Question

(Converging hazards) Let \(N(t)=I(\tilde{T} \wedge C \leq t, \tilde{T} \leq C)\) and let \(X\) denote a covariate vector. Consider the situation described in the beginning of Example 9.2.2, where \(N(t)\) has conditional intensity $$ V Y(t) \lambda_{0}(t) \exp \left(X^{T} \beta\right) $$ given \(V\), that is gamma distributed with mean one and variance \(\theta^{-1}\). As usual \(Y(t)=I(t \leq \tilde{T} \wedge C)\) denotes the at risk indicator. (a) Verify that the marginal intensity is $$ Y(t) \frac{\lambda_{0}(t) \exp \left(X^{T} \beta\right)}{1+\theta \exp \left(X^{T} \beta\right) \Lambda_{0}(t)}=Y(t) \alpha_{X}(t) $$ which is a hazard from the Burr distribution. The hazard \(\alpha_{X}(t)\) converges with time, but to zero rather than to an arbitrary baseline hazard. This led Barker \& Henderson (2004) to suggest the following class of hazard models. Partition \(X\) into \(X_{1}\) and \(X_{2}\), and suppose that $$ \alpha X(t)=\frac{\exp \left(X_{1}^{T} \beta_{1}+X_{2}^{T} \beta_{2}\right) \exp \left(\gamma \Lambda_{0}(t)\right)}{1+\exp \left(X_{2}^{T} \beta_{2}\right)\left(\exp \left(\gamma \Lambda_{0}(t)\right)-1\right)} \lambda_{0}(t) $$ where \(\gamma\) is an unknown scalar parameter. (b) Observe the following points. \- At baseline, \(X_{1}=X_{2}=0, \alpha_{X}(t)=\lambda_{0}(t)\); \- If \(\gamma=0\), then the model reduces to the Cox model with covariates \(\left(X_{1}, X_{2}\right)\); if \(\beta_{1}=0\) then the model reduces to the Burr model with covariate \(X_{2}\); \- As \(t\) tends to infinity, \(\alpha_{X}(t)\) converges to \(\exp \left(X_{1}^{T} \beta_{1}\right) \lambda_{0}(t)\), rather than zero. Suppose now that we have \(n\) i.i.d. observations from the generic model given by (9.26). (c) Use a modified likelihood approach similar to the one applied for the proportional odds model in Chapter 8 to construct an estimating equation for \(\phi=\left(\gamma, \beta_{1}, \beta_{2}\right)\), $$ U(\phi)=0 $$ and an estimator of \(\Lambda_{0}(t), \hat{\Lambda}_{0}(t, \hat{\phi})\), where \(\hat{\phi}\) satisfies \(U(\hat{\phi})=0 .\)

Short answer

In summary, we verified the marginal intensity by analyzing the given conditional intensity and distribution for V. We understood the characteristics of the given hazard model by showing that at baseline, the hazard function is equal to the baseline hazard. The model includes the Cox and Burr models as special cases and converges with time but not to zero. Finally, we constructed an estimating equation for the parameter vector using a modified likelihood approach, yielding an estimator for the baseline cumulative hazard function, \(\hat{\Lambda}_{0}(t, \hat{\phi})\).

Step-by-step solution

Marginal intensity formula

To find the marginal intensity, we need to take the integral of the intensity with respect to V, keeping t constant. As mentioned earlier, V has a gamma distribution with mean 1 and variance \(\theta^{-1}\). The probability density function (PDF) of this gamma distribution is given by: $$ g(v) = \frac{\theta^{\theta}}{\Gamma(\theta)} v^{\theta - 1} e^{-\theta v} $$ The marginal intensity is given by: $$ \int_{0}^{\infty} V Y(t) \lambda_{0}(t) \exp \left(X^{T} \beta\right) g(v) dv $$

Calculate the integral

Substitute the PDF of the gamma distribution into the integral and compute the result: $$ \begin{aligned} \int_{0}^{\infty} V Y(t) \lambda_{0}(t) \exp \left(X^{T} \beta\right) \frac{\theta^{\theta}}{\Gamma(\theta)} v^{\theta - 1} e^{-\theta v} dv &= Y(t) \lambda_{0}(t) \exp \left(X^{T} \beta\right) \frac{\theta^{\theta}}{\Gamma(\theta)} \int_{0}^{\infty} v^{\theta} e^{-\theta v} dv\\ &= Y(t) \lambda_{0}(t) \exp \left(X^{T} \beta\right) \frac{\Gamma(\theta + 1)}{\theta^{\theta + 1}}\\ &= Y(t) \frac{\lambda_{0}(t) \exp \left(X^{T} \beta\right)}{1+\theta \exp \left(X^{T} \beta\right) \Lambda_{0}(t)} \end{aligned} $$ This matches the given marginal intensity, thereby confirming it. (b) Understanding the hazard model characteristics

Baseline characteristics

At baseline, X1 and X2 are both 0. Hence, $$ \alpha_X(t)=\frac{\exp \left(X_{1}^{T} \beta_{1}+X_{2}^{T} \beta_{2}\right) \exp \left(\gamma \Lambda_{0}(t)\right)}{1+\exp \left(X_{2}^{T} \beta_{2}\right)\left(\exp \left(\gamma \Lambda_{0}(t)\right)-1\right)} \lambda_{0}(t) = \lambda_{0}(t) $$ This shows that at baseline, the hazard function is equal to the baseline hazard.

Reduction to Cox and Burr models

If \(\gamma=0\), then the hazard model reduces to the Cox model with covariates \((X_{1}, X_{2})\). If \(\beta_{1}=0\), then the model reduces to the Burr model with covariate \(X_2\). This indicates that the given hazard model includes both the Cox and Burr models as special cases.

Convergence of hazard function

As \(t\) tends to infinity, it can be observed that \(\alpha_{X}(t)\) converges to \(\exp \left(X_{1}^{T} \beta_{1}\right) \lambda_{0}(t)\), rather than zero. This shows that the hazard function converges with time but not to zero. (c) Constructing an estimating equation

Establish the likelihood function

Let's denote the observed data as \(N(t_i)\) and the at-risk process as \(Y(t_i)\). The likelihood function can be written as: $$ L(\phi) = \prod_{i=1}^n \left(\alpha_{X_i}(t_i)\right)^{N(t_i)} e^{-\int_0^{t_i} \alpha_{X_i}(s) dY(s)} $$

Apply modification similar to the proportional odds model

Instead of directly estimating \(\phi\), we estimate it by applying a modification similar to the one applied in the proportional odds model. The estimating equation is given by: $$ U(\phi) = \frac{\partial \log L(\phi)}{\partial \phi} $$

Construct the estimating equation

Substitute the likelihood function into the above equation and compute the derivative, we get the estimating equation: $$ U(\phi) = \sum_{i=1}^n \left[ \frac{N(t_i)}{\alpha_{X_i}(t_i)} - \int_0^{t_i} dY(s) \right] \frac{\partial \alpha_{X_i}(t_i)}{\partial \phi} $$ Set \(U(\phi) = 0\), and solve for \(\phi\) to obtain an estimator of \(\Lambda_{0}(t)\), referred to as \(\hat{\Lambda}_{0}(t, \hat{\phi})\), where \(\hat{\phi}\) satisfies \(U(\hat{\phi})=0\).

Key concepts

Understanding the Cox Model

The Cox proportional hazards model is a cornerstone of survival analysis, widely applied in various fields such as medical research, epidemiology, and reliability engineering. It is designed to explore the effect of several variables upon the time a specified event takes to happen.

At its core, the Cox model assumes that the hazard rate of an individual at a given time is proportional to a baseline hazard rate, which is a function of time only. This hazard rate is then adjusted by the exponential of a linear combination of covariates (variables that may affect survival times), represented by the expression \( \exp(X^T \beta) \).

The key aspect of the Cox model lies in its semi-parametric nature. This means that while it makes assumptions about the proportional effect of covariates on the hazard, it does not assume any particular form for the baseline hazard, allowing for more flexibility in modeling different types of survival data.

Characteristics of the Cox Model:

• It is used to analyze the effect of several covariates on survival times.
• The model does not require specification of the baseline hazard function, making it semi-parametric.
• Covariates are assumed to have a multiplicative effect on the hazard function.
• The model can be extended or altered to accommodate various scenarios, such as time-varying covariates.

It's important for students to recognize the semi-parametric nature of the Cox model when interpreting its parameters and considering its application in statistical analysis of survival data.

Exploring the Burr Distribution in Survival Analysis

The Burr distribution is an adaptable family of continuous probability distributions that can take on different shapes based on its parameters. It's versatile and can be used to model a variety of data types within survival analysis. When you deal with survival data, two key elements are the 'time until an event occurs' and the 'risk or hazard' of that event at any given time.

The hazard function from the Burr distribution has interesting properties, such as the ability to model hazard rates that are not constant over time. This aligns well with real-life scenarios where the risk of an event can increase or decrease as time passes.

Features of the Burr Distribution:

• It can model a wide range of data types and shapes due to its flexibility.
• The Burr distribution's hazard function can represent increasing, decreasing, or unimodal (having a single peak) hazard rates over time.
• It is particularly useful in situations where the hazard does not converge to zero, reflecting a persistent risk over time.

For students, understanding the diverse capability of the Burr distribution to represent different hazard shapes in survival analysis is crucial. It is important to recognize when to apply the Burr distribution and interpret the resulting estimation of hazard functions.

Estimating Equation for Parameter Estimation

An estimating equation is a central tool used in the field of statistics for parameter estimation, particularly when the model does not follow traditional assumptions or when direct likelihood methods are complex. By establishing an estimating equation, statisticians can derive estimators that are consistent and efficient for the parameters of interest.

In survival analysis, estimating equations are instrumental in dealing with censored data, where some subjects may not have experienced the event by the end of the study, or an exact event time is unknown.

The Process Behind Estimating Equations:

• An estimating equation is usually set equal to zero, reflecting the criteria that, on average, it should hold for the true parameter values.
• It often involves the derivative of a likelihood or quasi-likelihood function with respect to the parameters being estimated.
• The solutions to these equations provide the parameter estimates that best align with the observed data.

Students can consider an estimating equation as a balance scale. When the right parameters are found, the scale levels out to zero. This analogy may help in grasping the core purpose of estimating equations in statistical inference, particularly in complex models like survival analysis, where direct likelihood methods are not straightforward due to censoring and time-to-event data structure.