{C_{00}}(T_{j,k}^\ast)} I\{u \le T_{j,k}^\ast \le t\}\right \} \frac{d\widehat M_i^{C_{00}}(u)}{Y_{\bullet}^{C_{00}}(u)} \mbox{ , for }k=1,2. \end{aligned} $$

For the variance of $P_{03}^{\mathrm {AIPCW}}(0,t)$, since

$$\displaystyle \begin{aligned} & \sqrt{n}\left \{\widehat{P}_{03}^{\mathrm{AIPCW}}(0,t)- P_{03}(0,t) \right \} \\[.2cm] &=\frac{1}{\sqrt n}\sum_{i=1}^n \left \{\frac{R_{i,3}(t)}{G_{C_{00}}(T_{i,3}^\ast) G_{C_{33}}(t)}-P_{03}(0,t)\right \}\\ &+ \frac{1}{\sqrt n}\sum_{i=1}^n \frac{R_{i,3}(t)}{\widehat G_{C_{33}}(t)} \left \{\frac{1}{\widehat{G}_{C_{00}}(T_{i,3}^\ast)} - \frac{1}{G_{C_{00}}(T_{i,3}^\ast)}\right \}\\ &+ \frac{1}{\sqrt n}\sum_{i=1}^n \frac{R_{i,3}(t)}{G_{C_{00}}(T_{i,3}^\ast)} \left \{\frac{1}{\widehat{G}_{C_{33}}(t)} - \frac{1}{G_{C_{33}}(t)}\right \}\\ &\approx_p \frac{1}{\sqrt n}\sum_{i=1}^n \left \{\frac{R_{i,3}(t)}{G_{C_{00}}(T_{i,3}^\ast) G_{C_{33}}(t)}-P_{03}(0,t)\right \}\\ &+\frac{1}{\sqrt n}\sum_{i=1}^n \int_0^t \left \{\sum_{j=1}^n \frac{R_{j,3}(t)}{G_{C_{00}}(T_{j,3}^\ast) G_{C_{33}}(t)} I\{u \le T_{j,3}^\ast \le t\}\right \} \frac{dM_i^{C_{00}}(u)}{Y_{\bullet}^{C_{00}}(u)}\\ &+\frac{1}{\sqrt n}\sum_{i=1}^n \left \{\sum_{j=1}^n \frac{R_{j,3}(t)}{G_{C_{00}}(T_{j,3}^\ast) G_{C_{33}}(t)} \right \} \int_0^t \frac{d M_i^{C_{33}}(u)}{Y_{\bullet}^{C_{33}}(u)}, \end{aligned} $$

where $M_i^{C_{33}}(u)$ can be estimated by

$$\displaystyle \begin{aligned} d\widehat{M}_i^{C_{33}}(u)&=dN_{i}^{C_{33}}(u) - Y_i^{C_{33}}(u) d\widehat \varLambda^{C_{33}}(u)\\ d\widehat{\varLambda}^{C_{33}}(u)&=\sum_j \frac{d N_{j}^{C_{33}}(u)}{Y_{\bullet}^{C_{33}}(u)}. \end{aligned} $$

Then, variance of $\left \{ \widehat {P}_{03}^{\mathrm {AIPCW}}(0,t)- P_{03}(0,t) \right \}$ can be estimated by

$$\displaystyle \begin{aligned} \begin{array}{rcl} \widehat{\varSigma}_{P_{03}}(t)=\frac{1}{n^2}\sum_{i=1}^n \left\{\widehat{W}_{i,03}(t)\right \}^2,{} \end{array} \end{aligned} $$

(5)

where

$$\displaystyle \begin{aligned} \widehat{W}_{i,03}(t)&= \left\{\frac{R_{i,3}(t)}{\widehat G_{C_{00}}(T_{i,3}^\ast) \widehat G_{C_{33}}(t)}- \widehat P_{03}^{\mathrm{AIPCW}}(0,t)\right \}\\ &+\int_0^t \left\{\sum_{j=1}^n \frac{R_{j,3}(t)}{\widehat G_{C_{00}}(T_{j,3}^\ast) \widehat G_{C_{33}}(t)} I\{u \le T_{j,3}^\ast \le t\}\right \} \frac{d\widehat M_i^{C_{00}}(u)}{Y_{\bullet}^{C_{00}}(u)}\\ &+\left\{\sum_{j=1}^n \frac{R_{j,3}(t)}{\widehat G_{C_{00}}(T_{j,3}^\ast) \widehat G_{C_{33}}(t)} \right \} \int_0^t \frac{d \widehat M_i^{C_{33}}(u)}{Y_{\bullet}^{C_{33}}(u)}. \end{aligned} $$

Similarly for k = 4, 5, variance of $\left \{\widehat {P}_{0k}^{\mathrm {AIPCW}}(0,t)- P_{0k}(0,t) \right \}$ can be estimated by

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \widehat{\varSigma}_{P_{0k}}(t)=\frac{1}{n^2}\sum_{i=1}^n \left \{\widehat{W}_{i,0k}(t)\right \}^2, \end{array} \end{aligned} $$

(6)

where

$$\displaystyle \begin{aligned} \widehat{W}_{i,0k}(t)&= \left \{\frac{R_{i,k}(t)}{\widehat G_{C_{00}}(T_{i,3}^\ast) \widehat G_{C_{33}}(T_{i,45}^\ast)}- \widehat P_{0k}^{\mathrm{AIPCW}}(0,t)\right \}\\ &+\int_0^t \left \{\sum_{j=1}^n \frac{R_{j,k}(t)}{\widehat G_{C_{00}}(T_{j,3}^\ast) \widehat G_{C_{33}}(T_{j,45}^\ast)} I\{u \le T_{j,3}^\ast \le t\}\right \}\frac{d\widehat M_i^{C_{00}}(u)}{Y_{\bullet}^{C_{00}}(u)}\\ &+\int_0^t \left \{\sum_{j=1}^n \frac{R_{j,k}(t)}{\widehat G_{C_{00}}(T_{j,3}^\ast) \widehat G_{C_{33}}(T_{j,45}^\ast)} I\{u \le T_{j,45}^\ast \le t\}\right \} \frac{d\widehat M_i^{C_{33}}(u)}{Y_{\bullet}^{C_{33}}(u)}. \end{aligned} $$

The derived moment-type variance estimates have two parts. For example, there are two parts in $\widehat {W}_{i,00}(t)=\widehat {W}_{i1,00}(t)+\widehat {W}_{i2,00}(t)$ for estimating variance of $P_{00}^{\mathrm {AIPCW}}(0,t)$. The second part

$$\displaystyle \begin{aligned} \widehat{W}_{i2,00}(t)=\left \{\sum_{j=1}^n \frac{R_{j,0}(t)}{\widehat G_{C_{00}}(t)}\right \}\int_0^t \frac{d\widehat M_i^{C_{00}}(u)}{Y_{\bullet}^{C_{00}}(u)}\end{aligned} $$

(7)

is needed due to the estimation of the censoring distributions. In other words, when the censoring distributions are known, $\widehat {W}_{i2,00}(t)$ becomes zero, and $\widehat {W}_{i,00}(t)$ reduces to

$$\displaystyle \begin{aligned} \widehat{W}_{i1,00}(t)=\left \{\frac{R_{i,0}(t)}{\widehat G_{C_{00}}(t)} - \widehat P_{00}^{\mathrm{AIPCW}}(0,t)\right \}.\end{aligned} $$

(8)

In most situations, $\widehat {W}_{i2,00}(t)$ is negligible. The variance estimates estimated from $\widehat {W}_{i,00}(t)$ are close to that estimated from $\widehat {W}_{i1,00}(t)$ alone, but not necessarily larger, because the covariance of W _i1,00(t) and W _i2,00(t) are not necessarily positive. Similar arguments are true for the variances of transition probabilities to other states. We presented both variance estimates in the simulation studies.

Appendix 2

For a four-level twelve-state model, we here only present the variance estimates of P ₀₉(t), P _0,10(t), P _0,11(t), one can refer to Appendix 1 for the variance of other transition probabilities. We have

$$\displaystyle \begin{aligned} &\sqrt{n}\left\{\widehat{P}_{09}^{\mathrm{AIPCW}}(0,t) - P_{09}(0,t) \right\}\\ &=\frac{1}{\sqrt n}\sum_{i=1}^n \left\{\frac{R_{i,39}(t)}{\widehat G_{C_{00}}(T_{i,3}^\ast) \widehat G_{C_{33}}(T_{i,9}^\ast) \widehat G_{C_{99,3}}(t)}+\frac{R_{i,49}(t)}{\widehat G_{C_{00}}(T_{i,4}^\ast) \widehat G_{C_{44}}(T_{i,9}^\ast)\widehat G_{C_{99,4}}(t)}\right.\\ &\qquad \left. -P_{09}(0,t) \right\}\\ &=\frac{1}{\sqrt n}\sum_{i=1}^n \left\{\frac{R_{i,39}(t)}{G_{C_{00}}(T_{i,3}^\ast)G_{C_{33}}(T_{i,9}^\ast)G_{C_{99,3}}(t)}+\frac{R_{i,49}(t)}{G_{C_{00}}(T_{i,4}^\ast) G_{C_{44}}(T_{i,9}^\ast)G_{C_{99,4}}(t)}\right.\\ &\qquad \left.-P_{09}(0,t) \right\}\\ &+\frac{1}{\sqrt n}\sum_{i=1}^n \left\{\frac{R_{i,39}(t)}{\widehat G_{C_{00}}(T_{i,3}^\ast) \widehat G_{C_{33}}(T_{i,9}^\ast) \widehat G_{C_{99,3}}(t)}-\frac{R_{i,39}(t)}{G_{C_{00}}(T_{i,3}^\ast) G_{C_{33}}(T_{i,9}^\ast) G_{C_{99,3}}(t)} \right\}\\ & +\frac{1}{\sqrt n}\sum_{i=1}^n\left\{\frac{R_{i,49}(t)}{\widehat G_{C_{00}}(T_{i,4}^\ast) \widehat G_{C_{44}}(T_{i,9}^\ast)\widehat G_{C_{99,4}}(t)}-\frac{R_{i,49}(t)}{G_{C_{00}}(T_{i,4}^\ast) G_{C_{44}}(T_{i,9}^\ast) G_{C_{99,4}}(t)} \right\}\\ &\approx_p\frac{1}{\sqrt n}\sum_{i=1}^n\left\{\frac{R_{i,39}(t)}{G_{C_{00}}(T_{i,3}^\ast)G_{C_{33}}(T_{i,9}^\ast)G_{C_{99,3}}(t)} + \frac{R_{i,49}(t)}{G_{C_{00}}(T_{i,4}^\ast)G_{C_{44}}(T_{i,9}^\ast)G_{C_{99,4}}(t)}\right.\\ &\left.\qquad -P_{09}(0,t) \right\}\\ &+\frac{1}{\sqrt n}\sum_{i=1}^n \int_0^t \left\{ \sum_{j=1}^n \frac{R_{j,39}(t)}{G_{C_{00}}(T_{j,3}^\ast)G_{C_{33}}(T_{j,9}^\ast)G_{C_{99,3}}(t)} I\{u \leq T_{j,3}^\ast \leq t\} \right\} \frac{dM_i^{C_{00}}(u)}{Y_{\bullet}^{C_{00}}(u)}\\ &+\frac{1}{\sqrt n}\sum_{i=1}^n \int_0^t \left\{ \sum_{j=1}^n \frac{R_{j,39}(t)}{G_{C_{00}}(T_{j,3}^\ast)G_{C_{33}}(T_{j,9}^\ast)G_{C_{99,3}}(t)} I\{u \leq T_{j,9}^\ast \leq t, \eta_{j,3}=1\} \right\} \\ &\qquad \frac{dM_i^{C_{33}}(u)}{Y_{\bullet}^{C_{33}}(u)}\\ &+\frac{1}{\sqrt n}\sum_{i=1}^n \left\{ \sum_{j=1}^n \frac{R_{j,39}(t)}{G_{C_{00}}(T_{j,3}^\ast)G_{C_{33}}(T_{j,9}^\ast)G_{C_{99,3}}(t)} I\{u \leq T_{j,9}^\ast \leq t, \eta_{j,3}=1\} \right\}\\ &\qquad \int_0^t\frac{dM_i^{C_{99,3}}(u)}{Y_{\bullet}^{C_{99,3}}(u)}\\[.2cm] &+\frac{1}{\sqrt n}\sum_{i=1}^n \int_0^t \left\{ \sum_{j=1}^n \frac{R_{j,49}(t)}{G_{C_{00}}(T_{j,4}^\ast)G_{C_{44}}(T_{j,9}^\ast)G_{C_{99,4}}(t)} I\{u \leq T_{j,4}^\ast \leq t\} \right\}\\ &\qquad \frac{dM_i^{C_{00}}(u)}{Y_{\bullet}^{C_{00}}(u)}\\ &+\frac{1}{\sqrt n}\sum_{i=1}^n \int_0^t \left\{ \sum_{j=1}^n \frac{R_{j,49}(t)}{G_{C_{00}}(T_{j,4}^\ast)G_{C_{44}}(T_{j,9}^\ast)G_{C_{99,4}}(t)} I\{u \leq T_{j,9}^\ast \leq t, \eta_{j,4}=1\} \right\} \\ &\qquad \frac{dM_i^{C_{44}}(u)}{Y_{\bullet}^{C_{44}}(u)}\\ &+\frac{1}{\sqrt n}\sum_{i=1}^n \left\{ \sum_{j=1}^n \frac{R_{j,49}(t)}{G_{C_{00}}(T_{j,4}^\ast)G_{C_{44}}(T_{j,9}^\ast)G_{C_{99,4}}(t)} I\{u \leq T_{j,9}^\ast \leq t, \eta_{j,4}=1\} \right\}\\ &\qquad \int_0^t \frac{dM_i^{C_{99,4}}(u)}{Y_{\bullet}^{C_{99,4}}(u)}, \end{aligned} $$

where $M_i^{C_{99,s}}(u)$, for s = 3, 4 can be estimated by

$$\displaystyle \begin{aligned} d\widehat{M}_i^{C_{99,s}}(u) &=dN_i^{C_{99,s}}(u) - Y_i^{C_{99,s}}(u) d\widehat{\varLambda}^{C_{99,s}}(u)\\[.2cm] d\widehat{\varLambda}^{C_{99,s}}(u)&=\frac{d N_{\bullet}^{C_{99,s}}(u)}{Y_{\bullet}^{C_{99,s}}(u)}. \end{aligned} $$

Thus, the variance of $\sqrt {n}\left \{\widehat {P}_{09}^{\mathrm {AIPCW}}(0,t)- P_{09}(0,t) \right \}$ can be estimated by

$$\displaystyle \begin{aligned} \begin{aligned} \widehat{\varSigma}_{P_{09}}(t)=\frac{1}{n}\sum_{i=1}^n \left\{\widehat{W}_{i,09}(t)\right\}^2, \end{aligned} \end{aligned} $$

(9)

where

$$\displaystyle \begin{aligned} \widehat{W}_{i,09}(t)&=\left\{\frac{R_{i,39}(t)}{\widehat G_{C_{00}}(T_{i,3}^\ast)\widehat G_{C_{33}}(T_{i,9}^\ast)\widehat G_{C_{99,3}}(t)} +\frac{R_{i,49}(t)}{\widehat G_{C_{00}}(T_{i,4}^\ast)\widehat G_{C_{44}}(T_{i,9}^\ast)\widehat G_{C_{99,4}}(t)}\right.\\ &\qquad \left. -\widehat P_{09}^{\mathrm{AIPCW}}(0,t) \right\}\\ &+\int_0^t\left\{ \sum_{j=1}^n \frac{R_{j,39}(t)}{\widehat G_{C_{00}}(T_{j,3}^\ast)\widehat G_{C_{33}}(T_{j,9}^\ast) \widehat G_{C_{99,3}}(t)} I\{u \leq T_{j,3}^\ast \leq t\} \right\} \frac{d\widehat{M}_i^{C_{00}}(u)}{Y_{\bullet}^{C_{00}}(u)}\\ &+\int_0^t\left\{\sum_{j=1}^n \frac{R_{j,39}(t)}{\widehat G_{C_{00}}(T_{j,3}^\ast)\widehat G_{C_{33}}(T_{j,9}^\ast)\widehat G_{C_{99,3}}(t)} I\{u \leq T_{j,9}^\ast \leq t, \eta_{j,3}=1\} \right\}\\ &\qquad \frac{d\widehat M_i^{C_{33}}(u)}{Y_{\bullet}^{C_{33}}(u)}\\ &+\left\{\sum_{j=1}^n\frac{R_{j,39}(t)}{\widehat G_{C_{00}}(T_{j,3}^\ast)\widehat G_{C_{33}}(T_{j,9}^\ast) \widehat G_{C_{99,3}}(t)} I\{u \leq T_{j,9}^\ast \leq t, \eta_{j,3}=1\} \right\}\\ &\qquad \int_0^t\frac{d\widehat M_i^{C_{99,3}}(u)}{Y_{\bullet}^{C_{99,3}}(u)}\\ &+\int_0^t \left\{ \sum_{j=1}^n \frac{R_{j,49}(t)}{\widehat G_{C_{00}}(T_{j,4}^\ast)\widehat G_{C_{44}}(T_{j,9}^\ast)\widehat G_{C_{99,4}}(t)} I\{u \leq T_{j,4}^\ast \leq t\} \right\}\\ & \qquad \frac{d\widehat M_i^{C_{00}}(u)}{Y_{\bullet}^{C_{00}}(u)}\\ &+ \int_0^t \left\{ \sum_{j=1}^n \frac{R_{j,49}(t)}{\widehat G_{C_{00}}(T_{j,4}^\ast)\widehat G_{C_{44}}(T_{j,9}^\ast)\widehat G_{C_{99,4}}(t)} I\{u \leq T_{j,9}^\ast \leq t, \eta_{j,4}=1\} \right\}\\ &\qquad \frac{d\widehat M_i^{C_{44}}(u)}{Y_{\bullet}^{C_{44}}(u)}\\ &+\left\{\sum_{j=1}^n\frac{R_{j,49}(t)}{\widehat G_{C_{00}}(T_{j,4}^\ast)\widehat G_{C_{44}}(T_{j,9}^\ast)\widehat G_{C_{99,4}}(t)} I\{u \leq T_{j,9}^\ast \leq t, \eta_{j,4}=1\} \right\}\\ &\qquad \int_0^t\frac{d\widehat M_i^{C_{99,4}}(u)}{Y_{\bullet}^{C_{99,4}}(u)} \end{aligned} $$

Similarly, for k = 10, 11, the variance of $\sqrt {n}\left \{\widehat {P}_{0k}^{\mathrm {AIPCW}}(0,t)- P_{0k}(0,t) \right \}$ can be estimated by:

$$\displaystyle \begin{aligned} \widehat{\varSigma}_{P_{0k}}(t)=\frac{1}{n}\sum_{i=1}^n \left\{\widehat{W}_{i,0k}(t)\right\}^2. {} \end{aligned}$$

where

$$\displaystyle \begin{aligned} \widehat{W}_{i,0k}(t)&=\left\{\frac{R_{i,3k}(t)}{\widehat G_{C_{00}}(T_{i,3}^\ast)\widehat G_{C_{33}}(T_{i,9}^\ast) \widehat G_{C_{99,3}}(T_{i,L_4}^\ast)} \right.\\ &\qquad \left.+ \frac{R_{i,4k}(t)}{\widehat G_{C_{00}}(T_{i,4}^\ast)\widehat G_{C_{44}}(T_{i,9}^\ast)\widehat G_{C_{99,4}}(T_{i,L_4}^\ast)} - \widehat P_{0k}^{\mathrm{AIPCW}}(0,t) \right\}\\ &+\int_0^t\left\{ \sum_{j=1}^n \frac{R_{j,3k}(t)}{\widehat G_{C_{00}}(T_{j,3}^\ast)\widehat G_{C_{33}}(T_{j,9}^\ast) \widehat G_{C_{99,3}}(T_{j,L_4}^\ast)} I\{u \leq T_{j,3}^\ast \leq t\} \right\}\\ &\qquad \frac{d\widehat M_i^{C_{00}}(u)}{Y_{\bullet}^{C_{00}}(u)}\\ &+\int_0^t\left\{ \sum_{j=1}^n \frac{R_{j,3k}(t)}{\widehat G_{C_{00}}(T_{j,3}^\ast)\widehat G_{C_{33}}(T_{j,9}^\ast) \widehat G_{C_{99,4}}(T_{j,L_4}^\ast)} I\{u \leq T_{j,9}^\ast \leq t, \eta_{j,3}=1\} \right\}\\ &\qquad \frac{d\widehat M_i^{C_{33}}(u)}{Y_{\bullet}^{C_{33}}(u)}\\ &+\int_0^t\left\{\sum_{j=1}^n\frac{R_{j,3k}(t)}{\widehat G_{C_{00}}(T_{j,3}^\ast)\widehat G_{C_{33}}(T_{j,9}^\ast) \widehat G_{C_{99,3}}(T_{j,L_4}^\ast)}I\{u \leq T_{j,L_4}^\ast \leq t, \eta_{j,3}=1\}\right\}\\ &\qquad \frac{d\widehat M_i^{C_{99,3}}(u)}{Y_{\bullet}^{C_{99,3}}(u)}\\ &+\int_0^t\left\{ \sum_{j=1}^n \frac{R_{j,4k}(t)}{\widehat G_{C_{00}}(T_{j,4}^\ast)\widehat G_{C_{44}}(T_{j,9}^\ast) \widehat G_{C_{99,4}}(T_{j,L_4}^\ast)} I\{u \leq T_{j,4}^\ast \leq t\} \right\}\\ &\qquad \frac{d\widehat M_i^{C_{00}}(u)}{Y_{\bullet}^{C_{00}}(u)}\\[.2cm] &+\int_0^t\left\{ \sum_{j=1}^n \frac{R_{j,4k}(t)}{\widehat G_{C_{00}}(T_{j,4}^\ast)\widehat G_{C_{44}}(T_{ij,9}^\ast) \widehat G_{C_{99,4}}(T_{j,L_4}^\ast)} I\{u \leq T_{j,9}^\ast \leq t, \eta_{j,4}=1\} \right\}\\ &\qquad \frac{d\widehat M_i^{C_{44}}(u)}{Y_{\bullet}^{C_{44}}(u)}\\ &+\int_0^t\left\{\sum_{j=1}^n\frac{R_{j,4k}(t)}{\widehat G_{C_{00}}(T_{j,4}^\ast)\widehat G_{C_{44}}(T_{j,9}^\ast) \widehat G_{C_{99,4}}(T_{j,L_4}^\ast)} I\{u \leq T_{j,L_4}^\ast \leq t, \eta_{j,4}=1\}\right\}\\ &\qquad \frac{d\widehat M_i^{C_{99,4}}(u)}{Y_{\bullet}^{C_{99,4}}(u)}. \end{aligned} $$