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# 统计代写|生存模型代考Survival Models代写|Cox’s Proportional Hazards Model

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## 统计代写|生存模型代考Survival Models代写|Cox’s Proportional Hazards Model

The Cox ${ }^1$ model proposes that the hazard rates of individuals are related via the relationship
$$h(t ; z)=h_0(t) \exp (\beta \cdot z)$$
where $\beta \in \mathbb{R}^p$ is a vector of regression parameters. Equation (12) shows a multiplicative influence on the hazard of any deviation away from zero in each of the $p$ covariates in $z$. Here $h_0(t)$ is known as the baseline hazard and represents the hazard of a (possibly hypothetical) individual with $z=0$.
The survivor function and density follow:
\begin{aligned} & S(t ; z)=S_0(t)^{\exp (\beta \cdot z)}, \ & f(t ; z)=\exp (\beta \cdot z) S_0(t)^{\exp (\beta \cdot z)-1} f_0(t), \end{aligned}
where $S_0(t)$ and $f_0(t)$ are the baseline survivor and density functions corresponding to $h_0(t)$

In $(12)$, only $h_0$, not $z$ or $\beta$, depends on time but the model can also be formulated with time-dependent covariates.

Under the Cox model, the hazard functions of two individuals with covariates $z_1, z_2$ are in constant proportion at all times; that is,
$$\frac{h\left(t ; z_1\right)}{h\left(t ; z_2\right)}=\frac{\exp \left(\beta \cdot z_1\right)}{\exp \left(\beta \cdot z_2\right)}=\exp \left{\beta \cdot\left(z_1-z_2\right)\right},$$
giving rise to the name proportional hazards.
Note that in general we can have any positive function $\psi$ of the covariates leading to $h(t ; z)=h_0(t) \psi(z ; \beta)$ and a ‘proportional hazards’ structure.

However, Cox’s model easily ensures that the hazard is always positive, and gives a linear model for the log-hazard which is very convenient in theory and practice.

## 统计代写|生存模型代考Survival Models代写|Model checking

In a practical survival problem several possible explanatory variables might present themselves as possibly affecting the survival time.

Part of the modelling procedure is to assess which (if any) are relevant to the model. That is, which ones have a significant effect on the distribution of $T$.
The fact that the partial likelihood behaves like a full likelihood allows us to make use of the test statistics from $\S 4.3$.

Suppose we currently have a model with $p$ covariates and we wish to consider the inclusion of an extra $q$ covariates.

We can fit both models, one with $p$ and the other with the $p+q$ covariates, by maximising the partial likelihoods. Let $\ell_p$ and $\ell_{p+q}$ be the maximised loglikelihoods of the two nested models. Note that $\ell_{p+q} \geq \ell_p$.
The likelihood ratio statistic is
$$2\left(\ell_{p+q}-\ell_p\right)$$
and has an asymptotic $\chi_q^2$ distribution under the null hypothesis that the extra $q$ covariates have no additional effect in the presence of the original $p$ explanatory variables.
$$H_0: \beta_{p+1}=\ldots=\beta_{p+q}=0 \text { vs. } H_1:\left(\beta_{p+1}, \ldots, \beta_{p+q}\right) \in \mathbb{R}^q$$

# 生存模型代考

## 统计代写|生存模型代考Survival Models代写|Cox’s Proportional Hazards Model

$$h(t ; z)=h_0(t) \exp (\beta \cdot z)$$

$$S(t ; z)=S_0(t)^{\exp (\beta \cdot z)}, \quad f(t ; z)=\exp (\beta \cdot z) S_0(t)^{\exp (\beta \cdot z)-1} f_0(t),$$

## 统计代写|生存模型代考Survival Models代写|Model checking

$$2\left(\ell_{p+q}-\ell_p\right)$$

$$H_0: \beta_{p+1}=\ldots=\beta_{p+q}=0 \text { vs. } H_1:\left(\beta_{p+1}, \ldots, \beta_{p+q}\right) \in \mathbb{R}^q$$

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。