My question refers to the paper Borgan et al (2000). In short, consider this rather general form of the pseudo-likelihood for estimating coefficients of a Cox model in a case-cohort design (equation from the paper):

$$\tilde{\mathcal{L}}(\boldsymbol{\beta}) = \prod_{t_j} \Bigg[ \frac{\exp(\boldsymbol{\beta}^\text{T} \mathbf{Z}_{i_j}(t_j)) w_{i_j}(t_j)}{\sum_{f \in \tilde{\mathcal{R}}(t_j)} Y_k (t_j) \exp(\boldsymbol{\beta}^\text{T} \mathbf{Z}_k(t_j)) w_k(t_j)} \Bigg].$$

The proposed Estimator III set the weights and at-risk sets to be:

$$\begin{align} w_k(t_j) &= \frac{n_{s(k)}}{m_{s(k)}}, \\[12pt] \tilde{\mathcal{R}}(t_j) &= \begin{cases} \tilde{\mathcal{C}} & \text{if } i_j \in \tilde{\mathcal{C}}, \\[6pt] \tilde{\mathcal{C}} \cup \{ i_j \} \setminus \{ J_{s(i_j)} \} & \text{if } i_j \notin \tilde{\mathcal{C}}, \\ \end{cases} \end{align}$$

where $J_l$ is a randomly selected (at the start of the estimation) subject among the subcohort members from stratum $l$ and $s(i_j)$ is the stratum to which the subject $i_j$ belongs. Therefore, for each run of estimation, $J_{s(i_j)}$ is different for each stratum, and therefore the estimates of interests, $\hat{\boldsymbol{\beta}}$, etc., will be different.

Question: When I want to compare estimates from this estimator with randomness and those from estimators without randomness, given all sample information (times to event, covariates, strata memberships, etc.), should I run Estimator III multiple times and compare the average of estimates to the estimate from one run of other estimators without randomness? Or just pick the estimate from one run of Estimator III?



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