## 统计代写|生物统计代写biostatistics代考|MPH701

statistics-lab™ 为您的留学生涯保驾护航 在代写生物统计biostatistics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。我们的专家在代写生物统计biostatistics代写方面经验极为丰富，各种生物统计biostatistics相关的作业也就用不着说。

• 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|生物统计代写biostatistics代考|Extension to the Regression Case

We want to extend the methodology of Sect. $3.2$ to the regression setting where the location parameter varies across observations as a linear function of a set of $p$, say, explanatory variables, which are assumed to include the constant term, as it is commonly the case. If $x_{i}$ is the vector of covariates pertaining to the $i$ th subject, observation $y_{i}$ is now assumed to be drawn from ST $\left(\xi_{i}, \omega, \lambda, \nu\right)$ where
$$\xi_{i}=x_{i}^{\top} \beta, \quad i=1, \ldots, n,$$
for some $p$-dimensional vector $\beta$ of unknown parameters; hence now the parameter vector is $\theta=\left(\beta^{\top}, \omega, \lambda, v\right)^{\top}$. The assumption of independently drawn observations is retained.

The direct extension of the median as an estimate of location, which was used in Sect. 3.2, is an estimate of $\beta$ obtained by median regression, which corresponds to adoption of the least absolute deviations fitting criterion instead of the more familiar least squares. This can also be viewed as a special case of quantile regression, when the quantile level is set at $1 / 2$. A classical treatment of quantile regression

is Koenker (2005) and corresponding numerical work can be carried out using the $R$ package quantreg, see Koenker (2018), among other tools.

Use of median regression delivers an estimate $\tilde{\tilde{\beta}}^{m}$ of $\beta$ and a vector of residual values, $r_{i}=y_{i}-x_{i}^{\top} \tilde{\beta}^{m}$ for $i=1, \ldots, n$. Ignoring $\beta$ estimation errors, these residuals are values sampled from $\mathrm{ST}\left(-m_{0}, \omega^{2}, \lambda, v\right)$, where $m_{0}$ is a suitable value, examined shortly, which makes the distribution to have 0 median, since this is the target of the median regression criterion. We can then use the same procedure of Sect. 3.2, with the $y_{i}$ ‘s replaced the $r_{i}$ ‘s, to estimate $\omega, \lambda, v$, given that the value of $m_{0}$ is irrelevant at this stage.

The final step is a correction to the vector $\tilde{\beta}^{m}$ to adjust for the fact that $y_{i}-x_{i}^{\top} \beta$ should have median $m_{0}$, that is, the median of ST $(0, \omega, \lambda, v)$, not median 0 . This amounts to increase all residuals by a constant value $m_{0}$, and this step is accoomplishéd by sêtting a vectoor $\tilde{\beta}$ with all components equal tō $\tilde{\beta}^{m}$ except that the intercept term, $\beta_{0}$ say, is estimated by
$$\tilde{\beta}{0}=\tilde{\beta}{0}^{m}-\tilde{\omega} q_{2}^{\mathrm{ST}}$$
similarly to $(10)$

## 统计代写|生物统计代写biostatistics代考|Extension to the Multivariate Case

Consider now the case of $n$ independent observations from a multivariate $Y$ variable with density (6), hence $Y \sim \mathrm{ST}{d}(\xi, \Omega, \alpha, v)$. This case can be combined with the regression setting of Sect. 3.3, so that the $d$-dimensional location parameter varies for each observation according to $$\xi{i}^{\top}=x_{i}^{\top} \beta, \quad i=1, \ldots, n,$$
where now $\beta=\left(\beta_{\cdot 1}, \ldots, \beta_{\cdot d}\right)$ is a $p \times d$ matrix of parameters. Since we have assumed that the explanatory variables include a constant term, the regression case subsumes the one of identical distribution, when $p=1$. Hence we deal with the regression case directly, where the $i$ th observation is sampled from $Y_{i} \sim$ $\mathrm{ST}{d}\left(\xi{i}, \Omega, \alpha, v\right)$ and $\xi_{i}$ is given by (12), for $i=1, \ldots, n$.

Arrange the observed values in a $n \times d$ matrix $y=\left(y_{i j}\right)$. Application of the procedure presented in Sects. $3.2$ and $3.3$ separately to each column of $y$ delivers estimates of $d$ univariate models. Specifically, from the $j$ th column of $y$, we obtain estimates $\tilde{\theta}{j}$ and corresponding ‘normalized’ residuals $\tilde{z}{i j}$ :
$$\tilde{\theta}{j}=\left(\tilde{\beta}{\cdot j}^{\top}, \tilde{\omega}{j}, \tilde{\lambda}{j}, \tilde{v}{j}\right)^{\top}, \quad \tilde{z}{i j}=\tilde{\omega}{j}^{-1}\left(y{i j}-x_{i}^{\top} \tilde{\beta}_{\cdot j}\right)$$

where it must be recalled that the ‘normalization’ operation uses location and scale parameters, but these do not coincide with the mean and the standard deviation of the underlying random variable.

Since the meaning of expression (12) is to define a set of univariate regression modes with a common design matrix, the vectors $\tilde{\beta}{-1}, \ldots, \tilde{\beta}{\cdot d}$ can simply be arranged in a $p \times d$ matrix $\tilde{\beta}$ which represents an estimate of $\beta$.

The set of univariate estimates in (13) provide $d$ estimates for $v$, while only one such a value enters the specification of the multivariate ST distribution. We have adopted the median of $\tilde{v}{1}, \ldots, \tilde{v}{d}$ as the single required estimate, denoted $\tilde{v}$.

The scale quantities $\tilde{\omega}{1}, \ldots, \tilde{\omega}{d}$ estimate the square roots of the diagonal elements of $\Omega$, but off-diagonal elements require a separate estimation step. What is really required to estimate is the scale-free matrix $\bar{\Omega}$. This is the problem examined next.

If $\omega$ is the diagonal matrix formed by the squares roots of $\Omega_{11}, \ldots, \Omega_{\text {cld }}$, all variables $\omega^{-1}\left(Y_{i}-\xi_{i}\right)$ have distribution $\mathrm{ST}{d}(0, \bar{\Omega}, \alpha, v)$, for $i=1, \ldots, n$. Denote by $Z=\left(Z{1}, \ldots, Z_{d}\right)^{\top}$ the generic member of this set of variables. We are concerned with the distribution of the products $Z_{j} Z_{k}$, but for notational simplicity we focus on the specific product $W=Z_{1} Z_{2}$, since all other products are of similar nature.

We must then examine the distribution of $W=Z_{1} Z_{2}$ when $\left(Z_{1}, Z_{2}\right)$ is a bivariate ST variable. This looks at first to be a daunting task, but a major simplification is provided by consideration of the perturbation invariance property of symmetrymodulated distributions, of which the ST is an instance. For a precise exposition of this property, see for instance Proposition $1.4$ of Azzalini and Capitanio (2014), but in the present case it says that, since $W$ is an even function of $\left(Z_{1}, Z_{2}\right)$, its distribution does not depend on $\alpha$, and it coincides with the distribution of the case $\alpha=0$, that is, the case of a usual bivariate Student’s $t$ distribution, with dependence parameter $\bar{\Omega}_{12}$.

## 统计代写|生物统计代写biostatistics代考|Simulation Work to Compare Initialization Procedures

Several simulations runs have been performed to examine the performance of the proposed methodology. The computing environment was $\mathrm{R}$ version 3.6.0. The reference point for these evaluations is the methodology currently in use, as provided by the publicly available version of $R$ package $s n$ at the time of writing, namely version 1.5-4; see Azzalini (2019). This will be denoted ‘the current method’ in the following. Since the role of the proposed method is to initialize the numerical MLE search, not the initialization procedure per se, we compare the new and the current method with respect to final MLE outcome. However, since the numerical optimization method used after initialization is the same, any variations in the results originate from the different initialization procedures.

We stress again that in a vast number of cases the working of the current method is satisfactory and we are aiming at improvements when dealing with ‘awkward samples’. These commonly arise with ST distributions having low degrees of freedom, about $v=1$ or even less, but exceptions exist, such as the second sample in Fig. $2 .$

The primary aspect of interest is improvement in the quality of data fitting. This is typically expressed as an increase of the maximal achieved log-likelihood, in its penalized form. Another desirable effect is improvement in computing time.

The basic set-up for such numerical experiments is represented by simple random samples, obtained as independent and identically distributed values drawn from a named ST $(\xi, \omega, \lambda, v)$. In all cases we set $\xi=0$ and $\omega=1$. For the other ingredients, we have selected the following values:
$\lambda: 0, \quad 2, \quad 8$,
$v: 1,3,8$,
$n: 50,100,250,500$
and, for each combination of these values, $N=2000$ samples have been drawn.
The smallest examined sample size, $n=50$, must be regarded as a sort of ‘sensible lower bound’ for realistic fitting of flexible distributions such as the ST. In this respect, recall the cautionary note of Azzalini and Capitanio (2014, p. 63) about the fitting of a SN distribution with small sample sizes. Since the ST involves an additional parameter, notably one having a strong effect on tail behaviour, that annotation holds a fortiori here.

For each of the $3 \times 3 \times 4 \times 2000=72,000$ samples so generated, estimation of the parameters $(\xi, \omega, \lambda, \nu)$ has been carried out using the following methods.

## 统计代写|生物统计代写biostatistics代考|Extension to the Regression Case

X一世=X一世⊤b,一世=1,…,n,

b~0=b~0米−ω~q2小号吨

## 统计代写|生物统计代写biostatistics代考|Extension to the Multivariate Case

X一世⊤=X一世⊤b,一世=1,…,n,

θ~j=(b~⋅j⊤,ω~j,λ~j,在~j)⊤,和~一世j=ω~j−1(是一世j−X一世⊤b~⋅j)

（13）中的一组单变量估计提供d估计为在，而只有一个这样的值进入多元 ST 分布的规范。我们采用了 $\tilde{v} {1}、\ldots、\tilde{v} {d}的中位数一个s吨H和s一世nGl和r和q在一世r和d和s吨一世米一个吨和,d和n○吨和d\波浪号 {v}$。

λ:0,2,8,

n:50,100,250,500

## 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|生物统计代写biostatistics代考|STA 310

statistics-lab™ 为您的留学生涯保驾护航 在代写生物统计biostatistics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。我们的专家在代写生物统计biostatistics代写方面经验极为丰富，各种生物统计biostatistics相关的作业也就用不着说。

• 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|生物统计代写biostatistics代考|Numerical Aspects and Some Illustrations

Since, on the computational side, we shall base our work the R package sn, described by Azzalini (2019), it is appropriate to describe some key aspects of this package. There exists a comprehensive function for model fitting, called selm, but the actual numerical work in case of an ST model is performed by functions st. mple and mst. mple, in the univariate and the multivariate case, respectively. To numerical efficiency, we shall be using these functions directly, rather than via selm. As their names suggest, st. mple and mst. mple perform MPLE, but they can be used for classical MLE as well, just by omitting the penalty function. The rest of the description refers to st. mple, but mst. mple follows a similar scheme.
In the univariate case, denote by $\theta=(\xi, \omega, \alpha, \nu)^{\top}$ the parameters to be cstimatcd, or possibly $\theta=\left(\beta^{\top}, w, \alpha, v\right)^{\top}$ when a lincar regrcssion mudel is introduced for the location parameter, in which case $\beta$ is a vector of $p$ regression coefficients. Denote by $\log L(\theta)$ the log-likelihood function at point $\theta$. If no starting values are supplied, the first operation of st.mple is to fit a linear model to the available explanatory variables; this reduces to the constant covariate value 1 if $p=1$. For the residuals from this linear fit, sample cumulants of order up to four are computed, hence including the sample variance. An inversion from these

values to $\theta$ may or may not be possible, depending on whether the third and fourth sample cumulants fall in the feasible region for the ST family. If the inversion is successful, initial values of the parameters are so obtained; if not, the final two components of $\theta$ are set at $(\alpha, v)=(0,10)$, retaining the other components from the linear fit. Starting from this point, MLE or MPLE is searched for using a general numerical optimization procedure. The default procedure for performing this step is the $\mathrm{R}$ function nlminb, supplied with the score functions besides the log-likelihood function. We shall refer, comprehensively, to this currently standard procedure as ‘method M0’.

In all our numerical work, method M0 uses st. mple, and the involved function nlminb, with all tuning parameters kept at their default values. The only activated option is the one switching between MPLE and MLE, and even this only for the work of the present section. Later on, we shall always use MPLE, with penalty function Openalty which implements the method proposed in Azzalini and Arellano-Valle (2013).

We start our numerical work with some illustrations, essentially in graphical form, of the log-likelihood generated by some simulated datasets. The aim is to provide a direct perception, although inevitably limited, of the possible behaviour of the log-likelihood and the ensuing problems which it poses for MLE search and other inferential procedures. Given this aim, we focus on cases which are unusual, in some way or another, rather than on ‘plain cases’.

The type of graphical display which we adopt is based on the profile loglikelihood function of $(\alpha, v)$, denoted $\log L_{p}(\alpha, v)$. This is obtained, for any given $(\alpha, v)$, by maximizing $\log L(\theta)$ with respect to the remaining parameters. To simplify readability, we transform $\log L_{p}(\alpha, v)$ to the likelihood ratio test statistic, also called ‘deviance function’:
$$D(\alpha, v)=2\left{\log L_{p}(\hat{\alpha}, \hat{v})-\log L_{p}(\alpha, v)\right}$$
where $\log L_{p}(\hat{\alpha}, \hat{v})$ is the overall maximum value of the log-likelihood, equivalent to $\log L(\hat{\theta})$. The concept of deviance applies equally to the penalized log-likelihood.
The plots in Fig. 2 displays, in the form of contour level plots, the behaviour of $D(\alpha, v)$ for two artificially generated samples, with $v$ expressed on the logarithmic scale for more convenient readability. Specifically, the top plots refer to a sample of size $n=50$ drawn from the $\operatorname{ST}(0,1,1,2)$; the left plot, refers to the regular log-likelihood, while the right plot refers to the penalized log-likelihood. The plots include marks for points of special interest, as follows:
$\Delta$ the true parameter point;
o the point having maximal (penalized) log-likelihood on a $51 \times 51$ grid of points spanning the plotted area;

• the MLE or MPLE point selected by method M0;
• the preliminary estimate to be introduced in Sect. 3.2, later denoted M1;
$\times$ the MLE or MPLE point selected by method M2 presented later in the text.

## 统计代写|生物统计代写biostatistics代考|Preliminary Remarks and the Basic Scheme

We have seen in Sect. 2 the ST log-likelihood function can be problematic; it is then advisable to select carefully the starting point for the MLE search. While contrasting the risk of landing on a local maximum, a connected aspect of interest is to reduce the overall computing time. Here are some preliminary considerations about the stated target.

Since these initial estimates will be refined by a subsequent step of log-likelihood maximization, there is no point in aiming at a very sophisticate method. In addition, we want to keep the involved computing header as light as possible. Therefore, we want a method which is simple and quick to compute; at the same time, it should be reasonably reliable, hopefully avoiding nonsensical outcomes.

Another consideration is that we cannot work with the methods of moments, or some variant of it, as this would impose a condition $v>4$, bearing in mind the constraints recalled in Sect. 1.2. Since some of the most interesting applications of ST-based models deal with very heavy tails, hence with low degrees of freedom, the condition $v>4$ would be unacceptable in many important applications. The implication is that we have to work with quantiles and derived quantities.

To ease exposition, we begin by presenting the logic in the basic case of independent observations from a common univariate distribution $\mathrm{ST}\left(\xi, \omega^{2}, \lambda, v\right)$. The first step is to select suitable quantile-based measures of location, scale,

asymmetry and tail-weight. The following list presents a set of reasonable choices; these measures can be equally referred to a probability distribution or to a sample, depending on the interpretation of the terms quantile, quartile and alike.

Location The median is the obvious choice here; denote it by $q_{2}$, since it coincides with the second quartile.

Scale A commonly used measure of scale is the semi-interquartile difference, also called quartile deviation, that is
$$d_{q}=\frac{1}{2}\left(q_{3}-q_{1}\right)$$
where $q_{j}$ denotes the $j$ th quartile; see for instance Kotz et al. (2006, vol. 10, p. 6743).

Asymmetry A classical non-parametric measure of asymmetry is the so-called Bowley’s measure
$$G=\frac{\left(q_{3}-q_{2}\right)-\left(q_{2}-q_{1}\right)}{q_{3}-q_{1}}=\frac{q_{3}-2 q_{2}+q_{1}}{2 d_{q}}$$
see Kotz et al. (2006, vol. 12, p. 7771-3). Since the same quantity, up to an inessential difference, had previously been used by Galton, some authors attribute to him its introduction. We shall refer to $G$ as the Galton-Bowley measure.

Kurtosis A relatively more recent proposal is the Moors measure of kurtosis, presented in Moors (1988),
$$M=\frac{\left(e_{7}-e_{5}\right)+\left(e_{3}-e_{1}\right)}{e_{6}-e_{2}}$$
where $e_{j}$ denotes the $j$ th octile, for $j=1, \ldots, 7$. Clearly, $e_{2 j}=q_{j}$ for $j=$ $1,2,3$

## 统计代写|生物统计代写biostatistics代考|Inversion of Quantile-Based Measures to ST Parameters

For the inversion of the parameter set $Q=\left(q_{2}, d_{q}, G, M\right)$ to $\theta=(\xi, \omega, \lambda, v)$, the first stage considers only the components $(G, M)$ which are to be mapped to $(\lambda, v)$, exploiting the invariance of $G$ and $M$ with respect to location and scale. Hence, at this stage, we can work assuming that $\xi=0$ and $\omega=1$.

Start by computing, for any given pair $(\lambda, v)$, the set of octiles $e_{1}, \ldots, e_{7}$ of $\mathrm{ST}(0,1, \lambda, v)$, and from here the corresponding $(G, M)$ values. Operationally, we have computed the ST quantiles using routine qst of package sn. Only nonnegative values of $\lambda$ need to be considered, because a reversal of the $\lambda$ sign simply reverses the sign of $G$, while $M$ is unaffected, thanks to the mirroring property of the ST quantiles when $\lambda$ is changed to $-\lambda$.

Initially, our numerical exploration of the inversion process examined the contour level plots of $G$ and $M$ as functions of $\lambda$ and $v$, as this appeared to be the more natural approach. Unfortunately, these plots turned out not to be useful, because of the lack of a sufficiently regular pattern of the contour curves. Therefore these plots are not even displayed here.

A more useful display is the one adopted in Fig. 3, where the coordinate axes are now $G$ and $M$. The shaded area, which is the same in both panels, represents the set of feasible $(G, M)$ points for the ST family. In the first plot, each of the black lines indicates the locus of points with constant values of $\delta$, defined by (4), when $v$ spans the positive half-line; the selected $\delta$ values are printed at the top of the shaded area, when feasible without clutter of the labels. The use of $\delta$ instead of $\lambda$ simply yields a better spread of the contour lines with different parameter values, but it is conceptually irrelevant. The second plot of Fig. 3 displays the same admissible region with superimposed a different type of loci, namely those corresponding to specified values of $v$, when $\delta$ spans the $[0,1]$ interval; the selected $v$ values are printed on the left side of the shaded area.

Details of the numerical calculations are as follows. The Galton-Bowley and the Moors measures have been evaluated over a $13 \times 25$ grid of points identified by the selected values
\begin{aligned} \delta^{}=&(0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,0.95,0.99,1) \ v^{}=&(0.30,0.32,0.35,0.40,0.45,0.50,0.60,0.70,0.80,0.90,1,1.5,2\ &3,4,5,7,10,15,20,30,40,50,100, \infty) \end{aligned}

## 统计代写|生物统计代写biostatistics代考|Numerical Aspects and Some Illustrations

D(\alpha, v)=2\left{\log L_{p}(\hat{\alpha}, \hat{v})-\log L_{p}(\alpha, v)\right}D(\alpha, v)=2\left{\log L_{p}(\hat{\alpha}, \hat{v})-\log L_{p}(\alpha, v)\right}

Δ真正的参数点；
o 在 a 上具有最大（惩罚）对数似然的点51×51跨越绘图区域的点网格；

• 方法 M0 选择的 MLE 或 MPLE 点；
• 将在 Sect 中介绍的初步估计。3.2，后面记为M1；
×文中稍后介绍的方法 M2 选择的 MLE 或 MPLE 点。

## 统计代写|生物统计代写biostatistics代考|Preliminary Remarks and the Basic Scheme

dq=12(q3−q1)

G=(q3−q2)−(q2−q1)q3−q1=q3−2q2+q12dq

## 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|生物统计代写biostatistics代考|BIOL 220

statistics-lab™ 为您的留学生涯保驾护航 在代写生物统计biostatistics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。我们的专家在代写生物统计biostatistics代写方面经验极为丰富，各种生物统计biostatistics相关的作业也就用不着说。

• 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|生物统计代写biostatistics代考|Flexible Distributions: The Skew-t Case

In the context of distribution theory, a central theme is the study of flexible parametric families of probability distributions, that is, families allowing substantial variation of their behaviour when the parameters span their admissible range.

For brevity, we shall refer to this domain with the phrase ‘flexible distributions’. The archetypal construction of this logic is represented by the Pearson system of curves for univariate continuous variables. In this formulation, the density function is regulated by four parameters, allowing wide variation of the measures of skewness and kurtosis, hence providing much more flexibility than in the basic case represented by the normal distribution, where only location and scale can be adjusted.

Since Pearson times, flexible distributions have remained a persistent theme of interest in the literature, with a particularly intense activity in recent years. A prominen feature of newer developments is the increased sonsideration for multivariate distributions, reflecting the current availability in applied work of larger datasets, both in sample size and in dimensionality. In the multivariate setting, the various formulations often feature four blocks of parameters to regulate location, scale, skewness and kurtosis.

While providing powerful tools for data fitting, flexible distributions also pose some challenges when we enter the concrete estimation stage. We shall be working with maximum likelihood estimation (MLE) or variants of it, but qualitatively similar issues exist for other criteria. Explicit expressions of the estimates are out of the question; some numerical optimization procedure is always involved and this process is not so trivial because of the larger number of parameters involved, as compared with fitting simpler parametric models, such as a Gamma or a Beta distribution. Furthermore, in some circumstances, the very flexibility of these parametric families can lead to difficulties: if the data pattern does not aim steadily towards a certain point of the parameter space, there could be two or more such points which constitute comparably valid candidates in terms of log-likelihood or some other estimation criterion. Clearly, these problems are more challenging with small sample size, later denoted $n$, since the log-likelihood function (possibly tuned by a prior distribution) is relatively more flat, but numerical experience has shown that they can persist even for fairly large $n$, in certain cases.

## 统计代写|生物统计代写biostatistics代考|The Skew-t Distribution: Basic Facts

Before entering our actual development, we recall some basic facts about the ST parametric family of continuous distributions. In its simplest description, it is obtained as a perturbation of the classical Student’s $t$ distribution. For a more specific description, start from the univariate setting, where the components of the family are identified by four parameters. Of these four parameters, the one denoted $\xi$ in the following regulates the location of the distribution; scale is regulated by the positive parameter $\omega$; shape (representing departure from symmetry) is regulated by $\lambda$; tail-weight is regulated by $v$ (with $v>0$ ), denoted ‘degrees of freedom’ like for a classical $t$ distribution.

It is convenient to introduce the distribution in the ‘standard case’, that is, with location $\xi=0$ and scale $\omega=1$. In this case, the density function is
$$t(z ; \lambda, v)=2 t(z ; v) T\left(\lambda z \sqrt{\frac{v+1}{v+z^{2}}} ; v+1\right), \quad z \in \mathbb{R}$$

where
$$t(z ; v)=\frac{\Gamma\left(\frac{1}{2}(v+1)\right)}{\sqrt{\pi v} \Gamma\left(\frac{1}{2} v\right)}\left(1+\frac{z^{2}}{v}\right)^{-(v+1) / 2}, \quad z \in \mathbb{R}$$
is the density function of the classical Student’s $t$ on $v$ degrees of freedom and $T(\cdot ; v)$ denotes its distribution function; note however that in (1) this is evaluated with $v+1$ degrees of freedom. Also, note that the symbol $t$ is used for both densities in (1) and (2), which are distinguished by the presence of either one or two parameters.

If $Z$ is a random variable with density function (1), the location and scale transform $Y=\xi+\omega Z$ has density function
$$t_{Y}(x ; \theta)=\omega^{-1} t(z ; \lambda, v), \quad z=\omega^{-1}(x-\xi),$$
where $\theta=(\xi, \omega, \lambda, v)$. In this case, we write $Y \sim \operatorname{ST}\left(\xi, \omega^{2}, \lambda, v\right)$, where $\omega$ is squared for similarity with the usual notation for normal distributions.

When $\lambda=0$, we recover the scale-and-location family generated by the $t$ distribution (2). When $v \rightarrow \infty$, we obtain the skew-normal (SN) distribution with parameters $(\xi, \omega, \lambda)$, which is described for instance by Azzalini and Capitanio (2014, Chap. 2). When $\lambda=0$ and $v \rightarrow \infty$, (3) converges to the $\mathrm{N}\left(\xi, \omega^{2}\right)$ distribution.

Some instances of density (1) are displayed in the left panel of Fig. 1. If $\lambda$ was replaced by $-\lambda$, the densities would be reflected on the opposite side of the vertical axis, since $-Y \sim \operatorname{ST}\left(-\xi, \omega^{2},-\lambda, \nu\right)$.

## 统计代写|生物统计代写biostatistics代考|Basic General Aspects

The high flexibility of the ST distribution makes it particularly appealing in a wide range of data fitting problems, more than its companion, the SN distribution. Reliable techniques for implementing connected MLE or other estimation methods are therefore crucial.

From the inference viewpoint, another advantage of the ST over the related SN distribution is the lack of a stationary point at $\lambda=0$ (or $\alpha=0$ in the multivariate case), and the implied singularity of the information matrix. This stationary point of the SN is systematic: it occurs for all samples, no matter what $n$ is. This peculiar aspect has been emphasized more than necessary in the literature, considering that it pertains to a single although important value of the parameter. Anyway, no such problem exists under the ST assumption. The lack of a stationary point at the origin was first observed empirically and welcomed as ‘a pleasant surprise’ by Azzalini and Capitanio (2003), but no theoretical explanation was given. Additional numerical evidence in this direction has been provided by Azzalini and Genton (2008). The theoretical explanation of why the SN and the ST likelihood functions behave differently was finally established by Hallin and Ley (2012).

Another peculiar aspect of the SN likelihood function is the possibility that the maximum of the likelihood function occurs at $\lambda=\pm \infty$, or at $|\alpha| \rightarrow \infty$ in the multivariate case. Note that this happens without divergence of the likelihood function, but only with divergence of the parameter achieving the maximum. In this respect the SN and the ST model are similar: both of them can lead to this pattern.
Differently from the stationarity point at the origin, the phenomenon of divergent estimates is transient: it occurs mostly with small $n$, and the probability of its occurrence decreases very rapidly when $n$ increases. However, when it occurs for the $n$ available data, we must handle it. There are different views among statisticians on whether such divergent values must be retained as valid estimates or they must be rejected as unacceptable. We embrace the latter view, for the reasons put forward by Azzalini and Arellano-Valle (2013), and adopt the maximum penalized likelihood estimate (MPLE) proposed there to prevent the problem. While the motivation for MPLE is primarily for small to moderate $n$, we use it throughout for consistency.
There is an additional peculiar feature of the ST log-likelihood function, which however we mention only for completeness, rather than for its real relevance. In cases when $v$ is allowed to span the whole positive half-line, poles of the likelihood function must exist near $v=0$, similarly to the case of a Student’s $t$ with unspecified degrees of freedom. This problem has been explored numerically by Azzalini and Capitanio (2003, pp. 384-385), and the indication was that these poles must exist at very small values of $v$, such as $\hat{v}=0.06$ in one specific instance.

This phenomenon is qualitatively similar to the problem of poles of the likelihood function for a finite mixture of continuous distributions. Even in the simple case of univariate normal components, there always exist $n$ poles on the boundary of the parameter space if the standard deviations of the components are unrestricted; see for instance Day (1969, Section 7). The problem is conceptually interesting, in both settings, but in practice it is easily dealt with in various ways. In the ST setting, the simplest solution is to impose a constraint $v>v_{0}>0$ where $v_{0}$ is some very small value, such as $v_{0}=0.1$ or $0.2$. Even if fitted to data, a $t$ or ST density with $v<0.1$ would be an object hard to use in practice.

## 统计代写|生物统计代写biostatistics代考|Basic General Aspects

ST 分布的高度灵活性使其在广泛的数据拟合问题中特别有吸引力，超过了它的同伴 SN 分布。因此，实现互联 MLE 或其他估计方法的可靠技术至关重要。

SN 似然函数的另一个特殊方面是似然函数的最大值出现在λ=±∞， 或|一个|→∞在多变量情况下。请注意，这种情况在似然函数没有发散的情况下发生，但只有在参数的发散达到最大值的情况下才会发生。在这方面，SN 和 ST 模型是相似的：它们都可以导致这种模式。

ST 对数似然函数还有一个额外的特殊功能，但是我们仅出于完整性而不是其真正相关性而提及它。在某些情况下在允许跨越整个正半线，似然函数的极点必须存在于附近在=0，类似于学生的情况吨具有未指定的自由度。Azzalini 和 Capitanio (2003, pp. 384-385) 对这个问题进行了数值研究，表明这些极点必须以非常小的值存在在， 如在^=0.06在一个特定的情况下。

## 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|生物统计代写biostatistics代考| DETERMINING THE SAMPLE SIZE

statistics-lab™ 为您的留学生涯保驾护航 在代写生物统计biostatistics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。我们的专家在代写生物统计biostatistics代写方面经验极为丰富，各种生物统计biostatistics相关的作业也就用不着说。

• 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|生物统计代写biostatistics代考|The Sample Size for Simple and Systematic Random Samples

In a simple random sample or a systematic random sample, the sample size required to produce a prespecified bound on the error of estimation for estimating the mean is based on the number of units in the population $(N)$, and the approximate variance of the population $\sigma^{2}$. Moreover, given the values of $N$ and $\sigma^{2}$, the sample size required for estimating a mean $\mu$ with bound on the error of estimation $B$ with a simple or systematic random sample is
$$n=\frac{N \sigma^{2}}{(N-1) D+\sigma^{2}}$$
where $D=\frac{B^{2}}{4}$. Note that this formula will not generally return a whole number for the sample size $n$; when the formula does not return a whole number for the sample size, the sample size should be taken to be the next largest whole number.
Example 3.11
Suppose a simple random sample is going to be taken from a population of $N=5000$ units with a variance of $\sigma^{2}=50$. If the bound on the error of estimation of the mean is supposed to be $B=1.5$, then the sample size required for the simple random sample selected from this population is
$$n=\frac{5000(50)}{4999\left(\frac{1.5^{2}}{4}\right)+50}=87.35$$
Since $87.35$ units cannot be sampled, the sample size that should be used is $n=88$. Also, $n=$ 88 would be the sample size required for a systematic random sample from this population when the desired bound on the error of estimation for estimating the mean is $B=1.5$. In this case, the systematic random sample would be a 1 in 56 systematic random sample since $\frac{5000}{88} \approx 56$.

In many research projects, the values of $N$ or $\sigma^{2}$ are often unknown. When either $N$ or $\sigma^{2}$ is unknown, the formula for determining the sample size to produce a bound on the error of estimation for a simple random sample can still be used as long as the approximate values of $N$ and $\sigma^{2}$ are available. In this case, the resulting sample size will produce a bound on the error of estimation that is close to $B$ provided the approximate values of $N$ and $\sigma^{2}$ are reasonably accurate.

The proportion of the units in the population that are sampled is $n / N$, which is called the sampling proportion. When a rough guess of the size of the population cannot be reasonably made, but it is clear that the sampling proportion will be less than $5 \%$, then an alternative formula for determining the sample size is needed. In this case, the sample size required for a simple random sample or a systematic random sample having bound on the error of estimation $B$ for estimating the mean is approximately
$$n=\frac{4 \sigma^{2}}{B^{2}}$$

## 统计代写|生物统计代写biostatistics代考|The Sample Size for a Stratified Random Sample

Recall that a stratified random sample is simply a collection of simple random samples selected from the subpopulations in the target population. In a stratified random sample, there are two sample size considerations, namely, the overall sample size $n$ and the allocation of $n$ units over the strata. When there are $k$ strata, the strata sample sizes will be denoted by $n_{1}, n_{2}, n_{3}, \ldots, n_{k}$, where the number to be sampled in strata 1 is $n_{1}$, the number to be sampled in strata 2 is $n_{2}$, and so on.

There are several different ways of determining the overall sample size and its allocation in a stratified random sample. In particular, proportional allocation and optimal allocation are two commonly used allocation plans. Throughout the discussion of these two allocation plans, it will be assumed that the target population has $k$ strata, $N$ units, and $N_{j}$ is the number of units in the $j$ th stratum.

The sample size used in a stratified random sample and the most efficient allocation of the sample will depend on several factors including the variability within each of the strata, the proportion of the target population in each of the strata, and the costs associated with sampling the units from the strata. Let $\sigma_{i}$ be the standard deviation of the $i$ th stratum, $W_{i}=N_{i} / N$ the proportion of the target population in the $i$ th stratum, $C_{0}$ the initial cost of sampling, $C_{i}$ the cost of obtaining an observation from the $i$ th stratum, and $C$ is the total cost of sampling. Then, the cost of sampling with a stratified random sample is
$$C=C_{0}+C_{1} n_{1}+C_{2} n_{2}+\cdots+C_{k} n_{k}$$
The process of determining the sample size for a stratified random sample requires that the allocation of the sample be determined first. The allocation of the sample size $n$ over the $k$ strata is based on the sampling proportions that are denoted by $w_{1}, w_{2}, \ldots w_{k}$. Once the sampling proportions and the overall sample size $n$ have been determined, the $i$ th stratum sample size is $n_{i}=n \times w_{i}$.

The simplest allocation plan for a stratified random sample is proportional allocation that takes the sampling proportions to be proportional to the strata sizes. Thus, in proportional allocation, the sampling proportion for the $i$ th stratum is equal to the proportion of the population in the ith stratum. That is, the sampling proportion for the $i$ th stratum is
$$w_{i}=\frac{N_{i}}{N}$$
The overall sample size for a stratified random sample based on proportional allocation that will have bound on error of estimation for estimating the mean equal to $B$ is
$$n=\frac{N_{1} \sigma_{1}^{2}+N_{2} \sigma_{2}^{2}+\cdots+N_{k} \sigma_{k}^{2}}{N\left[\frac{B^{2}}{4}\right]+\frac{1}{N}\left(N_{1} \sigma_{1}^{2}+N_{2} \sigma_{2}^{2}+\cdots+N_{k} \sigma_{k}^{2}\right)}$$
The sample size for the simple random sample that will be selected from the $i$ th stratum according to proportional allocation is
$$n \times w_{i}=n \times \frac{N_{i}}{N}$$

## 统计代写|生物统计代写biostatistics代考|Bar and Pie Charts

In the case of qualitative or discrete data, the graphical statistics that are most often used to summarize the data in the observed sample are the bar chart and the pie chart since the

important parameters of the distribution of a qualitative variable are population proportions. Thus, for a qualitative variable the sample proportions are the values that will be displayed in a bar chart or a pie chart.

In Chapter 2, the distribution of a qualitative variable was often presented in a bar chart in which the height of a bar represented the proportion or the percentage of the population having each quality the variable takes on. With an observed sample, bar charts can be used to represent the sample proportions or percentages for each of the qualities the variable takes on and can be used to make statistical inferences about the population distribution of the variable.

There are many types of bar charts including simple bar charts, stacked bar charts, and comparative side-by-side bar charts. An example of a simple bar chart for the weight classification for babies, which takes on the values normal and low, in the Birth Weight data set is shown in Figure 4.1.

Note that a bar chart represents the category percentages or proportions with bars of height equal to the percentage or proportion of sample observations falling in a particular category. The widths of the bars should be equal and chosen so that an appealing chart is produced. Bar charts may be drawn with either horizontal or vertical bars, and the bars in a bar chart may or may not be separated by a gap. An example of a bar chart with horizontal bars is given in Figure $4.2$ for the weight classification of babies in the Birth Weight data set.
In creating a bar chart it is important that

1. the proportions or percentages in each bar can be easily determined to make the bar chart easier to read and interpret.
2. the total percentage represented in the bar chart should be 100 since a distribution contains $100 \%$ of the population units.
3. the qualities associated with an ordinal variable are listed in the proper relative order! With a nominal variable the order of the categories is not important.
4. the bar chart has the axes of the bar chart clearly labeled so that it is clear whether the bars represent a percentage or a proportion.
5. the bar chart has either a caption or a title that clearly describes the nature of the bar chart.

## 统计代写|生物统计代写biostatistics代考|The Sample Size for Simple and Systematic Random Samples

n=ñσ2(ñ−1)D+σ2

n=5000(50)4999(1.524)+50=87.35

n=4σ2乙2

## 统计代写|生物统计代写biostatistics代考|The Sample Size for a Stratified Random Sample

C=C0+C1n1+C2n2+⋯+Cķnķ

n=ñ1σ12+ñ2σ22+⋯+ñķσķ2ñ[乙24]+1ñ(ñ1σ12+ñ2σ22+⋯+ñķσķ2)

n×在一世=n×ñ一世ñ

## 统计代写|生物统计代写biostatistics代考|Bar and Pie Charts

1. 可以轻松确定每个条形中的比例或百分比，以使条形图更易于阅读和解释。
2. 条形图中表示的总百分比应为 100，因为分布包含100%人口单位。
3. 与序数变量相关的质量以正确的相对顺序列出！对于名义变量，类别的顺序并不重要。
4. 条形图清楚地标记了条形图的轴，以便清楚条形是代表百分比还是比例。
5. 条形图具有清楚地描述条形图性质的标题或标题。

## 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|生物统计代写biostatistics代考| RANDOM SAMPLING

statistics-lab™ 为您的留学生涯保驾护航 在代写生物统计biostatistics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。我们的专家在代写生物统计biostatistics代写方面经验极为丰富，各种生物统计biostatistics相关的作业也就用不着说。

• 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|生物统计代写biostatistics代考|OBTAINING REPRESENTATIVE DATA

The purpose of sampling is to get a sufficient amount of data that is representative of target population so that statistical inferences can be made about the distribution and the parameters of the target population. Because a sample is only a subset of the units in the target population, it is generally impossible to guarantee that the sample data are representative of the target population; however, with a well-designed sampling plan, it will be unlikely to select a sample that is not representative of the target population. To ensure the likelihood that the sample data will be representative of the target population, the following components of the sampling process must be considered:

• Target Population The target population must be well defined, accessible, and the researcher should have a good understanding of the structure of the population. In particular, the researcher should be able to identify the units of the population, the approximate number of units in the population, subpopulations, the approximate shape of the distributions of the variables being studied, and the relevant parameters that need to be estimated.
• Sampling Units The Sampling units are the units of the population that will be sampled. A sampling unit may or may not be a unit of the population. In fact, in some sampling plans, the sampling unit is a collection of population units. The sampling unit is also the smallest unit in the target population that can be selected.
• Sampling Element A sampling element is an object on which measurements will be made. A sampling element may or may not be a sampling unit. When the sampling unit consists of several population units, it is called a cluster of units. If each

population unit in a cluster will be measured, then the sampling elements are the population units within the sampled clusters. In this case, the sampling element is a subunit of the sampling unit.

• Sampling Frame The sampling frame is the list of sampling units that are available for sampling. The sampling frame should be nearly equal to the target population. When the sampling frame is significantly different from the target population, it makes it less unlikely that a sample representative of the target population will be obtained, even with a well-designed sampling plan. Sampling frames that fail to include all of the units of the target population are said to undercover the target population and may lead to biased samples.
• Sample Size The sample size is the number of sampling units that will be selected. The sample size will be denoted by $n$ and must be sufficiently large to ensure the reliability of the statistical analysis. The variability in the target population plays a key role in determining the sample size necessary for the desired level of reliability associated with a statistical analysis.

## 统计代写|生物统计代写biostatistics代考|Probability Samples

The statistical theory that provides the foundation for the estimation or testing of research hypotheses about the parameters of a population is based on the sampling structure known as probability sampling. A probability sample is a sample that is selected in a random fashion according to some probability model. In particular, a probability sample is a sample chosen so that each of the possible samples is known in advance and the probability of drawing each sampling unit is known. Random samples are samples that arise through a sampling plan based on probability sampling.

Probability sampling allows flexibility in the sampling plan and can be designed specifically for the target population being studied. That is, a probability sampling plan allows a sample to be designed so that it will be unlikely to produce a sample that is not representative of the target population. Furthermore, probability samples allow for confidence statements and hypothesis tests to be made from the observed sample with a high degree of reliability.

Samples of convenience are samples that are not based on probability samples and are also referred to as nonprobability samples. The statistical theory that justifies the use of confidence statements and tests of hypotheses does not apply to nonprobability samples; therefore, confidence statements and test of the research hypotheses based on nonprobability samples are erroneous applications of statistics and should not be trusted.
In a random sample, the chance that a particular unit of the population will be selected is known prior to sampling, and the units available for sampling are selected at random according to these probabilities. The procedure for drawing a random sample is outlined below.

## 统计代写|生物统计代写biostatistics代考|Simple Random Sampling

The first sampling plan that will be discussed is the simple random sample. A simple random sample of size $n$ is a sample consisting of $n$ sampling units selected in a fashion that every possible sample of $n$ units has the same chance of being selected. In a simple random sample, every possible sample has the same chance of being selected, and moreover, each sampling unit has the same chance of being drawn in a sample. Simple random sampling is a reasonable sampling plan for sampling homogeneous or heterogeneous populations that do not have distinct subpopulations that are of interest to the researcher.
Example 3.3
Simple random sampling might be a reasonable sampling plan in the following scenarios:
a. A pharmaceutical company is checking the quality control issues of the tablet form of a new drug. Here, the company might take a random sample of tablets from a large pool of available drug tablets it has recently manufactured.
b. The Federal Food and Drug Administration (FDA) may take a simple random sample of a particular food product to check the validity of the information on the nutrition label.
c. A state might wish to take a simple random sample of medical doctors to review whether or not the state’s continuing education requirements are being satisfied.
d. A federal or state environment agency may wish to take a simple random sample of homes in a mining town to investigate the general health of the town’s inhabitants and contamination problems in the homes resulting from the mining operation.

The number of possible simple random samples of size $n$ selected from a sampling frame listing of $N$ sampling units is
$$\left(\begin{array}{l} N \ n \end{array}\right)=\frac{N !}{n !(N-n) !}$$
The probability that any one of the possible simple random samples of $n$ units selected from a sampling frame of $N$ units is
$$\frac{1}{\frac{N !}{n !(N-n) !}}=\frac{n !(N-n) !}{N !}$$

## 统计代写|生物统计代写biostatistics代考|OBTAINING REPRESENTATIVE DATA

• 目标人群 目标人群必须明确定义、易于访问，并且研究人员应对人群结构有很好的了解。特别是，研究人员应该能够识别总体单位、总体中单位的大致数量、亚总体、所研究变量分布的大致形状以及需要估计的相关参数。
• 抽样单位 抽样单位是要抽样的总体单位。抽样单位可能是也可能不是人口的单位。事实上，在一些抽样计划中，抽样单位是人口单位的集合。抽样单位也是目标人群中可以选择的最小单位。
• 采样元件 采样元件是将对其进行测量的对象。采样元件可以是也可以不是采样单元。当抽样单位由若干人口单位组成时，称为单位群。如果每个

• 抽样框架 抽样框架是可用于抽样的抽样单位列表。抽样框架应该几乎等于目标人群。当抽样框架与目标人群显着不同时，即使采用精心设计的抽样计划，也不太可能获得代表目标人群的样本。未能包括目标总体的所有单位的抽样框架被称为隐藏目标总体，并可能导致样本有偏差。
• 样本大小 样本大小是要选择的抽样单位的数量。样本大小将表示为n并且必须足够大以确保统计分析的可靠性。目标人群的可变性在确定与统计分析相关的所需可靠性水平所需的样本量方面起着关键作用。

## 统计代写|生物统计代写biostatistics代考|Simple Random Sampling

：一家制药公司正在检查一种新药片剂的质量控制问题。在这里，该公司可能会从其最近生产的大量可用药片中随机抽取片剂样本。

C。一个州可能希望对医生进行简单的随机抽样，以审查该州的继续教育要求是否得到满足。
d。联邦或州环境机构可能希望对采矿城镇中的房屋进行简单的随机抽样，以调查该镇居民的总体健康状况以及采矿作业导致的房屋污染问题。

(ñ n)=ñ!n!(ñ−n)!

1ñ!n!(ñ−n)!=n!(ñ−n)!ñ!

## 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|生物统计代写biostatistics代考| PROBABILITY MODELS

statistics-lab™ 为您的留学生涯保驾护航 在代写生物统计biostatistics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。我们的专家在代写生物统计biostatistics代写方面经验极为丰富，各种生物统计biostatistics相关的作业也就用不着说。

• 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|生物统计代写biostatistics代考|The Binomial Probability Model

The binomial probability model can be used for modeling the number of times a particular event occurs in a sequence of repeated trials. In particular, a binomial random variable is a discrete variable that is used to model chance experiments involving repeated dichotomous trials. That is, the binomial model is used to model repeated trials where the outcome of each trial is one of the two possible outcomes. The conditions under which the binomial probability model can be used are given below.

A random variable satisfying the above conditions is called a binomial random variable. Note that a binomial random variable $X$ simply counts the number of successes that occurred in $n$ trials. The probability distribution for a binomial random variable $X$ is given by the mathematical expression
$$p(x)=\frac{n !}{x !(n-x) !} p^{x}(1-p)^{n-x} \quad \text { for } x=0,1, \ldots, n$$
where $p(x)$ is the probability that $X$ is equal to the value $x$. In this formula

• $\frac{n !}{x !(n-x) !}$ is the number of ways for there to be $x$ successes in $n$ trials,
• $n !=n(n-1)(n-2) \cdots 3 \cdot 2 \cdot 1$ and $0 !=1$ by definition,
• $p$ is the probability of a success on any of the $n$ trials,
• $p^{x}$ is the probability of having $x$ successes in $n$ trials,
• $1-p$ is the probability of a failure on any of the $n$ trials,
• $(1-p)^{n-x}$ is the probability of getting $n-x$ failures in $n$ trials.
Examples of the binomial distribution are given in Figure 2.24. Note that a binomial distribution will have a longer tail to the right when $p<0.5$, a longer tail to the left when $p>0.5$, and is symmetric when $p=0.5$.

Because the computations for the probabilities associated with a binomial random variable are tedious, it is best to use a statistical computing package such as MINITAB for computing binomial probabilities.

## 统计代写|生物统计代写biostatistics代考|The Normal Probability Model

The choice of a probability model for continuous variables is generally based on historical data rather than a particular set of conditions. Just as there are many discrete probability models, there are also many different probability models that can be used to model the distribution of a continuous variable. The most commonly used continuous probability model in statistics is the normal probability model.

The normal probability model is often used to model distributions that are expected to be unimodal and symmetric, and the normal probability model forms the foundation for many of the classical statistical methods used in biostatistics. Moreover, the distribution of many natural phenomena can be modeled very well with the normal distribution. For example, the weights, heights, and IQs of adults are often modeled with normal distributions.

The standard normal, which will be denoted by $Z$, is a normal distribution having mean 0 and standard deviation 1. The standard normal is used as the reference distribution from which the probabilities and percentiles associated with any normal distribution will be determined. The cumulative probabilities for a standard normal are given in Tables A.1 and A.2; because $99.95 \%$ of the standard normal distribution lies between the values $-3.49$ and $3.49$, the standard normal values are only tabulated for $z$ values between $-3.49$ and $3.49$. Thus, when the value of a standard normal, say $z$, is between $-3.49$ and $3.49$, the tabled value for $z$ represents the cumulative probability of $z$, which is $P(Z \leq z)$ and will be denoted by $\Phi(z)$. For values of $z$ below $-3.50, \Phi(z)$ will be taken to be 0 and for values of $z$ above $3.50, \Phi(z)$ will be taken to be 1. Tables A.1 and A.2 can be used to compute all of the probabilities associated with a standard normal.

The values of $z$ are referenced in Tables A.1 and A.2 by writing $z=a . b c$ as $z=a . b+0.0 c$. To locate a value of $z$ in Table A.1 and A.2, first look up the value $a . b$ in the left-most column of the table and then locate $0.0 c$ in the first row of the table. The value cross-referenced by $a . b$ and $0 . c$ in Tables A.1 and A.2 is $\Phi(z)=P(Z \leq z)$. The rules for computing the probabilities for a standard normal are given below.

## 统计代写|生物统计代写biostatistics代考|Z Scores

The result of converting a non-standard normal value, a raw value, to a $Z$-value is a $Z$ score. A $Z$ score is a measure of the relative position a value has within its distribution. In particular, a $Z$ score simply measures how many standard deviations a point is above or below the mean. When a $Z$ score is negative the raw value lies below the mean of its distribution, and when a $Z$ score is positive the raw value lies above the mean. $Z$ scores are unitless measures of relative standing and provide a meaningful measure of relative standing only for mound-shaped distributions. Furthermore, $Z$ scores can be used to compare the relative standing of individuals in two mound-shaped distributions.
Example 2.41
The weights of men and women both follow mound-shaped distributions with different means and standard deviations. In fact, the weight of a male adult in the United States is approximately normal with mean $\mu=180$ and standard deviation $\sigma=30$, and the weight of a female adult in the United States is approximately normal with mean $\mu=145$ and standard deviation $\sigma=15$. Given a male weighing $215 \mathrm{lb}$ and a female weighing $170 \mathrm{lb}$, which individual weighs more relative to their respective population?

The answer to this question can be found by computing the $Z$ scores associated with each of these weights to measure their relative standing. In this case,
$$z_{\text {male }}=\frac{215-180}{30}=1.17$$
and
$$z_{\text {female }}=\frac{170-145}{15}=1.67$$
Since the female’s weight is $1.67$ standard deviations from the mean weight of a female and the male’s weight is $1.17$ standard deviations from the mean weight of a male, relative to their respective populations a female weighing $170 \mathrm{lb}$ is heavier than a male weighing $215 \mathrm{lb}$.

## 统计代写|生物统计代写biostatistics代考|The Binomial Probability Model

p(X)=n!X!(n−X)!pX(1−p)n−X 为了 X=0,1,…,n

• n!X!(n−X)!是有多少种方式X成功n试验，
• n!=n(n−1)(n−2)⋯3⋅2⋅1和0!=1根据定义，
• p是任何一个成功的概率n试验，
• pX是拥有的概率X成功n试验，
• 1−p是任何一个失败的概率n试验，
• (1−p)n−X是得到的概率n−X失败n试验。
图 2.24 给出了二项分布的示例。请注意，当二项分布的右尾较长时p<0.5, 一条较长的尾巴在左边时p>0.5, 并且是对称的p=0.5.

## 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|生物统计代写biostatistics代考|The Coefficient of Variation

statistics-lab™ 为您的留学生涯保驾护航 在代写生物统计biostatistics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。我们的专家在代写生物统计biostatistics代写方面经验极为丰富，各种生物统计biostatistics相关的作业也就用不着说。

• 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|生物统计代写biostatistics代考|The Coefficient of Variation

The standard deviations of two populations resulting from measuring the same variable can be compared to determine which of the two populations is more variable. That is, when one standard deviation is substantially larger than the other (i.e., more than two times as large), then clearly the population with the larger standard deviation is much more variable than the other. It is also important to be able to determine whether a single population is highly variable or not. A parameter that measures the relative variability in a population is the coefficient of variation. The coefficient of variation will be denoted by CV and is defined to be
$$\mathrm{CV}=\frac{\sigma}{|\mu|}$$
The coefficient of variation is also sometimes represented as a percentage in which case
$$\mathrm{CV}=\frac{\sigma}{|\mu|} \times 100 \%$$

The coefficient of variation compares the size of the standard deviation with the size of the mean. When the coefficient of variation is small, this means that the variability in the population is relatively small compared to the size of the mean of the population. On the other hand, when the coefficient of variation is large, this indicates that the population varies greatly relative to the size of the mean. The standard for what is a large coefficient of variation differs from one discipline to another, and in some disciplines a coefficient of variation of less than $15 \%$ is considered reasonable, and in other disciplines larger or smaller cutoffs are used.

Because the standard deviation and the mean have the same units of measurement, the coefficient of variation is a unitless parameter. That is, the coefficient is unaffected by changes in the units of measurement. For example, if a variable $X$ is measured in inches and the coefficient of variation is $\mathrm{CV}=2$, then coefficient of variation will also be 2 when the units of measurement are converted to centimeters. The coefficient of variation can also be used to compare the relative variability in two different and unrelated populations; the standard deviation can only be used to compare the variability in two different populations based on similar variables.

## 统计代写|生物统计代写biostatistics代考|Parameters for Bivariate Populations

In most biomedical research studies, there are many variables that will be recorded on each individual in the study. A multivariate distribution can be formed by jointly tabulating, charting, or graphing the values of the variables over the $N$ units in the population. For example, the bivariate distribution of two variables, say $X$ and $Y$, is the collection of the ordered pairs
$$\left(X_{1}, Y_{1}\right),\left(X_{2}, Y_{2}\right),\left(X_{3}, Y_{3}\right), \ldots,\left(X_{N}, Y_{N}\right)$$
These $N$ ordered pairs form the units of the bivariate distribution of $X$ and $Y$ and their joint distribution can be displayed in a two-way chart, table, or graph.

When the two variables are qualitative, the joint proportions in the bivariate distribution are often denoted by $p_{a b}$, where
$$p_{a b}=\text { proportion of pairs in population where } X=a \text { and } Y=b$$
The joint proportions in the bivariate distribution are then displayed in a two-way table or two-way bar chart. For example, according to the American Red Cross, the joint distribution of blood type and Rh factor is given in Table $2.7$ and presented as a bar chart in Figure $2.21$.

## 统计代写|生物统计代写biostatistics代考|Basic Probability Rules

Determining the probabilities associated with complex real-life events often requires a great deal of information and an extensive scientific understanding of the structure of the chance experiment being studied. In fact, even when the sample space and event are easily identified, the determination of the probability of an event can be an extremely difficult task. For example, in studying the side effects of a drug, the possible side effects can generally be anticipated and the sample space will be known. However, because humans react differently to drugs, the probabilities of the occurrence of the side effects are generally unknown. The probabilities of the side effects are often estimated in clinical trials.

The following basic probability rules are often useful in determining the probability of an event.

1. When the outcomes of a random experiment are equally likely to occur, the probability of an event $A$ is the number of outcomes in $A$ divided by the number of simple events in $\mathcal{S}$. That is,
$$P(A)=\frac{\text { number of simple events in } A}{\text { number of simple events in } \mathcal{S}}=\frac{N(A)}{N(\delta)}$$
2. For every event $A$, the probability of $A$ is the sum of the probabilities of the outcomes comprising $A$. That is, when an event $A$ is comprised of the outcomes $O_{1}, O_{2}, \ldots, O_{k}$, the probability of the event $A$ is
$$P(A)=P\left(O_{1}\right)+P\left(O_{2}\right)+\cdots+P\left(O_{k}\right)$$
3. For any two events $A$ and $B$, the probability that either event $A$ or event $B$ occurs is
$$P(A \text { or } B)=P(A)+P(B)-P(A \text { and } B)$$
4. The probability that the event $A$ does not occur is 1 minus the probability that the event $A$ does occur. That is,
$$P(A \text { does not occur })=1-P(A)$$

C在=σ|μ|

C在=σ|μ|×100%

## 统计代写|生物统计代写biostatistics代考|Parameters for Bivariate Populations

(X1,是1),(X2,是2),(X3,是3),…,(Xñ,是ñ)

p一个b= 人口中对的比例 X=一个 和 是=b

## 统计代写|生物统计代写biostatistics代考|Basic Probability Rules

1. 当随机实验的结果同样可能发生时，事件发生的概率一个是结果的数量一个除以简单事件的数量小号. 那是，
磷(一个)= 简单事件的数量 一个 简单事件的数量 小号=ñ(一个)ñ(d)
2. 对于每一个事件一个, 的概率一个是结果的概率之和，包括一个. 也就是说，当一个事件一个由结果组成○1,○2,…,○ķ, 事件的概率一个是
磷(一个)=磷(○1)+磷(○2)+⋯+磷(○ķ)
3. 对于任意两个事件一个和乙, 任一事件的概率一个或事件乙发生是
磷(一个 或者 乙)=磷(一个)+磷(乙)−磷(一个 和 乙)
4. 事件发生的概率一个不发生是 1 减去事件发生的概率一个确实发生。那是，
磷(一个 不发生 )=1−磷(一个)

## 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|生物统计代写biostatistics代考|Describing a Population with Parameters

statistics-lab™ 为您的留学生涯保驾护航 在代写生物统计biostatistics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。我们的专家在代写生物统计biostatistics代写方面经验极为丰富，各种生物统计biostatistics相关的作业也就用不着说。

• 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|生物统计代写biostatistics代考|Proportions and Percentiles

Populations are often summarized by listing the important percentages or proportions associated with the population. The proportion of units in a population having a particular characteristic is a parameter of the population, and a population proportion will be denoted by $p$. The population proportion having a particular characteristic, say characteristic $A$, is defined to be
$$p=\frac{\text { number of units in population having characteristic A }}{N}$$
Note that the percentage of the population having characteristic A is $p \times 100 \%$. Population proportions and percentages are often associated with the categories of a qualitative variable or with the values in the population falling in a specific range of values. For example, the distribution of a qualitative variable is usually displayed in a bar chart with the height of a bar representing either the proportion or percentage of the population having that particular value.
Example 2.12
The distribution of blood type according to the American Red Cross is given in Table $2.4$ in terms of proportions.

An important proportion in many biomedical studies is the proportion of individuals having a particular disease, which is called the prevalence of the disease. The prevalence of a disease is defined to be
Prevalence $=$ The proportion of individuals in a well-defined population having the disease of interest
For example, according to the Centers for Disease Control and Prevention (CDC) the prevalence of smoking among adults in the United States in January through June 2005 was $20.9 \%$. Proportions also play important roles in the study of survival and cure rates, the occurrence of side effects of new drugs, the absolute and relative risks associated with a disease, and the efficacy of new treatments and drugs.

## 统计代写|生物统计代写biostatistics代考|Parameters Measuring Centrality

The two parameters in the population of values of a quantitative variable that summarize how the variable is distributed are the parameters that measure the typical or central values in the population and the parameters that measure the spread of the values within the population. Parameters describing the central values in a population and the spread of a population are often used for summarizing the distribution of the values in a population; however, it is important to note that most populations cannot be described very well with only the parameters that measure centrality and the spread of the population.

Measures of centrality, location, or the typical value are parameters that lie in the “center” or “middle” region of a distribution. Because the center or middle of a distribution is not easily determined due to the wide range of different shapes that are possible with a distribution, there are several different parameters that can be used to describe the center of a population. The three most commonly used parameters for describing the center of a population are the mean, median, and mode. For a quantitative variable $X$.

• The mean of a population is the average of all of the units in the population, and will be denoted by $\mu$. The mean of a variable $X$ measured on a population consisting of $N$ units is
$$\mu=\frac{\text { sum of the values of } X}{N}=\frac{\sum X}{N}$$
• The median of a population is the 50 th percentile of the population, and will be denoted by $\tilde{\mu}$. The median of a population is found by first listing all of the values of the variable $X$, including repeated $X$ values, in ascending order. When the number of units in the population (i.e., $N$ ) is an odd number, the median is the middle observation in the list of ordered values of $X$; when $N$ is an even number, the median will be the average of the two observations in the middle of the ordered list of $X$ values.
• The mode of a population is the most frequent value in the population, and will be denoted by $M$. In a graph of the probability density function, the mode is the value of $X$ under the peak of the graph, and a population can have more than one mode as shown in Figure 2.8.

The mean, median, and mode are three different parameters that can be used to measure the center of a population or to describe the typical values in a population. These three parameters will have nearly the same value when the distribution is symmetric or mound shaped. For long-tailed distributions, the mean, median, and mode will be different, and the difference in their values will depend on the length of the distribution’s longer tail. Figures $2.12$ and $2.13$ illustrate the relationships between the values of the mean, median, and mode for long-tail right and long-tail left distributions.

## 统计代写|生物统计代写biostatistics代考|Measures of Dispersion

While the mean, median, and mode of a population describe the typical values in the population, these parameters do not describe how the population is spread over its range of values. For example, Figure $2.16$ shows two populations that have the same mean, median, and mode but different spreads.

Even though the mean, median, and mode of these two populations are the same, clearly, population I is much more spread out than population II. The density of population II is greater at the mean, which means that population II is more concentrated at this point than population I.

When describing the typical values in the population, the more variation there is in a population the harder it is to measure the typical value, and just as there are several ways of measuring the center of a population there are also several ways to measure the variation in a population. The three most commonly used parameters for measuring the spread of a population are the variance, standard deviation, and interquartile range. For a quantitative variable $X$

• the variance of a population is defined to be the average of the squared deviations from the mean and will be denoted by $\sigma^{2}$ or $\operatorname{Var}(X)$. The variance of a variable $X$

measured on a population consisting of $N$ units is
$$\sigma^{2}=\frac{\text { sum of all(deviations from } \mu)^{2}}{N}=\frac{\sum(X-\mu)^{2}}{N}$$

• the standard deviation of a population is defined to be the square root of the variance and will be denoted by $\sigma$ or $\operatorname{SD}(X)$.
$$\operatorname{SD}(X)=\sigma=\sqrt{\sigma^{2}}=\sqrt{\operatorname{Var}(X)}$$
• the interquartile range of a population is the distance between the 25 th and 75 th percentiles and will be denoted by IQR.
$$\mathrm{IQR}=75 \text { th percentile }-25 \text { th percentile }=X_{75}-X_{25}$$

## 统计代写|生物统计代写biostatistics代考|Proportions and Percentiles

p= 人口中具有特征 A 的单位数 ñ

## 统计代写|生物统计代写biostatistics代考|Parameters Measuring Centrality

• 总体的平均值是总体中所有单位的平均值，表示为μ. 变量的平均值X在由以下人员组成的总体上测量ñ单位是
μ= 的值的总和 Xñ=∑Xñ
• 人口的中位数是人口的第 50 个百分位，表示为μ~. 通过首先列出变量的所有值来找到总体的中位数X，包括重复X值，按升序排列。当人口中的单位数（即，ñ) 是奇数，中位数是 的有序值列表中的中间观察值X; 什么时候ñ是偶数，中位数将是有序列表中间的两个观察值的平均值X价值观。
• 人口的众数是人口中出现频率最高的值，记为米. 在概率密度函数图中，众数是X如图 2.8 所示，一个总体可以有多个众数。

## 统计代写|生物统计代写biostatistics代考|Measures of Dispersion

• 总体的方差定义为与均值的平方偏差的平均值，并表示为σ2或者曾是⁡(X). 变量的方差X

σ2= 所有的总和（偏离 μ)2ñ=∑(X−μ)2ñ

• 总体的标准差定义为方差的平方根，表示为σ或者标清⁡(X).
标清⁡(X)=σ=σ2=曾是⁡(X)
• 人口的四分位距是第 25 和第 75 个百分位数之间的距离，用 IQR 表示。
我问R=75 百分位数 −25 百分位数 =X75−X25

## 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|生物统计代写biostatistics代考|POPULATIONS AND VARIABLES

statistics-lab™ 为您的留学生涯保驾护航 在代写生物统计biostatistics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。我们的专家在代写生物统计biostatistics代写方面经验极为丰富，各种生物统计biostatistics相关的作业也就用不着说。

• 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|生物统计代写biostatistics代考|Qualitative Variables

Qualitative variables take on nonnumeric values and are usually used to represent a distinct quality of a population unit. When the possible values of a qualitative variable have no intrinsic ordering, the variable is called a nominal variable; when there is a natural ordering of the possible values of the variable, then the variable is called an ordinal variable. An example of a nominal variable is Blood Type where the standard values for blood type are $\mathrm{A}, \mathrm{B}, \mathrm{AB}$, and $\mathrm{O}$. Clearly, there is no intrinsic ordering of these blood types, and hence, Blood Type is a nominal variable. An example of an ordinal variable is the variable Pain where a subject is asked to describe their pain verbally as

• No pain,
• Mild pain,
• Discomforting pain,
• Distressing pain,
• Intense pain,
• Excruciating pain.
In this case, since the verbal descriptions describe increasing levels of pain, there is a clear ordering of the possible values of the variable Pain levels, and therefore, Pain is an ordinal qualitative variable.
Example 2.2
In the Framingham Heart Study of coronary heart disease, the following two nominal qualitative variables were recorded:
$$\text { Smokes }=\left{\begin{array}{l} \text { Yes } \ \text { No } \end{array}\right.$$
• and
• $$• \text { Diabetes }=\left{\begin{array}{l} • \text { Yes } \ • \text { No } • \end{array}\right. •$$
• Example $2.3$
• An example of an ordinal variable is the variable Baldness when measured on the Norwood-Hamilton scale for male-pattern baldness. The variable Baldness is measured according to the seven categories listed below:
• I Full head of hair without any hair loss.
• II Minor recession at the front of the hairline.
• III Further loss at the front of the hairline, which is considered “cosmetically significant.”
• IV Progressively more loss along the front hairline and at the crown.
• V Hair loss extends toward the vertex.
• VI Frontal and vertex balding areas merge into one and increase in size.
• VII All hair is lost along the front hairline and crown.
• Clearly, the values of the variable Baldness indicate an increasing degree of hair loss, and thus, Baldness as measured on the Norwood-Hamilton scale is an ordinal variable. This variable is also measured on the Offspring Cohort in the Framingham Heart Study.

## 统计代写|生物统计代写biostatistics代考|A quantitative variable

A quantitative variable is a variable that takes only numeric values. The values of a quantitative variable are said to be measured on an interval scale when the difference between two values is meaningful; the values of a quantitative variable are said to be measured on a ratio scale when the ratio of two values is meaningful. The key difference between a variable measured on an interval scale and a ratio scale is that on a ratio scale there is a “natural zero” representing absence of the attribute being measured, while there is no natural zero for variables measured on only an interval scale. Some scales of measurement will have natural zero and some will not. When a measurement scale has a natural zero, then the ratio of two measurements is a meaningful measure of how many times larger one value is than the other. For example, the variable Fat that represents the grams of fat in a food product is measured on a ratio scale because the value Fat $=0$ indicates that the unit contained absolutely no fat. When a scale of measurement does not have a natural zero, then only the difference between two measurements is a meaningful comparison of the values of the two measurements. For example, the variable Body Temperature is measured on a scale that has no natural zero since Body Temperature $=0$ does not indicate that the body has no temperature.

Since interval scales are ordered, the difference between two values measures how much larger one value is than another. A ratio scale is also an interval scale but has the additional property that the ratio of two values is meaningful. Thus, for a variable measured on an interval scale the difference of two values is the meaningful way to compare the values, and for a variable measured on a ratio scale both the difference and the ratio of two values are meaningful ways to compare difference values of the variable. For example, body temperature in degrees Fahrenheit is a variable that is measured on an interval scale so that it is meaningful to say that a body temperature of $98.6$ and a body temperature of $102.3$ differ by $3.7$ degrees; however, it would not be meaningful to say that a temperature of $102.3$ is $1.04$ times as much as a temperature of $98.6$. On the other hand, the variable weight in pounds is measured on a ratio scale, and therefore, it would be proper to say that a weight of $210 \mathrm{lb}$ is $1.4$ times a weight of $150 \mathrm{lb}$; it would also be meaningful to say that a weight of $210 \mathrm{lb}$ is $60 \mathrm{lb}$ more than a weight of $150 \mathrm{lb}$.

## 统计代写|生物统计代写biostatistics代考|Multivariate Data

In most research problems, there will be many variables that need to be measured. When the collection of variables measured on each unit consists of two or more variables, a data set is called a multivariate data set, and a multivariate data set consisting of only two variables is called a bivariate data set. In a multivariate data set, there is usually one variable that is of primary interest to a research question that is believed to be explained by some of the other variables measured in the study. The variable of primary interest is called a response variable and the variables believed to cause changes in the response are called explanatory variables or predictor variables. The explanatory variables are often referred to as the input variables and the response variable is often referred to as the output variable. Furthermore, in a statistical model, the response variable is the variable that is being modeled; the explanatory variables are the input variables in the model that are believed to cause or explain differences in the response variable. For example, in studying the survival of melanoma patients, the response variable might be Survival Time that is expected to be influenced by the explanatory variables Age, Gender, Clark’s Stage, and Tumor Size. In this case, a model relating Survival Time to the explanatory variables Age, Gender, Clark’s Stage, and Tumor Size might be investigated in the research study.

A multivariate data set often consists of a mixture of qualitative and quantitative variables. For example, in a biomedical study, several variables that are commonly measured are a subject’s age, race, gender, height, and weight. When data have been collected, the multivariate data set is generally stored in a spreadsheet with the columns containing the data on each variable and the rows of the spreadsheet containing the observations on each subject in the study.

In studying the response variable, it is often the case that there are subpopulations that are determined by a particular set of values of the explanatory variables that will be important in answering the research questions. In this case, it is critical that a variable be included in the data set that identifies which subpopulation each unit belongs to. For example, in the National Health and Nutrition Examination Survey (NHANES) study, the distribution of the weight of female children was studied. The response variable in this study was weight and some of the explanatory variables measured in this study were height, age, and gender. The result of this part of the NHANES study was a distribution of the weights of females over a certain range of age. The resulting distributions were summarized in the chart given in Figure $2.2$ that shows the weight ranges for females for several different ages.

## 统计代写|生物统计代写biostatistics代考|Qualitative Variables

• 不痛，
• 轻微的疼痛，
• 令人不适的疼痛，
• 让人心疼的痛，
• 剧烈的疼痛，
• 难以忍受的疼痛。
在这种情况下，由于口头描述描述了疼痛程度的增加，因此变量疼痛水平的可能值有一个明确的顺序，因此，疼痛是一个有序的定性变量。
例 2.2
在冠心病的弗雷明汉心脏研究中，记录了以下两个名义上的定性变量：
$$\text { Smokes }=\left{ 是的 不 \正确的。$$
• $$• \text { 糖尿病 }=\left{\begin{array}{l} • \文本{是} \ • \文本{没有} • \end{数组}\对。 •$$
• 例子2.3
• 序数变量的一个例子是变量 Baldness，当用 Norwood-Hamilton 量表测量男性型秃发时。变量秃头根据以下列出的七个类别进行测量：
• 我满头的头发没有任何脱发。
• II 发际线前部的轻微后退。
• III 发际线前部的进一步损失，这被认为是“具有美容意义的”。
• IV 沿着前发际线和头顶逐渐减少。
• V 脱发向顶点延伸。
• VI 前额和头顶秃发区域合并为一个并增加大小。
• VII 所有的头发都沿着前发际线和头顶脱落。
• 显然，变量秃头的值表明脱发程度的增加，因此，在诺伍德-汉密尔顿量表上测量的秃头是一个序数变量。这个变量也在弗雷明汉心脏研究的后代队列中测量。

## 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|生物统计代写biostatistics代考|DESCRIBING POPULATIONS

statistics-lab™ 为您的留学生涯保驾护航 在代写生物统计biostatistics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。我们的专家在代写生物统计biostatistics代写方面经验极为丰富，各种生物统计biostatistics相关的作业也就用不着说。

• 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|生物统计代写biostatistics代考|The Phases of a Clinical Trial

Clinical research is often conducted in a series of steps, called phases. Because a new drug, medicine, or treatment must be safe, effective, and manufactured at a consistent quality, a series of rigorous clinical trials are usually required before the drug, medicine, or treatment can be made available to the general public. In the United States the FDA regulates and oversees the testing and approval of new drugs as well as dietary supplements, cosmetics, medical devices, blood products, and the content of health claims on food labels. The approval of a new drug by the FDA requires extensive testing and evaluation of the drug through a series of four clinical trials, which are referred to as phase $I, I I, I I I$, and $I V$ trials.
Each of the four phases is designed with a different purpose and to provide the necessary information to help biomedical researchers answer several different questions about

a new drug, treatment, or biomedical procedure. After a clinical trial is completed, the researchers use biostatistical methods to analyze the data collected during the trial and make decisions and draw conclusions about the meaning of their findings and whether further studies are needed. After each phase in the study of a new drug or treatment, the research team must decide whether to proceed to the next phase or stop the investigation of the drug/treatment. Formal approval of a new drug or biomedical procedure generally cannot be made until a phase III trial is completed and there is strong evidence that the drug/treatment is safe and effective.

The purpose of a phase $I$ clinical trial is to investigate the safety, efficacy, and side effects of a new drug or treatment. Phase I trials usually involve a small number of subjects and take place at a single or only a few different locations. In a drug trial, the goal of a phase I trial is often to investigate the metabolic and pharmacologic actions of the drug, the efficacy of the drug, and the side effects associated with different dosages of the drug. Phase I drug trials are also referred to as dose finding trials.

## 统计代写|生物统计代写biostatistics代考|POPULATIONS AND VARIABLES

In a properly designed biomedical research study, a well-defined target population and a particular set of research questions dictate the variables that should be measured on the units being studied in the research project. In most research problems, there are many variables

that must be measured on each unit in the population. The outcome variables that are of primary interest are called the response variables, and the variables that are believed to explain the response variables are called the explanatory variables or predictor variables. For example, in a clinical trial designed to study the efficacy of a specialized treatment designed to reduce the size of a malignant tumor, the following explanatory variables might be recorded for each patient in the study: age, gender, race, weight, height, blood type, blood pressure, and oxygen uptake. The response variable in this study might be change in the size of the tumor.

Variables come in a variety of different types; however, each variable can be classified as being either quantitative or qualitative in nature. A variable that takes on only numeric values is a quantitative variable, and a variable that takes on non-numeric values is called a qualitative variable or a categorical variable. Note that a variable is a quantitative or qualitative variable based on the possible values the variable can take on.
Example $2.1$
In a study of obesity in the population of children aged 10 or less in the United States, some possible quantitative variables that might be measured include age, height, weight, heart rate, body mass index, and percent body fat; some qualitative variables that might be measured on this population include gender, eye color, race, and blood type. A likely choice for the response variable in this study would be the qualitative variable Obese defined by
$$\text { Obese }= \begin{cases}\text { Yes } & \text { for a body mass index of }>30 \ \text { No } & \text { for a body mass index of } \leq 30\end{cases}$$

## 统计代写|生物统计代写biostatistics代考|Qualitative Variables

Qualitative variables take on nonnumeric values and are usually used to represent a distinct quality of a population unit. When the possible values of a qualitative variable have no intrinsic ordering, the variable is called a nominal variable; when there is a natural ordering of the possible values of the variable, then the variable is called an ordinal variable. An example of a nominal variable is Blood Type where the standard values for blood type are $\mathrm{A}, \mathrm{B}, \mathrm{AB}$, and $\mathrm{O}$. Clearly, there is no intrinsic ordering of these blood types, and hence, Blood Type is a nominal variable. An example of an ordinal variable is the variable Pain where a subject is asked to describe their pain verbally as

• No pain,
• Mild pain,
• Discomforting pain,
• Distressing pain,
• Intense pain,
• Excruciating pain.
In this case, since the verbal descriptions describe increasing levels of pain, there is a clear ordering of the possible values of the variable Pain levels, and therefore, Pain is an ordinal qualitative variable.
Example 2.2
In the Framingham Heart Study of coronary heart disease, the following two nominal qualitative variables were recorded:
$$\text { Smokes }=\left{\begin{array}{l} \text { Yes } \ \text { No } \end{array}\right.$$
• and
• $$• \text { Diabetes }=\left{\begin{array}{l} • \text { Yes } \ • \text { No } • \end{array}\right. •$$

## 统计代写|生物统计代写biostatistics代考|POPULATIONS AND VARIABLES

肥胖 ={ 是的  对于体重指数 >30  不  对于体重指数 ≤30

## 统计代写|生物统计代写biostatistics代考|Qualitative Variables

• 不痛，
• 轻微的疼痛，
• 令人不适的疼痛，
• 让人心疼的痛，
• 剧烈的疼痛，
• 难以忍受的疼痛。
在这种情况下，由于口头描述描述了疼痛程度的增加，因此变量疼痛水平的可能值有一个明确的顺序，因此，疼痛是一个有序的定性变量。
例 2.2
在冠心病的弗雷明汉心脏研究中，记录了以下两个名义上的定性变量：
$$\text { Smokes }=\left{ 是的 不 \正确的。$$
• $$• \text { 糖尿病 }=\left{\begin{array}{l} • \文本{是} \ • \文本{没有} • \end{数组}\对。 •$$

## 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。