### 统计代写|广义线性模型代写generalized linear model代考|STAT3015

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|广义线性模型代写generalized linear model代考|Negative Binomial

Given a series of independent trials, each with probability of success $p$, let $Z$ be the number of trials until the $k^{h t h}$ success. Then:
$$P(Z=z)=\left(\begin{array}{l} z-1 \ k-1 \end{array}\right) p^{k}(1-p)^{z-k} \quad z=k, k+1, \ldots$$
The negative binomial can arise naturally in several ways. Imagine a system that can withstand $k$ hits before failing. The probability of a hit in a given time period is $p$ and we count the number of time periods until failure. The negative binomial also arises from a generalization of the Poisson where the parameter $\lambda$ is gamma distributed. The negative binomial also comes up as a limiting distribution for urn schemes that can be used to model contagion.

We get a more convenient parameterization if we let $Y=Z-k$ and $p=(1+\alpha)^{-1}$ so that:
$$P(Y=y)=\left(\begin{array}{c} y+k-1 \ k-1 \end{array}\right) \frac{\alpha^{y}}{(1+\alpha)^{y+k}}, \quad y=0,1,2, \ldots$$
then $E Y=\mu=k \alpha$ and var $Y=k \alpha+k \alpha^{2}=\mu+\mu^{2} / k$
The log-likelihood is then:
$$\sum_{i=1}^{n}\left(y_{i} \log \frac{\alpha}{1+\alpha}-k \log (1+\alpha)+\sum_{j=0}^{y_{i}-1} \log (j+k)-\log \left(y_{i} !\right)\right)$$
The most convenient way to link the mean response $\mu$ to a linear combination of the predictors $X$ is:
$$\eta=x^{T} \beta=\log \frac{\alpha}{1+\alpha}=\log \frac{\mu}{\mu+k}$$
We can regard $k$ as fixed and determined by the application or as anditional parameter to be estimated. More on regression models for negative binomial responses may be found in Cameron and Trivedi (1998) and Lawless (1987).

## 统计代写|广义线性模型代写generalized linear model代考|Zero Inflated Count Models

Sometimes we see count response data where the number of zeroes appearing is significantly greater than the Poisson or negative binomial models would predict. Consider the number of arrests for criminal offenses incurred by individuals. A large number of people have never been arrested by the police while a smaller number have been detained on multiple occasions. Modifying the Poisson by adding a dispersion parameter does not adequately model this divergence from the standard count distributions.

We consider a sample of 915 biochemistry graduate students as analyzed by Long $(1990)$. The response is the number of articles produced during the last three years of the PhD. We are interested in how this is related to the gender, marital status, number of children, prestige of the department and productivity of the advisor of the student. The dataset may be found in the pscl package of Zeileis et al. (2008) which also provides the new model fitting functions needed in this section. We start by fitting a Poisson regression model:
$n=915 p-6$
Deviance $=1634.371$ Null Deviance $=1817.405$ (Difference $=183.034$ )
We can see that deviance is significantly larger than the degrees of freedom. Some experimentation reveals that this cannot be solved by using a richer linear predictor or by eliminating some outliers. We might consider a dispersed Poisson model or negative binomial but some thought suggests that there are good reasons why a student might produce no articles at all. We count and predict how many students produce between zero and seven articles. Very few students produce more than seven articles so we ignore these. The predprob function produces the predicted probabilities for each case. By summing these, we get the expected number for each article count.

## 统计代写|广义线性模型代写generalized linear model代考|Two-by-Two Tables

The data shown in Table $6.1$ were collected as part of a quality improvement study at a semiconductor factory. A sample of wafers was drawn and cross-classified according to whether a particle was found on the die that produced the wafer and whether the wafer was good or bad. More details on the study may be found in Hall (1994). The data might have arisen under several possible sampling schemes:

1. We observed the manufacturing process for a certain period of time and observed 450 wafers. The data were then cross-classified. We could use a Poisson model.
2. We decided to sample 450 wafers. The data were then cross-classified. We could use a multinomial model.
3. We selected 400 wafers without particles and 50 wafers with particles and then recorded the good or bad outcome. We could use a binomial model.
4. We selected 400 wafers without particles and 50 wafers with particles that also included, by design, 334 good wafers and 116 bad ones. We could use a hypergeometric model.

The first three sampling schemes are all plausible. The fourth scheme seems less likely in this example, but we include it for completeness. Such a scheme is more attractive when one level of each variable is relatively rare and we choose to oversample both levels to ensure some representation.

The main question of interest concerning these data is whether the presence of particles on the wafer affects the quality outcome. We shall see that all four sampling schemes lead to exactly the same conclusion. First, let’s set up the data in a convenient form for analysis:
$y<-\mathrm{c}(320,14,80,36)$ particle <- gl $(2,1,4$, labelsmc (“no”, “yes”) quality $<-\mathrm{g}(2,2$, labelsmc (“good”, “bad”)) (wafer <- data. frame (y, particle, quality)) y particle quality $\begin{array}{llll}1 & 320 & \text { no } & \text { good } \ 2 & 14 & \text { yes } & \text { good } \ 3 & 80 & \text { no } & \text { bad } \ 4 & 36 & \text { yes } & \text { bad }\end{array}$.

## 统计代写|广义线性模型代写generalized linear model代考|Negative Binomial

∑一世=1n(是一世日志⁡一个1+一个−ķ日志⁡(1+一个)+∑j=0是一世−1日志⁡(j+ķ)−日志⁡(是一世!))

n=915p−6

## 统计代写|广义线性模型代写generalized linear model代考|Two-by-Two Tables

1. 我们观察了一段时间的制造过程，观察了450个晶圆。然后对数据进行交叉分类。我们可以使用泊松模型。
2. 我们决定对 450 个晶圆进行采样。然后对数据进行交叉分类。我们可以使用多项式模型。
3. 我们选择了 400 个没有颗粒的晶圆和 50 个有颗粒的晶圆，然后记录了结果的好坏。我们可以使用二项式模型。
4. 我们选择了 400 个没有颗粒的晶圆和 50 个有颗粒的晶圆，按照设计，还包括 334 个好晶圆和 116 个坏晶圆。我们可以使用超几何模型。

## 有限元方法代写

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## MATLAB代写

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