### 统计代写|生物统计学作业代写Biostatistics代考|Relative Risk

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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|Relative Risk

One of the most often used ratios in epidemiological studies is relative risk, a concept for the comparison of two groups or populations with respect to a certain unwanted event (e.g., disease or death). The traditional method of expressing it in prospective studies is simply the ratio of the incidence rates:
$$\text { relative risk }=\frac{\text { disease incidence in group } 1}{\text { disease incidence in group } 2}$$
However, the ratio of disease prevalences as well as follow-up death rates can also be formed. Usually, group 2 is under standard conditions-such as nonexposure to a certain risk factor-against which group 1 (exposed) is measured. A relative risk greater than $1.0$ indicates harmful effects, whereas a relative risk below $1.0$ indicates beneficial effects. For example, if group 1 consists of smokers and group 2 of nonsmokers, we have a relative risk due to smoking. Using the data on end-stage renal disease (ESRD) of Example 1.12, we can obtain the relative risks due to diabetes (Table 1.10). All three numbers are greater than 1 (indicating higher mortality for diabetics) and form a decreasing trend with increasing age.

## 统计代写|生物统计学作业代写Biostatistics代考|Odds and Odds Ratio

The relative risk, also called the risk ratio, is an important index in epidemiological studies because in such studies it is often useful to measure the increased risk (if any) of incurring a particular disease if a certain factor is present. In cohort studies such an index is obtained readily by observing the experience of groups of subjects with and without the factor, as shown above. In a casecontrol study the data do not present an immediate answer to this type of question, and we now consider how to obtain a useful shortcut solution.

Suppose that each subject in a large study, at a particular time, is classified as positive or negative according to some risk factor, and as having or not having a certain disease under investigation. For any such categorization the population may be enumerated in a $2 \times 2$ table (Table 1.11). The entries $A, B$, $C$ and $D$ in the table are sizes of the four combinations of disease presence/ absence and factor presence/absence, and the number $N$ at the lower right corner of the table is the total population size. The relative risk is
\begin{aligned} \mathrm{RR} &=\frac{A}{A+B} \div \frac{C}{C+D} \ &=\frac{A(C+D)}{C(A+B)} \end{aligned}
In many situations, the number of subjects classified as disease positive is small compared to the number classified as disease negative; that is,
\begin{aligned} &C+D \simeq D \ &A+B \simeq B \end{aligned}
and therefore the relative risk can be approximated as follows:
\begin{aligned} \mathrm{RR} & \simeq \frac{A D}{B C} \ &=\frac{A / B}{C / D} \ &=\frac{A / C}{B / D} \end{aligned}
where the slash denotes division. The resulting ratio, $A D / B C$, is an approximate relative risk, but it is often referred to as an odds ratio because:

1. $A / B$ and $C / D$ are the odds in favor of having disease from groups with or without the factor.
1. $A / C$ and $B / D$ are the odds in favor of having been exposed to the factors from groups with or without the disease. These two odds can easily be estimated using case-control data, by using sample frequencies. For example, the odds $A / C$ can be estimated by $a / c$, where $a$ is the number of exposed cases and $c$ the number of nonexposed cases in the sample of cases used in a case-control design.

For the many diseases that are rare, the terms relative risk and odds ratio are used interchangeably because of the above-mentioned approximation. Of course, it is totally acceptable to draw conclusions on an odds ratio without invoking this approximation for disease that is not rare. The relative risk is an important epidemiological index used to measure seriousness, or the magnitude of the harmful effect of suspected risk factors. For example, if we have
$$\mathrm{RR}=3.0$$
we can say that people exposed have a risk of contracting the disease that is approximately three times the risk of those unexposed. A perfect $1.0$ indicates no effect, and beneficial factors result in relative risk values which are smaller than 1.0. From data obtained by a case-control or retrospective study, it is impossible to calculate the relative risk that we want, but if it is reasonable to assume that the disease is rare (prevalence is less than $0.05$, say), we can calculate the odds ratio as a stepping stone and use it as an approximate relative risk (we use the notation $\simeq$ for this purpose). In these cases, we interpret the odds ratio calculated just as we would do with the relative risk.

## 统计代写|生物统计学作业代写Biostatistics代考|Generalized Odds for Ordered 2D k Tables

In this section we provide an interesting generalization of the concept of odds ratios to ordinal outcomes which is sometime used in biomedical research. Readers, especially beginners, may decide to skip it without loss of continuity; if so, corresponding exercises should be skipped accordingly: $1.24(\mathrm{~b}), 1.25(\mathrm{c})$, $1.26(\mathrm{~b}), 1.27(\mathrm{~b}, \mathrm{c}), 1.35(\mathrm{c}), 1.38(\mathrm{c})$, and $1.45(\mathrm{~b})$.

We can see this possible generalization by noting that an odds ratio can be interpreted as an odds for a different event. For example, consider again the same $2 \times 2$ table as used in Section 1.3.2 (Table 1.11). The number of casecontrol pairs with different exposure histories is $(A D+B C)$; among them, $A D$ pairs with an exposed case and $B C$ pairs with an exposed control. Therefore $A D / B C$, the odds ratio of Section 1.3.2, can be seen as the odds of finding a pair with an exposed case among discordant pairs (a discordant pair is a casecontrol pair with different exposure histories).

The interpretation above of the concept of an odds ratio as an odds can be generalized as follows. The aim here is to present an efficient method for use with ordered $2 \times k$ contingency tables, tables with two rows and $k$ columns having a certain natural ordering. The figure summarized is the generalized odds formulated from the concept of odds ratio. Let us first consider an example concerning the use of seat belts in automobiles. Each accident in this example is classified according to whether a seat belt was used and to the severity of injuries received: none, minor, major, or death (Table 1.13).

To compare the extent of injury from those who used seat belts with those who did not, we can calculate the percent of seat belt users in each injury group that decreases from level “none” to level “death,” and the results are:
None: $\quad \frac{75}{75+65}=54 \%$
Minor: $\quad \frac{160}{160+175}=48 \%$
Major: $\frac{100}{100+135}=43 \%$
Death: $\quad \frac{15}{15+25}=38 \%$
What we are seeing here is a trend or an association indicating that the lower the percentage of seat belt users, the more severe the injury.

We now present the concept of generalized odds, a special statistic specifically formulated to measure the strength of such a trend and will use the same example and another one to illustrate its use. In general, consider an ordered $2 \times k$ table with the frequencies shown in Table 1.14.

## 统计代写|生物统计学作业代写Biostatistics代考|Relative Risk

相对风险 = 群体发病率 1 群体发病率 2

## 统计代写|生物统计学作业代写Biostatistics代考|Odds and Odds Ratio

RR=一种一种+乙÷CC+D =一种(C+D)C(一种+乙)

C+D≃D 一种+乙≃乙

RR≃一种D乙C =一种/乙C/D =一种/C乙/D

1. 一种/乙和C/D是有或没有该因素的群体患疾病的几率。
2. 一种/C和乙/D是有利于暴露于患有或未患有疾病的群体的因素的几率。通过使用样本频率，可以使用病例对照数据轻松估计这两个几率。例如，赔率一种/C可以估计为一种/C， 在哪里一种是暴露病例的数量和C病例对照设计中使用的病例样本中未暴露病例的数量。

RR=3.0

## 广义线性模型代考

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