Let \(Z_{i} = (1, \tilde{Z_{i} ^{T}})^{T}\) and \(\beta _{0,k} = (log(\eta _{0,k}) , \tilde{\beta_{0,k} ^{T}})^{T}\), by (1), (2) and \(E(W_{i} | \tilde {Z} _{i} , \xi _{i} = k) = 1\) we can get \[ E[N_{i} ^{*} (t) | \xi_{i} = k, Z_{i} \} = \mu_{0} (t) exp(Z_{i} ^{T} \beta _{0,k}) \tag{1} \] Where: \(\mu _{0} (t) = \int _{0} ^{t} \lambda _{0} (s) ds\)
Given \(C_{i}\) and \((N_{i} ^{*} , \xi _{i})\) are independent given \(Z_{i}\), so we have \(E[\frac{N_{i} ^{*} (C _{i})}{\mu_{0} (C_{i})} | \xi _{i} = k, Z_{i}] = exp(Z^{T} _{i} \beta _{0,k})\)
Thus: \[ E[ I(\xi _{i} = k) Z_{i} \{ \frac{N_{i} ^{*} (C _{i})}{\mu_{0} (C_{i})} - exp(Z_{i} ^{T} \beta _{0,k}) \}] = 0 \tag{2} \]
To build estimating equation for \(\beta _{0,k}\), we need to deal with unobserved \(\xi_{i}\)
The authors recover the missing information on \(I(\xi _{i} =k)\) by conditioning it on the observed \(Z_{i}, C_{i}\) and \(D_{i} = N_{i}(C_{i})\)
Let \(\tau_{i,k} = E[I(\xi _{i} = k) | Z_{i}, D_{i}, C_{i}]\), we have \[ \tau _{i,k} = \frac{P(D_{i} = d_{i} | \xi _{i} = k, Z _{i}, C_{i}) P(\xi _{i} = k | Z_{i}, C_{i})}{\sum _{l = 1} ^{K} P(D_{i} = d_{i} | \xi _{i} = l, Z _{i}, C_{i}) P(\xi _{i} = l | Z_{i}, C_{i})} \tag{3} \]
- \(\xi_{i}\) can be modeled by multinomial logistic regression, so we have:
\[ P(\xi _{i} = k | Z_{i}, C_{i}) = p_{k}(\alpha _{0} , \tilde{Z} _{i}) = \frac{exp(\tilde{Z} _{i} ^{T} \alpha _{0,k})}{\sum _{l = 1} ^{K} exp(\tilde{Z} _{i} ^{T} \alpha _{0,l})} \tag{4} \]
-
As shown in model assumption (SLCARE section), \(N_{i} ^{*} (t)\) can be treated as a non-homogeneous Poisson process with mean \(exp(Z^{T}\beta _{k})W\mu_{0}(t)\), thus \(\{ \mu_{0}(T^{(i)}) \}\) can be viewed as random variables generated from a Poisson process with mean \(exp(Z^{T}\beta _{k})Wt\). Therefore, Let \(T^{(0)} = 0\), \(\{ \mu_{0}(T^{(j)}) - \mu_{0}(T^{(j-1)}) \}\) , \(\{ \mu_{0}(T^{(d)}) \}\) follows exponential, gamma distribution.
As \(P ( D _{i} = d_{i} | \xi _{i} = k , Z _{i}, C _{i}) = P ( \mu _{0}(T^{(d)}), \mu_{0} (C) | \xi _{i} = k , Z _{i}, C _{i})\), we have:
\[ P ( D _{i} = d_{i} | \xi _{i} = k , Z _{i}, C _{i}) = \int _{0} ^{\infty} \frac{ \{ exp(Z_{i} ^{T} \beta _{0,k}) w \cdot \mu_{0}(C _{i}) \} ^{d_{i}} }{d_{i} !} exp \{ - exp(Z_{i} ^{T} \beta _{0,k}) w \cdot \mu _{0} (C _{i}) \} \cdot f _{W} (w) dw \tag{5} \]
By (2) - (5)
we can have 1st estimating equation: \[ S_{1,n,k}(\alpha, \beta, \mu_{0}) = \frac{1}{n}\sum _{i = 1} ^{n} \tau_{ik}(\alpha, \beta, \mu_{0}) Z_{i} \{ \frac{N_i ^{*} (C _{i})}{\mu_{0} (C_{i})} - exp(Z_{i} ^{T} \beta_{k}) \} = 0,~~~k = 1, \cdots , K \tag{6} \]
Based on logistic regression model, the score equation when \(\xi _{i}\) are observed is: \[ \sum _{ i = 1} ^{n} \sum _{k = 1} ^{K} I(\xi _{i} = k) \frac{\partial }{\partial \alpha} log ~p_{k}(\alpha, \tilde{Z_{i}}) = 0 \tag{7} \] Again by conditional score, we can have our 2nd estimating equation: \[ S_{2,n,k}(\alpha, \beta, \mu_{0}) = \frac{1}{n}\sum _{i = 1} ^{n} \tau_{ik}(\alpha, \beta, \mu_{0}) ( \tilde{Z_{i}} - \frac{exp(\tilde{Z} _{i} ^{T} \alpha _{k}) \tilde{Z_{i}}}{\sum _{j = 1} ^{K} exp(\tilde{Z} _{i} ^{T} \alpha _{j})} ) = 0,~~~k = 1, \cdots , K \tag{8} \] In (10) and (11), \(\mu _{0}(t)\) can be evaluated by a Nelson-Aalen type estimator, under the assumed multiplicative intensity model. We can have $ (t) = exp{ (t) } $ with \(\hat{H}(t) = - \int _{t} ^{\upsilon ^{*}} \frac{\sum _{i = 1} ^{n} d N_{i}(s)}{\sum _{i = 1} ^{n} I(C_{i} \geq s) N_{i}(s)}\).(9)
Finally, the authors proposed the estimating equations for \(\alpha _{0}\) and \(\beta _{0}\): \[ n^{1/2} S_{1,n}(\alpha, \beta, \hat{\mu}) = 0 \tag{10} \]
\[ n^{1/2} S_{2,n}(\alpha, \beta, \hat{\mu}) = 0 \tag{11} \]
where \(S_{j,n} (\alpha, \beta, \hat{\mu}) = ( S_{j,n,1} (\alpha, \beta, \hat{\mu})^{T}, \cdots , S_{j,n,K} (\alpha, \beta, \hat{\mu})^{T} )^{T}\), \(j = 1,2\)