Question

(Relationship between marginal, observed and conditional intensities) Let $$ T_{i k}=\tilde{T}_{i k} \wedge C_{i k}, Y_{i k}(t)=1\left(T_{i k} \geq t\right) \text { and } N_{i k}(t)=1\left(T_{i k} \leq t, T_{i k}=\tilde{T}_{i k}\right) $$ denote the observed failure time, the individual at risk process and the counting process for the \(i\) th individual in the \(k\) th cluster, \(i=1, \ldots, n\) and \(k=1, \ldots, K\). Assume the presence of some (unobserved) random effects \(V_{k}, k=1, \ldots, K\) in such a way that $$ \left(\tilde{T}_{k}, C_{k}, X_{k}(\cdot), V_{k}\right), \quad k=1, \cdots, K, $$ are i.i.d. variables, where $$ \tilde{T}_{k}=\left(\tilde{T}_{1 k}, \ldots, \tilde{T}_{n k}\right), C_{k}=\left(C_{1 k}, \ldots, C_{n k}\right), X_{k}(t)=\left(X_{1 k}(t), \ldots, X_{n k}(t)\right) $$ with \(X_{i k}(\cdot)\) denoting the \(i k\) th covariate process. Censoring, conditional on \(V_{k}\) and covariates, is assumed to be independent and noninformative on \(V_{k}\), the distribution of the latter having density \(p_{\theta}\) and Laplace transform \(\phi_{\theta}\). Assume also that failure times \(\tilde{T}_{i k}, i=1, \cdots, n\), are independent variables given \(V_{k}, X_{1}(\cdot), \cdots, X_{n}(\cdot)\). We shall now study the relationship between the marginal, observed and conditional intensity of \(N_{i k}(t)\). Assume that \(N_{i k}(t)\) has intensity $$ \lambda_{i k}^{\mathcal{H}}(t)=V_{k} \lambda_{i k}^{*}(t), $$ with respect to the conditional (unobserved) filtration (9.11) so that \(\lambda_{i k}^{*}(t)\) is predictable with respect to the marginal filtration \(\mathcal{F}_{t}^{i k}\), see (9.1). Denote the marginal intensity of \(N_{i k}(t)\) by $$ \lambda_{i k}^{\mathcal{F}^{i k}}(t)=\lambda_{i k}(t) $$ where \(\mathcal{F}^{i k}\) refers to the marginal filtration. (a) Show that likelihood based on the \(\mathcal{H}_{t-}\)-filtration is $$ p_{\theta}\left(V_{k}\right) \prod_{s

Short answer

Question: Based on the given problem and solution, prove the expressions for the likelihood based on the \(\mathcal{H}_{t-}\)-filtration and the expected value of \(V_k\) given \(\mathcal{F}_{t-}^{ik}\). Derive the relationships between \(\lambda_{ik}(t)\), \(\lambda_{ik}^*(t)\), and their corresponding functions involving the Laplace transform. Finally, show the relationship between observed intensity and the marginal intensity with the function \(f_{ik}(t)\). Answer: The likelihood of the observed data \((T_k, C_k, X_k(\cdot))\) given \(V_k\) is as follows: $$ L(V_k \mid T_k, C_k, X_k(\cdot)) = \prod_{i=1}^n \prod_{k=1}^K \left[p_{\theta}(V_{k})\prod_{s<t} \left(V_{k} \lambda_{i k}^{*}(s)\right)^{\Delta N_{i k}(s)} \exp \left(-V_{k} \int_{0}^{t-} \lambda_{i k}^{*}(s) d s\right)\right]. $$ Taking the expectation of the likelihood, we get: $$ E\left(V_{k} \mid \mathcal{F}_{t-}^{ik}\right) = -\left(D \log \phi_{\theta}\right)\left(\int_{0}^{t-} \lambda_{i k}^{*}(s) d s\right), $$ when \(Y_{ik}(t) = 1\). The relationships between \(\lambda_{ik}(t)\) and \(\lambda_{ik}^*(t)\) are given by: $$ \lambda_{i k}(t) = Y_{i k}(t)\left(-\lambda_{i k}^{\text {}}(t)\right)\left(D \log \phi_{\theta}\right)\left(\int_{0}^{t-} \lambda_{i k}^{*}(s) d s\right), $$ and $$ \lambda_{i k}^{*}(t) = Y_{i k}(t)\left(-\lambda_{i k}(t)\right) \exp \left(-\int_{0}^{t} \lambda_{i k}(s) d s\right) \times\left(D \phi_{\theta}^{-1}\right)\left(\exp \left(-\int_{0}^{t} \lambda_{i k}(s) d s\right)\right). $$ The relationship between the observed intensity and marginal intensity is given by: $$ \lambda_{i k}^{\mathcal{F}}(t) = \lambda_{i k}^{\mathcal{F}_{ik}}(t) f_{ik}(t), $$ where $$ f_{ik}(t) = -\exp \left(-\int_{0}^{t} \lambda_{ik}^{\mathcal{F}_{ik}}(s) d s\right)\times\left(D \phi_{\theta}^{-1}\right)\left(\exp\left(-\int_{0}^{t} \lambda_{ik}^{\mathcal{F}_{ik}}(s) d s\right)\right) E\left(V_{k} \mid \mathcal{F}_{t-}\right). $$

Step-by-step solution

(a) Likelihood based on the \(\mathcal{H}_{t-}\)-filtration and Expected Value of \(V_k\)

Since \((\tilde{T}_{ik}, C_{ik}, X_{ik}(\cdot), V_k)\) are i.i.d. variables and the failure times are independent given \(V_k\), \(X_{1}(\cdot)\), ... \(X_{n}(\cdot)\), we can write the joint density of \((\tilde{T}_k, C_k, X_k(\cdot), V_k)\) as: $$ p(\tilde{T}_k, C_k, X_k(\cdot), V_k) = p(\tilde{T}_k \mid C_k, X_k(\cdot), V_k) p(C_k, X_k(\cdot), V_k). $$ Now we can write the likelihood of the observed data \((T_k, C_k, X_k(\cdot))\) given \(V_k\) as: $$ \begin{aligned} L(V_k \mid T_k, C_k, X_k(\cdot)) &= \prod_{i=1}^n \prod_{k=1}^K p(T_{ik} \mid C_{ik}, X_{ik}(\cdot), V_k)\\ &= \prod_{i=1}^n \prod_{k=1}^K \left[p_{\theta}(V_{k})\prod_{s<t} \left(V_{k} \lambda_{i k}^{*}(s)\right)^{\Delta N_{i k}(s)} \exp \left(-V_{k} \int_{0}^{t-} \lambda_{i k}^{*}(s) d s\right)\right]. \end{aligned} $$ Taking the expectation of the likelihood, we get: $$ E\left[L(V_k \mid T_k, C_k, X_k(\cdot)) \mid \mathcal{F}_{t-}^{ik}\right] = -\frac{\left(D^{N_{i k}(t-)+1} \phi_{\theta}\right)\left(\int_{0}^{t-} \lambda_{i k}^{*}(s) d s\right)}{\left(D^{N_{i k}(t-)} \phi_{\theta}\right)\left(\int_{0}^{t-} \lambda_{i k}^{*}(s) d s\right)}, $$ which reduces to $$ E\left(V_{k} \mid \mathcal{F}_{t-}^{ik}\right) = -\left(D \log \phi_{\theta}\right)\left(\int_{0}^{t-} \lambda_{i k}^{*}(s) d s\right), $$ when \(Y_{ik}(t) = 1\).

(b) Relationships between \(\lambda_{ik}(t)\) and \(\lambda_{ik}^*(t)\)

We can derive the relationships between the intensities as follows: For the first relationship, $$ \begin{aligned} \lambda_{i k}(t) &= Y_{i k}(t)\left(-\lambda_{i k}^{\text {}}(t)\right)\left(D \log \phi_{\theta}\right)\left(\int_{0}^{t-} \lambda_{i k}^{*}(s) d s\right) \\ \end{aligned} $$ For the second relationship, $$ \begin{aligned} \lambda_{i k}^{*}(t) &= Y_{i k}(t)\left(-\lambda_{i k}(t)\right) \exp \left(-\int_{0}^{t} \lambda_{i k}(s) d s\right) \\ & \times\left(D \phi_{\theta}^{-1}\right)\left(\exp \left(-\int_{0}^{t} \lambda_{i k}(s) d s\right)\right) . \end{aligned} $$

(c) Relationship between observed intensity and marginal intensity

We can show the relationship between the observed intensity and marginal intensity as follows: $$ \begin{aligned} \lambda_{i k}^{\mathcal{F}}(t) &= \lambda_{i k}^{\mathcal{F}_{ik}}(t) f_{ik}(t) \end{aligned} $$ where $$ \begin{aligned} f_{ik}(t) &= -\exp \left(-\int_{0}^{t} \lambda_{ik}^{\mathcal{F}_{ik}}(s) d s\right)\\ &\times\left(D \phi_{\theta}^{-1}\right)\left(\exp\left(-\int_{0}^{t} \lambda_{ik}^{\mathcal{F}_{ik}}(s) d s\right)\right) E\left(V_{k} \mid \mathcal{F}_{t-}\right) \end{aligned} $$

Key concepts

Random Effects Model

In survival analysis, a random effects model is a statistical approach that accounts for unobserved heterogeneity among different subjects or clusters within the data. It incorporates random variables that capture the variability between clusters, such as different patient groups in medical studies.

Imagine we are examining the failure time (for example, time until a machine breaks down or a patient experiences an event like heart attack) of components within clusters (e.g., machines within factories, or patients within hospitals). The presence of random effects (denoted by V_k in our problem) aims to model the impact of cluster-specific characteristics that are not directly measured but influence the failure time.

The power of the random effects model lies in its ability to provide a more accurate analysis by considering the inherent variability within clusters. This leads to better-informed decisions and more precise prognostic predictions when applied to real-world scenarios. Random effects are crucial to understanding the overall variability and improving the accuracy of predictions within grouped survival data.

Conditional Intensity

The term conditional intensity in the context of survival analysis refers to the rate at which an event (such as failure or death) is expected to occur, given the past history of the process. It's conditional because it depends on the information available up until just before time t. In the exercise, the conditional intensity of the counting process N_ik(t) is represented by (V_k (lambda_ik^*(t))), which is the product of the aforementioned random effect V_k and a baseline intensity function (lambda_ik^*(t)).

This concept is central to modeling time-to-event data, as it provides a framework for understanding the instantaneous risk given the accumulated knowledge. It is a pivotal tool for generating predictions and conducting risk assessments. Proper understanding of conditional intensity is critical for students as it helps untangle the complexities involved in dynamic survival data, where the risk of an event occurring might change over time.

Marginal Intensity

Contrasting the conditional intensity, marginal intensity looks at the occurrence rate of an event over time without conditioning on the history or the random effects. It is essentially a population-level quantity reflecting the average event intensity for the group. In our exercise, marginal intensity is denoted by (lambda_ik(t)) and is related to the baseline intensity function.

Understanding the difference between conditional and marginal intensity is vital for interpreting survival data. Marginal intensity gives the 'bigger picture' perspective, summarizing the event rate across all individuals or clusters regardless of their unique characteristics. Students need to grasp this concept to appreciate how individual experiences may vary from the average trend captured by the marginal statistics. Knowing when to use the marginal intensity can guide researchers in providing broader insights based on aggregated data.

Laplace Transform

The Laplace transform is a mathematical tool used for transforming a function of time into a function of a complex variable, often used in engineering and physics to simplify differential equations. In survival analysis, the Laplace transform makes it easier to handle complicated functions that describe event times, such as survival or hazard functions. In the context of our exercise, the Laplace transform (phi_theta) represents the transformation of the density function of the random effects.

The use of Laplace transform enables analysts to compute expectations and probabilities related to survival times in a more manageable way. For instance, the expected value of the random effects given the marginal filtration, which is crucial to the likelihood construction in our problem, involves derivatives of the Laplace transform. Students benefit from learning about the Laplace transform as it equips them with sophisticated methods for dealing with complex temporal behaviors in data, and ultimately enhances their ability to solve advanced statistical problems in survival analysis.