91Ó°ÊÓ

Let \(Y\) be a positive continuous random variable with survivor and hazard functions \(\mathcal{F}(y)\) and \(h(y)\). Let \(\psi(x)\) and \(\chi(x)\) be arbitrary continuous positive functions of the covariate \(x\), with \(\psi(0)=\chi(0)=1\). In a proportional hazards model, the effect of a non-zero covariate is that the hazard function becomes \(h(y) \psi(x)\), whereas in an accelerated life model, the survivor function becomes \(\mathcal{F}\\{y \chi(x)\\}\). Show that the survivor function for the proportional hazards model is \(\mathcal{F}(y)^{\psi(x)}\), and deduce that this model is also an accelerated life model if and only if $$ \log \psi(x)+G(\tau)=G\\{\tau+\log \chi(x)\\} $$ where \(G(\tau)=\log \left\\{-\log \mathcal{F}\left(e^{\tau}\right)\right\\}\). Show that if this holds for all \(\tau\) and some non-unit \(\chi(x)\), we must have \(G(\tau)=\kappa \tau+\alpha\), for constants \(\kappa\) and \(\alpha\), and find an expression for \(\chi(x)\) in terms of \(\psi(x) .\) Hence or otherwise show that the classes of proportional hazards and accelerated life models coincide if and only if \(Y\) has a Weibull distribution.

Step by step solution

01

Identify the Problem

We need to show that the survivor function for a proportional hazards model is \( \mathcal{F}(y)^{\psi(x)} \) and deduce under what conditions this model is also an accelerated life model.

02

Proportional Hazards Model Survivor Function

In a proportional hazards model, the relationship is described by the hazard function becoming \( h(y) \psi(x) \). The survival function is related to the hazard function by \( S(y) = \exp(-\int_0^y h(y') \, dy') \). When the hazard function is \( h(y) \psi(x) \), the integral becomes \( \int_0^y h(y') \psi(x) \, dy' \), resulting in the survivor function \( \exp(-\psi(x) \int_0^y h(y') \, dy') = S(y)^{\psi(x)} = \mathcal{F}(y)^{\psi(x)} \).

03

Relation to Accelerated Life Model

In an accelerated life model, the survivor function is \( \mathcal{F}(y \chi(x)) \). To equate this with \( \mathcal{F}(y)^{\psi(x)} \), both survivor functions need to describe the same function under varying \( x \). Given the equation involving \( G(\tau) \), we are tasked to identify when the proportional hazard model is also an accelerated life model.

04

Differentiating the Functional Equation

The condition \( \log \psi(x) + G(\tau) = G(\tau + \log \chi(x)) \) allows us to equate the forms of the survivor function in both models. Differentiating both sides with respect to \( \tau \) yields \( G'(\tau) = G'(\tau + \log \chi(x)) \). This suggests \( G' \) is constant if this holds for all \( \tau \), meaning \( G(\tau) = \kappa \tau + \alpha \).

05

Determining the Function \( \chi(x) \)

Using the equation \( \log \psi(x) + G(\tau) = G(\tau + \log \chi(x)) \) and the linear form \( G(\tau) = \kappa \tau + \alpha \), we deduce \( \kappa (\log \chi(x)) = \log \psi(x) \), leading to \( \chi(x) = \psi(x)^{1/\kappa} \).

06

Conclusion and Weibull Distribution

A Weibull distribution satisfies \( G(\tau) = \kappa \tau + \alpha \), allowing both the proportional hazards and accelerated life models to coincide. Therefore, this coincidence implies that \( Y \) must have a Weibull distribution.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Proportional Hazards Model

The Proportional Hazards Model is a statistical tool widely used in survival analysis. Imagine a situation where we want to understand how different factors (covariates) impact the risk of an event, such as equipment failure or time to disease recovery.

In this model, the effect of a covariate on the hazard function is multiplicative. This means we modify the baseline hazard function by multiplying it with a function of the covariate. If the original hazard function is denoted by \( h(y) \), after considering the covariate impact, it becomes \( h(y) \psi(x) \), where \( \psi(x) \) is a specific function describing the covariate's influence.

This modification also influences the survivor function, which measures the probability of surviving past a certain time \( y \). Due to this covariate-modified hazard function, the survivor function changes to \( \mathcal{F}(y)^{\psi(x)} \).

• The baseline hazard function characterizes the risk without any influence from covariates.
• A positive covariate function \( \psi(x) \) enhances the understanding of how external factors increase or decrease hazard.
• Overall, this model helps analyze the relationship between survival and covariates over time.

Accelerated Life Model

The Accelerated Life Model (ALM) is another approach commonly used in survival analysis. Unlike the Proportional Hazards Model, the ALM alters the timeline of the event occurrence due to covariates. This implies that covariates can speed up or slow down the life process.

In the ALM framework, the survivor function is altered to \( \mathcal{F}(y \chi(x)) \), where \( \chi(x) \) is the function describing how covariates scale time. Essentially, the covariate shifts the timeline by modifying the time scale directly and hence, affects the rate at which failures occur.

• The model reflects changes in life duration caused by covariates.
• It propositionalizes the time acceleration approach where each covariate adjusts the scale of time.
• Understanding ALM helps clarify which factors contribute to life span shortening or prolongation.

Interestingly, the Proportional Hazards Model coincides with the Accelerated Life Model in very specific conditions, which leads us into discussing the Weibull distribution.

Weibull Distribution

The Weibull Distribution is a versatile model often used to model time-to-failure data. It provides the mathematical basis needed for the coherence of both Proportional Hazards and Accelerated Life Models.

For a random variable \( Y \) to follow a Weibull distribution, its logarithm of the survivor function must have the form \( G(\tau) = \kappa \tau + \alpha \). This linear relationship allows the surviving probability functions in both PHM and ALM to match when expanded.

• Characterized by its ability to model diverse shapes of hazard functions, providing immense flexibility in survival analysis.
• The distribution function's scalability allows for the agreement between time-scale adjustment and the proportional hazard mechanisms.
• Understanding the Weibull distribution is key to insightfully deploying both hazard and accelerated model structures effectively.

The real power of the Weibull distribution is the succinct form it presents, aligning both models fundamentally, making it indispensable in reliability engineering and life data analysis.