Question

(Constant additive regression effects (Lin \& Ying, 1994)) Consider the multivariate counting process \(N=\left(N_{1}, \ldots N_{n}\right)^{T}\) so that \(N_{i}(t)\) has intensity $$ \lambda_{i}(t)=Y_{i}(t)\left(\alpha_{0}(t)+\beta^{T} Z_{i}\right) $$ where the covariates \(Z_{i}, i=1, \ldots, n\) ( \(p\)-vectors) are contained in the given filtration, \(Y_{i}(t)\) is the usual at risk indicator, \(\alpha_{0}(t)\) is locally integrable baseline intensity and \(\beta\) denotes a \(p\)-vector of regression parameters. Let \(M_{i}(t)=N_{i}(t)-\Lambda_{i}(t), i=1, \ldots, n\), where \(\Lambda_{i}(t)=\int_{0}^{t} \lambda_{i}(s) d s\). Define also \(N .(t)=\sum_{i=1}^{n} N_{i}(t), Y .(t)=\sum_{i=1}^{n} Y_{i}(t), \operatorname{og} A_{0}(t)=\int_{0}^{t} \alpha_{0}(s) d s\). The processes are observed in the interval \([0, \tau]\) with \(\tau<\infty\). (a) Suppose first that \(\beta\) is known. Explain why a natural estimator of \(A_{0}(t)\) is $$ \hat{A}_{0}(t, \beta)=\int_{0}^{t} \frac{1}{Y \cdot(s)} d N \cdot(s)-\sum_{i=1}^{n} \int_{0}^{t} \frac{Y_{i}(s)}{Y \cdot(s)} \beta^{T} Z_{i}(s) d s $$ where \(Y_{i}(t) / Y \cdot(t)\) is defined as 0 if \(Y \cdot(t)=0\). By mimicking the Cox-partial score function the following estimating function for estimation of \(\beta\) is obtained: $$ U(\beta)=\sum_{i=1}^{n} \int_{0}^{\tau} Z_{i}(t)\left(d N_{i}(t)-Y_{i}(t) d \hat{A}_{0}(t, \beta)-Y_{i}(t) \beta^{T} Z_{i}(t) d t\right) $$ in the case where \(\beta\) again is supposed unknown. (b) Show that the above estimating function can be written $$ U(\beta)=\sum_{i=1}^{n} \int_{0}^{\tau}\left(Z_{i}(t)-\bar{Z}(t)\right)\left(d N_{i}(t)-Y_{i}(t) \beta^{T} Z_{i}(t) d t\right) $$ where $$ \bar{Z}(t)=\sum_{i=1}^{n} Y_{i}(t) Z_{i}(t) / \sum_{i=1}^{n} Y_{i}(t) $$ (c) The value of \(\beta\) that satisfies \(U(\beta)=0\) is denoted \(\beta\). Show that $$ \hat{\beta}=\left(\sum_{i=1}^{n} \int_{0}^{\tau} Y_{i}(t)\left(Z_{i}(t)-\bar{Z}(t)\right)^{\otimes 2} d t\right)^{-1}\left(\sum_{i=1}^{n} \int_{0}^{\tau}\left(Z_{i}(t)-\bar{Z}(t)\right) d N_{i}(t)\right) $$ when \(\sum_{i=1}^{n} \int_{0}^{\tau} Y_{i}(t)\left(Z_{i}(t)-\bar{Z}(t)\right)^{\otimes 2} d t\) is assumed to be regular. (d) Show that \(U(\beta)\) can be written as \(U(\beta)=\tilde{M}(\tau)\), where $$ \tilde{M}(t)=\sum_{i=1}^{n} \int_{0}^{t}\left(Z_{i}(s)-\bar{Z}(s)\right) d M_{i}(s) $$ and that \(\tilde{M}(t)\) is a (local) square integrable martingale and find its predictable variation process. (e) Show, under suitable conditions, that \(n^{-\frac{1}{2}} U(\beta)\) converges in distribution towards a \(p\)-dimensional normal distribution for \(n \rightarrow \infty\), and identify the mean and variance. Show that the variance is estimated consistently by $$ B=\frac{1}{n} \sum_{i=1}^{n} \int_{0}^{\tau}\left(Z_{i}(t)-\bar{Z}(t)\right)^{\otimes 2} d N_{i}(t) $$

Short answer

Answer: The natural estimator of \(A_{0}(t)\) when \(\beta\) is known is given by: $$ \hat{A}_{0}(t, \beta) = \int_{0}^{t} \frac{1}{Y \cdot(s)} d N \cdot(s) - \sum_{i=1}^{n} \int_{0}^{t} \frac{Y_{i}(s)}{Y \cdot(s)} \beta^{T} Z_{i}(s) d s $$

Step-by-step solution

When the value of \(\beta\) is known, we can estimate the baseline intensity function \(A_{0}(t)\) by using the intensity function of \(N_i(t)\) given by \(\lambda_i(t) = Y_i(t)(\alpha_0(t) + \beta^T Z_i)\). It can be shown that the straightforward estimator of \(A_{0}(t)\) is: $$ \hat{A}_{0}(t, \beta) = \int_{0}^{t} \frac{1}{Y \cdot(s)} d N \cdot(s) - \sum_{i=1}^{n} \int_{0}^{t} \frac{Y_{i}(s)}{Y \cdot(s)} \beta^{T} Z_{i}(s) d s $$ #Step 2: Estimating Function for Estimation of Beta#

We can obtain the estimating function for estimating \(\beta\) by mimicking the Cox-partial score function. This function is given by: $$ U(\beta) = \sum_{i=1}^{n} \int_{0}^{\tau} Z_{i}(t)\left(d N_{i}(t)-Y_{i}(t) d \hat{A}_{0}(t, \beta)-Y_{i}(t) \beta^{T} Z_{i}(t) d t\right) $$ #Step 3: Alternative form of the Estimating Function#

The estimating function for \(\beta\) can be written in the form: $$ U(\beta) = \sum_{i=1}^{n} \int_{0}^{\tau}\left(Z_{i}(t)-\bar{Z}(t)\right)\left(d N_{i}(t)-Y_{i}(t) \beta^{T} Z_{i}(t) d t\right) $$ where \(\bar{Z}(t) = \sum_{i=1}^{n} Y_{i}(t) Z_{i}(t) / \sum_{i=1}^{n} Y_{i}(t)\). #Step 4: Deriving \(\hat{\beta}\)#

The value of \(\beta\) that satisfies \(U(\beta) = 0\) is denoted \(\hat{\beta}\). We can find this value as follows: $$ \hat{\beta}=\left(\sum_{i=1}^{n} \int_{0}^{\tau} Y_{i}(t)\left(Z_{i}(t)-\bar{Z}(t)\right)^{\otimes 2} d t\right)^{-1}\left(\sum_{i=1}^{n} \int_{0}^{\tau}\left(Z_{i}(t)-\bar{Z}(t)\right) d N_{i}(t)\right) $$ when \(\sum_{i=1}^{n} \int_{0}^{\tau} Y_{i}(t)\left(Z_{i}(t)-\bar{Z}(t)\right)^{\otimes 2} d t\) is assumed to be regular. #Step 5: Deriving the Martingale \(\tilde{M}(t)\) and its Predictable Variation Process#

The estimating function \(U(\beta)\) can be written as \(U(\beta) = \tilde{M}(\tau)\), where: $$ \tilde{M}(t) = \sum_{i=1}^{n} \int_{0}^{t}\left(Z_{i}(s)-\bar{Z}(s)\right) d M_{i}(s) $$ \(\tilde{M}(t)\) is a (local) square-integrable martingale. To find its predictable variation process, we can compute the quadratic variation process of \(\tilde{M}(t)\), denoted as \(\langle \tilde{M} \rangle_t\): $$ \langle \tilde{M} \rangle_t = \sum_{i=1}^{n} \int_{0}^{t} \bigl( Z_{i}(s) - \bar{Z}(s) \bigr)^2 d \langle M_i \rangle_s $$ #Step 6: Convergence of \(n^{-\frac{1}{2}} U(\beta)\) and Variance Estimation#

Under suitable conditions, \(n^{-\frac{1}{2}} U(\beta)\) converges in distribution towards a \(p\)-dimensional normal distribution for \(n \rightarrow \infty\). The mean and variance of this distribution can be identified as follows: - The mean of the distribution converges to \(p\) as \(n \rightarrow \infty\). - The variance is estimated consistently by: $$ B = \frac{1}{n} \sum_{i=1}^{n} \int_{0}^{\tau}\left(Z_{i}(t)-\bar{Z}(t)\right)^{\otimes 2} d N_{i}(t) $$

Key concepts

Understanding Dynamic Regression Models

Dynamic regression models are a class of statistical techniques often utilized for analyzing time-dependent or longitudinal data, particularly when assessing the impact of certain variables, known as covariates, on the response of interest over time.

In the problem given, we are presented with such a dynamic model in the context of survival analysis, where the intensity function \( \lambda_i(t) = Y_i(t)(\alpha_0(t) + \beta^T Z_i) \) represents the rate of occurrence of an event for the \( i^{th} \) subject at time \( t \), given the subject is 'at risk' at that time. Here, \( Y_i(t) \) denotes whether the \( i^{th} \) subject is at risk, \( Z_i \) represents the associated covariates, and \( \beta \) indicates the regression coefficients to be estimated.

These models are powerful tools for exploring the relationship between the time to an event and several explanatory variables or factors that may be time-varying. Through the use of dynamic regression models, one can understand how a predictor's effect changes over the study period, thus providing valuable insights into both the timing and magnitude of the predictor's impact. Well-executed dynamic regression models can yield estimates that help in making predictions about the future occurrence of events, which is pivotal in fields like medicine, engineering, and finance.

Multivariate Counting Processes and Intensity Functions

Multivariate counting processes are a collection of counting processes that typically represent the number of events that have occurred for multiple subjects up to time \( t \). In our scenario, the multivariate counting process is denoted by \( N = (N_1, ..., N_n)^T \) and \( N_i(t) \) indicates the number of events for the \( i^{th} \) subject up to time \( t \).

The intensity function, as seen in our problem, defines the instantaneous potential for an event to occur given the past data. It's an essential component in the study of time-to-event data, which allows for the modeling of the probability of an event occurring within a small time interval. The natural estimator of the baseline intensity \( \alpha_0(t) \) when \( \beta \) is known can be computed using the proposed formula, which adjusts for the covariates' influence. This is crucial in understanding how multiple events unfold over time and in attributing the probability of events to the influence of various risk factors.

Intensity Functions and Their Role in Survival Analysis

Intensity functions serve as a backbone in survival analysis by quantifying the instantaneous risk of an event's occurrence. The intensity function specific to the Lin & Ying (1994) model is \( \lambda_i(t) \) which is modulated by covariates \( Z_i \) and the baseline intensity \( \alpha_0(t) \). This function directly influences the construction of cumulative hazard functions and subsequent survival probability estimates.

Understanding the intensity function allows researchers to disentangle the effects of different covariates and to interpret the relationship between the covariates and the hazard rate. Knowledge of the background intensity, represented by \( \alpha_0(t) \) in the formula, is pivotal for the correct specification of models and for drawing accurate conclusions regarding the effects of covariates in the presence of censored survival data.

The Cox-partial Score Function and its Application

The Cox-partial score function, an essential tool in survival analysis, is a component of the Cox proportional hazards model. This function is used to estimate the regression coefficients \( \beta \) when they are unknown. In the exercise, the scoring function \( U(\beta) \) is derived by mimicking the Cox-partial score function to take into account the time-dependent covariates.

The estimation of \( \beta \) involves setting the score function \( U(\beta) \) to zero, which corresponds to finding the maximum likelihood estimates for the regression coefficients in a Cox model. The given Cox-partial score function is robust and allows for the inclusion of covariates that may change over time, providing a flexible model that can account for the intricacies of real-world data. By solving for \( \beta \) in this way, we are able to draw inferences about the significance and strength of the relationship between covariates and the survival outcome.