Posted on Categories:生存模型, 统计代写, 统计代考

# 统计代写|生存模型代考Survival Models代写|Empirical survivor function

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## 统计代写|生存模型代考Survival Models代写|Empirical survivor function

Suppose we have gathered some survival time data for $n$ individuals drawn from a population with common survivor function $S(t)$. Assume, for now, that there are no censored observations. Then the empirical survivor function
$$\hat{S}(t)=\frac{\text { number of observations }>t}{n}$$
provides an estimate of $S(t)$ derived purely from the data. Notice that $\hat{S}(t)$ is an antitonic step function with jumps at the death times. $(\hat{S}(t)$ is a non-parametric estimator of $S(t)$. In Chapter 5 we shall see how to estimate $S$ in the presence of censoring.)
Since $S(t)=\exp {-H(t)}$, the simple transformation
$$\hat{H}(t)=-\log \hat{S}(t)$$
then provides a similar, data-based estimate of the cumulative hazard.

We can use a plot of $\hat{H}(t)$ to see how the overall shape of the function compares with those dictated by some different distributions. In particular,

• $H(t)$ vs. $t$ is linear for the exponential;
• $\log H(t)$ vs. $\log t$ is linear for the Weibull;
• $\log H(t)$ vs. $t$ approaches linearity for large $t$ for the Gompertz-Makeham.
After plotting $\hat{H}(t)$ in these ways we can decide which assumption is closest. This provides us with a crude but useful method of selecting a parametric model for a given set of data.

## 统计代写|生存模型代考Survival Models代写|Maximum Likelihood Estimation

Assume that the data $\left(t_1, \ldots, t_n\right)$ are $n$ independent, possibly censored realisations from the distribution $F(t ; \theta)$, where the form of $F$ is known (e.g. Weibull) but the parameters $\theta$ are unknown.

Definition: The likelihood is the joint probability of the observed data, regarded as a function of the unknown parameters $\theta$.

The likelihood principle states that all of the information about the parameters in a sample of data is contained in the likelihood function.

The law of likelihood extends this principle to state that the degree to which the data supports one parameter value over another is given by the ratio of their likelihoods.

The Maximum likelihood estimate (MLE) of $\theta$ is based solely upon the likelihood of the observed data. Maximum likelihood estimation thus provides a principled and general method of estimating parameters in parametric distributions using observed data.

As the name suggests the MLE is the value of the parameters that maximises the probability of the observed data,
$$\begin{gathered} \hat{\theta}=\underset{\theta \in \Theta}{\arg \sup } L(\theta), \ L(\theta)=\prod_{i=1}^n \operatorname{Pr}\left(t_i \mid F(; \theta)\right), \end{gathered}$$
where $L$ is the likelihood function and $\operatorname{Pr}\left(t_i \mid F(; \theta)\right)$ is the probability of observing the $i^{\text {th }}$ observation given the distribution $F$ with parameters $\theta$.

If none of the observations are censored we find,
$$L(\theta)=\prod_{i=1}^n f\left(t_i ; \theta\right) .$$
For each censored observation the contribution to the likelihood depends on the type of censoring. Recall, an observation is left (right) censored if we only have an upper (lower) bound for $T$, or interval censored if we only know that $T$ lies in a specified interval.

• $\operatorname{Pr}\left(t_i \mid F(; \theta)\right)=S\left(t_i ; \theta\right)$ for an observation right-censored at $t_i$;
• $\operatorname{Pr}\left(t_i \mid F(; \theta)\right)=F\left(t_i ; \theta\right)$ for an observation left-censored at $t_i$;
• $\operatorname{Pr}\left(t_i \mid F(; \theta)\right)=F\left(t_i^{(u)} ; \theta\right)-F\left(t_i^{(I)} ; \theta\right)$ for an observation interval-censored within $\left[t_i^{(I)}, t_i^{(u)}\right]$.

# 生存模型代考

## 统计代写|生存模型代考Survival Models代写|Empirical survivor function

$$\hat{S}(t)=\frac{\text { number of observations }>t}{n}$$

$$\hat{H}(t)=-\log \hat{S}(t)$$

• $H(t)$ 对比 $t$ 对于指数是线性的；
• $\log H(t)$ 对比 $\log t$ 对于 Weibull 是线性的;
• $\log H(t)$ 对比 $t$ 接近线性大 $t$ 对于 Gompertz-Makeham。
绘图后 $\hat{H}(t)$ 通过这些方式，我们可以决定哪个假设最接近。这为我们提供了一种为给定数据集选择参数 模型的粗略但有用的方法。

## 统计代写|生存模型代考Survival Models代写|Maximum Likelihood Estimation

$$\hat{\theta}=\underset{\theta \in \Theta}{\arg \sup } L(\theta), L(\theta)=\prod_{i=1}^n \operatorname{Pr}\left(t_i \mid F(; \theta)\right),$$

$$L(\theta)=\prod_{i=1}^n f\left(t_i ; \theta\right)$$

• $\operatorname{Pr}\left(t_i \mid F(; \theta)\right)=S\left(t_i ; \theta\right)$ 对于右删失的观察 $t_i ;$
• $\operatorname{Pr}\left(t_i \mid F(; \theta)\right)=F\left(t_i ; \theta\right)$ 对于左截尾的观察 $t_i$;
• $\operatorname{Pr}\left(t_i \mid F(; \theta)\right)=F\left(t_i^{(u)} ; \theta\right)-F\left(t_i^{(I)} ; \theta\right)$ 对于在以下范围内删失的观察区间 $\left[t_i^{(I)}, t_i^{(u)}\right]$.

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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。