# 统计代写|线性回归分析代写linear regression analysis代考|Response Transformations for Experimental Design

## 统计代写|线性回归分析代写linear regression analysis代考|Response Transformations for Experimental Design

A model for an experimental design is $Y_i=E\left(Y_i\right)+e_i$ for $i=1, \ldots, n$ where the error $e_i=Y_i-E\left(Y_i\right)$ and $E\left(Y_i\right) \equiv E\left(Y_i \mid \boldsymbol{x}i\right)$ is the expected value of the response $Y_i$ for a given vector of predictors $\boldsymbol{x}_i$. Many models can be fit with least squares (OLS or LS) and are linear models of the form $$Y_i=x{i, 1} \beta_1+x_{i, 2} \beta_2+\cdots+x_{i, p} \beta_p+e_i=\boldsymbol{x}i^T \boldsymbol{\beta}+e_i$$ for $i=1, \ldots, n$. Often $x{i, 1} \equiv 1$ for all $i$. In matrix notation, these $n$ equations become
$$\boldsymbol{Y}=\boldsymbol{X} \boldsymbol{\beta}+\boldsymbol{e}$$
where $\boldsymbol{Y}$ is an $n \times 1$ vector of dependent variables, $\boldsymbol{X}$ is an $n \times p$ design 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. If the fitted values are $\hat{Y}_i=\boldsymbol{x}_i^T \hat{\boldsymbol{\beta}}$, then $Y_i=\hat{Y}_i+r_i$ where the residuals $r_i=Y_i-\hat{Y}_i$.

The applicability of an experimental design model can be expanded by allowing response transformations. An important class of response transformation models adds an additional unknown transformation parameter $\lambda_o$, such that
$$Y_i=t_{\lambda_o}\left(Z_i\right) \equiv Z_i^{\left(\lambda_o\right)}=E\left(Y_i\right)+e_i=\boldsymbol{x}i^T \boldsymbol{\beta}+e_i .$$ If $\lambda_o$ was known, then $Y_i=t{\lambda_o}\left(Z_i\right)$ would follow the linear model for the experimental design.

Definition 5.18. Assume that all of the values of the “response” $Z_i$ are positive. A power transformation has the form $Y=t_\lambda(Z)=Z^\lambda$ for $\lambda \neq 0$ and $Y=t_0(Z)=\log (Z)$ for $\lambda=0$ where $\lambda \in \Lambda_L={-1,-1 / 2,0,1 / 2,1}$.

## 统计代写|线性回归分析代写linear regression analysis代考|One Way Anova in ARC

5.17. To get in ARC, you need to find the ARC icon. Suppose the ARC icon is in a math progs folder. Move the cursor to the math progs folder, click the right mouse button twice, move the cursor to ARC, double click, move the cursor to ARC, double click. These menu commands will be written “math progs $>$ ARC $>$ ARC.” To quit ARC, move cursor to the $\mathrm{x}$ in the northeast corner and click.

This Cook and Weisberg (1999a, p. 289) data set contains IQ scores on 27 pairs of identical twins, one raised by foster parents $I Q f$ and the other by biological parents $I Q b . C$ gives the social class of the biological parents: $C=1$ for upper class, 2 for middle class and 3 for lower class. Hence the Anova test is for whether mean IQ depends on class.
a) Activate twins.lsp dataset with the menu commands “File $>$ Load $>$ Data $>$ twins.lsp.”
b) Use the menu commands “Twins $>$ Make factors,” select $C$ and click on $O K$. The line ” ${\mathrm{F}} \mathrm{C}$ Factor 27 Factor-first level dropped” should appear on the screen.
c) Use the menu commands “Twins>Description” to see a description of the data.
d) Enter the menu commands “Graph\&Fit>Fit linear LS” and select ${F} C$ as the term and IQb as the response. Highlight the output by pressing the left mouse key and dragging the cursor over the output. Then use the menu commands “Edit> Copy.” Enter Word and use the menu command “Paste.”

e) Enter the menu commands “Graph\&Fit>Boxplot of” and enter $I Q b$ in the selection box and $C$ in the Condition on box. Click on $O K$. When the boxplots appear, click on the Show Anova box. Click on the plot, hit the Ctrl and $c$ keys at the same time. Enter Word and use the menu command “Paste.” Include the output in Word. Notice that the regression and Anova $F$ statistic and p-value are the same.
f) Residual plot: Enter the menu commands “Graph\&Fit>Plot of,” select “L1:Fit-Values” for the “H” box and “L1:Residuals” for the “V” box, and click on “OK.” Click on the plot, hit the Ctrl and $c$ keys at the same time. Enter Word and use the menu command “Paste.”
g) Response plot: Enter the menu commands “Graph\&Fit>Plot of,” select “L1:Fit-Values” for the “H” box and “IQb” for the “V” box, and click on “OK.” When the plot appears, move the OLS slider bar to 1 to add the identity line. Click on the plot, hit the Ctrl and $c$ keys at the same time. Enter Word and use the menu command “Paste.”
h) Perform the 4 step test for Ho: $\mu_1=\mu_2=\mu_3$.

# 线性回归代考

## 统计代写|线性回归分析代写linear regression analysis代考|Response Transformations for Experimental Design

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

## 统计代写|线性回归分析代写linear regression analysis代考|One Way Anova in ARC

5.17. 要进入 ARC，您需要找到 ARC 图标。假设 ARC 图标位于 math progs 文件夹中。 将光标移动到math progs文件夹，双击鼠标右键，将光标移动到ARC，双击，将光标移 动到ARC，双击。这些菜单命令将被写成“math progs $>$ 弧>弧。” 要退出 ARC，请将光 标移动到x在东北角，然后单击。

a) 使用菜单命令“文件“激活 twins.lsp 数据集>加载>数据>双胞胎。Isp。”
b) 使用菜单命令“Twins>制造因素，”选择 $C$ 然后点击 $O K$. 行”FCFactor 27 Factor-first level dropped”应该出现在屏幕上。
c) 使用菜单命令“Twins>Description”查看数据的描述。
d) 进入菜单命令”Graph\&Fit>Fit linear LS”，选择 $F C$ 作为术语， $I Q b$ 作为响应。通过 按下鼠标左键并将光标拖到输出上来突出显示输出。然后使用菜单命令“编辑>复制”。 输入 Word 并使用菜单命令”粘贴”。
e) 进入菜单命令”Graph\&Fit>Boxplot of”，输入IQb在选择框和 $C$ 在框中的条件。点击 $O K$. 当箱线图出现时，单击 Show Anova 框。单击图，按 Ctrl 和c同时键。输入 Word 并使用菜单命令”粘贴”在 Word 中包含输出。请注意，回归和方差分析 $F$ 统计量和 $\mathrm{p}$ 值是相同的。
f) 残差图: 进入菜单命令”Graph\&Fit>Plot of”， “H”框选择”L1:Fit-Values”， “V”框选择 “L1:Residuals”，点击”OK” ” 单击图， 按 Ctrl 和 $c$ 同时键。输入 Word 并使用菜单命令”粘 贴”。
g) 响应图: 输入菜单命令”Graph\&Fit>Plot of”, “H”框选择”L1:Fit-Values”, “V”框选择 “IQb”，点击“OK”。出现图时，将 OLS 滑块移动到 1 以添加恒等线。单击图，按 Ctrl 和 $c$ 同时键。输入 Word 并使用菜单命令”粘贴”。
h) 对 Ho 执行 4 步测试: $\mu_1=\mu_2=\mu_3$.

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