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# 统计代写|统计推断代考Statistical Inference代写|Hierarchical Models

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## 统计代写|统计推断代考Statistical Inference代写|Hierarchical Models

One prominent example is hierarchical regression models, also known as random coefficient models. These are two- (or three-, etc.) level models, where the coefficients at one level are random variables which vary as a function of still other variables. Thus, for example, in a longitudinal model, coefficients of a response over time (e.g., linear) can be modeled as varying as a function of some third variable, e.g., group membership. Or, putting time in the position of the third variable, we can see whether a correlation changes over time. Hierarchical models have become very popular in the last few decades.

Random effects models were known as far back as the 1930 s, before any explicit introduction of Bayesian statistics. Hays (1963, 1973) was one of the few textbook authors to give them a serious presentation. It was a nontrivial topic, because the algebra was more complicated than for the more usual fixed effects. The practical rationale for their use was not compelling. Most of the categorical variables used as independent variables by psychologists are fixed, anyway (e.g., sex); even for the textbook examples of random effects, like teachers, the assumption of random sampling was as strained as it was for the selection of participants.

The exposition of random effects models entailed an interesting delicacy: Mathematically there is no difference between a normal distribution of a random effect and a normal distribution of belief around a specified value. So frequentists discussing random effects often insisted that there was nothing Bayesian about their work. It was natural, in any case, that most of the distribution work on random effects was done by Bayesians. Such work (e.g., Lindley \& Smith, 1972), however, remained theoretical until a way was found for estimation of the parameters. When that problem was solved (e.g., Dempster, Rubin, \& Tsutakawa, 1981), and statistical programs were written to incorporate the new methods, hierarchical models won quick acceptance. And the distinctive Bayesian terminology (e.g., “shrinkage”) caused no particular concern.

In the incorporation into the mainstream of another Bayesian technique-multiple imputation of missing data-the fact that the mathematics was less well understood has had serious implications for practice, though they have not generally been recognized.

## 统计代写|统计推断代考Statistical Inference代写|Multiple Imputation of Missing Data

The multiple imputation of missing data has always been treated as a Bayesian procedure, though there is no reason on the face of it why it should be intrinsically Bayesian. Schafer (1997) gives the cryptic explanation that the frequentist approach requires specification of the missingness mechanism and the Bayesian approach does not. But this is illogical: It is surely necessary for Bayesian as well as frequentist statistics. To see the source of this claim, we have to exhibit the mathematics.
If we assume that $\theta$ is the parameter of interest, that the variable $y$ is distributed according to $f$, and that $m$ is a missing data indicator distributed according to $g$, then the likelihood of the data is
$$L(\theta)=g(m \mid \varphi) f(y \mid \theta)$$
In Bayesian statistics if the prior probability of $\theta$ is $f^{\prime}(\theta)$, the posterior probability is
$$f^{\prime \prime}(\theta \mid y)=\frac{f^{\prime}(\theta) g(m \mid \varphi) f(y \mid \theta)}{\int f^{\prime}(\theta) g(m \mid \varphi) f(y \mid \theta) d \theta} .$$
If $\varphi$ is independent of $\theta$, in other words if the missingness parameter is independent of the parameter being estimated-a condition that Rubin (1976) has defined as missingness at random – then the missingness mechanism $g$ factors out of the integral, and cancels out of the ratio. Thus under a Bayesian solution the missingness mechanism is irrelevant, so long as the data are missing at random. This happy circumstance is an artifact of Bayesian posterior probabilities being conditional probabilities and therefore ratios. As in this case a constant multiplier, of numerator and denominator, the missing data mechanism does not affect the ratio, as it does the absolute probability of the frequentist inference. There is nothing wrong with this result, so long as the conditional nature of the posterior probability is kept in mind; in practice, of course, it is not and is treated like any other absolute frequentist probability.

# 统计推断代写

## 统计代写|统计推断代考Statistical Inference代写|Multiple Imputation of Missing Data

$$L(\theta)=g(m \mid \varphi) f(y \mid \theta)$$

$$f^{\prime \prime}(\theta \mid y)=\frac{f^{\prime}(\theta) g(m \mid \varphi) f(y \mid \theta)}{\int f^{\prime}(\theta) g(m \mid \varphi) f(y \mid \theta) d \theta} .$$

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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。