### 统计代写|广义线性模型代写generalized linear model代考|STAT6175

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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|Model for the average treatment effect

As mentioned in Section $1.1$, the primary interest of $S: T$ design, especially in confirmatory trials in the later phases of drug development, is to estimate the overall or average treatment effect during the evaluation period, leading to sample size calculation in the design stage (see Chapter 12 for details). So, let us consider here the case where we would like to estimate the average treatment effect during the evaluation period even though the treatment effect is not always expected to be constant over the evaluation period. However, in such a situation, it is important to select the evaluation period during which the treatment effect could be expected to be stable to a certain extent.

Here also, let us introduce two models, a random intercept model and a random intercept plus slope model. The former model is
$$\begin{array}{rlr} g\left{E\left(y_{i j} \mid b_{0 i}\right)\right} & =g\left{\mu_{i j} \mid b_{0 i}\right} & \ & = \begin{cases}\beta_0+\beta_1 x_{1 i}+b_{0 i}+\boldsymbol{w}i^t \boldsymbol{\xi} & \text { for } j \leq 0 \ \beta_0+\beta_1 x{1 i}+b_{0 i}+\beta_2+\beta_3 x_{1 i}+\boldsymbol{w}i^t \boldsymbol{\xi} & \text { for } j \geq 1\end{cases} \ b{0 i} & \sim N\left(0, \sigma_{B 0}^2\right), \ i & =1, \ldots, n_1(\text { control group }), & \ & =n_1+1, \ldots, n_1+n_2(\text { new treatment group }), \ j & =-(S-1),-(S-2), \ldots, 0,1, \ldots, T, & \end{array}$$
and the latter is
\begin{aligned} g\left{E\left(y_{i j} \mid \boldsymbol{b}i\right)\right} &= \begin{cases}\beta_0+\beta_1 x{1 i}+b_{0 i}+\boldsymbol{w}i^t \boldsymbol{\xi} & \text { for } j \leq 0 \ \beta_0+\beta_1 x{1 i}+b_{0 i}+b_{1 i}+\beta_2+\beta_3 x_{1 i}+\boldsymbol{w}i^t \boldsymbol{\xi} & \text { for } j \geq 1 \ \boldsymbol{b}_i & =\left(b{0 i}, b_{1 i}\right)^t \sim N(\mathbf{0}, \Phi),\end{cases} \end{aligned}
where both a random intercept $b_{0 i}$ and a random slope $b_{1 i}$ denote the same meanings as those introduced in the previous section. It should be noted here also that the random slope introduced in the above model does not mean the slope on the time or a linear time trend, which will be considered in the next section.

## 统计代写|广义线性模型代写generalized linear model代考|Model for the treatment by linear time interaction

In the previous subsection, we focused on the generalized linear mixed models where we are primarily interested in comparing the average treatment effect during the evaluation period. However, there are other study designs where the primary interest is not the average treatment effect but the treatment-bytime interaction (Diggle et al., 2002; Fitzmaurice et al., 2011). For example, in the National Institute of Mental Health Schizophrenia Collaboratory Study, the primary interest is testing the drug by linear time interaction, i.e., testing whether the differences between treatment groups are linearly increasing over time (Hedeker and Gibbons, 2006; Roy et al., 2007; Bhumik et al., 2008).
So, let us consider here a random intercept and a random intercept plus slope model with a random slope on the time (a linear time trend model) for testing the null hypothesis that the rates of change or improvement over time are the same in the new treatment group compared with the control group. The former model is
\begin{aligned} g\left{E\left(y_{i j} \mid b_{0 i}\right)\right} &=g\left{\mu_{i j} \mid b_{0 i}\right} \ &=\beta_0+b_{0 i}+\beta_1 x_{1 i}+\left(\beta_2+\beta_3 x_{1 i}\right) t_j+\boldsymbol{w}i^t \boldsymbol{\xi} \ b{0 i} & \sim N\left(0, \sigma_{B 0}^2\right) \ i &=1, \ldots, n_1(\text { control group }) \ &=n_1+1, \ldots, n_1+n_2(\text { new treatment group }) \ j &=-(S-1),-(S-2), \ldots, 0,1, \ldots, T \end{aligned}

where $t_j$ denotes the $j$ th measurement time and we have
$$t_{-S+1}=t_{-S+2}=\cdots=t_0=0 .$$
Interpretation of the fixed-effects parameters $\boldsymbol{\beta}$ of interest are as follows:

1. $\beta_2$ denotes the rate of change over time in the control group.
2. $\beta_2+\beta_3$ denotes the same quantity in the new treatment group.
3. $\beta_3$ denotes the difference in these two rates of change over time, i.e., the treatment effect of the new treatment compared with the control treatment.

## 统计代写|广义线性模型代写generalized linear model代考|Model for the treatment by linear time interaction

$$t_{-S+1}=t_{-S+2}=\cdots=t_0=0 .$$

1. $\beta_2$ 表示对照组随时间的变化率。
2. $\beta_2+\beta_3$ 表示新治疗组中的相同数量。
3. $\beta_3$ 表示这两种变化率随时间的差异，即新治疗与对照治疗相比的治疗效果。

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