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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|广义线性模型代写generalized linear model代考|Change from baseline

Let $C F B_{k i j}$ denote the change from baseline at time $j$ for the $i$ th subject in the $k$ th treatment group; then we have
$$C \hat{F} B_{k i j}=y_{k i j}-y_{k i 0}=\left(\hat{\beta}j-\hat{\beta}_0\right)+\left(\hat{\gamma}{k j}-\hat{\gamma}{k 0}\right) .$$ Namely, the mean change from baseline at time $j$ in the $k$ th treatment group is $$\frac{1}{n_k} \sum{i=1}^{n_k} C F B_{k i j}=\bar{y}{k+j}-\bar{y}{k+0}=\left(\hat{\beta}j-\hat{\beta}_0\right)+\left(\hat{\gamma}{k j}-\hat{\gamma}{k 0}\right) .$$ Therefore, the difference in mean changes from baseline at time $j$ of the $k$ th treatment group compared with the $m$ th treatment group is expressed as \begin{aligned} \hat{\tau}{k m}^{(j)} &=\left(\frac{1}{n_k} \sum_{i=1}^{n_k} C F B_{k i j}-\frac{1}{n_m} \sum_{i=1}^{n_m} C F B_{m i j}\right) \ &=\left(\hat{\gamma}{k j}-\hat{\gamma}{k 0}\right)-\left(\hat{\gamma}{m j}-\hat{\gamma}{m 0}\right)=\hat{\gamma}{k j}-\hat{\gamma}{m j} . \end{aligned}
When the trial consists of two treatment groups, the experimental group and the control group, the difference in mean changes from baseline at time $j$ of the experimental group $(k=2)$ compared with the control group $(k=1)$ is
$$\hat{\tau}{21}^{(j)}=\hat{\gamma}{2 j}-\hat{\gamma}{1 j}=\hat{\gamma}{2 j}(j=1, \ldots, T) .$$
Namely, $\hat{\gamma}{k j}-\hat{\gamma}{m j}$ denotes the estimate of the effect size of the $k$ th treatment compared with the $m$ th treatment at time $j$.

The overall estimate of the relative effect size of the $k$ th treatment compared with the $m$ th treatment over the measurement period $(1 \leq j \leq T)$ is
\begin{aligned} \hat{\tau}{k m} &=\frac{1}{T} \sum{j=1}^T \hat{\tau}{k m}^{(j)} \ &=\frac{1}{T} \sum{j=1}^T\left(\frac{1}{n_k} \sum_{i=1}^{n_k} C F B_{k i j}-\frac{1}{n_m} \sum_{i=1}^{n_m} C F B_{m i j}\right) \ &=\frac{1}{T} \sum_{j=1}^T\left{\left(\hat{\gamma}{k j}-\hat{\gamma}{k 0}\right)-\left(\hat{\gamma}{m j}-\hat{\gamma}{m 0}\right)\right} \end{aligned}

## 统计代写|广义线性模型代写generalized linear model代考|Split-plot design

The principle of the split-plot design is as follows. Let us consider some agricultural field experiment where the field is divided into $N=G n$ “main plots” to compare the levels of one factor A such as the addition of different ameliorants by allocating them at random to the main plots. Then, each main plot is divided into $T+1$ “subplots” to compare the levels of the other factor, B, such as, the addition of different dressing, by allocating them at random to subplots within a main plot. A naive ANOVA model for this design will be
\begin{aligned} y_{k i j}=& \mu+\alpha_k+\beta_j+\gamma_{k j}+\epsilon_{k i j}, \ & \epsilon_{k i j} \sim N\left(0, \sigma_E^2\right) \ & k=1, \ldots, G ; i=1, \ldots, n ; j=0,1,2, \ldots, T, \end{aligned}
where $y_{k i j}$ denotes the yield of the $j$ th level of factor $\mathrm{B}$ within the $i$ th main plot allocated to the $k$ th level of factor $\mathrm{A}, \alpha_k$ denotes the fixed effects of the $k$ th level of factor $\mathrm{A}, \beta_j$ denotes the fixed effects of the $j$ th level of factor $\mathrm{B}$, $\gamma_{k j}$ denotes the fixed effects of the interaction between the $k$ th level of factor A and the $j$ th level of factor $B$, and $\epsilon_{k i j}$ is an random error assumed to be distributed with normal distribution with mean 0 and variance $\sigma_E^2$. It should be noted that the subscript $j$ starts from 0 , not from 1 , because the split-plot design is due to extend to the repeated measures design.

However, the ANOVA model (3.17) assumes that there is no difference in fertilities between the original main plots, which is unlikely in practice. So, to take this variability into account, the following ANOVA model must be considered:
\begin{aligned} y_{k i j}=& \mu+\alpha_k+\beta_j+\gamma_{k j}+b_{k i}+\epsilon_{k i j} \ & \epsilon_{k i j} \sim N\left(0, \sigma_E^2\right), b_{k i} \sim N\left(0, \sigma_B^2\right) \ & k=1, \ldots, G ; i=1, \ldots, n ; j=0,1,2, \ldots, T \end{aligned}

# 广义线性模型代考

## 统计代写|广义线性模型代写generalized linear model代考|Change from baseline

$$C \hat{F} B_{k i j}=y_{k i j}-y_{k i 0}=\left(\hat{\beta} j-\hat{\beta}0\right)+(\hat{\gamma} k j-\hat{\gamma} k 0) .$$ 即，从基线到时间的平均变化 $j$ 在里面 $k$ 治疗组是 $$\frac{1}{n_k} \sum i=1^{n_k} C F B{k i j}=\bar{y} k+j-\bar{y} k+0=\left(\hat{\beta} j-\hat{\beta}0\right)+(\hat{\gamma} k j-\hat{\gamma} k 0) .$$ 因此，时间平均变化与基线的差异 $j$ 的 $k$ 治疗组与 $m$ 治疗组表示为 $$\hat{\tau} k m^{(j)}=\left(\frac{1}{n_k} \sum{i=1}^{n_k} C F B_{k i j}-\frac{1}{n_m} \sum_{i=1}^{n_m} C F B_{m i j}\right) \quad=(\hat{\gamma} k j-\hat{\gamma} k 0)-(\hat{\gamma} m j-\hat{\gamma} m 0)=\hat{\gamma} k j-\hat{\gamma} m j .$$

$$\hat{\tau} 21^{(j)}=\hat{\gamma} 2 j-\hat{\gamma} 1 j=\hat{\gamma} 2 j(j=1, \ldots, T) .$$

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## 统计代写|广义线性模型代写generalized linear model代考|Split-plot design

$$y_{k i j}=\mu+\alpha_k+\beta_j+\gamma_{k j}+\epsilon_{k i j}, \quad \epsilon_{k i j} \sim N\left(0, \sigma_E^2\right) k=1, \ldots, G ; i=1, \ldots, n ; j=0,1,2, \ldots, T,$$

$$y_{k i j}=\mu+\alpha_k+\beta_j+\gamma_{k j}+b_{k i}+\epsilon_{k i j} \quad \epsilon_{k i j} \sim N\left(0, \sigma_E^2\right), b_{k i} \sim N\left(0, \sigma_B^2\right) k=1, \ldots, G ; i=1, \ldots, n ; j=0,1,2, \ldots, T$$

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