### 统计代写|生物统计学作业代写Biostatistics代考| SPECIAL CASE OF BINARY DATA

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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|生物统计学作业代写Biostatistics代考|SPECIAL CASE OF BINARY DATA

Observations or measurements may be made on different scales. If each element of a data set may lie at only a few isolated points, we have a discrete data set. A special case of discrete data are binary data, where each outcome has only two possible values; examples are gender and an indication of whether a treatment is a success or a failure. If each element of this set may theoretically lie anywhere on a numerical scale, we have a continuous data set; examples are blood pressure and cholesterol level. Chapter 1 deals with the summarization and description of discrete data, especially binary data; the primary statistic was proportion. In this chapter the emphasis so far has been on continuous measurements, where, for example, we learn to form sample mean and use it as

a measure of location, a typical value representing the data set. In addition, the variance and/or standard deviation is formed and used to measure the degree of variation or dispersion of data around the mean. In this short section we will see that binary data can be treated as a special case of continuous data.

Many outcomes can be classified as belonging to one of two possible categories: presence and absence, nonwhite and white, male and female, improved and not improved. Of course, one of these two categories is usually identified as being of primary interest; for example, presence in the presence and absence classification, or nonwhite in the white and nonwhite classification. We can, in general, relabel the two outcome categories as positive $(+)$ and negative $(-)$. An outcome is positive if the primary category is observed and is negative if the other category is observed. The proportion is defined as in Chapter 1 :
$$p=\frac{x}{n}$$
where $x$ is the number of positive outcomes and $n$ is the sample size. However, it can also be expressed as
$$p=\frac{\sum x_{i}}{n}$$
where $x_{i}$ is ” 1 ” if the $i$ th outcome is positive and ” 0 “s otherwise. In other words, a sample proportion can be viewed as a special case of sample means where data are coded as 0 or 1 . But what do we mean by variation or dispersion, and how do we measure it?

Let us write out the variance $s^{2}$ using the shortcut formula of Section $2.2$ but with the denominator $n$ instead of $n-1$ (this would make little difference because we almost always deal with large samples of binary data):
$$s=\sqrt{\frac{\sum x_{i}^{2}-\left(\sum x_{i}\right)^{2} / n}{n}}$$
Since $x_{i}$ is binary, with ” 1 ” if the $i$ th outcome is positive and ” 0 ” otherwise, we have
$$x_{i}^{2}=x_{i}$$
and therefore,
\begin{aligned} s^{2} &=\frac{\sum x_{i}-\left(\sum x_{i}\right)^{2} / n}{n} \ &=\frac{\sum x_{i}}{n}\left(1-\frac{\sum x_{i}}{n}\right) \ &=p(1-p) \end{aligned}

In other words, the statistic $p(1-p)$ can be used in place of $s^{2}$ as a measure of variation; the logic can be seen as follows. First, the quantity $p(1-p)$, with $0 \leq p \leq 1$, attains its maximum value when $p=0.5$. For example,
\begin{aligned} (0.1)(0.9) &=0.09 \ & \vdots \ (0.4)(0.6) &=0.24 \ (0.5)(0.5) &=0.25 \ (0.6)(0.4) &=0.24 \ & \vdots \ (0.9)(0.1) &=0.09 \end{aligned}
The values of $p(1-p)$ are greatest in the vicinity of $p=0.5$ and decrease as we go toward both ends $(0$ and 1$)$ of the range of $p$. If we are performing a cointossing experiment or conducting an election; the result would be most unpredictable when the chance to obtain the outcome wanted is in the vicinity of $p=0.5$. In other words, the quantity $p(1-p)$ is a suitable statistic to measure the volatility, dispersion, and variation. The corresponding statistic for standard deviation is $\sqrt{p(1-p)}$.

## 统计代写|生物统计学作业代写Biostatistics代考|COEFFICIENTS OF CORRELATION

Methods discussed in this chapter have been directed to the analyses of data where a single continuous measurement was made on each element of a sample. However, in many important investigations we may have two measurements made: where the sample consists of pairs of values and the research objective is concerned with the association between these variables. For example, what is the relationship between a mother’s weight and her baby’s weight? In Section $1.3$ we were concerned with the association between dichotomous variables. For example, if we want to investigate the relationship between a disease and a certain risk factor, we could calculate an odds ratio to represent the strength of the relationship. In this section we deal with continuous measurements, and the method is referred to as correlation analysis. Correlation is a concept that carries the common colloquial implication of association, such as “height and weight are correlated.” The statistical procedure will give the word a technical meaning; we can actually calculate a number that tells the strength of the association.

When dealing with the relationship between two continuous variables, we first have to distinguish between a deterministic relationship and a statistical relationship. For a deterministic relationship, values of the two variables are related through an exact mathematical formula. For example, consider the

relationship between hospital cost and number of days in hospital. If the costs are $\$ 100$for admission and$\$150$ per day, we can easily calculate the total cost given the number of days in hospital, and if any set of data is plotted, say cost versus number of days, all data points fall perfectly on a straight line. Unlike a deterministic relationship, a statistical relationship is not perfect. In general, the points do not fall perfectly on any line or curve.

Table $2.12$ gives the values for the birth weight $(x)$ and the increase in weight between days 70 and 100 of life, expressed as a percentage of the birth weight $(y)$ for 12 infants. If we let each pair of numbers $(x, y)$ be represented by a dot in a diagram with the $x$ ‘s on the horizontal axis, we have Figure 2.13. The dots do not fall perfectly on a straight line, but rather, scatter around a line, very typical for statistical relationships. Because of this scattering of dots, the diagram is called a scatter diagram. The positions of the dots provide some information about the direction as well as the strength of the association under the investigation. If they tend to go from lower left to upper right, we have a positive association; if they tend to go from upper left to lower right, we have a negative association. The relationship becomes weaker and weaker as the dis-tribution of the dots clusters less closely around the line, and becomes virtually no correlation when the distribution approximates a circle or oval (the method is ineffective for measuring a relationship that is not linear).

## 统计代写|生物统计学作业代写Biostatistics代考|Pearson’s Correlation Coe‰cient

Consider the scatter diagram shown in Figure $2.14$, where we have added a vertical and a horizontal line through the point $(\bar{x}, \bar{y})$ and label the four quarters as I, II, III, and IV. It can be seen that

• In quarters I and III,
$$(x-\bar{x})(y-\bar{y})>0$$
so that for positive association, we have
$$\sum(x-\bar{x})(y-\bar{y})>0$$
Furthermore, this sum is large for stronger relationships because most of the dots, being closely clustered around the line, are in these two quarters.
• Similarly, in quarters II and IV,
$$(x-\bar{x})(y-\bar{y})<0$$

$$\sum(x-\bar{x})(y-\bar{y})<0$$
for negative association.
With proper standardization, we obtain
$$r=\frac{\sum(x-\bar{x})(y-\bar{y})}{\sqrt{\left[\sum(x-\bar{x})^{2}\right]\left[\sum(y-\bar{y})^{2}\right]}}$$
so that
$$-1 \leq r \leq 1$$
This statistic, $r$, called the correlation coefficient, is a popular measure for the strength of a statistical relationship; here is a shortcut formula:
$$r=\frac{\sum x y-\left(\sum x\right)\left(\sum y\right) / n}{\sqrt{\left[\sum x^{2}-\left(\sum x\right)^{2} / n\right]\left[\sum y^{2}-\left(\sum y\right)^{2} / n\right]}}$$
Meanningful interpretation of the correlation coefficient $r$ is rather complicated at this level. We will revisit the topic in Chapter 8 in the context of regression analysis, a statistical method that is closely connected to correlation. Generally:

• Values near 1 indicate a strong positive association.
• Values near $-1$ indicate a strong negative association.
• Values around 0 indicate a weak association.
Interpretation of $r$ should be made cautiously, however. It is true that a scatter plot of data that results in a correlation number of $+1$ or $-1$ has to lie in a perfectly straight line. But a correlation of 0 doesn’t mean that there is no association; it means that there is no linear association. You can have a correlation near 0 and yet have a very strong association, such as the case when the data fall neatly on a sharply bending curve.

## 统计代写|生物统计学作业代写Biostatistics代考|SPECIAL CASE OF BINARY DATA

p=Xn

p=∑X一世n

s=∑X一世2−(∑X一世)2/nn

X一世2=X一世

s2=∑X一世−(∑X一世)2/nn =∑X一世n(1−∑X一世n) =p(1−p)

(0.1)(0.9)=0.09 ⋮ (0.4)(0.6)=0.24 (0.5)(0.5)=0.25 (0.6)(0.4)=0.24 ⋮ (0.9)(0.1)=0.09

## 统计代写|生物统计学作业代写Biostatistics代考|Pearson’s Correlation Coe‰cient

• 在第一和第三季度，
(X−X¯)(是−是¯)>0
所以对于正关联，我们有
∑(X−X¯)(是−是¯)>0
此外，这个总和对于更牢固的关系来说很大，因为大多数点都紧密地聚集在这条线上，都在这两个季度中。
• 同样，在第二和第四季度，
(X−X¯)(是−是¯)<0
导致

∑(X−X¯)(是−是¯)<0

r=∑(X−X¯)(是−是¯)[∑(X−X¯)2][∑(是−是¯)2]

−1≤r≤1

r=∑X是−(∑X)(∑是)/n[∑X2−(∑X)2/n][∑是2−(∑是)2/n]

• 接近 1 的值表示强正相关。
• 值接近−1表示强烈的负相关。
• 0 附近的值表示弱关联。
的解释r但是，应该谨慎进行。确实，数据的散点图会导致相关数+1或者−1必须在一条完美的直线上。但相关性为 0 并不意味着没有关联；这意味着没有线性关联。您可以在 0 附近建立相关性，但关联性非常强，例如数据整齐地落在急剧弯曲的曲线上的情况。

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## MATLAB代写

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