### 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|OLET5610

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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|多元统计分析代写Multivariate Statistical Analysis代考|Log-Linear Models for Contingency Tables

Consider a $(J \times K)$ two-way table, where $y_{j k}$ is the number of observations having the nominal value $j$ for the first qualitative character and nominal value $k$ for the second character. Since the total number of observations is fixed $n=$ $\sum_{j=1}^J \sum_{k=1}^K y_{j k}$, there are $J K-1$ free cells in the table. The multinomial likelihood can be written as in (8.6)
$$L=\frac{n !}{\prod_{j=1}^J \prod_{k=1}^K y_{j k} !} \prod_{j=1}^J \prod_{k=1}^K\left(\frac{m_{j k}}{n}\right)^{y_{j k}},$$
where we now introduce a log-linear structure to analyse the role of the rows and the columns to determine the parameters $m_{j k}=\mathrm{E}\left(y_{j k}\right)$ (or $p_{j k}$ ).

1. Model without interaction
Suppose that there is no interaction between the rows and the columns: this corresponds to the hypothesis of independence between the two qualitative characters. In other words, $p_{j k}=p_j p_k$ for all $j, k$. This implies the log-linear model:
$$\log m_{j k}=\mu+\alpha_j+\gamma_k \text { for } j=1, \ldots, J, k=1, \ldots, K,$$
where, as in ANOVA models for identification purposes $\sum_{j=1}^J \alpha_j=\sum_{k=1}^K \gamma_k=$ 0 . Using the same coding devices as above, the model can be written as
$$\log m=\mathcal{X} \beta .$$

For a $(2 \times 3)$ table we have:
$$\log m=\left(\begin{array}{l} \log m_{11} \ \log m_{12} \ \log m_{13} \ \log m_{21} \ \log m_{22} \ \log m_{23} \end{array}\right), \mathcal{X}=\left(\begin{array}{rrrr} 1 & 1 & 1 & 0 \ 1 & 1 & 0 & 1 \ 1 & 1 & -1 & -1 \ 1 & -1 & 1 & 0 \ 1 & -1 & 0 & 1 \ 1 & -1 & -1 & -1 \end{array}\right), \beta=\left(\begin{array}{l} \beta_0 \ \beta_1 \ \beta_2 \ \beta_3 \end{array}\right)$$
where the first column of $\mathcal{X}$ is for the constant term, the second column is the coded column for the 2-levels row effect and the two last columns are the coded columns for the 3-levels column effect. The estimation is obtained by maximising the log-likelihood which is equivalent to maximising the function $L(\beta)$ in $\beta$ :
$$L(\beta)=\sum_{j=1}^J \sum_{k=1}^K y_{j k} \log m_{j k} .$$

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Three-Way Tables

The models presented above for two-way tables can be extended to higher order tables but at a cost of notational complexity. We show how to adapt to threeway tables. This deserves special attention due to the presence of higher-order interactions in the saturated model.

A $(J \times K \times L)$ three-way table may be constructed under multinomial sampling as follows: each of the $n$ observations falls in one, and only one, category of each of three categorical variables having $J, K$ and $L$ modalities respectively. We end up with a three-dimensional table with $J K L$ cells containing the counts $y_{j k \ell}$ where $n=\sum_{j, k, \ell} y_{j k \ell}$. The expected counts depend on the unknown probabilities $p_{j k \ell}$ in the usual way:
$$m_{j k \ell}=n p_{j k \ell}, j=1, \ldots, J, k=1, \ldots, K, \ell=1, \ldots, L$$

1. The saturated model
A full saturated log-linear model reads as follows:
\begin{aligned} \log m_{j k \ell}= & \mu+\alpha_j+\beta_k+\gamma \ell+(\alpha \beta){j k}+(\alpha \gamma){j \ell}+(\beta \gamma){k \ell}+(\alpha \beta \gamma){j k \ell}, \ j & =1, \ldots, J, k=1, \ldots, K, \ell=1, \ldots, L . \end{aligned}
The restrictions are the following (using the “dot” notation for summation on the corresponding indices):
\begin{aligned} & \alpha_{(\bullet)}=\beta_{(\bullet)}=\gamma_{(\bullet)}=0 \ & (\alpha \beta){j \bullet}=(\alpha \gamma){j \bullet}=(\beta \gamma){k \bullet}=0 \ & (\alpha \beta){\bullet k}=(\alpha \gamma){\bullet \ell}=(\beta \gamma){\bullet \ell}=0 \ & (\alpha \beta \gamma){j k \bullet}=(\alpha \beta \gamma){j \bullet \ell}=(\alpha \beta \gamma){\bullet k \ell}=0 \end{aligned} The parameters $(\alpha \beta){j k},(\alpha \gamma){j \ell},(\beta \gamma){k \ell}$ are called first-order interactions. The second-order interactions are the parameters $(\alpha \beta \gamma){j k \ell}$, they allow to take into account heterogeneities in the interactions between two of the three variables. For instance, let $\ell$ stand for the two gender categories $(L=2)$, if we suppose that $(\alpha \beta \gamma){j k 1}=-(\alpha \beta \gamma)_{j k 2} \neq 0$. we mean that the interactions between the variable $J$ and $K$ are not the same for both gender categories.

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Log-Linear Models for Contingency Tables

$$L=\frac{n !}{\prod_{j=1}^J \prod_{k=1}^K y_{j k} !} \prod_{j=1}^J \prod_{k=1}^K\left(\frac{m_{j k}}{n}\right)^{y_{j k}}$$

1. 没有交互作用
的模型假设行和列之间没有交互作用：这对应于两个定性特征之间的独立性假设。换句话说， $p_{j k}=p_j p_k$ 对所有人 $j, k$. 这意味着对数线性模型:
$$\log m_{j k}=\mu+\alpha_j+\gamma_k \text { for } j=1, \ldots, J, k=1, \ldots, K,$$
其中，与用于识别目的的 ANOVA 模型一样 $\sum_{j=1}^J \alpha_j=\sum_{k=1}^K \gamma_k=0$ 。使用与上述相同的编码设 备，模型可以写成
$$\log m=\mathcal{X} \beta$$
为一个 $(2 \times 3)$ 我们有表:
其中第一列 $\mathcal{X}$ 是常数项，第二列是 2 级行效应的编码列，最后两列是 3 级列效应的编码列。估计是通过最 大化对数似然得到的，这相当于最大化函数 $L(\beta)$ 在 $\beta$ :
$$L(\beta)=\sum_{j=1}^J \sum_{k=1}^K y_{j k} \log m_{j k} .$$

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Three-Way Tables

$$m_{j k \ell}=n p_{j k \ell}, j=1, \ldots, J, k=1, \ldots, K, \ell=1, \ldots, L$$

1. 饱和模型
一个完整的饱和对数线性模型如下:
$$\log m_{j k \ell}=\mu+\alpha_j+\beta_k+\gamma \ell+(\alpha \beta) j k+(\alpha \gamma) j \ell+(\beta \gamma) k \ell+(\alpha \beta \gamma) j k \ell, j=1, \ldots, J$$
限制如下（使用“点”符号对相应索引求和）：
$$\alpha_{(\bullet)}=\beta_{(\bullet)}=\gamma_{(\bullet)}=0 \quad(\alpha \beta) j \bullet=(\alpha \gamma) j \bullet=(\beta \gamma) k \bullet=0(\alpha \beta) \bullet k=(\alpha \gamma) \bullet \ell=(\beta \gamma) \bullet \ell$$
参数 $(\alpha \beta) j k,(\alpha \gamma) j \ell,(\beta \gamma) k \ell$ 称为一阶相互作用。二阶相互作用是参数 $(\alpha \beta \gamma) j k \ell$ ，它们允许考虑 三个变量中两个变量之间相互作用的异质性。例如，让 $\ell$ 代表两个性别类别 $(L=2)$ ，如果我们假设 $(\alpha \beta \gamma) j k 1=-(\alpha \beta \gamma)_{j k 2} \neq 0$. 我们的意思是变量之间的相互作用 $J$ 和 $K$ 两种性别类别的情况并不相同。

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