## 统计代写|广义线性模型代写generalized linear model代考|Heteroscedastic models

So far, we have considered the homoscedastic model (4.7)
$$\boldsymbol{\Sigma}_i=\sigma_E^2 \boldsymbol{I}+\boldsymbol{Z} \boldsymbol{\Phi} \boldsymbol{Z}^t,$$
where it is assumed that

1. the residuals $\epsilon_{i j}$ are independent and identically normally distributed with mean 0 and variance $\sigma_E^2$ common to all the treatment groups and
2. the random effects $b_i$ are normally distributed with mean 0 and covariance $\boldsymbol{\Phi}$ common to all the treatment groups.

In this section, let us check these homoscedastic assumptions for Model Vb. To do this in PROC MIXED, first, we have to define treatment as a numeric factor as follows:
class subject visit treatment/ref=first ;
Then, change the RANDOM statements and/or the REPEATED statements shown in Program $7.2$ by adding the option group = treatment as follows (some of them have already been introduced in Section 3.6):

1. For the model with the heteroscedastic covariance $\boldsymbol{\Phi}$ : random intercept post/ type=un subject= subject g gcorr group=treatment ;

## 统计代写|广义线性模型代写generalized linear model代考|Model VI: Random intercept model

The random intercept model with a random intercept $b_{0 i}$ is expressed as
$$\begin{gathered} E\left(y_{i 0} \mid b_{0 i}\right) \sim\left{\begin{array}{l} \beta_0+b_{0 i}+\boldsymbol{w}i^t \boldsymbol{\xi}, i=1, \ldots, n_1 \text { (TAU group) } \ \beta_0+b{0 i}+\beta_1+\boldsymbol{w}i^t \boldsymbol{\xi}, i=n_1+1, \ldots, N \text { (BtheB group) } \end{array}\right. \ E\left(y{i j} \mid b_{0 i}\right) \sim\left{\begin{array}{l} \beta_0+b_{0 i}+\beta_2 t_j+\boldsymbol{w}i^t \boldsymbol{\xi}, i=1, \ldots, n_1(\text { TAU group) } \ \beta_0+b{0 i}+\beta_1+\left(\beta_2+\beta_3\right) t_j+\boldsymbol{w}_i^t \boldsymbol{\xi}, i=n_1+1, \ldots, N \ \text { (BtheB group) } \end{array}\right. \ j=1, \ldots, 4, \end{gathered}$$
where:

1. $t_j$ denotes the $j$ th measurement time, i.e., $t_0=0, t_1=2, t_2=3, t_3=5$ and $t_4=8$.
2. $\beta_0$ denotes the mean BDI score at the baseline period in the TAU group.
3. $\beta_0+b_{i 0}$ denotes the subject-specific mean BDI score at the baseline period in the TAU group.
4. $\beta_1$ denotes the difference in means of the BDI score between the two treatment groups at the baseline period.
5. $\beta_2$ denotes the average rate of change of the BDI score per month during the evaluation period in the TAU group.
6. $\beta_2+\beta_3$ denotes the average rate of change of the BDI score per month during the evaluation period in the BtheB group.
7. $\beta_3$ denotes the difference in average rates of change per month, i.e., the effect size of the treatment by linear time interaction.

## 统计代写|广义线性模型代写generalized linear model代考|Heteroscedastic models

$$\boldsymbol{\Sigma}_i=\sigma_E^2 \boldsymbol{I}+\boldsymbol{Z} \Phi \boldsymbol{Z}^t,$$

1. 残差 $\epsilon_{i j}$ 是独立的且同正态分布，均值为 0 ，方差为 $\sigma_E^2$ 对所有治疗組和
2. 随机效应 $b_i$ 正态分布，均值为 0 , 协方差 $\Phi$ 所有治疗组共有。
在本节中，让我们检查模型 $\mathrm{Vb}$ 的这些同方差假设。要在 PROC MIXED 中做到这一点，首先，我们必须将 治疗定义为一个数字因子，如下所示:
class subject visit treatment/ref=first;
然后，更改程序中显示的 RANDOM 语句和/或 REPEATED 语句 $7.2$ 通过添加选项 group = treatment 如下 (其中一些已经在第 $3.6$ 节中介绍过) :
3. 对于具有异方差协方差的模型 $\boldsymbol{\Phi}$ : 随机截取 post/type=un subject= subject g gcorr group=treatment;

## 统计代写|广义线性模型代写generalized linear model代考|Model VI: Random intercept model

$\$ \$$\backslash begin{gathered E left(y_{i 0} \backslash mid b_{0 i}\right) \backslash sim \backslash left { \beta_0+b_{0 i}+\boldsymbol{w} i^t \boldsymbol{\xi}, i=1, \ldots, n_1 (TAU group) \beta_0+b 0 i+\beta_1+\boldsymbol{w} i^t \boldsymbol{\xi}, i=n_1+1, \ldots, N (BtheB group) \beta_0+b_{0 i}+\beta_2 t_j+\boldsymbol{w} i t \boldsymbol{\xi}, i=1, \ldots, n_1 (TAU group) \beta_0+b 0 i+\beta_1+\left(\beta_2+\beta_3\right) t_j+\boldsymbol{w}_i^t \boldsymbol{\xi}, i=n_1+1, \ldots, N 、正确的。। j=1, \Idots, 4 , \end{最集} } \ \$$

1. $t_j$ 表示 $j$ 测量时间，即 $t_0=0, t_1=2, t_2=3, t_3=5$ 和 $t_4=8$.
2. $\beta_0$ 表示 $\mathrm{TAU}$ 组基线期的平均 $\mathrm{BDI}$ 分数。
3. $\beta_0+b_{i 0}$ 表示 $\mathrm{TAU}$ 组基线期的受试者特定平均 $\mathrm{BDI}$ 分数。
4. $\beta_1$ 表示基线期两个治疗组之间 BDI 评分平均值的差异。
5. $\beta_2$ 表示 $\mathrm{TAU}$ 组在评估期间每月 BDI 分数的平均变化率。
6. $\beta_2+\beta_3$ 表示 BtheB 组在评估期间每月 BDI 分数的平均变化率。
7. $\beta_3$ 表示每月平均变化率的差异，即线性时间交互作用的治疗效果大小。

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