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

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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

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