### 统计代写|生物统计学作业代写Biostatistics代考| PROBABILITY AND PROBABILITY MODELS

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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|PROBABILITY AND PROBABILITY MODELS

Most of Chapter 1 dealt with proportions. A proportion is defined to represent the relative size of the portion of a population with a certain (binary) characteristic. For example, disease prevalence is the proportion of a population with a disease. Similarly, we can talk about the proportion of positive reactors to a certain screening test, the proportion of males in colleges, and so on. A proportion is used as a descriptive measure for a target population with respect to a binary or dichotomous characteristic. It is a number between 0 and 1 (or $100 \%$; the larger the number, the larger the subpopulation with the chacteristic [e.g., $70 \%$ male means more males (than $50 \%$ )].

Now consider a population with certain binary characteristic. A random selection is defined as one in which each person has an equal chance of being selected. What is the chance that a person with the characteristic will be selected (e.g., the chance of selecting, say, a diseased person)? The answer depends on the size of the subpopulation to which he or she belongs (i.e., the proportion). The larger the proportion, the higher the chance (of such a person being selected). That chance is measured by the proportion, a number between 0 and 1, called the probability. Proportion measures size; it is a descriptive statistic. Probability measures chance. When we are concerned about the outcome (still uncertain at this stage) with a random selection, a proportion (static, no action) becomes a probability (action about to be taken). Think of this simple example about a box containing 100 marbles, 90 of them red and the other 10 blue. If the question is: “Are there red marbles in the box?”, someone who saw the box’s contents would answer ” $90 \%$ ” But if the question is: “If I take one marble at random, do you think I would have a red one?”, the answer would be ” $90 \%$ chance.” The first $90 \%$ represents a proportion; the second $90 \%$ indicates the probability. In addition, if we keep taking random selections (called repeated sampling), the accumulated long-term relative frequency with which an event occurs (i.e., characteristic to be observed) is equal to the proportion of the subpopulation with that characteristic. Because of this observation, proportion and probability are sometimes used interchangingly. In the following sections we deal with the concept of probability and some simple applications in making health decisions.

## 统计代写|生物统计学作业代写Biostatistics代考|Certainty of Uncertainty

Even science is uncertain. Scientists are sometimes wrong. They arrive at different conclusions in many different areas: the effects of a certain food ingredient or of low-level radioactivity, the role of fats in diets, and so on. Many studies are inconclusive. For example, for decades surgeons believed that a radical mastectomy was the only treatment for breast cancer. More recently, carefully designed clinical trials showed that less drastic treatments seem equally effective.

Why is it that science is not always certain? Nature is complex and full of unexplained biological variability. In addition, almost all methods of observation and experiment are imperfect. Observers are subject to human bias and error. Science is a continuing story; subjects vary; measurements fluctuate. Biomedical science, in particular, contains controversy and disagreement; with the best of intentions, biomedical data-medical histories, physical examinations, interpretations of clinical tests, descriptions of symptoms and diseasesare somewhat inexact. But most important of all, we always have to deal with incomplete information: It is either impossible, or too costly, or too time consuming, to study the entire population; we often have to rely on information gained from a sample – that is, a subgroup of the population under investigation. So some uncertainty almost always prevails. Science and scientists cope with uncertainty by using the concept of probability. By calculating probabilities, they are able to describe what has happened and predict what should happen in the future under similar conditions.

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

The target population of a specific research effort is the entire set of subjects at which the research is aimed. For example, in a screening for cancer in a community, the target population will consist of all persons in that community who are at risk for the disease. For one cancer site, the target population might be all women over the age of 35 ; for another site, all men over the age of 50 .

The probability of an event, such as a screening test being positive, in a target population is defined as the relative frequency (i.e., proportion) with which the event occurs in that target population. For example, the probability of

having a disease is the disease prevalence. For another example, suppose that out of $N=100,000$ persons of a certain target population, a total of 5500 are positive reactors to a certain screening test; then the probability of being positive, denoted by $\operatorname{Pr}($ positive), is
\begin{aligned} \operatorname{Pr}(\text { positive }) &=\frac{5500}{100,000} \ &=0.055 \text { or } 5.5 \% \end{aligned}
A probability is thus a descriptive measure for a target population with respect to a certain event of interest. It is a number between 0 and 1 (or zero and $100 \%$; the larger the number, the larger the subpopulation. For the case of continuous measurement, we have the probability of being within a certain interval. For example, the probability of a serum cholesterol level between 180 and $210(\mathrm{mg} / 100 \mathrm{~mL})$ is the proportion of people in a certain target population who have cholesterol levels falling between 180 and $210(\mathrm{mg} / 100 \mathrm{~mL})$. This is measured, in the context of the histogram of Chapter 2 , by the area of a rectangular bar for the class $(180-210)$. Now of critical importance in the interpretation of probability is the concept of random sampling so as to associate the concept of probability with uncertainty and chance.

Let the size of the target population be $N$ (usually, a very large number), a sample is any subset-say, $n$ in number $(n<N)$ – of the target population. Simple random sampling from the target population is sampling so that every possible sample of size $n$ has an equal chance of selection. For simple random sampling:

1. Each individual draw is uncertain with respect to any event or characteristic under investigation (e.g., having a disease), but
2. In repeated sampling from the population, the accumulated long-run relative frequency with which the event occurs is the population relative frequency of the event.

The physical process of random sampling can be carried out as follows (or in a fashion logically equivalent to the following steps).

1. A list of all $N$ subjects in the population is obtained. Such a list is termed a frame of the population. The subjects are thus available to an arbitrary numbering (e.g., from 000 to $N=999$ ). The frame is often based on a directory (telephone, city, etc.) or on hospital records.
2. A tag is prepared for each subject carrying a number $1,2, \ldots, N$.
3. The tags are placed in a receptacle (e.g., a box) and mixed thoroughly.
4. A tag is drawn blindly. The number on the tag then identifies the subject from the population; this subject becomes a member of the sample.

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

1. 对于正在调查的任何事件或特征（例如，患有疾病），每次抽签都不确定，但是
2. 在从总体中重复抽样中，事件发生的累积长期相对频率就是事件的总体相对频率。

1. 所有的清单ñ获得人口中的受试者。这样的列表被称为总体框架。因此，主题可用于任意编号（例如，从 000 到ñ=999）。该框架通常基于目录（电话、城市等）或医院记录。
2. 为每个带有编号的受试者准备一个标签1,2,…,ñ.
3. 将标签放置在容器（例如盒子）中并彻底混合。
4. 盲目地绘制一个标签。然后标签上的数字从人群中识别出受试者；该主题成为样本的成员。

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## MATLAB代写

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