## 统计代写|实验设计作业代写experimental design代考|ASSESSING THE TREATMENT MEANS

statistics-lab™ 为您的留学生涯保驾护航 在代写实验设计experimental designatistical Modelling方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写实验设计experimental design代写方面经验极为丰富，各种代写实验设计experimental design相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等楖率论
• 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

## 统计代写|实验设计作业代写experimental design代考|ASSESSING THE TREATMENT MEANS

In the last chapter we desoribed the linear model for a simple experiment, found estimates of the parameters, and described a test for the hypothesis that there $1 s$ no overall treatment effect. In this chapter we cover the next step of examining more closely the pattern of differences among the treatment means. There are a number of approaches, One extreme is to test only the hypotheses framed before the experiment was carried out, but this approach wastes much of the information from the experiment. On the other hand, to carry out conventional hypothesis tests on every effect that looks interesting can be very misleading, for reasons which we now examine.
There are three main difflculties. First, two tests based on the same experiment are unlikely to be independent. Tests will usually involve the same estimate of variance, and if this estimate happens to be too small, every test will be too significant. Further, any comparisons involving the same means will be affected the same way by chance differences between estimate and parameter. As an example consider a case where there are three treatments assessing a new drug. Treatment “A” is the placebo, treatment ” $B^{H}$ the drug administered in one big dose and treatment “C” the orug administered in two half doses. If chance variation happens to make the number of cures on the experimental units using the placebo (A) rather low, the differences between $A$ and $B$, and $A$ and $C$ will both be overstated in the same way. Therefore two significant t-tests, one between $A$ and $B$, the other between $A$ and $C$, cannot be taken as independent corroboration of the effectiveness of the drug.

## 统计代写|实验设计作业代写experimental design代考|SPECIFIC HYPOTHESES

Any experiment should be designed to answer specif ic questions. If these questions are stated clearly it will be possible to construct a single linear function of the treatment means which answers each question. It can be a difficult for the statistician to discover what these questions are, but this type of problem is beyond the scope of this book. We will present some examples.
Example 6.2+1 Drug comparison example
In the drug comparison experiment mentioned in Section 1 one question might be, is the drug effective? Rather than doing two separate

tests ( $A \vee B$ and A $\vee$ C), a single test of $A$ against the average of $B$ and $C$ gives an unambiguous answer which uses all the relevant data. That is use
$\bar{y}{A}-\left(\bar{y}{B}+\bar{y}{C}\right) / 2$ with variance $\left[1 / r{A}+\left(1 / r_{B}+1 / r_{B}\right) / 4\right] \sigma^{2}$
Having decided that the drug has an effect the next question may be, how much better is two half doses than one complete dose. This will be estimated by the difference between treatment means for $B$ and C. For inferences remember that $\sigma^{2}$ is estimated by $s^{2}$, and this appears with its degrees of freedom in the ANOVA table.

## 统计代写|实验设计作业代写experimental design代考|Exper imentwise Error Rate

The above are examples of inferences to answer specific questions. Each individual inference will be correot, but there are several inferences being made on each experiment. If all four suggested comparisons were made on the fertilizer experiment, the probability of making at least one type I error will be much higher than the signif icance level of an individual test. If the traditional $5 \%$ level is used, and there are no treatment effeots at all, and the individual tests were independent, the number of significant results from the experiment would be a binomial random variable with $n=4$ and $p=.05$. The probability of no significant results will be $(1-0.05)^{4}$, so that the probability of at least one will be $1-(1-0.05)^{4}=$ 0.185. If one really wanted to have the error rate per experiment equal to $0.05$ each individual test would have to use a significance level, $p$, satisfying
\begin{aligned} 1-(1-p)^{4} &=0.05 \ \text { or } & p=0.013 \end{aligned}

Unfortunately, the underlying assumptions are false because, as we noted in Section 1, each inference is not independent. The correlation between test statistics will usually be positive because each depends on the same variance estimate, and so the probability of all four being nonsignificant w1ll be greater than that calculated above and so the value of p given above will be too low. If the error rate per experiment is important, the above procedure at least provides a lower bound. Usually though it is suffleient to be suspicious of experiments producing many significant results, particularly if the var fance estimate $1 s$ based on rather few degrees of freedom and is smaller than is usually found in similar experiments. Experimenters should not necessarily be congratulated on obtaining many significant results.

In section 1, another source of dependence was mentioned. This results from the same treatment means being used in different conparisons. If the questions being asked are themselves not independent, the inferences cannot be either. However, it is possible to design a treatment structure so that independent questions can be assessed independently. This will be the topic of the next section.

## 统计代写|实验设计作业代写experimental design代考|Exper imentwise Error Rate

1−(1−p)4=0.05  或者 p=0.013

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 统计代写|实验设计作业代写experimental design代考|THE EXPERIMENTAL DESIGN MODEL

statistics-lab™ 为您的留学生涯保驾护航 在代写实验设计experimental designatistical Modelling方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写实验设计experimental design代写方面经验极为丰富，各种代写实验设计experimental design相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等楖率论
• 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

## 统计代写|实验设计作业代写experimental design代考|THE EXPERIMENTAL DESIGN MODEL

Back in Chapter 1, Section 1, it was explained that the aim of all the work so far is to fit models to a population using data from a sample. The model so fitted shows what values of $y$ are 1 kely to be associated with any given values of the $x$ ‘s. Just as any elementary statistics text will warn that a correlation does not imply causation, so a well fltting regression model does not imply that changes in the $x$ values will cause changes in the y variable in’ accordance the with model. If the model has been constructed by passive observation of sets of $y^{\prime} s$ and $x^{\prime} s$, any intervention to change the $x$ values to values which do not occur naturally will change the population to be something different from what the model describes, and so the model no longer applies.
In an experiment a population is not passively observed, but the $x$ values are the result of intervention by the experimenter. The aim then is to intervene to choose $x$ values in such a way as to give accurate estimates of the model parameters with a minimum of effort, and to ensure that the relationships described by the model really do describe the way changes in $y$ are caused by changes in $x$.

## 统计代写|实验设计作业代写experimental design代考|WHAT MAKES AN EXPERIMENT?

One of the 1 argest experiments ever conducted makes an interesting example, partly because it was almost a disaster. The experiment was designed to test the effectiveness of the Salk polio vacoine, and was conducted in the United States in 1954 . A good account of the experiment is in Freedman, Pisani, and Purves, 1978 . There were two difficulties with the experiment. First, polio is a relatively uncommon disease, The vacoine can only be tested if 1 t is tried on a group large enough to contain a reasonable number of polio victims. For the common cold a fer hundred people would be plent y, but for pol1o several hundred thousand were required. Second, polio is a disease which comes and goes from one season to the next. With the common cold, which most people catch most years, if the incidence drops in a test group after administering a vaccine, the drop can probably be attributed to the vacoine. But not so with polio = 1 t may well have been a year in which the disease would have waned anyway.
The original experimental plan for the polio vaccine was to vaccinate second year school children and compare their incidence of polio with that of first and third year children. The intention is clear. The children being compared should be as alike as possible, and this plan would ensure that the geographical area and the time period would be the same for both groups. Of course they would be of different ages, but since the ages of the untreated group brackets that of the treated group one might not expect this to matter too much. A large number of experts must have considered that these remaining differences would not matter because this experiment was put into effect. 221,998 children were vaceinated, 725, 173 here not, and the rate of polio following vaccination was about twice as great $1 n$ the unvaceinated group as in the vaccinated group, $(54$ per 100,000 instead of 25 ).

## 统计代写|实验设计作业代写experimental design代考|Experimental Unit

We will define an experimental unit as that object or group of objects to which a treatment has been randomly assigned. Some examples will make this definftion clear. An educationalist may be comparing teaching methods. Each child is given a test after being taught by the method assigned to $1 \mathrm{t}$, so there is one observation per child. However the child is only an experimental unit if each child were individually assigned to a teaching method. If the children were first divided into small groups, and groups were assigned randomly to teaching method, then the group would become an experimental unft and the observation on it would be the average of all the observations on the individual children. If the two classes were used, each randomly assigned to a teaching method, a class would be an experimental unit, and because there would be no replication, no assessment would be possible. In an experiment comparing weedicides, various combinations of weedicides are randomly assigned to $5 \mathrm{~m} \times 2 \mathrm{~m}$ areas. Within each of these areas $f$ ive $+25 \mathrm{~m} x+25 \mathrm{~m}$ quadrats,

positioned at random, are examined for percent weed cover. This is a very common type of biological experiment, and one where our definition of experimental units is important. The weedicides were assigned to the $5 \mathrm{~m} \times 2 \mathrm{~m}$ areas, and so these are the experimental units, not the $.25 \mathrm{~m} \times .25 \mathrm{~m}$ quadrats.
As a very general and 1 mportant principle, remember that any treatment comparison must be based on one number per experimental unit. This number might often be a mean of several observations, but one must beware of pretending that by increasing the number of subsamples one can increase the replication. It is experimental units which must be replicated.

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 统计代写|实验设计作业代写experimental design代考| AGGREGATION OF DATA

statistics-lab™ 为您的留学生涯保驾护航 在代写实验设计experimental designatistical Modelling方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写实验设计experimental design代写方面经验极为丰富，各种代写实验设计experimental design相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等楖率论
• 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

## 统计代写|实验设计作业代写experimental design代考|AGGREGATION OF DATA

The dummy variable approach of the previous section provides one method of combining together different sets of data. In other situations $1 \mathrm{t}$ may be appropriate to merely sum or average data sets, but it is wise to tread warily for the pattern followed by an individual data set may be quite different from that of the aggregate of the data sets. Aggregation sometimes gives rather disturbing and nonsensical results. Consider Figure $3.9 .1$ in which there is a slight overall downwards trend in the $y$ values as the $x$ values inorease. If there are, as shown, three identifiable subgroups in the data, they may, in fact, suggest a positive trend within each group. In this example, it would be spurious to aggregate the groups. One overall model could be used provided that dummy variables were included to distinguish the $y$-intercepts of each group.
For the lactation records of Appendix $C 3$, it would be useful to fit an overall model to the aggregated data of the five cows. One approach may be to fit a model to each cow’s yield and then to average the estimated coefficients. Problems remain, however, for the model with these averaged coefficients may not fit well any of the individual data sets. Another approach is to average the sets of values of the dependent variable. With different patterns of ylelds, one must ask whether this aggregation is a reasonable thing to do. For example, the maximum yield for cow no, 1 occurred in the sixth week, but for cow no. 5 it occurred in the ninth week. Perhaps the data for each cow should be lagged so that the maxima correspond, and then the appropriate milk yields added (that is $16.30$ of cow no. 1 added to $31.27$ of cow no. 2 etc.). This would ensure that the maxima correspond, but it does not take into account the different shapes of the graphs or that one cow may give milk for a longer time than another. (For convenience, the lactation records here were all truncated to 38 weeks even though some actually gave milk for a longer period).

## 统计代写|实验设计作业代写experimental design代考|PECULIARITIES OF OBSERVATIONS

In Chapter 3 we considered the relationship between variables $\mathbf{y}$ and $X=\left(x_{1}, x_{2}, \ldots, x_{k}\right)$, that $1 s$, the relationship between the column vectors. In this chapter, we turn our attention to the rows or individual data points
$$\left(x_{11}, x_{2 i}, \ldots, x_{k i}, y_{1}\right)$$
We have already seen that the variances of the predicted values and of the residuals depend on the particular values of the predictor variables, $x$. Peculiar values of the $x^{\prime} s$ could be termed sensitive, or high leverage, points as will be explained in Section 2. On the other hand, the observed value of $y$ may be unusual for a given set of $x$ values and y may then be termed an outlier as explained in Section $3 .$
Also, in Sections 6 through 8 , the emphasis is again on the variables in the model and perhaps these topios, more logically, should fall into Chapter 3. They have been added for completeness as they are topics often referred to in other texts.

## 统计代写|实验设计作业代写experimental design代考|OUTLIERS

Both the prediotor and dependent variables will have their parts to play $1 n$ deciding whether an observation $1 s$ unusual. The predictor variables determine whether a point has hfgh leverage. The value of the dependent variable, $y$, for a given set of $x$ values will determine whether the point is an outlier.

Finst, we consider the residuals, or preferably the studentized residuals. As explained in section 1.3, it is good practice to plot the residuals against the predioted values of $y$ and the predictor variables which are already in the model or which are being considered for inelusion $1 n$ the model. A studentized residual of large absolute value may suggest that an error of measurement or of coding or some such has occurred in the response variable. If the sample size is reasonably large, the observation could be deleted from the analys1s. It is always worthwhile, though, to consider such outliers

very carefully for they may suggest conditions under whioh the model is not valid.

It should be noted that the size of the residuals depends on the model which is fitted. If more, or different, predictor variables are included in the model then it is likely that different points will show up as being potential outliers. It is not possible, then, to completely divorce the detection of outliers from the search for the best model.

Outliers may also be obscured by the presence of points of high leverage for these tend to constrain the prediction curve to pass close to their associated y values. These interrelated effects should warn us to tread cautiously as there is no guaranteed failsafe approach to the problem. Many solutions have been suggested and the interested reader may consult Hoaglin and Welsch (1978). We shall not consider the tests in detail which are contained in this article. To decide whether the i-th observation is an outlier, a fruitful approach is to see the effect that would result from the omission of the 1 -th row of the data. In particular, how would this omission affect the residual at the point and how would it affect the slope of the prediction line?

## 统计代写|实验设计作业代写experimental design代考|PECULIARITIES OF OBSERVATIONS

(X11,X2一世,…,Xķ一世,是的1)

## 统计代写|实验设计作业代写experimental design代考|OUTLIERS

prediotor 和因变量都将发挥作用1n决定是否观察1s异常。预测变量确定一个点是否具有 hfgh 杠杆。因变量的值，是的, 对于给定的一组X值将确定该点是否为异常值。

Finst，我们考虑残差，或者最好是学生化的残差。如第 1.3 节所述，将残差与是的以及已经在模型中或正在考虑排除的预测变量1n该模型。绝对值较大的学生化残差可能表明响应变量中出现了测量或编码错误等。如果样本量相当大，则可以从分析中删除观察值。不过，考虑这些异常值总是值得的

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 统计代写|实验设计作业代写experimental design代考| Forward Selection

statistics-lab™ 为您的留学生涯保驾护航 在代写实验设计experimental designatistical Modelling方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写实验设计experimental design代写方面经验极为丰富，各种代写实验设计experimental design相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等楖率论
• 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

## 统计代写|实验设计作业代写experimental design代考|Forward Selection

We shall use the heart data of the last two sections to illustrate this. In section 3.5, this data is written in correlation form. If the model is to include only one predictor variable, then $B$ would be chosen as it gives the highest SSR which is also the correlation coefficient with $y$. Before B is placed in the model, we test that it has a significant effect on $y$ by using an $F$-test, or equivalently, a t-test.
We test $\mathrm{H}: B_{2}=0$ in the model $y=B_{0}+B_{2} x_{2}+\varepsilon$
$$F=(S S R / 1) /(\mathrm{SSE} / 44)=0.657 /(0.342 / 44)=84.7$$
Clearly, we reject $H$ and include $x_{2}$ in the model. We now try to add

another predictor variable to the model. We look for the variable which, with B, gives the nighest value of SSR. From Table 3.6.1, we see that SSR for $B$ and $C=0.715$ which $1 s$ greater than for $A B=$ $0.667, \mathrm{BD}=0.686, \mathrm{FB}=0.676$, and $\mathrm{BE}=0.659$. Does $\mathrm{C}$ add significantly to SSR over and above B itself? We use the method of reduced models to determine this.
Full model: $y=\beta_{0}+\beta_{2} x_{2}+B_{3} x_{3}+\varepsilon ; \quad$ SSE $=0.285$
Reduced model: $y=\beta_{0}+\beta_{2} x_{2}+\varepsilon ; \quad$ SSE $=0.342$
Difference $=0.05 ?$
$F=0.057 /(0.285 / 43)=8.6$
The tabulated $F$ at $5 \%$ level with 1,43 degrees of freedom = $4.190$ that we reject the reduced model in favor of the full model. We then attempt to add a third variable to the model. The vari= able which adds most to SSR in association with $B$ and $C$ is D as BCD gives an $\mathrm{SSR}=0.718 .$ An $\mathrm{F}$ statistic is evaluated to determine whether $D$ adds significantly to SSR over and above $B$ and $C$.

Ful1 mode1: $y=\beta_{0}+\beta_{2} x_{2}+B_{3} x_{3}+B_{4} x_{4}+\varepsilon ; S S E=0.282$
Reduced model: $y=\beta_{0}+\beta_{2} x_{2}+\beta_{3} x_{3}+E ; \quad$ SSE $=0,285$
Difference $=0.003$
$$F=0.003 /(0.282 / 42)=0.4$$
Clearly this is too small to reject the reduced model and we select as the optimal model that with B and $C$ as predictor variables.

## 统计代写|实验设计作业代写experimental design代考|Backward Elimination

Another approach is to commence with the full model of six prediotor variables and to attempt to remove variables sequentially. In Table 3.6.1, the five predictor model with the greatest SSR Is BCDEF $0.749$ or SSE $=0.251$. To declde if A should be removed, we conpare this SSE with that of the full model using the F statistio.
$$F=(0.251-0.247) /(0.247 / 39)=0.004 /(0.247 / 39)=0.6$$

Clearly, the effect of $A$ is not significant and can be removed. We look to remove one of these remaining five variables by considering the SSR for each of the four predictor models. These are:
$\mathrm{BCDE}=0.719, \quad \mathrm{CDEF}=0.703, \quad \mathrm{BDEF}=0.712, \quad \mathrm{FBCD}=0.721$
and $\mathrm{BEFC}=0.741$
We choose this last one with $D$ omitted and test whether this causes a significant reduction in SSR. The full model is now BCDEF and the reduced model is BCEF.
$$F=0.008 /(0.251 / 40)=1.3$$
This value is low compared with the $5 \%$ tabulated value for 1 and 40 degrees of freedom which equals $4.08$ so that we proceed to eliminate a further variable. The three variable sums of squares are
$$C E F=0.684, \quad F B C=0.717, \quad B E F=0.704, \quad E B C=0.717$$
For either of the models with SSR $=0.717$,
$$E=0.024 /(0.259 / 41)=3.8$$
This is slightly below the oritical value of $4.08$ so that we proceed and compare these two models, FBC and $\mathrm{EBC}$, with $\mathrm{BC}$ the last subset of these with two variables.
$$F=0.002 /(0.283 / 42)=0.3$$
We are then reduced to the model BC as in the forwand selection process.
There are a number of points to notice about these sequential methods.

## 统计代写|实验设计作业代写experimental design代考|QUALITATIVE (DUMMY) VARIABLES

It is of ten useful to introduce variables into a model to enable certain specifio effects to be revealed and tested. Usually these take the form of qualitative variables which show up the differences

between subgroups in the data. We shall use an example to explore these ideas.

In Example 1.5.1, we listed the value of an Australian stamp (1963 twopenny sepia coloured in the years 1972-1980). We could compare this with the listed value of another stamp, and for few obvious reasons, we have chosen the 1867 New Zealand fourpenny rose colored full face queen. We shall use the same transformation as before, namely
$$y_{1}=\ln v_{t}, y_{2}=\ln v_{t}$$
for the Australian and New zealand stamp respectively. The data is given in Table 3.8.1. We could $f$ it a separate model to each stamp, that is, for the Australian stamp
$$y_{1}=a_{1} 1+\alpha_{2} t_{1}+\varepsilon_{1}$$
and the New Zealand stamp
$$\boldsymbol{y}{2}=B{1} \mathbf{1}+B_{2} \mathrm{t}{2}+\varepsilon{2}$$
If the distributions of the deviations can be assumed to be the same, it will be adrisable to join these models into a single model.

## 统计代写|实验设计作业代写experimental design代考|Forward Selection

F=(小号小号R/1)/(小号小号和/44)=0.657/(0.342/44)=84.7

F=0.057/(0.285/43)=8.6

F=0.003/(0.282/42)=0.4

## 统计代写|实验设计作业代写experimental design代考|Backward Elimination

F=(0.251−0.247)/(0.247/39)=0.004/(0.247/39)=0.6

F=0.008/(0.251/40)=1.3

C和F=0.684,F乙C=0.717,乙和F=0.704,和乙C=0.717

F=0.002/(0.283/42)=0.3

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 统计代写|实验设计作业代写experimental design代考|CORRELATION FORM

statistics-lab™ 为您的留学生涯保驾护航 在代写实验设计experimental designatistical Modelling方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写实验设计experimental design代写方面经验极为丰富，各种代写实验设计experimental design相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等楖率论
• 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

## 统计代写|实验设计作业代写experimental design代考|CORRELATION FORM

When the main concern is to decide which variables to include in the model, a very useful transformation of the data is to scale each variable, predictors and dependent variables alike, so that the normal equations can be written in correlation form. This enables us to identify important variables which should be included in the model and it also reveals some of the dependenoles between the predictor variables.

As usual, we consider the variables to be in deviation form. The correlation coefficient between $x_{1}$ and $x_{2}$ is
$$\left.\left.r_{12}=s_{12} / \sqrt{\left(s_{11}\right.} s_{22}\right)=\sum x_{1} x_{2} / \sqrt{\left(s_{11}\right.} s_{22}\right)$$
If we divide each variable $x_{1}$ by $\sqrt{S}{11}$ and denote the result as $$x{1}^{}=x_{1} / \sqrt{s}{1 i}$$ then $x{i}^{}$ is said to be in correlation form. Notice that
$$\Sigma x_{i}^{}=0$$ $$\Sigma\left(x_{i}^{}\right)^{2}=1$$
$$\Sigma x_{i}^{} x_{j}^{}=r_{1 j}$$
We have transformed the model from

$$y=B_{1} x_{1}+B_{2} x_{2}+\varepsilon \text { to } y^{}=\alpha_{1} x_{1}^{}+\alpha_{2} x_{2}^{*}+\varepsilon$$
and the normal equations simplify from
\begin{aligned} &s_{11} b_{1}+s_{12} b_{2}=s_{y 1} \ &s_{12} b_{1}+s_{22} b_{2}=s_{y 2} \end{aligned} \text { to } \quad r_{12} a_{1}+r_{12}+a_{2}=r_{y 1}=r_{y 2} \quad \text { (3.5.3) }

## 统计代写|实验设计作业代写experimental design代考|VARIABLE SELECTION – ALL POSSIBLE REGRESSIONS

In many situations, researchers know which variables may be included in the predictor model. There is some advantage in reducing the number of predictor variables to form a more parsimonious model. One way to achieve this is to run all possible regressions and to consider such statistics as the coefficient of determination, $R^{2}=$ SSR/SST.
We will use the heart data of Section 3.5, again relabelling the variables as A through $F$. With the variables in correlation form, $R^{2}=S S R$, the sum of squares for regression, and this is given for each possible combination of predictor variables in Table $3.6 .1$.

To assist the choice of the best subset, C.L. Mallows suggested fitting all possible models and evaluating the statistic
$$C_{p}=S S E_{p} / s^{2}-(n-2 p)$$
Here, $n$ is the number of observations and $p$ is the number of predictor variables in the subset, including a constant term. For each subset, the value of Mallows’ statistio can be evaluated from the correponding value of SSR. The complete set of these statistics are listed in Table 3.6.2. For each subset we use the mean squared error, MSE, of the full model as an estimate of the variance.

Suppose that the true model has q predictor variables.

## 统计代写|实验设计作业代写experimental design代考|VARIABLE SELECTION – SEQUENTIAL METHODS

When the number of possible variables in a model is large, it may be inappropriate to run every possible regression and evaluate Mallows’ statistic for each one, even though short cuts can be taken to evaluate such statistios by adding or subtracting terms rather than by evaluating each one from scratch.

Another approach is to add, or remove, variables, sequentially. We have seen that adding a variable will increase SSR, the sum of squares for regression. From Section $3.4$ we could perform an F-test to decide if the increase in SSR is si gnificant. The first method we consider is that of forward selection.

## 统计代写|实验设计作业代写experimental design代考|CORRELATION FORM

r12=s12/(s11s22)=∑X1X2/(s11s22)

ΣX一世=0Σ(X一世)2=1
ΣX一世Xj=r1j

$$y=B_{1} x_{1}+B_{2} x_{2}+\varepsilon \text { to } y^{ }=\alpha_{1} x_{1}^{ }+\alpha_{ 2} x_{2}^{*}+\varepsilon 一种nd吨H和n这r米一种l和q在一种吨一世这nss一世米pl一世F是的Fr这米 s11b1+s12b2=s是的1 s12b1+s22b2=s是的2\text { to } \quad r_{12} a_{1}+r_{12}+a_{2}=r_{y 1}=r_{y 2} \quad \text { (3.5.3) }$$

## 统计代写|实验设计作业代写experimental design代考|VARIABLE SELECTION – ALL POSSIBLE REGRESSIONS

Cp=小号小号和p/s2−(n−2p)

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 统计代写|实验设计作业代写experimental design代考|WHICH VARIABLES SHOULD BE INCLUDED IN THE MODEL

statistics-lab™ 为您的留学生涯保驾护航 在代写实验设计experimental designatistical Modelling方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写实验设计experimental design代写方面经验极为丰富，各种代写实验设计experimental design相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等楖率论
• 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

## 统计代写|实验设计作业代写experimental design代考|INTRODUCTION

When a model can be formed by including some, or all, of the predictor variables, there is a problem in deciding how many variables to include. The decision we arrive at will depend to some extent on the purpose we have in mind. If we merely wish to explain the variation of the dependent variable in the sample, then $1 \mathrm{t}$ would seem obvious that as many predictor variables as possible should be included. This can be seen with the lactation curve of Example $2.11$. If enough powers of $w$ were added to the model the curve would pass through every observed value, but it would be so jagged and complicated it would be difficult to understand what was happening. On the other hand, a small model has the advantage that it is easy to understand the relationships between the variables. Further more, a small model will usually yield estimators which are less influenced by peculiarites of the sample and so are more stable. Another important decision which must be made is whether to use the original predictor variables or to transform them in some way, often by taking a linear combination. For example, the cost of a particular kind of fencing for a rectangular field may largely depend on the length and breadth of the field. If all the fields in the

sample are in the same proportions then only one variable (length or breadth) would be needed. Even if they are not in the same proportions, one variable may be sufficient, namely the sum of the length and the breadth or, indeed, the perimeter. This is our ideal solution, reducing the number of predictor variables from two to one and at the same time obtaining a predictor variable which has physi= cal meaning, With a particular data set, the predicted value of the cost may be $y=1.11+0.9$ b so that the best single variable would be the $r$ ight hand side with $1=l$ ength and $b=b r e a d t h$, but this particular linear combination would have no physical meaning. We need to keep both aspects in mind, balancing statistioal optimum against physical meaning.
In the first section we shall 11 mit our discussion to orthogonal predictor variables, Although this may seem an unnecessarily strong restriction to place on the model, orthogonal variables of ten exist in experimental design situations. Indeed the values of the variables in the sample are often deliberately chosen to be orthogonal. We explain the advantages of this in section $3.2$, while in section $3.4$ we show that 1 t is possible to transform variables, for any data set, so that they are orthogonal.

## 统计代写|实验设计作业代写experimental design代考|ORTHOGONAL PREDICTOR VARIABLES

If the variables in a model are expressed as deviations from their means and if there are $k$ predictor variables, the sum of squares for regression is given by
\begin{aligned} \mathrm{SSR} &=b_{1} s_{y 1}+b_{2} s_{y 2}+\cdots+b_{k} s_{y k} \ &=s_{y 1}^{2} / s_{11}+s_{y 2}^{2} / s_{22}+\cdots+s_{y k}^{2} / s_{k k} \end{aligned}
The total sum of squares is
$$\text { SST }-s_{y y}=\sum y_{i}^{2}$$
By subtraction, we find the sum of squares for error (residual) is

$$\mathrm{SSE}=S S T-S S R$$
In this seotion, we assume that the predicton variables are orthog= onal and explore the implications of the number of variables included in the model.

We consider now the effect of adding another variable, $x_{k+1}$, to the model and assume that this variable $1 s$ also orthogonal to the other predfotor variables. The SST will not be affected by adding $x_{k+1}$ to the model. We introduce the notation that SSR(k) is the sum of squares for negression when the variables $x_{1}, x_{2}, \cdots x_{k}$ are in the model. It is clear that
(i) $\operatorname{SSR}(k+1) \geq \operatorname{SSR}(k)$
This follows from $(3.2 .1)$ as each term in the sum cannot be negative so that adding a further variable cannot decrease the sum of squares for regression.
(ii) $\operatorname{SSE}(k+1) \leq \operatorname{SSE}(k)$
This is the other side of the coin and follows from $(3.2 .2)$.
(111)
$$R(k+1)^{2}=\operatorname{SSR}(k+1) / \operatorname{SST} \geq \mathrm{R}(k)^{2}=\operatorname{SSR}(k) / \mathrm{SST}$$
SSR $(k+1)$ can be thought of as the amount of variation in $y$ explained by the $(k+1)$ predictor variables, and $R(k+1)^{2}$ is the proportion of the variation in y explained by these variables. These monotone properties are illustrated by the diagrams in figure $3.2 .1$.

## 统计代写|实验设计作业代写experimental design代考|ADDING NONORTHOGONAL VARIABLES SEQUENTIALLY

Although orthogonal predictor variables are the ideal, they will rarely occur in practice with observational data. If some of the predictor variables are highly correlated, the matrix $X \mathrm{~T} X$ will be nearly singular. This could raise statistical and numerical problems, particularly if there is interest in estimating the coefficients of the model. We nave more to say on this in the next section and in a later section on Ridge Estimators.

Moderate correlations between predictor variables will cause few problems. While it is not essential to convert predictor variables to others which are orthogonal, it is instruotive to do so as it gives insight into the meaning of the coefficients and the tests of significance based on them.

In Problem 1.5, we considered predicting the outcome of a student in the mathematics paper 303 (which we denoted by y) by marks

recelved in the papers 201 and 203 (denoted by $x_{1}$ and $x_{2}$, respectively). The actual numbers of these papers are not relevant, but, for interest sake, the paper 201 was a calculus paper and 203 an al gebra paper, both at second year university level and 303 was a third year paper in algebra. The sum of squares for regression when $y$ is regressed singly and together on the $x$ variables (and the $F^{2}$ values) are:
$\begin{array}{lll}\text { SSR on } 201 \text { alone : } & 1433.6 & (.405) \ \text { SSR on } 203 \text { alone : } & 2129.2 & (.602) \ \text { SSR on } 201 \text { and } 203: & 2265.6 & (.641)\end{array}$
Clearly, the two $x$ variables are not orthogonal (and, in fact, the correlation coefficient between them is 0.622) as the individual sums of squares for regression do not add to that given by the model with both variables included. Once we have regressed the 303 marks on the 201 marks, the additional sum of squares due to $2031 \mathrm{~s}$ (2265.6 $1433.6)=832$. In this section we show how to adjust one variable for another so that they are orthogonal, and, as a consequence, their sums of squares for regression add to that given by the model with both variables included.
$\begin{array}{ll}\text { SSR for } 201 & =1433.6=\text { SSR for } x_{1} \ \text { SSR for } 203 \text { adjusted for } 201=832.0=\text { SSR for } z_{2} \ \text { SSR for } 201 \text { and } 203 & =2265.6\end{array}$

## 统计代写|实验设计作业代写experimental design代考|ORTHOGONAL PREDICTOR VARIABLES

SST −s是的是的=∑是的一世2

（一）固态硬盘⁡(ķ+1)≥固态硬盘⁡(ķ)

(二)上证所⁡(ķ+1)≤上证所⁡(ķ)

(111)
R(ķ+1)2=固态硬盘⁡(ķ+1)/SST≥R(ķ)2=固态硬盘⁡(ķ)/小号小号吨

## 统计代写|实验设计作业代写experimental design代考|ADDING NONORTHOGONAL VARIABLES SEQUENTIALLY

SSR 开启 201 独自的 ： 1433.6(.405)  SSR 开启 203 独自的 ： 2129.2(.602)  SSR 开启 201 和 203:2265.6(.641)

SSR 为 201=1433.6= SSR 为 X1  SSR 为 203 调整为 201=832.0= SSR 为 和2  SSR 为 201 和 203=2265.6

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 统计代写|实验设计作业代写experimental design代考|Adjustment for Degrees of Freedom

statistics-lab™ 为您的留学生涯保驾护航 在代写实验设计experimental designatistical Modelling方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写实验设计experimental design代写方面经验极为丰富，各种代写实验设计experimental design相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等楖率论
• 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

## 统计代写|实验设计作业代写experimental design代考|Adjustment for Degrees of Freedom

The coefficient $R^{2}$ measures the proportion of the variation in $y$ which is explained by the predictor variables. Actually, it overestimates this proportion and the adjustment suggested here aims to correct this.

If the $y$ values were entirely random in the space of $n d 1 m e n-$ sions (the deviations from their mean would then be in an $n-1$ dimensioned sub-space) the $k$ predictor variables would still explain some variation in $y$, on average $k /(n-1) . R^{2}$ is corrected for this random effect by subtracting $k /(n-1)$. This is then scaled to give a value of 1 (perfect explanation of $y$ ) when $R^{2}=1$. Finally
$\operatorname{adj} R^{2}=\left[R^{2}-k /(n-1)\right][n-1] /[n-k-1]$
This adjusted value could even be negative if $\mathrm{R}^{2}$ is small, which highlights the only problem with it. What would a negative value mean? On the other hand the unadjusted value has a clear interpretation as the proportion of the variance of $y$ explained by the predictor variable.

## 统计代写|实验设计作业代写experimental design代考|The Univariate Case

Consider again the simple model
$$y=\alpha 1+\beta x+E$$
with $x$ in deviation form and $\varepsilon=N\left(0, \sigma^{2} I\right)$. Notice that 1 is orthogonal to $x$ so that the least squares estimates of $\alpha$ and $\beta$ are the same as would be obtained by regressing $y$ separately on 1 and $x$.

$$a=\Sigma y_{1} / n, b=\Sigma x_{1} y_{1} / \Sigma x_{1}^{2}$$
For a given value of $x=x_{0}$, the predicted value of $y$ is
$$\hat{y}{0}=a+b x{0}$$
A number of points should be kept in mind relating to this expression.
(i) The $x_{0}$ need not be one of the $x$ in the sample, but there are obvious dangers in predicting outside of the range of the sample. For one thing, the relationships between $x$ and $y$ may be linear for the $x$ values of the sample but the relationship may change outside of these values, as in Figure $2.7 .1$. One example of this could be the cancer causing effects of low doses of radiation. The incidence of cancer in this situation would be so low that it would be difficult to measure, requiring a very large sample size. To facilitate the research one could work at higher dosage rates $(x)$ and hope that one could extrapolate down to lower dosage rates, but this procedure is fraught with danger.
(ii) The predicted value of $y$ depends on all the $y$ values in the sample. We shall comment further on this in Chapter 4 where we shall discuss sensitive or high leverage values of $x$, by which we mean those $x^{\prime} s$ which have a very large effect on the predicted values of $\mathrm{y}$.
(iii) $\hat{y}$ estimates the mean value of $y$ when $x=x_{0}$.
(iv) $y$ is a linear combination of the $y$ values in the sample.

## 统计代写|实验设计作业代写experimental design代考|RESIDUALS

If the fltted model is the correct one (or close to the correct one), we would expect the residuals to refzect the properties of the deviations. In this chapter, we are mainly concerned with checking the residuals to be assured that the model with its assumptions is reasonable for the data, Is the distribution of the residuals consonant with the assumed distribution of the deviations? Does it appear that the variance about the line is constant? Does it appear that the deviations are independent?

If we recall that the prediction equation is $\mathbf{y}=\hat{y}+e$ then, in a sense, the residual and the predicted y value are opposite faces of the same coin for when one is large, the other is small. We shall not have much to say at this point on the sizes of individual residuals. Large values may indicate that the points are outliers but we shall say more on this in Chapter $4 .$

## 统计代写|实验设计作业代写experimental design代考|The Univariate Case

(一)X0不必是其中之一X在样本中，但在样本范围之外进行预测存在明显的危险。一方面，之间的关系X和是的可能是线性的X样本的值，但关系可能会在这些值之外发生变化，如图2.7.1. 这方面的一个例子可能是低剂量辐射的致癌作用。在这种情况下，癌症的发病率将非常低，以至于难以测量，需要非常大的样本量。为了促进研究，可以在更高的剂量率下工作(X)并希望人们可以推断出更低的剂量率，但这个过程充满了危险。
(ii) 的预测值是的取决于所有是的样本中的值。我们将在第 4 章进一步评论这一点，我们将讨论敏感或高杠杆值X, 我们的意思是那些X′s对预测值有很大影响是的.
㈢是的^估计平均值是的什么时候X=X0.
(四)是的是一个线性组合是的样本中的值。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 统计代写|实验设计作业代写experimental design代考|GOODNESS OF FIT OF THE MODEL

statistics-lab™ 为您的留学生涯保驾护航 在代写实验设计experimental designatistical Modelling方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写实验设计experimental design代写方面经验极为丰富，各种代写实验设计experimental design相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等楖率论
• 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

## 统计代写|实验设计作业代写experimental design代考|AN EXAMPLE – VALUE OF A POSTAGE STAMP OVER TIME

The value of an Australian stamp ( $1963 \& 2$ sepia colored) given, in $\boldsymbol{L}$ sterling, by the Stanley Gibbons Catalogues is shown in Table $1.7 .1$ for the years 1972-1980. The aim is to $f$ it a model to the value of the stamp over time.

Let the time be $x=0$ for $1972, x=1$ for 1973 , etc. In this example, interest probably centres on the relative value of the stamp from one year to the next. Alternatively, if there is interest in the investment value of the stamp over the period 1972-1980, it may be useful to express the value as $y$, the value relative to that of 1972. The values of $y$ and $x$ are also shown above and are graphed in Figure 1.7.1. The relationship between $y$ and $x$ is not a 1 inear one. When $y$ is transformed by taking natural logarithms, a strong linear trend is apparent as shown by Figure 1.7.2. The predicted values of in $y$ and the residuals are shown in Table 1.7.2. The prediction equation is

Of course, the constant term could be omitted and the graph forced to pass through the orlgin.
The residuals fall into a reasonable horizontal band so that there is 1 ttle evidence to contradict the assumptions that the deviations are distributed with mean zero and constant variance.
However, there is a marked pattern in the residuals for their signs are:
Clearly, the residuals are positively correlated (as a positive residual is often followed by a positive residual, and a negative residual followed by a negative). With the small number of observations in the sample, we should hesitate to make dogmatio statements about the model but the pattern in the residuals may suggest that
(i) The population mean is not correct.
The residuals could indicate that a cubed term would remove.

## 统计代写|实验设计作业代写experimental design代考|INTRODUCTION

In Chapter 1, we considered fitting models to data. To answer the question of how well a model fits the data, we need to develop statistical tests and, to do so, we lean heavily on the assumption that the deviations have independent normal distributions. For our purposes, the main benefit of the normality assumption is that we can find the distribution of estimates and perform test of significance. The theorems listed in Appendix B will be of assistance in this. For the general linear model
$$y=1 \alpha+X_{\beta}+\varepsilon$$
where $\alpha$ is a constant term, $X$ is an $n \times k$ matri $x$ of known constants, deviations from their mean, and $\varepsilon$ are the deviations, assumed to be independent and normally distributed with mean zero and constant variance of $\sigma^{2}$, that is $\varepsilon-N\left(0, \sigma^{2} I\right)$. I is an $n \times n$ identity matrix with all diagonal elements equal to 1 and off diagonal elements equal to zero. Thus, the covariances are zero and the variances are all equal.

With these assumptions, we have, in effect, made assumptions about $y$. The mean, or expected value, of $y$ is $X$ a as $E(\varepsilon)$ is 0 . The variance of $y$ is the variance of $\varepsilon_{,} \delta^{2} I$.
2.2 COEEFICIENT ESTIMATES FOR UNIVARIATE REGRESSION
Consider the simple univariate regression with a constant term included, that is
$y=a 1+B x+\varepsilon$ With $\varepsilon \sim N\left(0, \sigma^{2} I\right)$
or $\boldsymbol{y} \sim \mathrm{N}\left(\alpha 1+\beta x, \theta^{2} I\right)$
The least squares estimate of $B$ is
$$b=\left(x^{T} x\right)^{-1} x^{T} y$$
Note that since the $x^{\prime} s$ are deviations from thefr means
\begin{aligned} b &=S_{x y} / S_{x x} \quad(\text { from } 1+6.2) \ \text { and } S_{x y} &=\sum x_{1}\left(y_{1}-y\right)=\sum x_{i} y_{1}=x^{T} y \end{aligned}
As $b 1 s$ a linear combination of the observed $y$ values, then Appendix B 1 can be invoked to give
$d^{2}$ can be estimated by the sample variance of the residuals, namely
$$s^{2}=\sum e_{1}^{2} /(n-2)$$
This has $n-2$ degrees of freedom as there are $n$ observations, but two parameters, $\alpha$ and $\beta$, to be estimated.

## 统计代写|实验设计作业代写experimental design代考|ANOVA TABLES

The analysis of variance table, ANOVA, represents the components of the variation of the dependent variable, $y$. In section $1.5$, we

showed that, for any number of predictor variables, we can divide the vector of observations, $y$, into two orthogonal parts consisting of the predicted values and the residuals, which we can represent by Eigure 1.5.1.
As the vectors e and $\boldsymbol{y}$ are orthogonal, e $\boldsymbol{9}$, then from Pythagoras’ Theorem we know that
$(\operatorname{length} y)^{2}=(\text { ength } \mathbf{y})^{2}+(1 \text { ength e })^{2}\left(r^{2}\right.$
$$y^{T} y=\boldsymbol{y}^{T} \boldsymbol{y}=e^{T} e$$
If there are k predictor terms in the model, but no constant (intercept) term, then the sums of squares and degrees of freedom, d.f., can be displayed as in Table $2.4 .1 .$
Almost invariably it is deviations from the mean which the model is required to explain, so a constant term is included in the model. We have seen from section $1.6 .2$, this has the effect of adjusting each var iable for its mean, and the sum of squares for the mean is given by
$$S S(M e a n)=y^{T} 1\left(1_{1}^{T}\right)^{-1} 1^{T} y=\left(\Sigma y_{j}\right)^{2} / n=n y^{2}$$

（i）总体均值不正确。

## 统计代写|实验设计作业代写experimental design代考|INTRODUCTION

2.2 单变量回归的系数估计

b=(X吨X)−1X吨是的

b=小号X是的/小号XX( 从 1+6.2)  和 小号X是的=∑X1(是的1−是的)=∑X一世是的1=X吨是的

d2可以通过残差的样本方差来估计，即
s2=∑和12/(n−2)

## 统计代写|实验设计作业代写experimental design代考|ANOVA TABLES

(长度⁡是的)2=( 长度 是的)2+(1 英语 )2(r2

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 统计代写|实验设计作业代写experimental design代考|TRANSFORMATIONS TO OBTAIN LINEARITY

statistics-lab™ 为您的留学生涯保驾护航 在代写实验设计experimental designatistical Modelling方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写实验设计experimental design代写方面经验极为丰富，各种代写实验设计experimental design相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等楖率论
• 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

## 统计代写|实验设计作业代写experimental design代考|TRANSFORMATIONS TO OBTAIN LINEARITY

Two variables, $x$ and $y$, may be closely related but the relationship may not be linear. Ideally, theoretical clues would be present wh1ch point to a particular relationship such as an exponential growth model which is common in blology. Without such clues, we could firstly examine a scatter plot of $y$ against $x$.
Sometimes we may recognize a mathematical model which $f$ its the data well. Otherwise, we try to choose a simple transformation such

as ralsing the variable to a power p as in Table 1.4.1. A power of 1 leaves the variable unchanged, that is as raw data. As we proceed up or down the table from 1 , the strength of the transformation increases; as we move up the table the transformation stretohes larger values relatively more than smaller ones. Although the exponential does not flt in very well, we nave included $1 t$ as $1 t 1 s$ the inverse of the logarithmic transformation. other fractional powers oould be used but they may be difficult to interpret.
It would be feasible to transform either $y$ or $x$, and, indeed, a transformation of $y$ would be equivalent to the inverse transformation of $x$. For example, squaring $y$ would be equivalent to taking the square root of $x$. If there are two or more predictor variables, it is often advisable to transform these in different ways nather than $y$, for if y is transformed to be linearly related to one predictor variable it may then not be 1 inear ly related to another.
In Eigure 1.4.1, it is clear that we should stretoh out the graph by increasing large $x$ values, or, alternatively; reduce large $y$ values. Thus, we could try changing $x$ to $x$-squared, or $y$ to the square root of $y_{*}$ One point to be kept in mind here is that for p>0, $y=0$ when $x=0$ so that it may be advisable before invoking the power transformation to change the origin; in partfoular, we could onange y to $(y-a)$, and a good guess may be a $=30$. In Figure $1.4 .2$, it seems that large $x$ values and large $y$ values should be reduced suggesting that a reciprooal transformation may be appropriate. This would require the $x$ and $y$ axes to be asymptotes which, in particular, would

mean that a constant, perhaps 14 , should be subtracted from $y$. We could try changing
\begin{aligned} &y \text { to }(y-14)^{-1} \ &\text { or } y \text { to }(y=14)^{-0.5} \ &\text { or } x \text { to } x{ }^{-1} \ &\text { or } x \text { to } x=0.5 \end{aligned}

## 统计代写|实验设计作业代写experimental design代考|FITTING A MODEL USING VECTORS AND MATRICES

Appendix A contains a review of vectors, vector spaces and matrices and some readers may wish to refer to that section while reading the following.

In regression, we consider the relationship between a string of values of the dependent variable, $y$, and one or more strings of corresponding values of the predictor variables, the $x$ ‘s. It is useful to think of each string as a vector for it turns out that the relationships of interest between the variables are encapsulated in the lengths of the vectors and the angles between them. For the simple Example $1.3 .1$, the $x$ and $y$ readings can be written as column vectors:

The simplest model for this example would be a line through the origin
$$\mathrm{y}{\mathrm{i}}=B \mathrm{X}{\mathrm{i}}+\mathrm{E}{i}$$ or $\mathbf{y}=\beta \mathbf{x}+\boldsymbol{\varepsilon}$ in vector terms The normal equation $1 \mathrm{~s}$ $b \sum x{1}^{2}=\sum x_{1}^{y_{1}}$
or $\left(x^{T} x\right) b=x^{T} y$
giving $b=\left(x^{T} \mathbf{y}\right) /\left(x^{T} \mathbf{x}\right)=0.973$
$(1.5 .2)$
For each value of $x$ we can calculate the predicted value of $y$ as
$$\mathbf{y}=b x \text { or } x b=\left|\begin{array}{l} 0.5 \ 1.0 \ 1.5 \ 2.0 \ 2.5 \end{array}\right| 0.973=\left|\begin{array}{l} 0.486 \ 0.973 \ 1.459 \ 1.945 \ 2.432 \end{array}\right|$$
The predicted value can also be written as
$$y=x b=x\left(x^{T} x\right)^{-1} x^{T} y=P y$$
The matrix $P=x\left(x^{T} \mathbf{x}\right)^{-1} x^{T}$ is termed the projection matrix. More is said about this in section 1.7. Notice that for this case with $n=5$, $P$ is a $5 \times 5$ matrix, namely.

## 统计代写|实验设计作业代写experimental design代考|DEVIATIONS FROM MEANS

It is common practice when fitting a model to use the original (also called raw) data and to include $1 n$ the model a $y=i n t e r c e p t ~ t e r m ~(a l s o ~$ called a constant, or general mean). Most computer programs would convert the raw data to deviations from the mean, as these are used in such statistics as the correlation coefficient. Converting to deviations has the advantage of removing a parameter from the model, making 1 t easier to man1 pulate. Sometimes an examination of the deviations shows up trends which are not as clearly noticeable in the raw data. Problem $2.1$ is an example where deviations from the mean prove useful. It turns out that the estimated coefficients will be the same for the raw data with constant term as with the data 1 n deviation form.

For this section we change our notation slightly to make 1 t clear whether we are referring to the raw data (which we indicate by capital letters, $X, Y$, etc) or deviations from means (lower case $x$, $y$, etc).
$1.6 .1$ Estimates
Ignoring subsor ipts for simplicity, we can write for the case of one predictor variable,
$$x=X=\bar{X} \text { and } y=Y-\bar{Y}$$
where $\vec{X}$ and $\vec{Y}$ are the sample means. For the model
$$\mathrm{y}=8 x+\varepsilon$$
we saw in $(1.2 .7)$ the least squares estimates are given by
\begin{aligned} b &=\sum_{1}^{y_{1} / 2 x_{1}^{2}} \quad(\text { from } 1.2 .7) \ &=\mathrm{S}{x y} / \mathrm{S}{x x} \end{aligned} \quad (def ined by 1.2.13)
$(1.6 .2)$
Notice that the predicted value of $y$ in this case is $\hat{y}=b x$

(1.5.2)

## 统计代写|实验设计作业代写experimental design代考|DEVIATIONS FROM MEANS

1.6.1估计

X=X=X¯ 和 是的=是的−是的¯

b=∑1是的1/2X12( 从 1.2.7) =小号X是的/小号XX（由 1.2.13 定义）
(1.6.2)

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 统计代写|实验设计作业代写experimental design代考|FITTING A MODEL TO DATA

statistics-lab™ 为您的留学生涯保驾护航 在代写实验设计experimental designatistical Modelling方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写实验设计experimental design代写方面经验极为丰富，各种代写实验设计experimental design相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等楖率论
• 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

## 统计代写|实验设计作业代写experimental design代考|FITTING A MODEL TO DATA

The title of this chapter could well be the title of this book. In the first four chapters, we consider problems associated with fitting a regression model and in the last four we consider experimental designs. Mathematically, the two topics use the same model. The term regression is used when the model is fltted to observational data, and experimental design is used when the data is carefully organized to give the model special properties. For some data, the distinction may not be at all clear or, indeed, relevant, he shall consider sets of data consisting of observations of a variable of Interest which we shall call $y$, and we shall assume that these observations are a random sample from a population, usually infinite, of possible values. It is this population which is of primary interest, and not the sample, for in trying to fit models to the data we are really trying to flt models to the population from which the sample is drawn. For each observation, $y$, the model will be of the fork
observed $y=$ population mean + deviation
$(1.1 .1)$
The population mean may depend on the corresponding values of a prem dictor variable which we often label as $x$. For this reason, y is

called the dependent variable. The deviation term indicates the individual peculiarity of the observation, $y$, which makes it differ from the population mean.

As an example, $\$ y$could be the price paid for a house in a certain oity. The population mean could be thought of as the mean price paid for houses in that city, presumably in a given time period. In this case the deviation term could be very large as house prices would vary greatly depending on a number of factors such as the size and condition of the house as well as its position in the oity. In New Zealand, each house is given a government valuation, GV, which is reconsidered on a five year cycle. The price paid for a house will depend to some extent on its GV. The regression model could then be written in terms of$\$x$, the $\mathrm{GV}$, as:

## 统计代写|实验设计作业代写experimental design代考|HON TO FIT A LINE

As the deviation term involves the unexplained variation in $y$, we try to minimise this in some way. Suppose we postulate that the mean value of $y$ is a function of $x$. That is
$$E(y)=f(x)$$
Then for a sample of $n$ pairs of $y^{\prime} s$ with their corresponding $x^{\prime} s$ we have

The above notation assumes that the $x^{\prime} s$ are not random variables but are fixed in advance. If the $x^{\prime} s$ were in fact random variables we should write

$$f\left(x_{1}\right)=E\left(y_{1} \mid x_{1}=x_{1}\right)$$
= mean of $Y_{i}$ given that $X_{i}=x_{i}$
which gives the same results. We wil1 therefore assume in future that the $x^{\prime} s$ are $f$ ixed.
The simplest example of a function $f$ would arise if $y$ was proportional to $x$. We could imagine a situation where an inspector of weights and measures set out to test the scales used by shopkeepers. In this case, the $x^{\prime} s$ would be the welghts of standand measures while $y^{\prime} s$ would be the corresponding weights indicated by the shopkeeper’s scales. The model would be
The mean value of $y$ when $x=x_{i}$ is given by
$$E\left(y_{i}\right)=B x_{i}=f\left(x_{i}\right)$$
This is called a regression curve. In this simple example we would expect the parameter $B$ to be 1 , or at least close to 1 . We think of the parameters as being fixed numbers which describe some attributes of the population.

## 统计代写|实验设计作业代写experimental design代考|Other Ways of Fitting a Curve

The main mroblem with the approach of least squares is that a large deviation will have an even larger square and this deviation may have an unduly large influence on the $f$ itted curve. To guard against such distortions we could try to isolate large deviations. We consider this in more detail in Chapter 4 under outliers and sensitive points. Alternatively, we could seek estimates which minimize a different function of the deviations.

If the model is expressed in terms of the population median of $y$, rather than its mean, another method of fitting a curve would be by minimizing $T$, the sum of the absolute values of deviations, that is
$$T=\sum_{i=1}^{n}\left|\varepsilon_{1}\right|$$
Although this is a sensible approach which works well, the actual mathematics is difficult when the distributions of estimates are sought. Hogg (1974) suggests minimizing
$T=\sum\left|\varepsilon_{i}\right|^{p} \quad$ with $\quad 1<p<2$
and $p=1.5$, in particular, may be a reasonable compromise. Again it is difficult to determine the exact distributions of the resulting estimates. If we are not so much interested in testing hypothesis as

estimating coefficients then this method provides estimates which are robust in the sense that they are not unduly affected by large dev1ations.

Notice that the deviations are the vertical distances from the regression line. It might, perhaps, seem more logical, or at least more symmetrical, to consider the perpendicular (orthogonal) distances from the regression line. However when our major concern is predicting $y$ from $x$ the vertical distances are more relevant because they represent the prediction error.

(1.1.1)

## 统计代写|实验设计作业代写experimental design代考|HON TO FIT A LINE

= 的平均值是的一世鉴于X一世=X一世

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。