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

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

The one way Anova model is used to compare $p$ treatments. Usually there is replication and Ho: $\mu_1=\mu_2=\cdots=\mu_p$ is a hypothesis of interest. Investigators may also want to rank the population means from smallest to largest.

Definition 5.6. Let $f_Z(z)$ be the pdf of $Z$. Then the family of pdfs $f_Y(y)=f_Z(y-\mu)$ indexed by the location parameter $\mu,-\infty<\mu<\infty$, is the location family for the random variable $Y=\mu+Z$ with standard $p d f f_Z(z)$

Definition 5.7. A one way fixed effects Anova model has a single qualitative predictor variable $W$ with $p$ categories $a_1, \ldots, a_p$. There are $p$ different distributions for $Y$, one for each category $a_i$. The distribution of
$$Y \mid\left(W=a_i\right) \sim f_Z\left(y-\mu_i\right)$$
where the location family has second moments. Hence all $p$ distributions come from the same location family with different location parameter $\mu_i$ and the same variance $\sigma^2$.

Definition 5.8. The one way fixed effects normal Anova model is the special case where $$Y \mid\left(W=a_i\right) \sim N\left(\mu_i, \sigma^2\right) .$$
Example 5.3. The pooled 2 sample $t$-test is a special case of a one way Anova model with $p=2$. For example, one population could be ACT scores for men and the second population ACT scores for women. Then $W=$ gender and $Y=$ score.

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

Definition 5.16. For the random effects one way Anova, the levels of the factor are a random sample of levels from some population of levels $\Lambda_F$. The cell means model for the random effects one way Anova is $Y_{i j}=\mu_i+e_{i j}$ for $i=1, \ldots, p$ and $j=1, \ldots, n_i$. The $\mu_i$ are randomly selected from some population $\Lambda$ with mean $\mu$ and variance $\sigma_\mu^2$, where $i \in \Lambda_F$ is equivalent to $\mu_i \in \Lambda$. The $e_{i j}$ and $\mu_i$ are independent, and the $e_{i j}$ are iid from a location family with pdf $f$, mean 0 , and variance $\sigma^2$. The $Y_{i j} \mid \mu_i \sim f\left(y-\mu_i\right)$, the location family with location parameter $\mu_i$ and variance $\sigma^2$. Unconditionally, $E\left(Y_{i j}\right)=\mu$ and $V\left(Y_{i j}\right)=\sigma_\mu^2+\sigma^2$

For the random effects model, the $\mu_i$ are independent random variables with $E\left(\mu_i\right)=\mu$ and $V\left(\mu_i\right)=\sigma_\mu^2$. The cell means model for fixed effects one way Anova is very similar to that for the random effects model, but the $\mu_i$ are fixed constants rather than random variables.

Definition 5.17. For the normal random effects one way Anova model, $\Lambda \sim N\left(\mu, \sigma_\mu^2\right)$. Thus the $\mu_i$ are independent $N\left(\mu, \sigma_\mu^2\right)$ random variables. The $e_{i j}$ are iid $N\left(0, \sigma^2\right)$ and the $e_{i j}$ and $\mu_i$ are independent. For this model, $Y_{i j} \mid \mu_i \sim N\left(\mu_i, \sigma^2\right)$ for $i=1, \ldots, p$. Note that the conditional variance $\sigma^2$ is the same for each $\mu_i \in \Lambda$. Unconditionally, $Y_{i j} \sim N\left(\mu, \sigma_\mu^2+\sigma^2\right)$.

The fixed effects one way Anova tested $H o: \mu_1=\cdots=\mu_p$. For the random effects one way Anova, interest is in whether $\mu_i \equiv \mu$ for every $\mu_i$ in $\Lambda$ where the population $\Lambda$ is not necessarily finite. Note that if $\sigma_\mu^2=0$, then $\mu_i \equiv \mu$ for all $\mu_i \in \Lambda$. In the sample of $p$ levels, the $\mu_i$ will differ if $\sigma_\mu^2>0$.

# 线性回归代考

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

Anova模型用于比较的一种方式 $p$ 治疗。通常有复制和Ho: $\mu_1=\mu_2=\cdots=\mu_p$ 是一 个有趣的假设。调查人员可能还想按从小到大的顺序对总体均值进行排序。

$$Y \mid\left(W=a_i\right) \sim f_Z\left(y-\mu_i\right)$$

$$Y \mid\left(W=a_i\right) \sim N\left(\mu_i, \sigma^2\right) .$$

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

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