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# 统计代写|回归分析代写Regression Analysis代考|Practical Versus Statistical Significance

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## 统计代写|回归分析代写Regression Analysis代考|Practical Versus Statistical Significance

While the Car Sales example above is a case where statistically significant $(p<0.05)$ curvature corresponds to practically significant (which means practically important) curvature, it is not always the case that statistical significance corresponds to practical significance. This can easily happen in data sets where $n$ is large (e.g., with “big data”), because with large data sets you have the ability to estimate even slight curvature very precisely, with small standard error.

The following simulation illustrates this situation: There is statistically significant ( $p=0.000166$ ) curvature, as shown by the hypothesis test for the quadratic term. However, the curvature is practically insignificant, as can be seen by graphing the linear and quadratic fitted functions. The sample size in this simulation is large, $n=1,000,000$, but not unusually large for “big data” applications.

4.5.1 Simulation Study to Demonstrate Practical vs. Statistical Significance
set.seed(54321) # For perfect replicability of the random simulation.
$\mathrm{x}=10+2 * \operatorname{rnorm}(1000000) ; \mathrm{xsq}=\mathrm{x}^{\wedge} 2$
$\mathrm{y}=2+.6 * \mathrm{x}+.003 * x$ sq $+4 * \operatorname{rnorm}(1000000)$ #beta2 $=.003$ does not equal 0 !
fit. quad $=\operatorname{lm}(\mathrm{y} \sim \mathrm{x}+\mathrm{xsq})$
summary(fit.quad) # Significant curvature: p-value $=0.000166$
## A . 1\% random sample from the data set is selected to make the scatterplot
## more legible. Otherwise, the points are too dense to view.
select $=\operatorname{runif}(1000000) ; \mathrm{x} 1=\mathrm{x}$ [select<.001] ; $\mathrm{y} 1=\mathrm{y}[$ select<.001]
plot (x1, y1, main = “Scatterplot of a $0.1 \%$ Subsample”)
abline(lsfit( $x, y)$, col=”gray”)

Figure 4.5 shows that the linear fit (solid line) is adequate, even though the true model is quadratic, and not linear.

## 统计代写|回归分析代写Regression Analysis代考|Evaluating the Constant Variance (Homoscedasticity) Assumption Using Graphical Methods

The first graph you should use to evaluate the constant variance assumption is the $\left(\hat{y}_i, e_i\right)$ scatterplot. Look for changes in the pattern of vertical variability of the $e_i$ for different $\hat{y}_i$. The most common indications of constant variance assumption violation are shapes that indicate either increasing variability of $Y$ for larger $\mathrm{E}(Y \mid X=x)$, or shapes that indicate decreasing variability of $Y$ for larger $\mathrm{E}(Y \mid X=x)$. Increasing variability of $Y$ for larger $\mathrm{E}(Y \mid X=x)$ is indicated by greater variability in the vertical ranges of the $e_i$ when $\hat{y}_i$ is larger.
Recall again that the constant variance assumption (like all assumptions) refers to the data-generating process, not the data. The statement “the data are homoscedastic” makes no sense. By the same logic, the statements “the data are linear” and “the data are normally distributed” also are nonsense. Thus, whichever pattern of variability that you decide to claim based on the $\left(\hat{y}_i, e_i\right)$ scatterplot, you should try to make sense of it in the context of the subject matter that determines the data-generating process. As one example, physical boundaries on data force smaller variance when the data are closer to the boundary. As another, when income increases, people have more choice as to whether or not they choose to purchase an item. Thus, there should be more variability in expenditures among people with more money than among people with less money. Whatever pattern you see in the $\left(\hat{y}_i, e_i\right)$ scatterplot should make sense to you from a subject matter standpoint.

While the LOESS smooth to the $\left(\hat{y}_i, e_i\right)$ scatterplot is useful for checking the linearity assumption, it is not useful for checking the constant variance assumption. Instead, you should use the LOESS smooth over the plot of $\left(\hat{y}_i,\left|e_i\right|\right)$. When the variability in the residuals is larger, they will tend to be farther from zero, giving larger mean absolute residuals $\left|e_i\right|$. An increasing trend in the $\left(\hat{y}_i,\left|e_i\right|\right)$ plot suggests larger variability in $Y$ for larger $\mathrm{E}(Y \mid X=x)$, and a flat trend line for the $\left(\hat{y}_i,\left|e_i\right|\right)$ plot suggests that the variability in $Y$ is nearly unrelated to $\mathrm{E}(Y \mid X=x)$. However, as always, do not over-interpret. Data are idiosyncratic (random), so even if homoscedasticity is true in reality, the LOESS fit to the $\left(\hat{y}_i,\left|e_i\right|\right)$ graph will not be a perfectly flat line, due to chance alone. To understand “chance alone” in this case you can simulate data from a homoscedastic model, construct the $\left(\hat{y}_i, \mid e_i\right)$ graph, and add the LOESS smooth. You will see that the LOESS smooth is not a perfect flat line, and you will know that such deviations are explained by chance alone.

The hypothesis test for homoscedasticity will help you to decide whether the observed deviation from a flat line is explainable by chance alone, but recall that the test does not answer the real question of interest, which is “Is the heteroscedasticity so bad that we cannot use the homoscedastic model?” (That question is best answered by simulating data sets having the type of heteroscedasticity you expect with your real data, then by performing the types of analyses you plan to perform on your real data, then by evaluating the performance of those analyses.)

## 统计代写|回归分析代写Regression Analysis代考|Practical Versus Statistical Significance

4.5.1模拟研究以证明实际意义与统计意义
set.seed(54321) ＃用于随机模拟的完美可复制性。
$\mathrm{x}=10+2 * \operatorname{rnorm}(1000000) ; \mathrm{xsq}=\mathrm{x}^{\wedge} 2$
$\mathrm{y}=2+.6 * \mathrm{x}+.003 * x$平方$+4 * \operatorname{rnorm}(1000000)$ ＃beta2 $=.003$不等于0 !

summary(fit.quad) ＃显著曲率:p值$=0.000166$
＃＃ a。从数据集中随机抽取1％的样本制作散点图
＃＃更清晰。否则，点太密集而无法查看。

plot (x1, y1, main =“$0.1 \%$子样本的散点图”)
Abline (lsfit($x, y)$, col=”gray”)

## 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。