# 统计代写|生物统计分析代写Biological statistic analysis代考|STAT201

#### Doug I. Jones

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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
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## 统计代写|生物统计分析代写Biological statistic analysis代考|Treatment, Unit, and Response Structures

The treatment structure of an experiment describes the treatment factors and their relationships. In our drug example, the experiment has a single treatment factor Drug with four levels $D 1, D 2, D 3$, and $D 4$. Other designs use several treatment factors, and each applied treatment is then a combination of one level from each treatment factor.

The unit (or design) structure describes the unit factors and their relationships. A unit factor logically organizes the experimental material, and our experiment has a single unit factor (Mouse) with 32 levels, each level corresponding to one mouse. Unit factors are of several basic types: the smallest subdivision of the experimental material to which levels of a treatment factor are randomly assigned is called the experimental unit of this treatment factor; it provides the residual variance for testing this treatment factor.

Groups of units are specified by a grouping factor, also known as a blocking factor; these are often non-specific and of no direct inferential interest, but are used to remove variation from comparisons or take account of units in the same group being more similar than units in different groups. A blocking factor can also be intrinsic and describe a non-randomizable property of another unit factor; a common example is the sex of an animal, which we cannot deliberately choose (so it is not a treatment), but which we need to keep track of in our inferences.

Treatment factors are often fixed factors with predetermined fixed levels, while unit factors are often random factors whose levels are a random sample from a population; in a replication of the experiment, the fixed factor levels would remain the same (we use the same four drugs again), while the random factor levels change (we do not use the same mice again). We denote treatment factors by an informative name written in bold and unit factors in italics; we denote random factors by parentheses: the treatment factor Drug is fixed for our experiment, while the unit factor (Mouse) is random.

The observations are recorded on the response unit factor, and we mainly consider experiments with a simple response structure where a single value is observed on one unit factor in the design, which we denote by underlining.

The treatment and unit structures are created by nesting and crossing of factors. A factor $A$ is crossed with another factor $B$ if each level of $A$ occurs together with each level of $B$ and vice versa. This implicitly defines a third interaction factor denoted by $A: B$, whose levels are the possible combinations of levels of $A$ with levels of $B$. In our paired design (Fig. 2.8), the treatment factor Vendor is crossed with (Mouse), since each kit (that is, each level of Vendor) is assigned to each mouse. We omitted the interaction factor, since it coincides with (Sample) in this case. The data layout for two crossed factors is shown in Fig. $4.3 \mathrm{~A}$; the cross-tabulation is completely filled.

## 统计代写|生物统计分析代写Biological statistic analysis代考|Constructing the Hasse Diagrams

To emphasize the two components of each experimental design, we draw separate Hasse diagrams for the treatment and unit structures, which we then combine into the experiment structure diagram by considering the randomization. The Hasse diagram visualizes the nesting/crossing relations between the factors. Each factor is represented by a node, shown as the factor name. If factor $B$ is nested in factor $A$, we write $B$ below $A$ and connect the two nodes with an edge. The diagram is thus ‘read’ from top to bottom. If $A$ and $B$ are crossed, we write them next to each other and connect each to the next factor that it is nested in. We then create a new factor denoted by $A: B$, whose levels are the combinations of levels of $A$ with levels of $B$, and draw one edge from $A$ and one edge from $B$ to this factor. Each diagram has a top node called $\mathbf{M}$ or $M$, which represents the grand mean, and all other factors are nested in this top node.

The Hasse diagrams for our drug example are shown in Fig. 4.4. The treatment structure contains the single treatment factor Drug, nested in the obligatory top node M (Fig. 4.4A). Similarly, the unit structure contains only the factor (Mouse) nested in the obligatory top node $M$ (Fig. 4.4B).

We construct the experiment structure diagram as follows: first, we merge the two top nodes $\mathbf{M}$ and $M$ of the treatment and unit structure diagram, respectively, into a single node $\mathbf{M}$. We then draw an edge from each treatment factor to its experimental unit factor. If necessary, we clean up the resulting diagram by removing unnecessary ‘shortcut’ edges: whenever there is a path $A-B-C$, we remove the edge $A-C$ if it exists since its nesting relation is already implied by the path.

In our example, we merge the two nodes $\mathbf{M}$ and $M$ into a single node $\mathbf{M}$. Both Drug and (Mouse) are now nested under the same top node. Since Drug is randomized on (Mouse), we write (Mouse) below Drug and connect the two nodes with an edge. This makes the edge from $\mathbf{M}$ to (Mouse) redundant and we remove it from the diagram (Fig. 4.4C).

# 生物统计分析代考

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