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

However, in practice, sensitivity analyses are recommended to consider a range of plausible alternative MNAR assumptions about the missing data because it is important to investigate the extent to which treatment effects are stable to departures from the MAR assumption (National Research Council, 2010).
For MNAR, we usually need a joint model of the data and the missing data mechanism called the pattern mixture model (Little, 1993):
$$p\left(\boldsymbol{y}_i, \boldsymbol{r}_i \mid \boldsymbol{\theta}, \boldsymbol{\xi}\right)=p\left(\boldsymbol{y}_i \mid \boldsymbol{r}_i, \boldsymbol{\theta}, \boldsymbol{\xi}\right) p\left(\boldsymbol{r}_i \mid \boldsymbol{\xi}\right)$$
which is a product of (1) the probability distribution of the data within each missing data pattern and (2) the probability of the missing data pattern. To do this, multiple imputation methods are useful and it is recommended to specify the numerical value of sensitivity parameter $\delta$ governing the degree of departure from the MAR assumption (e.g., Molenberghs and Kenward, 2007; Carpenter and Kenward, 2013). For example, specify a pattern mixture model including $\delta$ such that $\delta=0$ corresponds to the untestable assumption made in the primary analysis and $\delta \neq 0$ corresponds to plausible departures from that assumption. Then estimate the treatment effect over a plausible range of values of $\delta$. Regarding the details of methods for multiple imputations or textbooks (e.g., Schafer, 1997; Verbeke and Molenberghs, 2001; Diggle et al., 2002; Fitzmaurice et al., 2011; White et al., 2011; van Buuren, 2012; Carpenter and Kenward, 2013). Regarding the statistical software to perform the pattern-mixture model, the recently implemented MNAR statement in the SAS procedure PROC MI could be useful.

On the other hand, there are some ad hoc graphical procedures for assessing the missing data mechanism in longitudinal study. Especially, the procedure suggested by Carpenter et al. (2002) is very simple and just plot the repeated measurements at each time point, differentiating between two groups of patients: those who do and those who do not come in to their next scheduled visit. If you find any clear difference between the distributions of observed data for these two groups, it will be reasonable to assume that the missing data mechanism is not MCAR.

## 统计代写|广义线性模型代写generalized linear model代考|MMRM vs. LOCF

In this section, let us assume that (1) the Rat Data comes from a randomized clinical trial, (2) the primary endpoint is the CFB at week 3 and the primary statistical method is Student’s two-sample $t$-test. In handling missing data, the most frequently used method in the past was LOCF. In the LOCF analysis of the Rat Data shown in Table 6.2, the missing data at week 3 for subject No. 8 is replaced by the data $10.7$ at week 2 . However, the LOCF-based Student’s twosample $t$-test for the difference in means of CFB can be reasonably replaced by the linear mixed-effects model (5.23). Estimated treatment effects are

1. $-3.77 \pm 0.93,(p=0.0023)$ from the mixed-effects model for the original data set without missing data;
1. $-3.74 \pm 0.93,(p=0.0026)$ from the mixed-effects model for the data set with missing data; and
2. $-4.00 \pm 1.02(p=0.0029)$ from the LOCF-based Student’s two-sample $t$-test.

By the same token, if the analysis of covariance adjusting for the baseline data is defined as the primary statistical method in the trial protocol, then the traditional LOCF-based ANCOVA analysis can be reasonably replaced by one of the ANCUVA-type mixed-effects models $(3.34,3.35,4.16)$. The SAS program for the model (3.35) with the baseline effects changing across time modified to the style introduced in Section $5.3$ is shown in Program $6.4$ and part of the results are shown in Output 6.4.

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

$$p\left(\boldsymbol{y}_i, \boldsymbol{r}_i \mid \boldsymbol{\theta}, \boldsymbol{\xi}\right)=p\left(\boldsymbol{y}_i \mid \boldsymbol{r}_i, \boldsymbol{\theta}, \boldsymbol{\xi}\right) p\left(\boldsymbol{r}_i \mid \boldsymbol{\xi}\right)$$

## 统计代写|广义线性模型代写generalized linear model代考|MMRM vs. LOCF

1. $-3.77 \pm 0.93,(p=0.0023)$ 来自没有缺失数据的原始数据集的混合效应模型;
2. $-3.74 \pm 0.93,(p=0.0026)$ 来自具有缺失数据的数据集的混合效应模型；和 3. $-4.00 \pm 1.02(p=0.0029)$ 来自基于 LOCF 的学生的两个样本 $t$-测试。
同理，如果将基线数据的协方差调整分析定义为试验方案中的主要统计方法，那么传统的基于 LOCF 的 ANCOVA 分析可以合理地替换为 ANCUVA 型混合效应模型之一 $(3.34,3.35,4.16)$. 模型 (3.35) 的 SAS 程 序，其基线效应随时间变化，修改为第 1 节中介绍的样式5.3显示在程序中6.4部分结果显示在输出 $6.4$ 中。

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