## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Comparison of Two Mean Vectors

In many situations, we want to compare two groups of individuals for whom a set of $p$ characteristics has been observed. We have two random samples $\left{x_{i 1}\right}_{i=1}^{n_1}$ and $\left{x_{j 2}\right}_{j=1}^{n_2}$ from two distinct $p$-variate normal populations. Several testing issues can be addressed in this framework. In Test Problem 8 we will first test the hypothesis of equal mean vectors in the two groups under the assumption of equality of the two covariance matrices. This task can be solved by adapting Test Problem 2 .

In Test Problem 9 a procedure for testing the equality of the two covariance matrices is presented. If the covariance matrices differ, the procedure of Test Problem 8 is no longer valid. If the equality of the covariance matrices is rejected, an easy rule for comparing two means with no restrictions on the covariance matrices is provided in Test Problem $10 .$

Both samples provide the statistics $\bar{x}k$ and $\mathcal{S}_k, k=1,2$. Let $\delta=\mu_1-\mu_2$. We have $$\begin{gathered} \left(\bar{x}_1-\bar{x}_2\right) \sim N_p\left(\delta, \frac{n_1+n_2}{n_1 n_2} \Sigma\right) \ n_1 S_1+n_2 S_2 \sim W_p\left(\Sigma, n_1+n_2-2\right) . \end{gathered}$$ Let $\mathcal{S}=\left(n_1+n_2\right)^{-1}\left(n_1 S_1+n_2 S_2\right)$ be the weighted mean of $\mathcal{S}_1$ and $\mathcal{S}_2$. Since the two samples are independent and since $\mathcal{S}_k$ is independent of $\bar{x}_k$ (for $k=1,2$ ) it follows that $\mathcal{S}$ is independent of $\left(\bar{x}_1-\bar{x}_2\right)$. Hence, Theorem $5.8$ applies and leads to a $T^2$-distribution: $$\left.\frac{n_1 n_2\left(n_1+n_2-2\right)}{\left(n_1+n_2\right)^2}\left{\left(\bar{x}_1-\bar{x}_2\right)-\delta\right}^{\top} \mathcal{S}^{-1}\left{\left(\bar{x}_1-\bar{x}_2\right)-\delta\right}\right) \sim T{p, n_1+n_2-2}^2$$ or
$$\left{\left(\bar{x}1-\bar{x}_2\right)-\delta\right}^{\top} \mathcal{S}^{-1}\left{\left(\bar{x}_1-\bar{x}_2\right)-\delta\right} \sim \frac{p\left(n_1+n_2\right)^2}{\left(n_1+n_2-p-1\right) n_1 n_2} F{p, n_1+n_2-p-1}$$
This result, as in Test Problem 2, can be used to test $H_0: \delta=0$ or to construct a confidence region for $\delta \in \mathbb{R}^p$.

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Profile Analysis

Another useful application of Test Problem 6 is the repeated measurements problem applied to two independent groups. This problem arises in practice when we observe repeated measurements of characteristics (or measures of the same type under different experimental conditions) on the different groups which have to be compared. It is important that the $p$ measures (the “profile”) are comparable, and, in particular, are reported in the same units. For instance, they may be measures of blood pressure at $p$ different points in time, one group being the control group and the other the group receiving a new treatment. The observations may be the scores obtained from $p$ different tests of two different experimental groups. One is then interested in comparing the profiles of each group: the profile being just the vectors of the means of the $p$ responses (the comparison may be visualised in a two-dimensional graph using the parallel coordinate plot introduced in Sect. 1.7).
We are thus in the same statistical situation as for the comparison of two means:
$$\begin{array}{ll} X_{i 1} \sim N_p\left(\mu_1, \Sigma\right) & i=1, \ldots, n_1 \ X_{i 2} \sim N_p\left(\mu_2, \Sigma\right) & i=1, \ldots, n_2, \end{array}$$
where all variables are independent. Suppose the two population profiles look like in Fig. 7.1.
The following questions are of interest:

1. Are the profiles similar in the sense of being parallel (which means no interaction between the treatments and the groups)?
2. If the profiles are parallel, are they at the same level?
3. If the profiles are parallel, is there any treatment effect, i.e. are the profiles horizontal (profiles remain the same no matter which treatment received)?
The above questions are easily translated into linear constraints on the means and a test statistic can be obtained accordingly.

## 统计代写|多元统计分析代写多元统计分析代考|两个平均向量的比较

$$\left{\left(\bar{x}1-\bar{x}_2\right)-\delta\right}^{\top} \mathcal{S}^{-1}\left{\left(\bar{x}_1-\bar{x}_2\right)-\delta\right} \sim \frac{p\left(n_1+n_2\right)^2}{\left(n_1+n_2-p-1\right) n_1 n_2} F{p, n_1+n_2-p-1}$$

## 统计代写|多元统计分析代写多元统计分析代考|Profile分析

.

$$\begin{array}{ll} X_{i 1} \sim N_p\left(\mu_1, \Sigma\right) & i=1, \ldots, n_1 \ X_{i 2} \sim N_p\left(\mu_2, \Sigma\right) & i=1, \ldots, n_2, \end{array}$$
，其中所有变量都是独立的。假设两个种群剖面如图7.1所示。 . .

1. 在平行的意义上，这些剖面是否相似(这意味着治疗和组之间没有相互作用)?
2. 如果两个剖面是平行的，它们是否在同一水平线上?
3. 如果轮廓是平行的，是否有任何处理效果，即轮廓是水平的(无论接受哪种处理，轮廓保持相同)?
以上问题很容易转化为对均值的线性约束，可以得到相应的检验统计量。

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