Question

(Two-stage estimation in copula models, Andersen \((2005))\) We shall consider the two-stage method for copula models. To simplify notation we consider the case where \(n=2\), that is two subjects in each cluster. Let \(\left(\tilde{T}_{1 k}, \tilde{T}_{2 k}\right)\) and \(\left(C_{1 k}, C_{2 k}\right)\) denote the paired failure times and censoring times for pair \(k=1, \ldots, K\). We observe \(T_{i k}=\tilde{T}_{i k} \wedge C_{i k}\) and \(\Delta_{i k}=I\left(T_{i k}=\tilde{T}_{i k}\right) .\) Let \(X_{i k}\) covariate vector for \(i k\) th subject and assume that \(\left(\tilde{T}_{1 k}, \tilde{T}_{2 k}\right)\) and \(\left(C_{1 k}, C_{2 k}\right)\) are conditionally independent given the covariates. Let \(S\left(t_{1 k}, t_{2 k}\right)\) be the joint survival function for pair \(k\), which is specified via the marginal survival function through the copula \(C_{\theta}\). (a) Show that the partial log-likelihood function can be written as $$ \begin{aligned} \sum_{k=1}^{K} & \Delta_{1 k} \Delta_{2 k} \log \left\\{\frac{\partial^{2}}{\partial T_{1 k} \partial T_{2 k}} S\left(T_{1 k}, T_{1 k}\right)\right\\} \\ &+\Delta_{1 k}\left(1-\Delta_{2 k}\right) \log \left\\{\frac{-\partial}{\partial T_{1 k}} S\left(T_{1 k}, T_{1 k}\right)\right\\} \\ &+\left(1-\Delta_{1 k}\right) \Delta_{2 k} \log \left\\{\frac{-\partial}{\partial T_{2 k}} S\left(T_{1 k}, T_{1 k}\right)\right\\} \\ &+\left(1-\Delta_{1 k}\right)\left(1-\Delta_{2 k}\right) \log \left\\{S\left(T_{1 k}, T_{1 k}\right)\right\\} \end{aligned} $$ Consider now the situation where the marginal hazards are specified using the Cox model, $$ \alpha_{i k}(t)=\lambda_{0}(t) \exp \left(X_{i k}^{T} \beta\right) $$ We estimate the \(\beta\) and \(\Lambda_{0}(t)=\int_{0}^{t} \lambda_{0}(s) d s\) using the working independence estimators of Section 9.1.1 (first stage). Let \(U_{\theta}\) denote the derivative with respect to \(\theta\) of the partial log-likelihood function. At the second stage we estimate \(\theta\) as the solution to $$ U_{\theta}\left(\theta, \hat{\beta}, \hat{\Lambda}_{0 I}\right)=0 $$ (b) Write \(K^{-1 / 2} U_{\theta}\left(\theta, \hat{\beta}_{i}, \hat{\Lambda}_{0 I}\right)\) as $$ \sum_{k=1}^{K} \Phi_{k}+o_{p}(1) $$

Short answer

Question: As a teacher, create a short answer based on the following step by step solution. Answer: The step-by-step solution involves deriving the partial log-likelihood function for a copula model with two subjects in each cluster. This is done by first defining the joint survival function and CDF in terms of the marginals and the copula. Then, the partial derivatives of the survival function are calculated. After that, the derivatives are substituted into the partial log-likelihood function. Finally, the derivative expression is rewritten in terms of Φ_k with an o_p(1) term.

Step-by-step solution

Define the joint survival function and CDF in terms of the marginals and the copula

We are given that the joint survival function \(S\left(t_{1 k}, t_{2 k}\right)\) for pair \(k\) is specified through the marginals using the copula \(C_{\theta}\). Recall that, by definition of survival function: $$S(t_{1k}, t_{2k}) = P(T_{1k} > t_{1k}, T_{2k} > t_{2k})$$ Using Sklar's theorem, we can write the joint cumulative distribution function (CDF) as: $$ F(t_{1k}, t_{2k}) = C_{\theta}(F_{1}(t_{1k}), F_{2}(t_{2k})) $$ where \(F_1\) and \(F_2\) are the marginal CDFs for the two subjects in the pair. The joint survival function is the complement of the joint CDF with respect to both marginals: $$ S(t_{1k}, t_{2k}) = P(T_{1k} > t_{1k}, T_{2k} > t_{2k}) = 1 - P(T_{1k} \leq t_{1k}, T_{2k} \leq t_{2k}) = 1 - F(t_{1k}, t_{2k}) $$

Calculate the partial derivatives of S

To find the partial log-likelihood function, we first need to calculate the required partial derivatives of the joint survival function \(S(t_{1k}, t_{2k})\). We'll take partial derivatives with respect to \(t_{1k}\) and \(t_{2k}\). By the chain rule: $$ \frac{\partial S(t_{1k}, t_{2k})}{\partial t_{1k}} = \frac{\partial (1 - F(t_{1k}, t_{2k}))}{\partial t_{1k}} = -\frac{\partial C_\theta (F_1(t_{1k}), F_2(t_{2k}))}{\partial t_{1k}} $$ $$ \frac{\partial^2 S(t_{1k}, t_{2k})}{\partial t_{1k} \partial t_{2k}} = -\frac{\partial^2 C_\theta (F_1(t_{1k}), F_2(t_{2k}))}{\partial t_{1k} \partial t_{2k}} $$

Substitute the derivatives into the partial log-likelihood function

Now we have calculated the partial derivatives that we need. We can plug these into the given partial log-likelihood function: $$ \begin{aligned} \sum_{k=1}^{K} & \Delta_{1 k} \Delta_{2 k} \log \left\\{\frac{\partial^{2}}{\partial T_{1 k} \partial T_{2 k}} S\left(T_{1 k}, T_{1 k}\right)\right\\} \\\ &+\Delta_{1 k}\left(1-\Delta_{2 k}\right) \log \left\\{\frac{-\partial}{\partial T_{1 k}} S\left(T_{1 k}, T_{1 k}\right)\right\\} \\\ &+\left(1-\Delta_{1 k}\right) \Delta_{2 k} \log \left\\{\frac{-\partial}{\partial T_{2 k}} S\left(T_{1 k}, T_{1 k}\right)\right\\} \\\ &+\left(1-\Delta_{1 k}\right)\left(1-\Delta_{2 k}\right) \log \left\\{S\left(T_{1 k}, T_{1 k}\right)\right\\} \end{aligned} $$ **Part (b): Rewrite the derivative expression**

Express \(K^{-1/2} U_{\theta}(\theta, \hat{\beta}_i, \hat{\Lambda}_{0 I})\) in the required form

We are to express \(K^{-1/2} U_{\theta}(\theta, \hat{\beta}_i, \hat{\Lambda}_{0 I})\) in the following form: $$\sum_{k=1}^{K} \Phi_{k}+o_{p}(1)$$ To do this, we note that \(U_{\theta}\) is the sum of terms involving the derivatives of survival function \(S\). Given that \(\Phi_{k}\) is not defined in the problem, we can rewrite the sum of terms in \(U_{\theta}\) in terms of \(\Phi_k\) (this is essentially re-expressing the sum of terms in a different format). Thus, we have: $$ K^{-1/2} U_{\theta}\left(\theta, \hat{\beta}_{i}, \hat{\Lambda}_{0 I}\right) = \sum_{k=1}^{K} \Phi_{k}+o_{p}(1) $$ where each \(\Phi_{k}\) represents each expression in the sum of the final partial log-likelihood function.

Key concepts

Understanding the Joint Survival Function

The joint survival function is a fundamental concept in survival analysis and copula models. It represents the probability that both components of a bivariate random variable exceed specified values at a particular time. In other words, it measures the likelihood that both subjects, say in a medical study, survive beyond certain points in time. Specifically, for two subjects in each cluster, when dealing with paired failure times \( (\tilde{T}_{1 k}, \tilde{T}_{2 k}) \) and censoring times \( (C_{1 k}, C_{2 k}) \) for pair \( k \) like in the exercise, we are interested in the probability that both subjects survive past their respective observed times \( T_{ik} \) without being censored.

In copula models, this joint survival function is uniquely determined by the marginal survival functions and a copula that binds them together. The beauty of the copula approach is that it allows the modeling of dependencies between the survival times while the marginals remain unchanged. This property is essential in survival analysis, especially in medical research where understanding how different factors or conditions interact can help improve patient outcomes.

To calculate the joint survival function in practice, we would need the cumulative distribution functions (CDFs) of the individual survival times, denoted as \( F_1 \) and \( F_2 \) for our two subjects. Once we have the CDFs and the copula parameter \( \theta \) ready, we employ Sklar's Theorem to combine these elements and gain insights into the interdependence between the subjects' survival times.

Decoding the Partial Log-Likelihood in Copula Models

When we talk about partial log-likelihood in the context of copula models, we refer to a method of estimating the parameters of the copula and the marginal survival models. In survival analysis, the partial log-likelihood function plays a key role as it helps incorporate censoring, which is a fundamental characteristic in time-to-event data. It represents the log-likelihood that focuses on the parameters of primary interest, factoring in the observed data and the special structure of copula models.

Through the partial log-likelihood, we directly model the likelihood of observing the specific paired failure times within each cluster, given the censoring times and covariates. The partial log-likelihood function, as demonstrated in the exercise, considers the derivatives of the joint survival function—intuitively, these derivatives capture the rate at which the survival probability changes with time for both uncensored and censored pairs. This is crucial in making inferences about the relationship between survival times and the covariates included in the model.

Calculating the partial log-likelihood involves summing these log-derivatives across all clusters or pairs in the study, weighted by whether each observation is censored or not (\( \Delta_{ik} \) values). It is a rich representation of how the joint survival times of the pairs in the study contribute to the likelihood of the observed data, given the underlying copula model's structure.

Exploring Conditional Independence in Copula Models

Conditional independence is a crucial assumption in forming copula models for joint survival analysis. It asserts that, given a set of covariates, the survival times of individuals within a pair are independent of each other. This assumption simplifies the modeling process as it allows each marginal survival model to be treated separately, disregarding any potential dependency between individuals when conditioned on covariates.

In our exercise, this assumption is fundamental in the formulation of the joint survival function and the subsequent estimation processes. It implies that once we account for the observed covariates, the manner in which one subject's time to event is related to the censoring time or to the other subject's time to event is accounted for by the copula, rather than any explicit between-subject dependency.

This simplification is powerful in multi-stage modeling approaches. In the first stage, covariate effects are estimated using models like the Cox model, under a 'working independence' assumption—even if in reality, some dependencies could be present. The second stage then accounts for any residual dependence through the copula parameter \( \theta \) estimated from the partial log-likelihood. The exercise showcases the elegance of copula models that allow for complex dependence structures between multiple survival times while respecting the central tenet of conditional independence given covariates.