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# 统计代写|广义线性模型代写Generalized linear model代考|Overdispersion

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## 统计代写|广义线性模型代写Generalized linear model代考|Overdispersion

Overdispersion is a phenomenon that occurs when fitting data to discrete distributions, such as the binomial, Poisson, or negative binomial distribution. If the estimated dispersion after fitting is not near the assumed value, then the data may be overdispersed if the value is greater than expected or underdispersed if the value is less than expected. Overdispersion is far more common. For overdispersion, the simplest remedy is to assume a multiplicative factor in the usual implied variance. As such, the resulting covariance matrix will be inflated through multiplication by the estimated scale parameter. Care should be exercised because an inflated, estimated dispersion parameter may result from model misspecification rather than overdispersion, indicating that the model should be assessed for appropriateness by the researcher. Smith and Heitjan (1993) discuss testing and adjusting for departures from the nominal dispersion, Breslow (1990) discusses Poisson regression as well as other quasilikelihood models, and $\operatorname{Cox}(\underline{1983})$ gives a general overview of overdispersion. This topic is discussed in chapter 11.
Hilbe (2009) devotes an entire chapter to binomial overdispersion, and Hilbe (2011) devotes a chapter to extradispersed count models. Ganio and Schafer (1992), Lambert and Roeder (1995), and Dean and Lawless (1989) discuss diagnostics for overdispersion in GLMs.
A score test effectively compares the residuals with their expectation under the model. A test for overdispersion of the Poisson model, which compares its variance with the variance of a negative binomial, is given by
$$T_1^2=\frac{\left[\sum_{i=1}^n\left{\left(y_i-\widehat{\mu}i\right)^2-\left(1-\widehat{h}_i\right) \widehat{\mu}_i\right}\right]^2}{2 \sum{i=1}^n \widehat{\mu}_i^2} \sim \chi_1^2$$

## 统计代写|广义线性模型代写Generalized linear model代考|Assessing the link function

We may wish to investigate whether the link function is appropriate. For example, in Poisson regression, we may wish to examine whether the usual loglink (multiplicative effects) is appropriate compared with an identity link (additive effects). For binomial regression, we may wish to compare the logit link (symmetric about one half) with the complementary log-log link (asymmetric about one half).

Pregibon (1980) advocates the comparison of two link functions by embedding them in a parametric family of link functions. The Box-Cox family of power transforms
$$g(\mu ; \lambda)=\frac{\mu^\lambda-1}{\lambda}$$
and yields the log-link at $\lim {\lambda \rightarrow 0} g(\mu ; \lambda)$ and the identity link at $\lambda=1$. Likewise, the family $$g(\mu ; \lambda)=\ln \left{\frac{(1-\mu)^{-\lambda}-1}{\lambda}\right}$$ gives the logit link at $\lambda=1$ and the complementary log-log link at $\lim {\lambda \rightarrow 0} g(\mu ; \lambda)$.

## 统计代写|广义线性模型代写Generalized linear model代考|Overdispersion

Hilbe(2009)用了整整一章来讨论二项过分散，Hilbe(2011)用了一章来讨论外分散计数模型。Ganio和Schafer (1992)， Lambert和Roeder (1995)， Dean和Lawless(1989)讨论了glm中过度分散的诊断。

$$T_1^2=\frac{\left[\sum_{i=1}^n\left{\left(y_i-\widehat{\mu}i\right)^2-\left(1-\widehat{h}_i\right) \widehat{\mu}_i\right}\right]^2}{2 \sum{i=1}^n \widehat{\mu}_i^2} \sim \chi_1^2$$

## 统计代写|广义线性模型代写Generalized linear model代考|Assessing the link function

Pregibon(1980)主张通过将两个链接函数嵌入一个参数链接函数族来比较它们。Box-Cox家族的权力转换
$$g(\mu ; \lambda)=\frac{\mu^\lambda-1}{\lambda}$$

## 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。