## 统计代写|生物统计分析代写Biological statistic analysis代考|BIOL220

2022年10月11日

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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
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## 统计代写|生物统计分析代写Biological statistic analysis代考|An Example with Sub-sampling

In biological experimentation, experimental units are frequently sub-sampled, and the data contain several response values for each experimental unit. In our example, we might still randomize the drug treatments on the mice, but take four blood samples instead of one from each mouse and measure them independently. Then, the mice are still the experimental units for the treatment, but the blood samples now provide the response units. The Hasse diagrams in Fig. $4.5$ illustrate this design.

The treatment structure is identical to our previous example, and contains Drug as its only relevant factor. The unit structure now contains a new factor (Sample) with 128 levels, one for each measured enzyme level. It is the response factor that provides the observations. Since each sample belongs to one mouse, and each mouse has several samples, the factor (Sample) is nested in (Mouse). The observations are then partitioned first into 32 groups-one per mouse-and further into 128-one per sample per mouse. For the experiment structure, we randomize Drug on (Mouse), and arrive at the diagram in Fig. 4.5C.

The $F$-test for the drug effect again uses the mean squares for Drug on 3 degrees of freedom. Using our rule, we find that (Mouse) – and not (Sample) – is the experimental unit factor that provides the estimate of the variance for the $F$-denominator on 28 degrees of freedom. As far as this test is concerned, the 128 samples are technical replicates or pseudo-replicates. They do not reflect the biological variation against which we need to test the differences in enzyme levels for the four drugs, since drugs are randomized on mice and not on samples.

## 统计代写|生物统计分析代写Biological statistic analysis代考|The Linear Model

For a completely randomized design with $k$ treatment groups, we can write each datum $y_{i j}$ explicitly as the corresponding treatment group mean and a random deviation from this mean:
$$y_{i j}=\mu_i+e_{i j}=\mu+\alpha_i+e_{i j} .$$
The first model is called a cell means model, while the second, equivalent, model is a parametric model. If the treatments had no effect, then all $\alpha_i-\mu_i-\mu$ are zero and the data are fully described by the grand mean $\mu$ and the residuals $e_{i j}$. Thus, the parameters $\alpha_i$ measure the systematic difference of each treatment from the grand mean and are independent of the experimental units.

It is crucial for an analysis that the linear model fully reflects the structure of the experiment. The Hasse diagrams allow us to derive an appropriate model for any experimental design with comparative ease. For our example, the diagram in Fig.4.4C has three factors: M, Drug, and (Mouse), and these are reflected in the three sets of parameters $\mu, \alpha_i$, and $e_{i j}$. Note that there are four parameters $\alpha_i$ to produce the four group means, but given three and the grand mean $\mu$, the fourth parameter can be calculated; thus, there are four parameters $\alpha_i$, but only three can be independently estimated given $\mu$, as reflected by the three degrees of freedom for Drug. Further, the $e_{i j}$ are 32 random variables, and this is reflected in the fact that (Mouse) is a random factor. Given estimates for $\mu$ and $\alpha_i$, the $e_{i j}$ in each of the four groups must sum to zero and only 28 values are independent.
For the sub-sampling example in Fig.4.5, the linear model is
$$y_{i j k}=\mu+\alpha_i+m_{i j}+e_{i j k} \text {, }$$ where $m_{i j}$ is the average deviation of measurements of mouse $j$ in treatment group $i$ from the treatment group mean, and $e_{i j k}$ are the deviations of individual measurements of a mouse to its average. These terms correspond exactly to $\mathbf{M}$, Drug, (Mouse), and (Sample).

# 生物统计分析代考

## 统计代写|生物统计分析代写生物统计分析代考|线性模型

$$y_{i j}=\mu_i+e_{i j}=\mu+\alpha_i+e_{i j} .$$

$$y_{i j k}=\mu+\alpha_i+m_{i j}+e_{i j k} \text {, }$$，其中$m_{i j}$为处理组$i$中小鼠$j$的测量值与处理组平均值的平均偏差，$e_{i j k}$为单个小鼠测量值与其平均值的偏差。这些术语对应$\mathbf{M}$，药物，(老鼠)和(样本)。

## 有限元方法代写

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## MATLAB代写

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