## 统计代写|线性回归代写linear regression代考|Graphical Methods for Response Transformations

If the ratio of largest to smallest value of $y$ is substantial, we usually begin by looking at $\log y$.
Mosteller and Tukey (1977, p. 91)
The applicability of the multiple linear regression 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
$$Y_i=t_{\lambda_o}\left(Z_i\right) \equiv Z_i^{\left(\lambda_o\right)}=E\left(Y_i \mid \boldsymbol{x}i\right)+e_i=\boldsymbol{x}_i^T \boldsymbol{\beta}+e_i .$$ If $\lambda_o$ was known, then $Y_i=t{\lambda_o}\left(Z_i\right)$ would follow a multiple linear regression model with $p$ predictors including the constant. Here, $\boldsymbol{\beta}$ is a $p \times 1$ vector of unknown coefficients depending on $\lambda_o, \boldsymbol{x}$ is a $p \times 1$ vector of predictors that are assumed to be measured with negligible error, and the errors $e_i$ are assumed to be iid with zero mean.

Definition 3.2. 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
$$\lambda \in \Lambda_L={-1,-1 / 2,-1 / 3,0,1 / 3,1 / 2,1} .$$

## 统计代写|线性回归代写linear regression代考|Main Effects, Interactions, and Indicators

Section $1.4$ explains interactions, factors, and indicator variables in an abstract setting when $Y \Perp \boldsymbol{x} \mid \boldsymbol{x}^T \boldsymbol{\beta}$ where $\boldsymbol{x}^T \boldsymbol{\beta}$ is the sufficient predictor (SP). MLR is such a model. The Section $1.4$ interpretations given in terms of the $\mathrm{SP}$ can be given in terms of $E(Y \mid \boldsymbol{x})$ for MLR since $E(Y \mid \boldsymbol{x})=\boldsymbol{x}^T \boldsymbol{\beta}=S P$ for MLR.

Definition 3.5. Suppose that the explanatory variables have the form $x_2, \ldots, x_k, x_{j j}=x_j^2, x_{i j}=x_i x_j, x_{234}=x_2 x_3 x_4$, et cetera. Then the variables $x_2, \ldots, x_k$ are main effects. A product of two or more different main effects is an interaction. A variable such as $x_2^2$ or $x_7^3$ is a power. An $x_2 x_3$ interaction will sometimes also be denoted as $x_2: x_3$ or $x_2 * x_3$.

Definition 3.6. A factor $W$ is a qualitative random variable. Suppose $W$ has $c$ categories $a_1, \ldots, a_c$. Then the factor is incorporated into the MLR model by using $c-1$ indicator variables $x_{W_j}=1$ if $W=a_j$ and $x_{W_j}=0$ otherwise, where one of the levels $a_j$ is omitted, e.g. use $j=1, \ldots, c-1$. Each indicator variable has 1 degree of freedom. Hence the degrees of freedom of the $c-1$ indicator variables associated with the factor is $c-1$.

Rule of thumb 3.3. Suppose that the MLR model contains at least one power or interaction. Then the corresponding main effects that make up the powers and interactions should also be in the MLR model.

Rule of thumb $3.3$ suggests that if $x_3^2$ and $x_2 x_7 x_9$ are in the MLR model, then $x_2, x_3, x_7$, and $x_9$ should also be in the MLR model. A quick way to check whether a term like $x_3^2$ is needed in the model is to fit the main effects models and then make a scatterplot matrix of the predictors and the residuals, where the residuals $r$ are on the top row. Then the top row shows plots of $x_k$ versus $r$, and if a plot is parabolic, then $x_k^2$ should be added to the model. Potential predictors $w_j$ could also be added to the scatterplot matrix. If the plot of $w_j$ versus $r$ shows a positive or negative linear trend, add $w_j$ to the model. If the plot is quadratic, add $w_j$ and $w_j^2$ to the model. This technique is for quantitative variables $x_k$ and $w_j$.

## 统计代写|线性回归代写linear regression代考|Graphical Methods for Response Transformations

Mosteller 和 Tukey (1977, p. 91)

$$Y_i=t_{\lambda_o}\left(Z_i\right) \equiv Z_i^{\left(\lambda_o\right)}=E\left(Y_i \mid \boldsymbol{x} i\right)+e_i=\boldsymbol{x}i^T \boldsymbol{\beta}+e_i .$$ 如果 $\lambda_o$ 是已知的，那么 $Y_i=t \lambda_o\left(Z_i\right)$ 将道循多元线性回归模型 $p$ 包括常数的预测因子。这里， $\beta$ 是一个 $p \times 1$ 末知系数的向量取决于 $\lambda_o, \boldsymbol{x}$ 是一个 $p \times 1$ 假设以可忽略的误差测量的预测变量向量，以及误差 $e_i$ 被 假定为具有零均值的独立同分布。 定义 3.2。假设”响应”的所有值 $Z_i$ 是积极的。功率变换具有以下形式 $Y=t\lambda(Z)=Z^\lambda$ 为了 $\lambda \neq 0$ 和 $Y=t_0(Z)=\log (Z)$ 为了 $\lambda=0$ 在哪里
$$\lambda \in \Lambda_L=-1,-1 / 2,-1 / 3,0,1 / 3,1 / 2,1$$

## 统计代写|线性回归代写linear regression代考|Main Effects, Interactions, and Indicators

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