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## 统计代写|多尺度模型代写Multilevel Models代考|The adequacy of ordinary least squares estimates

When a variance partition coefficient is small, we can expect reasonably good agreement between the multilevel estimates and the simpler OLS ones. While it is difficult to give general guidelines about when OLS is an adequate alternative, we can readily derive an explicit formula for the balanced 2-level variance component model using a simple regression equation with an intercept and a single explanatory variable
$$y_{i j}=\beta_0+\beta_1 x_{i j}+u_j+e_{i j}$$
Write $\rho_y \rho_x$ for the intra-unit correlations for $Y, X$ respectively and $n$ for the number of level 1 units in each level 2 unit. To obtain an estimate of the correct standard error for the estimate of $\beta_1$, we multiply the usual OLS estimate of the standard error by the quantity
$$\left{1+\rho_y \rho_x(n-1)\right}^{1 / 2}$$

## 统计代写|多尺度模型代写Multilevel Models代考|A 2-level example using longitudinal educational achievement data

We shall fit the simple 2-level variance components model (2.7) to the JSP data with the maths score at age 11 as the response and a single explanatory variable, the maths score at age 8 , in addition to the constant term, equal to 1 and defining the intercept. The parameter values are displayed in Table $2.1$ with the ordinary least squares estimates given for comparison.

Comparing the OLS with the multilevel estimates, we see that the fixed coefficients are similar, but that there is a variance partition coefficient value of $0.14$. The estimate of the standard error of the between school variance is less than a third of the variance estimate, suggesting a value highly significantly different from zero ${ }^2$. This comparison, however, should be treated cautiously, since the variance estimate does not have a normal distribution and the standard error is only estimated, although the size of the sample here will make the latter caveat less important. It is generally preferable to carry out a likelihood ratio test by estimating the ‘deviance’ for the current model and the model omitting the level 2 variance (see McCullagh and Nelder, 1989) and the next section will deal more generally with inference procedures. The difference between the deviances is $63.1$. This value would normally be referred to tables of the chi-squared distribution with one degree of freedom, and is highly significant. In the present case, however, the null hypothesis of a zero variance is on the ‘boundary’ of the feasible parameter space; we do not envisage a negative variance. In this case, the P-value to be used is half the one obtained from the tables of the chi-squared distribution (Shapiro, 1985). Note that if we use the standard error estimate given in Table $2.1$ to judge significance we obtain the corresponding value of $(3.19 / 1.0)^2=10.2$ which, in this case, is very much smaller than the likelihood ratio test statistic. Note also that if we use this test we would again use half the nominal P-value since only positive departures are possible.

A similar issue arises when simultaneously testing several parameters where one or more is constrained in this way, such as a variance plus covariances. In this case, the appropriate test statistic is a weighted mixture of chi-squared distributions and details are given by Shapiro (1985). In the common case where we are testing a set of covariances and a single associated variance, this reduces to the following.

Suppose that we have $r$ covariance parameters and adopt a $5 \%$ significance level. Then the critical value, $c$, for judging significance is given by the following formula for a mixture of two chi-squared distributions
$$0.5 \times\left[\operatorname{pr}\left(\chi_r^2 \geq c\right)+\operatorname{pr}\left(\chi_{r+1}^2 \geq c\right)\right]=0.05$$

## 统计代写|多尺度模型代写Multilevel Models代考|The adequacy of ordinary least squares estimates

$$y_{i j}=\beta_0+\beta_1 x_{i j}+u_j+e_{i j}$$

\left 缺少或无法识别的分隔符

## 统计代写|多尺度模型代写Multilevel Models代考|A 2-level example using longitudinal educational achievement data

$$0.5 \times\left[\operatorname{pr}\left(\chi_r^2 \geq c\right)+\operatorname{pr}\left(\chi_{r+1}^2 \geq c\right)\right]=0.05$$

## 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。

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## 统计代写|多尺度模型代写Multilevel Models代考|Levels of aggregation and ecological fallacies

When studying relationships among variables, there has often been controversy about the appropriate ‘unit of analysis’. We have alluded to this already in the context of ignoring hierarchical data clustering and, as we have seen, the issue is resolved by explicit hierarchical modelling.

One of the best known early illustrations of what is often known as the ecological or aggregation fallacy was the study by Robinson (1950) of the relationship between literacy and ethnic background in the United States. When the mean literacy rates and mean proportions of Black Americans for each of nine census divisions are correlated the resulting value is $0.95$, whereas the individual-level correlation ignoring the grouping is $0.20$. Robinson was concerned to point out that aggregate-level relationships could not be used as estimates for the corresponding individual-level relationships and this point is now well understood. In Chapter 3 , we discuss some of the statistical consequences of modelling only at the aggregate level.

Sometimes the aggregate level is the principal level of interest, but nevertheless a multilevel perspective is useful. Consider the example (Derbyshire, 1987) of predicting the proportion of children socially ‘at risk’ in each local administrative area for the purpose of allocating central government expenditure on social services. Survey data are available for individual children with information on risk status so that a prediction can be made using area based variables as well as child and household based variables. The probability $(\pi)$ of a child being ‘at risk’ was estimated by the following (single level) equation
$$\operatorname{logit}(\pi)=-6.3+5.9 x_1+2.2 x_2+1.5 x_3$$
where $x_1$ is the proportion of children in the area in households with a lone parent, $x_2$ is the proportion of households in each area which have a density of more than $1.5$ persons per room and $x_3$ is the proportion of households whose ‘head’ was born in the British ‘New Commonwealth’ or Pakistan. All these explanatory variables are measured at the aggregate area level and the response is the proportion of children at risk in each area. Although we can regard this analysis as taking place entirely at the area level (with suitable weighting for the number of children in each area), there are advantages in thinking of it as a 2-level model with each child being a level 1 unit and the response variable being the binary response of whether or not the child is at risk.

## 统计代写|多尺度模型代写Multilevel Models代考|Causality

In the natural sciences, experimentation has a dominant position when making causal inferences. This is both because the units of interest can be manipulated experimentally, typically using random allocation, and because there is a widespread acceptance that the results of experiments are generalisable over space and time. The models described in this book can be applied to experimental or non-experimental data, but the final causal inferences can differ. Nevertheless, most of the examples used are from non-experimental studies in the human sciences and a few words on causal inferences from such data may be useful.

If we wish to answer questions about a possible causal relationship between, say, class size and educational achievement, an experimental study would need to assign different numbers of level 1 units (students) randomly to level 2 units (classes or teachers) and study the results over a time period of several years. This would be time consuming and could create ethical problems. In addition to such practical problems, any single study would be limited in time and place, and require extensive replication before results could be generalised confidently. The specific context of any study is important; for example, the state of the educational system and the resources available at the time of the study. The difficulty from an experimental viewpoint is that it is practically impossible to allocate randomly with respect to all such possible confounding factors.

A further limitation of randomised controlled trails (RCTs) is that they cannot necessarily deal with situations where the composition of a higher level unit interacts with the treatment of interest, to affect the responses of lower level units. Thus, in schooling studies the size of class may affect the progress of students only when the proportion of ‘low achieving’ students is above a certain threshold. Randomisation will tend to eliminate classes with extreme proportions so that such effects may not be discovered. Goldstein (1998) looks at this case in more detail.

## 统计代写|多尺度模型代写Multilevel Models代考|Levels of aggregation and ecological fallacies

$$\operatorname{logit}(\pi)=-6.3+5.9 x_1+2.2 x_2+1.5 x_3$$

## 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。

Posted on Categories:Multilevel Models, 多尺度模型, 数据科学代写, 统计代写, 统计代考

## avatest™帮您通过考试

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## 统计代写|多尺度模型代写Multilevel Models代考|Event history and survival models

Modelling the time spent in various states or situations is important in a number of areas. In industry the ‘time to failure’ of components is a key factor in quality control. In medicine, the survival time is a fundamental measurement in studying certain diseases. In economics, the duration of employment periods is of great interest. In education, researchers often study the time students spend on different tasks or activities.
In studying employment histories, any one individual will generally pass through several periods of employment or unemployment, while at the same time changing his characteristics, for example his level of qualifications. From a modelling point of view we need to consider the length of time spent in each type of employment, relating this both to constant factors such as an individual’s social origins or gender, and to changing or time dependent factors such as qualifications and age. In this case the multilevel structure is analogous to that for repeated measures data, with periods taking the place of occasions. Furthermore, generally we would have a further, higher level of the hierarchy, since individuals, which are the level 2 units, are themselves typically clustered into workplaces, which now constitute level 3 units. ${ }^1$ The structure may be even more complicated if these workplaces change from period to period; to include this level in our model, we need to consider cross-classifications of the units (see below). There are particular problems that arise when studying event duration data that are encountered when some information is ‘censored’ in the sense that instead of being able to observe the actual duration we only know that it is longer than some particular value, or in some cases less than a particular value (see Chapter 11).

## 统计代写|多尺度模型代写Multilevel Models代考|Discrete response data

Until now, we have assumed implicitly that our response or dependent variable is continuously distributed; for example, an exam score or anthropometric measure such as height. Many kinds of statistical models, however, deal with categorised responses, in the simplest case with proportions. Thus, we might be interested in a mortality rate, or an examination pass rate and how these vary from area to area or from school to school.

In studying mortality rates in a population, it is often of great concern to try to understand the factors associated with variations from area to area or community to community. This produces a basic 2-level structure with individuals at level 1 and communities at level 2. A typical study might record deaths over a given time period together with the characteristics of the individuals concerned, and level 2 characteristics of the communities, such as their sizes or social compositions. One analysis of interest would be to see whether any of these explanatory variables could explain betweencommunity variation. Another interest might be in studying whether mortality rate differences, say between men and women, varied from community to community.
Such models, part of the class known as generalised linear models, have been available for some time for single level data (McCullagh and Nelder, 1989), with associated software. In Chapter 4 we show how to fit multilevel models with different types of categorical response. Chapter 7 extends this to consider multivariate models with mixtures of different response types.

## 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。