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## 统计代写|线性回归代写Linear Regression代考|Latin Square Designs

Latin square designs have a lot of structure. The design contains a row block factor, a column block factor, and a treatment factor, each with $a$ levels. The two blocking factors, and the treatment factor are crossed, but it is assumed that there is no interaction. A capital letter is used for each of the $a$ treatment levels. So $a=3$ uses A, B, C while $a=4$ uses A, B, C, D.

Definition 7.5. In an $a \times a$ Latin square, each letter appears exactly once in each row and in each column. A standard Latin square has letters written in alphabetical order in the first row and in the first column.

Five Latin squares are shown below. The first, third, and fifth are standard. If $a=5$, there are 56 standard Latin squares.
$\begin{array}{lllllllllllll}\text { A B C } & \text { A B C } & \text { A B C D } & \text { A B C D E } & \text { A B C D E } \ \text { B C A } & \text { C A B } & \text { B A D C } & \text { E A B C D } & \text { B A E C D } \ \text { C A B } & \text { B C A } & \text { C D A B } & \text { D E A B C } & \text { C D A E B } \ & & & \text { D C B A } & \text { C D E A B } & \text { D E B A C } \ & & & & \text { B C D E A } & \text { E C D B A }\end{array}$
Definition 7.6. The model for the Latin square design is
$$Y_{i j k}=\mu+\tau_i+\beta_j+\gamma_k+e_{i j k}$$
where $\tau_i$ is the $i$ th treatment effect, $\beta_j$ is the $j$ th row block effect, $\gamma_k$ is the $k$ th column block effect with $i, j$, and $k=1, \ldots, a$. The errors $e_{i j k}$ are iid with 0 mean and constant variance $\sigma^2$. The $i$ th treatment mean $\mu_i=\mu+\tau_i$.

Shown below is an ANOVA table for the Latin square model given in symbols. Sometimes “Error” is replaced by “Residual,” or “Within Groups.” Sometimes rblocks and cblocks are replaced by the names of the blocking factors. Sometimes “p-value” is replaced by “P,” “Pr$(>F)$,” or “PR $>$ F.”

Rule of thumb 7.2. Let $p_{\text {block }}$ be $p_{\text {row }}$ or $p_{\text {col }}$. If $p_{\text {block }} \geq 0.1$, then blocking was not useful. If $0.05<p_{\text {block }}<0.1$, then the usefulness was borderline. If $p_{\text {block }} \leq 0.05$, then blocking was useful.

Be able to perform the 4 step ANOVA $F$ test for the Latin square design. This test is similar to the fixed effects one way ANOVA $F$ test.
i) Ho: $\mu_1=\mu_2=\cdots=\mu_a$ and $H_A$ : not Ho.
ii) Fo = MSTR/MSE is usually given by output.
iii) The pval $=\mathrm{P}\left(F_{a-1,(a-1)(a-2)}>F_o\right)$ is usually given by output.
iv) If the pval $\leq \delta$, reject Ho and conclude that the mean response depends on the factor level. Otherwise fail to reject Ho and conclude that the mean response does not depend on the factor level. (Or there is not enough evidence to conclude that the mean response depends on the factor level.) Give a nontechnical sentence. Use $\delta=0.05$ if $\delta$ is not given.

## 统计代写|线性回归代写Linear Regression代考|Factorial Designs

Factorial designs are a special case of the $k$ way Anova designs of Chapter 6 , and these designs use factorial crossing to compare the effects (main effects, pairwise interactions, $\ldots$, k-fold interaction) of the $k$ factors. If $A_1, \ldots, A_k$ are the factors with $l_i$ levels for $i=1, \ldots, k$ then there are $l_1 l_2 \cdots l_k$ treatments where each treatment uses exactly one level from each factor. The sample size $n=m \prod_{i=1}^k l_i \geq m 2^k$. Hence the sample size grows exponentially fast with $k$. Often the number of replications $m=1$.

Definition 8.1. An experiment has $n$ runs where a run is used to measure a response. A run is a treatment = a combination of $k$ levels. So each run uses exactly one level from each of the $k$ factors.

Often each run is expensive, for example, in industry and medicine. A goal is to improve the product in terms of higher quality or lower cost. Often the subject matter experts can think of many factors that might improve the product. The number of runs $n$ is minimized by taking $l_i=2$ for $i=1, \ldots, k$.
Definition 8.2. A $2^k$ factorial design is a $k$ way Anova design where each factor has two levels: low $=-1$ and high $=1$. The design uses $n=m 2^k$ runs. Often the number of replications $m=1$. Then the sample size $n=2^k$.

A $2^k$ factorial design is used to screen potentially useful factors. Usually at least $k=3$ factors are used, and then $2^3=8$ runs are needed. Often the units are time slots, and each time slot is randomly assigned to a run $=$ treatment. The subject matter experts should choose the two levels. For example, a quantitative variable such as temperature might be set at $80^{\circ} \mathrm{F}$ coded as -1 and $100^{\circ} F$ coded as 1 , while a qualitative variable such as type of catalyst might have catalyst $\mathrm{A}$ coded as -1 and catalyst $\mathrm{B}$ coded as 1 .

## 统计代写|线性回归代写Linear Regression代考|Latin Square Designs

ABC ABC ABCD ABCDE ABCDE BCA CAB BADC 定义 7.6。拉丁方设计的模型是
$$Y_{i j k}=\mu+\tau_i+\beta_j+\gamma_k+e_{i j k}$$

iv) 如果 $\mathrm{pval} \leq \delta$ ，拒绝 $\mathrm{Ho}$ 并得出平均响应取决于因子水平的结论。否则无法拒绝 $\mathrm{Ho}$ 并得出均值响应不依赖 于因子水平的结论。（或者没有足够的证据得出均值响应取决于因子水平的结论。）给出一个非技术性的句 子。使用 $\delta=0.05$ 如果 $\delta$ 没有给出。

## 统计代写|线性回归代写Linear Regression代考|Factorial Designs

$\mathrm{A} 2^k$ 析因设计用于筞选潜在有用的因素。通常至少 $k=3$ 使用因素，然后 $2^3=8$ 需要运行。通常单位是时隙， 每个时隙随机分配到一个运行=治疗。主题专家应选择这两个级别。例如，温度等定量变量可能设置为 $80^{\circ} \mathrm{F}$ 编 码为 -1 和 $100^{\circ} F$ 编码为 1 ，而催化剂类型等定性变量可能有催化剂 $A$ 编码为 -1 和催化剂 $B$ 编码为 1 。

## 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。

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## 统计代写|线性回归代写Linear Regression代考|Two Way Anova

Definition 6.1. The fixed effects two way Anova model has two factors $A$ and $B$ plus a response $Y$. Factor $A$ has $a$ levels and factor $B$ has $b$ levels. There are $a b$ treatments.

Definition 6.2. The cell means model for two way Anova is $Y_{i j k}=$ $\mu_{i j}+e_{i j k}$ where $i=1, \ldots, a ; j=1, \ldots, b$; and $k=1, \ldots, m$. The sample size $n=a b m$. The $\mu_{i j}$ are constants and the $e_{i j k}$ are iid from a location family with mean 0 and variance $\sigma^2$. Hence the $Y_{i j k} \sim f\left(y-\mu_{i j}\right)$ come from a location family with location parameter $\mu_{i j}$. The fitted values are $\hat{Y}{i j k}=\bar{Y}{i j 0}=\hat{\mu}{i j}$ while the residuals $r{i j k}=Y_{i j k}-\hat{Y}_{i j k}$.

For one way Anova models, the cell sizes $n_i$ need not be equal. For $K$ way Anova models with $K \geq 2$ factors, the statistical theory is greatly simplified if all of the cell sizes are equal. Such designs are called balanced designs.
Definition 6.3. A balanced design has all of the cell sizes equal: for the two way Anova model, $n_{i j} \equiv m$.

In addition to randomization of units to treatments, another key principle of experimental design is factorial crossing. Factorial crossing allows for estimation of main effects and interactions.

Definition 6.4. A two way Anova design uses factorial crossing if each combination of an $A$ level and a $B$ level is used and called a treatment. There are $a b$ treatments for the two way Anova model.

Experimental two way Anova designs randomly assign $m$ of the $n=m a b$ units to each of the $a b$ treatments. Observational studies take random samples of size $m$ from $a b$ populations.

Definition 6.5. The main effects are $A$ and $B$. The $A B$ interaction is not a main effect.

Remark 6.1. If $A$ and $B$ are factors, then there are 5 possible models.
i) The two way Anova model has terms $A, B$, and $A B$.
ii) The additive model or main effects model has terms $A$ and $B$.
iii) The one way Anova model that uses factor $A$.
iv) The one way Anova model that uses factor $B$.
v) The null model does not use any of the three terms $A, B$, or $A B$. If the null model holds, then $Y_{i j k} \sim f\left(y-\mu_{00}\right)$ so the $Y_{i j k}$ form a random sample of size $n$ from a location family, and the distribution of the response is the same for all $a b$ treatments. For models i)-iv), the distribution of the response is not the same for all $a b$ treatments.

## 统计代写|线性回归代写Linear Regression代考|K Way Anova Models

Use factorial crossing to compare the effects (main effects, pairwise interactions, $\ldots, K$-fold interaction if there are $K$ factors) of two or more factors. If $A_1, \ldots, A_K$ are the factors with $l_i$ levels for $i=1, \ldots, K$; then there are $l_1 l_2 \cdots l_K$ treatments where each treatment uses exactly one level from each factor.

On the previous page is a partial ANOVA table for a $K$ way Anova design with the degrees of freedom left blank. For $A$, use $H_0: \mu_{10 \cdots 0}=\cdots=\mu_{l_1 0 \cdots 0}$. The other main effects have similar null hypotheses. For interaction, use $H_0$ : no interaction.

These models get complex rapidly as $K$ and the number of levels $l_i$ increase. As $K$ increases, there are a large number of models to consider. For experiments, usually the 3 way and higher order interactions are not significant. Hence a full model that includes all $K$ main effects and $\left(\begin{array}{c}K \ 2\end{array}\right) 2$ way interactions is a useful starting point for response, residual, and transformation plots. The higher order interactions can be treated as potential terms and checked for significance. As a rule of thumb, significant interactions tend to involve significant main effects.

The sample size $n=m \prod_{i=1}^K l_i \geq m 2^K$ is minimized by taking $l_i=2$ for $i=1, \ldots, K$. Hence the sample size grows exponentially fast with $K$. Designs that use the minimum number of levels 2 are discussed in Section 8.1.

## 统计代写|线性回归代写Linear Regression代考|Two Way Anova

i) 方差分析模型有项的两种方式 $A, B$ ，和 $A B$.
ii) 加法模型或主效应模型有项 $A$ 和 $B$.
iii) 使用因子的单向 Anova 模型 $A$.
iv) 使用因子的单向 Anova 模型 $B$.
v) null 模型不使用这三个项中的任何一个 $A, B$ ，或者 $A B$. 如果空模型成立，则
$Y_{i j k} \sim f\left(y-\mu_{00}\right)$ 所以 $Y_{i j k}$ 形成大小随机样本 $n$ 来自一个位置族，并且响应的分布对所有人都是 相同的 $a b$ 治疗。对于模型 $\mathrm{i})-\mathrm{iv})$ ，所有响应的分布都不相同 $a b$ 治疗。

## 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。

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## avatest™帮您通过考试

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## 统计代写|线性回归代写Linear Regression代考|Random Vectors

The concepts of a random vector, the expected value of a random vector, and the covariance of a random vector are needed before covering generalized least squares. Recall that for random variables $Y_i$ and $Y_j$, the covariance of $Y_i$ and $Y_j$ is $\operatorname{Cov}\left(Y_i, Y_j\right) \equiv \sigma_{i, j}=E\left[\left(Y_i-E\left(Y_i\right)\right)\left(Y_j-E\left(Y_j\right)\right]=E\left(Y_i Y_j\right)-E\left(Y_i\right) E\left(Y_j\right)\right.$ provided the second moments of $Y_i$ and $Y_j$ exist.

Definition 4.1. $\boldsymbol{Y}=\left(Y_1, \ldots, Y_n\right)^T$ is an $n \times 1$ random vector if $Y_i$ is a random variable for $i=1, \ldots, n$. $\boldsymbol{Y}$ is a discrete random vector if each $Y_i$ is discrete, and $\boldsymbol{Y}$ is a continuous random vector if each $Y_i$ is continuous. A random variable $Y_1$ is the special case of a random vector with $n=1$.
Definition 4.2. The population mean of a random $n \times 1$ vector $\boldsymbol{Y}=$ $\left(Y_1, \ldots, Y_n\right)^T$ is
$$E(\boldsymbol{Y})=\left(E\left(Y_1\right), \ldots, E\left(Y_n\right)\right)^T$$
provided that $E\left(Y_i\right)$ exists for $i=1, \ldots, n$. Otherwise the expected value does not exist. The $n \times n$ population covariance matrix
$$\operatorname{Cov}(\boldsymbol{Y})=E\left[(\boldsymbol{Y}-E(\boldsymbol{Y}))(\boldsymbol{Y}-E(\boldsymbol{Y}))^T\right]=\left(\sigma_{i, j}\right)$$
where the $i j$ entry of $\operatorname{Cov}(\boldsymbol{Y})$ is $\operatorname{Cov}\left(Y_i, Y_j\right)=\sigma_{i, j}$ provided that each $\sigma_{i, j}$ exists. Otherwise $\operatorname{Cov}(\boldsymbol{Y})$ does not exist.

The covariance matrix is also called the variance-covariance matrix and variance matrix. Sometimes the notation $\operatorname{Var}(\boldsymbol{Y})$ is used. Note that $\operatorname{Cov}(\boldsymbol{Y})$ is a symmetric positive semidefinite matrix. If $\boldsymbol{Z}$ and $\boldsymbol{Y}$ are $n \times 1$ random vectors, $\boldsymbol{a}$ a conformable constant vector, and $\boldsymbol{A}$ and $\boldsymbol{B}$ are conformable constant matrices, then
$$E(\boldsymbol{a}+\boldsymbol{Y})=\boldsymbol{a}+E(\boldsymbol{Y}) \text { and } E(\boldsymbol{Y}+\boldsymbol{Z})=E(\boldsymbol{Y})+E(\boldsymbol{Z})$$

and
$$E(\boldsymbol{A} \boldsymbol{Y})=\boldsymbol{A} E(\boldsymbol{Y}) \text { and } E(\boldsymbol{A} \boldsymbol{Y} \boldsymbol{B})=\boldsymbol{A} E(\boldsymbol{Y}) \boldsymbol{B}$$
Also
$$\operatorname{Cov}(\boldsymbol{a}+\boldsymbol{A} \boldsymbol{Y})=\operatorname{Cov}(\boldsymbol{A} \boldsymbol{Y})=\boldsymbol{A} \operatorname{Cov}(\boldsymbol{Y}) \boldsymbol{A}^T .$$

## 统计代写|线性回归代写Linear Regression代考|GLS, WLS, and FGLS

Definition 4.3. Suppose that the response variable and at least one of the predictor variables is quantitative. Then the generalized least squares (GLS) model is
$$\boldsymbol{Y}=\boldsymbol{X} \boldsymbol{\beta}+\boldsymbol{e},$$
where $\boldsymbol{Y}$ is an $n \times 1$ vector of dependent variables, $\boldsymbol{X}$ is an $n \times p$ matrix of predictors, $\boldsymbol{\beta}$ is a $p \times 1$ vector of unknown coefficients, and $\boldsymbol{e}$ is an $n \times 1$ vector of unknown errors. Also $E(\boldsymbol{e})=\mathbf{0}$ and $\operatorname{Cov}(\boldsymbol{e})=\sigma^2 \boldsymbol{V}$ where $\boldsymbol{V}$ is a known $n \times n$ positive definite matrix.
Definition 4.4. The GLS estimator
$$\hat{\boldsymbol{\beta}}{G L S}=\left(\boldsymbol{X}^T \boldsymbol{V}^{-1} \boldsymbol{X}\right)^{-1} \boldsymbol{X}^T \boldsymbol{V}^{-1} \boldsymbol{Y} .$$ The fitted values are $\hat{\boldsymbol{Y}}{G L S}=\boldsymbol{X} \hat{\boldsymbol{\beta}}{G L S}$. Definition 4.5. Suppose that the response variable and at least one of the predictor variables is quantitative. Then the weighted least squares (WLS) model with weights $w_1, \ldots, w_n$ is the special case of the GLS model where $\boldsymbol{V}$ is diagonal: $\boldsymbol{V}=\operatorname{diag}\left(\mathrm{v}_1, \ldots, \mathrm{v}{\mathbf{n}}\right)$ and $w_i=1 / v_i$. Hence
$$\begin{gathered} \boldsymbol{Y}=\boldsymbol{X} \boldsymbol{\beta}+\boldsymbol{e}, \ E(\boldsymbol{e})=\mathbf{0} \text {, and } \operatorname{Cov}(\boldsymbol{e})=\sigma^2 \operatorname{diag}\left(\mathrm{v}1, \ldots, \mathrm{v}{\mathrm{n}}\right)=\sigma^2 \operatorname{diag}\left(1 / \mathrm{w}1, \ldots, 1 / \mathrm{w}{\mathrm{n}}\right) . \end{gathered}$$
Definition 4.6. The WLS estimator
$$\hat{\boldsymbol{\beta}}_{W L S}=\left(\boldsymbol{X}^T \boldsymbol{V}^{-1} \boldsymbol{X}\right)^{-1} \boldsymbol{X}^T \boldsymbol{V}^{-1} \boldsymbol{Y}$$

## 统计代写|线性回归代写Linear Regression代考|Random Vectors

$\operatorname{Cov}\left(Y_i, Y_j\right) \equiv \sigma_{i, j}=E\left[\left(Y_i-E\left(Y_i\right)\right)\left(Y_j-E\left(Y_j\right)\right]=E\left(Y_i Y_j\right)-E\left(Y_i\right) E\left(Y_j\right)\right.$ 提供的第二个 时刻 $Y_i$ 和 $Y_j$ 存在。

$$E(\boldsymbol{Y})=\left(E\left(Y_1\right), \ldots, E\left(Y_n\right)\right)^T$$

$$\operatorname{Cov}(\boldsymbol{Y})=E\left[(\boldsymbol{Y}-E(\boldsymbol{Y}))(\boldsymbol{Y}-E(\boldsymbol{Y}))^T\right]=\left(\sigma_{i, j}\right)$$

$$E(\boldsymbol{a}+\boldsymbol{Y})=\boldsymbol{a}+E(\boldsymbol{Y}) \text { and } E(\boldsymbol{Y}+\boldsymbol{Z})=E(\boldsymbol{Y})+E(\boldsymbol{Z})$$

$$E(\boldsymbol{A} \boldsymbol{Y})=\boldsymbol{A} E(\boldsymbol{Y}) \text { and } E(\boldsymbol{A} \boldsymbol{Y})=\boldsymbol{A} E(\boldsymbol{Y}) \boldsymbol{B}$$

$$\operatorname{Cov}(\boldsymbol{a}+\boldsymbol{A} \boldsymbol{Y})=\operatorname{Cov}(\boldsymbol{A} \boldsymbol{Y})=\boldsymbol{A} \operatorname{Cov}(\boldsymbol{Y}) \boldsymbol{A}^T$$

## 统计代写|线性回归代写Linear Regression代考|GLS, WLS, and FGLS

$$\boldsymbol{Y}=\boldsymbol{X} \boldsymbol{\beta}+\boldsymbol{e}$$

$$\hat{\boldsymbol{\beta}} G L S=\left(\boldsymbol{X}^T \boldsymbol{V}^{-1} \boldsymbol{X}\right)^{-1} \boldsymbol{X}^T \boldsymbol{V}^{-1} \boldsymbol{Y}$$

## 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。