Question

(Blocked Cox-Aalen) Let \(N(t)=I(T \leq t)\) have intensity on the Cox-Aalen form: $$ \lambda(t)=Y(t)\left(X^{T}(t) \alpha(t)\right) \exp \left(Z^{T}(t) \beta\right) $$ and assume that \(n\) i.i.d. replicas of the model are observed over the timeinterval \([0, \tau]\). Some effects are multiplicative and they can thus be interpreted as effect modifiers. The model assumes the effect of \(X_{1}\) and \(X_{2}\), say, is modified in the same way by \(Z\). One extension of the model allows different effect modification $$ \left.\lambda(t)=Y(t)\left(\sum_{k=1}^{K} X_{k}^{T}(t) \alpha_{k}(t)\right) \exp \left(Z^{T}(t) \gamma_{k}\right)\right) $$ where \(X=\left(X_{1}, X_{2}, \ldots, X_{K}\right)\) is grouped into \(K\) blocks. (a) By redefining the covariate \(Z\) the model can be written as $$ \lambda(t)=Y(t)\left(\sum_{k=1}^{K}\left(X_{k}^{T}(t) \alpha_{k}(t) \exp \left(Z_{k}^{T}(t) \bar{\gamma}\right)\right)\right. $$ Specify this construction in detail. (b) Derive a score equation for \(\gamma=\left(\gamma_{1}, \ldots, \gamma_{K}\right)\) or \(\tilde{\gamma}\) in the alternative parametrization. (c) Make the basic derivations to obtain the asymptotic properties.

Short answer

Question: Redefine the covariate \(Z_k\) in the Cox-Aalen model to rewrite the model in the desired form. Answer: \(Z_k = Z \cdot \exp(-Z^{T}\gamma_k)\)

Step-by-step solution

Redefine the Covariate Z

To achieve the desired form of the model: $$ \lambda(t)=Y(t)\left(\sum_{k=1}^{K}\left(X_{k}^{T}(t) \alpha_{k}(t) \exp \left(Z_{k}^{T}(t) \bar{\gamma}\right)\right)\right), $$ we define the new covariate \(Z_k\) as \(Z_k = Z \cdot \exp(-Z^{T}\gamma_k)\), such that \(Z_k^{T}(t) \bar{\gamma} = Z^{T}(t)\gamma_k\). Therefore, the model can be rewritten as: $$ \lambda(t) = Y(t)\left(\sum_{k=1}^{K} X_{k}^{T}(t) \alpha_{k}(t) \exp \left(Z_{k}^{T}(t) \bar{\gamma}\right)\right). $$

Derive the Score Equation for \(\gamma\) or \(\tilde{\gamma}\)

To derive the score equation, we first compute the partial derivative of the log-likelihood function with respect to \(\gamma\) or \(\tilde{\gamma}\). Let \(L(\gamma)\) be the log-likelihood function of the extended model. Thus, the score function is given by: $$ U(\gamma) = \frac{\partial L(\gamma)}{\partial \gamma}. $$ For each \(j=1, \ldots, K\), we derive: $$ U_j(\gamma) = \sum_{i=1}^{n}\left[\frac{\partial \log \lambda(T_i)}{\partial \gamma_j} - \sum_{k=1}^{K}\int_0^\tau \frac{\partial X^{T}_k(t)\alpha_k(t)}{\partial \gamma_j}\exp\left(Z_k^{T}(t) \bar{\gamma}\right) dN_i(t)\right], $$ where \(N(t)=I(T\leq t)\) with intensity on the Cox-Aalen form \(\lambda(t)\). Setting the system of equations \(U(\gamma) = 0\) leads to the score equations.

Obtain Asymptotic Properties

We want the asymptotic properties of the estimator \(\hat{\gamma}\). In particular, we need the variance of the estimator. To find the asymptotic properties, we start by finding the Fisher information matrix for the parameters \(\gamma\). Let \(I(\gamma)\) denote the Fisher information matrix, and the \(j\)-th and \(l\)-th elements of \(I(\gamma)\) is given by: $$ I_{j,l}(\gamma) = -E\left[\frac{\partial^2 L(\gamma)}{\partial \gamma_j \partial \gamma_l}\right]. $$ In our case, $$ I_{j,l}(\gamma) = E\left[\sum_{i=1}^n \int_0^\tau \frac{\partial^2 \log \lambda(T_i)}{\partial \gamma_j \partial \gamma_l} dN_i(t)\right]. $$ As the sample size \(n\) approaches infinity, the asymptotic properties can be described by the following theorem: under regularity conditions, if \(\hat{\gamma}\) is the maximum likelihood estimator (MLE) of \(\gamma\), then \(\hat{\gamma}\) is asymptotically normal with mean \(\gamma\) and variance \(\sigma^2 = I(\gamma)^{-1}\).

Key concepts

Survival Data Analysis

Survival data analysis, also referred to as 'time-to-event' analysis, is a branch of statistics that deals with the time until an event of interest occurs, such as death, relapse, or failure of a mechanical system. In this field, we often use survival models to estimate the survival function and hazard rate. The Cox-Aalen model is a type of survival model that accommodates the dynamic nature of covariates over time and allows for the assessment of time-dependent effects.

The key feature of survival data is 'censoring', which occurs when we have incomplete information about the survival time of an individual, due to either the study ending before the event occurs or the subject leaving the study. This leads to challenges in the analysis since traditional statistical methods can't be directly applied. Survival data analysis is equipped with techniques like the Cox-Aalen model to properly handle censored data.

Intensity Function

The intensity function, denoted by \(\lambda(t)\), is crucial in assessing the risk of an event happening at a particular time, given that the event has not yet happened. It is conceptually similar to the hazard function in survival analysis but accounts for covariates that may influence the risk. In the context of the Cox-Aalen model, the intensity function specifies how covariates affect the failure rate. The model expresses the intensity as a product of a baseline intensity, a time-dependent covariate effect, and exponential covariates.

Understanding the intensity function is vital in the Cox-Aalen model. It allows researchers to explore how various factors contribute to the risk over time, adapting the model to include both multiplicative and additive effects of covariates on survival times.

Asymptotic Properties

Asymptotic properties of estimators in statistical models are properties that describe the behavior of the estimators as the sample size becomes very large, technically referred to as 'infinity'. These properties are essential since they provide information about the distribution of an estimator when sampling from an infinite population.

In survival analysis, understanding the asymptotic properties of estimators like those obtained from the Cox-Aalen model provides us with assurances about their reliability. The two main asymptotic properties of concern are consistency (the ability of the estimator to converge to the true parameter values as the sample size grows) and normality (the distribution of the estimator approaching a normal distribution). The third property of interest is efficiency, which reflects the estimator's variance being as low as theoretically possible. The demonstration that an estimator is asymptotically normal, with mean equal to the true parameter value and a predictable variance, gives us confidence to use it in a practical sense for hypothesis testing and constructing confidence intervals.

Score Equation

In maximum likelihood estimation, the score equation is the derivative of the log-likelihood function with respect to the parameter being estimated. Setting the score equal to zero gives us the score equations, which are necessary for finding the maximum likelihood estimate of the parameter.

In the Cox-Aalen model, the score equation helps identify the best parameters that explain how covariates are associated with the hazard of an event. This can be achieved by taking the derivative of the log-likelihood function with respect to the parameter of interest and equating it to zero. The expression derived as a result is used to find estimates of the parameter that maximize the likelihood of observing the data. These steps are essential in the process of fitting the model to data in survival analysis.

Maximum Likelihood Estimation

Maximum likelihood estimation (MLE) is a statistical method for estimating the parameters of a model. In the context of survival analysis, it involves choosing the parameter values for the Cox-Aalen model that make the observed survival data most probable. The 'likelihood' is a function of the parameters given the data, and the estimation process involves finding the parameter values that maximize this likelihood.

The MLE is popular in survival analysis because of its desirable properties: it is consistent and asymptotically normal under certain conditions. Additionally, the computation of the MLE involves the creation of score equations and the use of numerical methods to solve for the parameter values that set these score equations to zero. It is these maximum likelihood parameter estimates that allow us to draw conclusions about the relationship between covariates and survival time.