第二十四篇 Influence and Effect
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本文翻译：chengguo0740，xy_persist
Know how to measure the effect a data point has on a statistic知道如何估计一个数据点在统计量上的影响
by Robert L. Mason and John C. Young
罗伯特L.梅森和约翰C.杨编写
Being able to determine the effect a data point has on summary statistics provides useful insight into the construction of better parameter estimators.
能确定一个数据点在概括统计的影响为更好的参数估计的构建提供了有用的见解
Consider, for example, a person with an annual income of more than $1 million in a room with nine others with annual incomes in the $25,000 to $50,000 range. The average income for this group of 10 is greater than $100,000, but this is not a useful summary statistic of the typical income of most members of the group.
比如，考虑一个年收入超过1百万美元的人和9个年收入在25,000和50,000美元之间的人在同一间屋子里，这个10人组的平均收入比100,000美元多很多，但这不是一个该组大多数成员代表性收入的有用的概括统计量。
From this example, you observe that the inclusion of an observation far removed from the bulk of the observations in a sample can have a great effect on the estimated overall mean. If included, the outlying observation actually pulls the average value of the group toward it.
从这个例子，你注意到在一个样本中那个远离绝大部分、包含了特殊数据的观察值会对总体的平均值有很大影响。如果被包括在内，这个偏离的观察值确实会拉动组的平均值靠近该值。
In another example, consider the group of bivariate observations contained in the circle in Figure 1. Such a circular region of data points indicates that the correlation between the two variables, x1 and x2, is close to zero; that is, no linear relationship exists between the two variables.
再举另外一个例子，考虑两变量的组被包含在图1的圆圈中。数据点的圆形区域指出两组变量的相互关系。X1和X2接近零点，也就是说，两个变量之间没有线性关系。
Figure 1 图表1 用一个单个的观察值来增加相互的关系
Observe the labeled point in the upper right-hand corner of the plot but outside of the circle. As the distance between this point and the mean of the group of points in the circle increases along the drawn 45° line, the correlation between the two variables will increase and approach its maximum value of one. Thus, this single outlying observation can distort the estimated value of the true correlation.
观察在圆的右上角但是在圆外部的那个标记的点。由于在这个点以及圆内点群的平均值之间的距离沿着图中45°线增加，这两个变量的相互关系也将增加并且接近两者之中的最大值。因此，单个的、偏离的观察值能歪曲真正关系的估计值。
The variance, σ2, of a variable x is defined as the average of the squared deviation of that variable from its population mean, µ. Consequently, the square of the distance that an outlying observation is from its mean, that is, (x – µ)2, can have a great impact on the estimated value of the variance parameter.
变量X的方差σ2被定义为X偏离其总体均值的平方。所以，一个偏离的观察值与其均值距离的平方，就是说（X-X均值）2会对方差参数估计值有很大的影响
For example, including the outlying point in Figure 1 with the circular group of points in the plot will increase the variances of x1 and x2. This occurs because the outlying point causes the data to be spread wider in both dimensions.
In two dimensions, scatter plots can be constructed to show how one or more data points can change the estimate of the means, the variances and the correlation coefficient between the variables.
例如，在图表1中包含了偏远点的圆形点群将会增加X1和X2的方差。这种情况的发生是因为远离的点导致数据向两个范围扩展。在这两个范围中，可以建立散点图来说明一个或多个数据点怎样改变均值的估计值、方差和两变量间的相关系数。
For example, Figure 2 contains four observations, labeled A, B, C and D, which are removed from the bulk of the data enclosed in the ellipse. The inclusion of points A or C will not affect the correlation coefficient because both support the linear trend in the data.
例如，图表2中包含4个观察值，标记为A, B, C 和D，这几个值与椭圆围住的绝大部分数据远离。包含了点A和C的数据不会影响相关系数，因为这两点都支持线性趋势。
Figure 2 图表2 四个观察值将会改变均值、方差和/或相互关系
Including point A, however, will increase the variances and decrease the means of x1 and x2, while including point C will increase the variances and the means of x1 and x2. The inclusion of points B or D will affect the correlation coefficient between the two variables because both lie in directions opposite of the linear trend of the data. In addition, including point B will decrease the mean but increase the variance of x1, while including point D will increase the mean and variance of x1.
包含点A将增加方差而且减少X1和X2的均值；然而包含点C将增加方差以及X1和X2的均值。包含点B或D将会影响两变量间的相关系数因为这两个点都在线性数据趋势的相反方向。另外，包含点B将减少均值但是增加X1的方差，包含点D将增加均值以及X1的方差。
The influence function 影响函数
Several mathematical procedures exist for determining the effect an observation has on these particular estimates. One popular procedure1 used in developing robust statistical estimators is based on developing an influence curve or influence function for use as a measure of the effect that an observation has on the parameter being estimated.
几种数学上传统的做法用来决定一个观察值在这些特定的估计上的影响。一种被用于发展健全统计估计的流行程序是建立在一条影响曲线或者影响函数，（这种曲线或函数）用来作为一个观察值对被估计参数影响的一种测量。
When applied to the mean, the influence function is exactly what you would expect: a measure of the difference between an observation and the mean:
当被运用于均值的时候，这种影响函数恰恰就是你所期望的：测量观察值和均值间的差异。
x – µ. Likewise, when applied to the variance, you obtain the expected answer that the influence function is the squared distance between an observation and the mean: (x – µ)2 – σ2.
x – µ同样地，当被运用于方差，你得到的期望中的答案是：影响函数是观察值和均值间差值的平方σ2
When applied to the sample correlation coefficient, r, between two variables, x1 and x2, the contours of the influence function are a set of hyperbolae given by the formula
当被运用于样本相关系数，在X1和X2两变量之间，影响函数的轮廓是一套公式规定的双曲线。
in which y1 and y2 are the studentized values of x1 and x2, and c is a chosen constant value. The selection of the value of c for drawing these contours is arbitrary (and chosen to include the bulk of the points), but nevertheless serves to identify observations removed from the data swarm.
在（上面的公式中）y1和y2是变量X1 和X2的观测值，c是一个常数。为了画出这些轮廓，c的数值的选择是随意的（被选择的值包括绝大部分数据点），仅仅为识别观察值远离数据群提供服务。
Superimposing these hyperbolic contours over the corresponding scatter plot for y1 and y2 allows you to determine which observations are having the greatest effect on the estimate of the correlation coefficient. Points inside the hyperbolae will influence function values greater that +c or less than –c. Points outside the hyperbolae will have influence function values between –c and +c. Figure 3 shows an example of these contours for a case in which c = ± 2.7 and r = 0.81.
将这些双曲线轮廓叠加在对应y1和y2的散点图上能使你确定哪些观察值对估计相关系数有最大的影响。双曲线内部
的点将影响函数值大于+c或者小于-c。双曲线外部的点将使函数值在-c和+c之间。图表3是一个当c = ± 2.7 及 r = 0.81时双曲线（轮廓）的例子展示：
Figure 3 相关系数的影响函数图
Detailed procedures exist for interpreting the data points in relation to the contour plots.2,3 Those points located on the side of the data swarm but inside the hyperbolae, such as point A in Figure 3, will decrease the value of the correlation coefficient. Those points located within the hyperbola on the ends of the data swarm, such as points B and C in Figure 3, will increase the correlation coefficient.
有详细的步骤来解释数据点与轮廓图的关系，这些点在数据群的一侧但是在双曲线的里面，例如图表3中的点A，就将减少相关系数值。那些位于双曲线内部、在数据群末端的点，例如图表中的点B和点C，将增加相关系数。
Influence functions also can be used for detecting outliers in a bivariate sample.4 For example, any point located within the hyperbolae, such as points A, B and C in Figure 3, are subject to removal. In addition, the influence function value in (1) can be computed for any other observation in the sample.
影响函数也能应用于查明在一个双变量样本中的远离点。例如，位于双曲线内部的任何一点，比如图表3中的点A、B和点C，（它们）倾向于远离（双曲线）。另外，在（1）中的影响函数值也能被用来估算在样本中的其他观察数据。
Influence function example 影响函数例子
Figure 4 contains the scatter plot of 212 observations selected at random from a bivariate normal distribution in which the variables are standardized with a correlation coefficient of 0.812. In this form, the correlation is the same as the covariance between the two variables. These observations are represented in Figure 4 by the points within the ellipse, excluding the one labeled point 2.
图表4包含了从一个双变量、方差是标准差、相关系数为0.812的标准分布中随机选取的212个观察值的散点图。在这种形式中，相关系数与两变量间的协方差相同。在图表4中这些观察值用椭圆内部的点表示，除了那个被标记出的点2.
Figure 4 图表4 数据散点图、椭圆和双曲线
For illustrative purposes, two additional observations, point 1 with coordinates (-2, 2) and point 2 with coordinates (1, 1), have been added to the plot in Figure 4. Point 1 is outside the data swarm and inside the hyperbolae, indicating it could be an outlier and could possibly have influence on the computation of sample statistics.作为解说性的目的，两个另外的观察值，坐标为(-2, 2)的点1及坐标为(1, 1)的点2也被加入到图表4中。点1在数据群的外侧但是在双曲线内侧，指出它是一个远离点并且可能对估算样本统计量有影响。
This is verified in Table 1 by comparing the sample correlation coefficient values obtained with and without this point (while ignoring point 2 Including point 1 decreases the pairwise correlation between the two variables from 0.812 to 0.776. In addition, Table 1 includes the effect on the sample means and variances of y1 and y2. For y1, the absolute value of the mean increases and the standard deviation slightly increases when point 1 is included. For y2, the absolute value of the mean decreases, but the standard deviation slightly increases when point 1 is included.
这个在列表1中通过对比含有这个点以及不含这个点的样本相关系数值证实。然而忽略点2而包括点1。两变量间成对的相互关系从0.812减少到0.776. 另外，列表1包含了对y1和y2样本均值和方差的影响。对于y1，当包括点1时，均值的绝对值增大且标准偏差略有增大。对于y2，均值的绝对值减小，但是当点1被包含的时候标准偏差略有增加。
Table 1 列表1 点2的概括统计量
The coordinates of point 1 are (-2, 2 )Including this point in the original sample, the value of the correlation coefficient is r = 0.776. These coordinates and the correlation coefficient are needed to compute the value of the influence function for point 1 using the earlier equation. The computed value is -7.1, which is less than the chosen value of c = -2.7. This result independently confirms what we see in Figure 4, namely that point 1 is inside the hyperbolae and has a decreasing effect on the correlation coefficient.
点1的坐标是（-2,2）原来的样本中包含这个点，相关系数值r = 0.776。这些坐标和相关系数对于用最早的方程式来计算点1的影响函数值是必需的。计算值是-7.1，这个数值小于选定的常数c = -2.7。这个结果独立地证明我们在列表4中看到的，即点1在双曲线内部并且对相关系数有减少方面的影响。
In contrast, point 2, with coordinates (1, 1), is contained within the data swarm (elliptical region) and outside of the hyperbolae in Figure 4. Thus, this point should have minimal effect on the sample estimates. This is confirmed when examining the results in Table 2.
相比之下，在图表4中，点2，坐标为(1, 1)，被包含在数据群内部（椭圆地带）且在双曲线外侧。因此这个点可能对样本估计有极微小的影响。这个会在列表2中检查结果的时候得到确认。
Table 2 列表2 点1的概括统计量
When the summary statistics and the correlations are computed with and without this point (while excluding point 1), small differences are noted in all of the statistics. These results also are confirmed by the small value of the influence function for this point. The computed value using the earlier equation is 0.2, which is between –c = -2.7 and +c = +2.7, and is therefore outside the hyperbolae in Figure 4.
当在包含和不包含这个点估计概括统计量和相互关系时（当不包括点1时），在所有的统计量中，小的差异是显著的。这些结果也被这个点影响函数的小的估计值所证明。用最早的方程式的计算值是0.2，这个值在–c = -2.7 和+c = +2.7之间，因此在图表4中双曲线的外部。
As can be seen from these examples, the influence function of a statistic is a key component in robust estimation because it helps you assess the influence that an observation has on the estimation of a statistic. It also is important in detecting outliers in the data.
从这些例子中我们可以看出，在完全估计中，统计量的影响函数是一个关键的组成部分。因为它帮助你评估一个观察值对统计量估计的影响。它在发现数据中远离的点上也是很重要的。
References 参考文献
Robert L. Mason is an institute analyst at Southwest Research Institute in San Antonio. He has a doctorate in statistics from Southern Methodist University in Dallas and is a fellow of ASQ and the American Statistical Association.
罗伯特L.曼森是圣地亚哥西南部研究机构的一位学会研究员。它在南卫理公会大学获得统计学博士学位，并且是ASQ和美国统计协会的研究员。
John C. Young is a retired statistics professor at McNeese State University in Lake Charles, LA. He received a doctorate in statistics from Southern Methodist University.
约翰C.杨是美国莱克查尔斯州麦克尼斯州立大学退休的一位统计学教授，他在南卫理公会大学获得统计学博士学位。
你好，我是小编H。请对以下文章有校稿兴趣的组员留下你的预计完成时间，并发短信息联系小编H，以便小编登记翻译者信息以及文章最终完成时的奖惩工作。
本文翻译：chengguo0740，xy_persist
Know how to measure the effect a data point has on a statistic知道如何估计一个数据点在统计量上的影响
by Robert L. Mason and John C. Young
罗伯特L.梅森和约翰C.杨编写
Being able to determine the effect a data point has on summary statistics provides useful insight into the construction of better parameter estimators.
能确定一个数据点在概括统计的影响为更好的参数估计的构建提供了有用的见解
Consider, for example, a person with an annual income of more than $1 million in a room with nine others with annual incomes in the $25,000 to $50,000 range. The average income for this group of 10 is greater than $100,000, but this is not a useful summary statistic of the typical income of most members of the group.
比如，考虑一个年收入超过1百万美元的人和9个年收入在25,000和50,000美元之间的人在同一间屋子里，这个10人组的平均收入比100,000美元多很多，但这不是一个该组大多数成员代表性收入的有用的概括统计量。
From this example, you observe that the inclusion of an observation far removed from the bulk of the observations in a sample can have a great effect on the estimated overall mean. If included, the outlying observation actually pulls the average value of the group toward it.
从这个例子，你注意到在一个样本中那个远离绝大部分、包含了特殊数据的观察值会对总体的平均值有很大影响。如果被包括在内，这个偏离的观察值确实会拉动组的平均值靠近该值。
In another example, consider the group of bivariate observations contained in the circle in Figure 1. Such a circular region of data points indicates that the correlation between the two variables, x1 and x2, is close to zero; that is, no linear relationship exists between the two variables.
再举另外一个例子，考虑两变量的组被包含在图1的圆圈中。数据点的圆形区域指出两组变量的相互关系。X1和X2接近零点，也就是说，两个变量之间没有线性关系。
Figure 1 图表1 用一个单个的观察值来增加相互的关系
Observe the labeled point in the upper right-hand corner of the plot but outside of the circle. As the distance between this point and the mean of the group of points in the circle increases along the drawn 45° line, the correlation between the two variables will increase and approach its maximum value of one. Thus, this single outlying observation can distort the estimated value of the true correlation.
观察在圆的右上角但是在圆外部的那个标记的点。由于在这个点以及圆内点群的平均值之间的距离沿着图中45°线增加，这两个变量的相互关系也将增加并且接近两者之中的最大值。因此，单个的、偏离的观察值能歪曲真正关系的估计值。
The variance, σ2, of a variable x is defined as the average of the squared deviation of that variable from its population mean, µ. Consequently, the square of the distance that an outlying observation is from its mean, that is, (x – µ)2, can have a great impact on the estimated value of the variance parameter.
变量X的方差σ2被定义为X偏离其总体均值的平方。所以，一个偏离的观察值与其均值距离的平方，就是说（X-X均值）2会对方差参数估计值有很大的影响
For example, including the outlying point in Figure 1 with the circular group of points in the plot will increase the variances of x1 and x2. This occurs because the outlying point causes the data to be spread wider in both dimensions.
In two dimensions, scatter plots can be constructed to show how one or more data points can change the estimate of the means, the variances and the correlation coefficient between the variables.
例如，在图表1中包含了偏远点的圆形点群将会增加X1和X2的方差。这种情况的发生是因为远离的点导致数据向两个范围扩展。在这两个范围中，可以建立散点图来说明一个或多个数据点怎样改变均值的估计值、方差和两变量间的相关系数。
For example, Figure 2 contains four observations, labeled A, B, C and D, which are removed from the bulk of the data enclosed in the ellipse. The inclusion of points A or C will not affect the correlation coefficient because both support the linear trend in the data.
例如，图表2中包含4个观察值，标记为A, B, C 和D，这几个值与椭圆围住的绝大部分数据远离。包含了点A和C的数据不会影响相关系数，因为这两点都支持线性趋势。
Figure 2 图表2 四个观察值将会改变均值、方差和/或相互关系
Including point A, however, will increase the variances and decrease the means of x1 and x2, while including point C will increase the variances and the means of x1 and x2. The inclusion of points B or D will affect the correlation coefficient between the two variables because both lie in directions opposite of the linear trend of the data. In addition, including point B will decrease the mean but increase the variance of x1, while including point D will increase the mean and variance of x1.
包含点A将增加方差而且减少X1和X2的均值；然而包含点C将增加方差以及X1和X2的均值。包含点B或D将会影响两变量间的相关系数因为这两个点都在线性数据趋势的相反方向。另外，包含点B将减少均值但是增加X1的方差，包含点D将增加均值以及X1的方差。
The influence function 影响函数
Several mathematical procedures exist for determining the effect an observation has on these particular estimates. One popular procedure1 used in developing robust statistical estimators is based on developing an influence curve or influence function for use as a measure of the effect that an observation has on the parameter being estimated.
几种数学上传统的做法用来决定一个观察值在这些特定的估计上的影响。一种被用于发展健全统计估计的流行程序是建立在一条影响曲线或者影响函数，（这种曲线或函数）用来作为一个观察值对被估计参数影响的一种测量。
When applied to the mean, the influence function is exactly what you would expect: a measure of the difference between an observation and the mean:
当被运用于均值的时候，这种影响函数恰恰就是你所期望的：测量观察值和均值间的差异。
x – µ. Likewise, when applied to the variance, you obtain the expected answer that the influence function is the squared distance between an observation and the mean: (x – µ)2 – σ2.
x – µ同样地，当被运用于方差，你得到的期望中的答案是：影响函数是观察值和均值间差值的平方σ2
When applied to the sample correlation coefficient, r, between two variables, x1 and x2, the contours of the influence function are a set of hyperbolae given by the formula
当被运用于样本相关系数，在X1和X2两变量之间，影响函数的轮廓是一套公式规定的双曲线。
in which y1 and y2 are the studentized values of x1 and x2, and c is a chosen constant value. The selection of the value of c for drawing these contours is arbitrary (and chosen to include the bulk of the points), but nevertheless serves to identify observations removed from the data swarm.
在（上面的公式中）y1和y2是变量X1 和X2的观测值，c是一个常数。为了画出这些轮廓，c的数值的选择是随意的（被选择的值包括绝大部分数据点），仅仅为识别观察值远离数据群提供服务。
Superimposing these hyperbolic contours over the corresponding scatter plot for y1 and y2 allows you to determine which observations are having the greatest effect on the estimate of the correlation coefficient. Points inside the hyperbolae will influence function values greater that +c or less than –c. Points outside the hyperbolae will have influence function values between –c and +c. Figure 3 shows an example of these contours for a case in which c = ± 2.7 and r = 0.81.
将这些双曲线轮廓叠加在对应y1和y2的散点图上能使你确定哪些观察值对估计相关系数有最大的影响。双曲线内部
的点将影响函数值大于+c或者小于-c。双曲线外部的点将使函数值在-c和+c之间。图表3是一个当c = ± 2.7 及 r = 0.81时双曲线（轮廓）的例子展示：
Figure 3 相关系数的影响函数图
Detailed procedures exist for interpreting the data points in relation to the contour plots.2,3 Those points located on the side of the data swarm but inside the hyperbolae, such as point A in Figure 3, will decrease the value of the correlation coefficient. Those points located within the hyperbola on the ends of the data swarm, such as points B and C in Figure 3, will increase the correlation coefficient.
有详细的步骤来解释数据点与轮廓图的关系，这些点在数据群的一侧但是在双曲线的里面，例如图表3中的点A，就将减少相关系数值。那些位于双曲线内部、在数据群末端的点，例如图表中的点B和点C，将增加相关系数。
Influence functions also can be used for detecting outliers in a bivariate sample.4 For example, any point located within the hyperbolae, such as points A, B and C in Figure 3, are subject to removal. In addition, the influence function value in (1) can be computed for any other observation in the sample.
影响函数也能应用于查明在一个双变量样本中的远离点。例如，位于双曲线内部的任何一点，比如图表3中的点A、B和点C，（它们）倾向于远离（双曲线）。另外，在（1）中的影响函数值也能被用来估算在样本中的其他观察数据。
Influence function example 影响函数例子
Figure 4 contains the scatter plot of 212 observations selected at random from a bivariate normal distribution in which the variables are standardized with a correlation coefficient of 0.812. In this form, the correlation is the same as the covariance between the two variables. These observations are represented in Figure 4 by the points within the ellipse, excluding the one labeled point 2.
图表4包含了从一个双变量、方差是标准差、相关系数为0.812的标准分布中随机选取的212个观察值的散点图。在这种形式中，相关系数与两变量间的协方差相同。在图表4中这些观察值用椭圆内部的点表示，除了那个被标记出的点2.
Figure 4 图表4 数据散点图、椭圆和双曲线
For illustrative purposes, two additional observations, point 1 with coordinates (-2, 2) and point 2 with coordinates (1, 1), have been added to the plot in Figure 4. Point 1 is outside the data swarm and inside the hyperbolae, indicating it could be an outlier and could possibly have influence on the computation of sample statistics.作为解说性的目的，两个另外的观察值，坐标为(-2, 2)的点1及坐标为(1, 1)的点2也被加入到图表4中。点1在数据群的外侧但是在双曲线内侧，指出它是一个远离点并且可能对估算样本统计量有影响。
This is verified in Table 1 by comparing the sample correlation coefficient values obtained with and without this point (while ignoring point 2 Including point 1 decreases the pairwise correlation between the two variables from 0.812 to 0.776. In addition, Table 1 includes the effect on the sample means and variances of y1 and y2. For y1, the absolute value of the mean increases and the standard deviation slightly increases when point 1 is included. For y2, the absolute value of the mean decreases, but the standard deviation slightly increases when point 1 is included.
这个在列表1中通过对比含有这个点以及不含这个点的样本相关系数值证实。然而忽略点2而包括点1。两变量间成对的相互关系从0.812减少到0.776. 另外，列表1包含了对y1和y2样本均值和方差的影响。对于y1，当包括点1时，均值的绝对值增大且标准偏差略有增大。对于y2，均值的绝对值减小，但是当点1被包含的时候标准偏差略有增加。
Table 1 列表1 点2的概括统计量
The coordinates of point 1 are (-2, 2 )Including this point in the original sample, the value of the correlation coefficient is r = 0.776. These coordinates and the correlation coefficient are needed to compute the value of the influence function for point 1 using the earlier equation. The computed value is -7.1, which is less than the chosen value of c = -2.7. This result independently confirms what we see in Figure 4, namely that point 1 is inside the hyperbolae and has a decreasing effect on the correlation coefficient.
点1的坐标是（-2,2）原来的样本中包含这个点，相关系数值r = 0.776。这些坐标和相关系数对于用最早的方程式来计算点1的影响函数值是必需的。计算值是-7.1，这个数值小于选定的常数c = -2.7。这个结果独立地证明我们在列表4中看到的，即点1在双曲线内部并且对相关系数有减少方面的影响。
In contrast, point 2, with coordinates (1, 1), is contained within the data swarm (elliptical region) and outside of the hyperbolae in Figure 4. Thus, this point should have minimal effect on the sample estimates. This is confirmed when examining the results in Table 2.
相比之下，在图表4中，点2，坐标为(1, 1)，被包含在数据群内部（椭圆地带）且在双曲线外侧。因此这个点可能对样本估计有极微小的影响。这个会在列表2中检查结果的时候得到确认。
Table 2 列表2 点1的概括统计量
When the summary statistics and the correlations are computed with and without this point (while excluding point 1), small differences are noted in all of the statistics. These results also are confirmed by the small value of the influence function for this point. The computed value using the earlier equation is 0.2, which is between –c = -2.7 and +c = +2.7, and is therefore outside the hyperbolae in Figure 4.
当在包含和不包含这个点估计概括统计量和相互关系时（当不包括点1时），在所有的统计量中，小的差异是显著的。这些结果也被这个点影响函数的小的估计值所证明。用最早的方程式的计算值是0.2，这个值在–c = -2.7 和+c = +2.7之间，因此在图表4中双曲线的外部。
As can be seen from these examples, the influence function of a statistic is a key component in robust estimation because it helps you assess the influence that an observation has on the estimation of a statistic. It also is important in detecting outliers in the data.
从这些例子中我们可以看出，在完全估计中，统计量的影响函数是一个关键的组成部分。因为它帮助你评估一个观察值对统计量估计的影响。它在发现数据中远离的点上也是很重要的。
References 参考文献
- Frank R. Hampel, "The Influence Curve and its Role in Robust Estimation," Journal of the American Statistical Association, 1974, pp. 383–393.
- 弗兰克R.汉姆普，“在完全估计中影响曲线和它的作用”，美国统计协会期刊，1974年，383-393页。
- Susan J. Devlin, Ramanathan Gnanadesikan and John R. Kettenring, "Robust Estimation and Outlier Detection With Correlation Coefficients," Biometrika, 1975, pp. 531–545.
- 苏珊J. 德弗林，罗曼森 Gnanadesikan和约翰R.凯特宁，“完全估计和远离点相关系数探测”，生物统计学，1975年 ,531-545页。
- Michael R. Chernick, "The Influence Function and its Application to Data Validation," American Journal of Mathematical and Management Sciences, 1982, pp. 263–288.
- 迈克尔R.柴内克，“影响函数及它在数据确认上的应用”，美国数学和管理科学杂志，1982年，263-288页。
- Chernick, "The Influence Function and its Application to Data Validation," American Journal of Mathematical and Management Sciences, see reference 3.
- 柴内克“影响函数及它在数据确认上的应用”，美国数学和管理科学杂志，参见引文3。
Robert L. Mason is an institute analyst at Southwest Research Institute in San Antonio. He has a doctorate in statistics from Southern Methodist University in Dallas and is a fellow of ASQ and the American Statistical Association.
罗伯特L.曼森是圣地亚哥西南部研究机构的一位学会研究员。它在南卫理公会大学获得统计学博士学位，并且是ASQ和美国统计协会的研究员。
John C. Young is a retired statistics professor at McNeese State University in Lake Charles, LA. He received a doctorate in statistics from Southern Methodist University.
约翰C.杨是美国莱克查尔斯州麦克尼斯州立大学退休的一位统计学教授，他在南卫理公会大学获得统计学博士学位。
本帖最后由 小编H 于 2011-7-19 14:27 编辑
本篇是由xy_persist翻译：
Know how to measure the effect a data point has on a statistic
了解如何测量一个数据点在统计中的影响
by Robert L. Mason and John C. Young
作者：Robert L. Mason，John C. Young
Being able to determine the effect a data point has on summary statistics provides useful insight into the construction of better parameter estimators.
能确定一个数据点的影响已经在总结统计提供了有用的洞察建设更好的参数估计。
Consider, for example, a person with an annual income of more than $1 million in a room with nine others with annual incomes in the $25,000 to $50,000 range. The average income for this group of 10 is greater than $100,000, but this is not a useful summary statistic of the typical income of most members of the group.
举例来说，一个收入超过一百万美元的人和另外九个收入在2.5万~5万美元之间的人，这组人员的十个人平均收入超过10万美元，但这并不是一项有效的统计，它并不能代表这组成员的平均状况，不具有典型性。
From this example, you observe that the inclusion of an observation far removed from the bulk of the observations in a sample can have a great effect on the estimated overall mean. If included, the outlying observation actually pulls the average value of the group toward it.
通过这个例子，你发现如果一个数据与组内的其他数据偏离很大的话，这一数值会对整组的平均值产生很大的影响，如果分析结果包含这一数据，输出的结果将会被拉动到偏向这一值的方向。
In another example, consider the group of bivariate observations contained in the circle in Figure 1. Such a circular region of data points indicates that the correlation between the two variables, x1 and x2, is close to zero; that is, no linear relationship exists between the two variables.
另外一个例子，还是图1中所包含的一组双变量观察数据，如图环形范围内的数据点表明两个变量（x1和x2）的相互关系接近于0，也就是说这两个变量没有线性关系。
Figure 1
Observe the labeled point in the upper right-hand corner of the plot but outside of the circle. As the distance between this point and the mean of the group of points in the circle increases along the drawn 45° line, the correlation between the two variables will increase and approach its maximum value of one. Thus, this single outlying observation can distort the estimated value of the true correlation.
观察图中右上角位于环形之外的数据点，连接该点与环内整组数据点的平均值大约呈45°角，两个变量之间的关系就会沿着这个方向变化并接近各自的最大值，这样一来，这个外部点就会影响真实的相关性结果，造成偏离。
The variance, σ2, of a variable x is defined as the average of the squared deviation of that variable from its population mean, µ. Consequently, the square of the distance that an outlying observation is from its mean, that is, (x – µ)2, can have a great impact on the estimated value of the variance parameter.
一个变量的方差σ2是指变量相对于平均值µ的平均偏差的平方，因此，一个外部点与平均值的距离的平方(x – µ)2能够极大的影响变量参数的平均。
For example, including the outlying point in Figure 1 with the circular group of points in the plot will increase the variances of x1 and x2. This occurs because the outlying point causes the data to be spread wider in both dimensions.
还是以图1中包含一个外部点的环形数据组为例，这个外部点会增加x1和x2之间的方差，这是由于外部点的存在导致数据在两个方向上的散布都增加了。
In two dimensions, scatter plots can be constructed to show how one or more data points can change the estimate of the means, the variances and the correlation coefficient between the variables.
在两个方向上，点图能够解释为什么一个点或多个点能够改变平均值，方差和变量间相关系数的评估。
For example, Figure 2 contains four observations, labeled A, B, C and D, which are removed from the bulk of the data enclosed in the ellipse. The inclusion of points A or C will not affect the correlation coefficient because both support the linear trend in the data.
如图2所示，图中包括四个观测值分别标记为A、B、C、D，这四个点都偏离于图中椭圆形范围内的其它点，图中点A或点C都位于图中点群的线性趋势上，因此并不影响相关系数。
Figure 2
图2 4个会影响平均值，方差或相关性的观测点
Including point A, however, will increase the variances and decrease the means of x1 and x2, while including point C will increase the variances and the means of x1 and x2. The inclusion of points B or D will affect the correlation coefficient between the two variables because both lie in directions opposite of the linear trend of the data. In addition, including point B will decrease the mean but increase the variance of x1, while including point D will increase the mean and variance of x1.
但是，如果包括点A会增加方差，并会减小x1和x2的平均值，而包括点C则会增大方差，也会增加x1和x2的平均值。点B和点D的会影响到两个变量的相关系数，这是因为这两个点位于数据组线性趋势的反方向。另外，如果分析结果时包括点B会使平均值变小，同时会增加x1的方差，包括点D会同时增加x1的平均值和方差。
The influence function
影响函数
Several mathematical procedures exist for determining the effect an observation has on these particular estimates. One popular procedure1 used in developing robust statistical estimators is based on developing an influence curve or influence function for use as a measure of the effect that an observation has on the parameter being estimated.
有一些数学方程能够用来定义这样的一个观测点对整体评估的影响。一个流行的方法就是用稳健统计理论，使用影响曲线或者影响函数作为分析方法来测评观测点对要评估的参数影响度。
When applied to the mean, the influence function is exactly what you would expect: a measure of the difference between an observation and the mean:
x – µ. Likewise, when applied to the variance, you obtain the expected answer that the influence function is the squared distance between an observation and the mean: (x – µ)2 – σ2.
当使用平均值时，影响函数就能给出我们所想看到的结果：观测值与平均值之间的差异x-μ。同样的，当我们运用方差，我们得到影响函数的结果是观测值与平均值之间距离的平方：(x – µ)2 – σ2.
When applied to the sample correlation coefficient, r, between two variables, x1 and x2, the contours of the influence function are a set of hyperbolae given by the formula
,
in which y1 and y2 are the studentized values of x1 and x2, and c is a chosen constant value. The selection of the value of c for drawing these contours is arbitrary (and chosen to include the bulk of the points), but nevertheless serves to identify observations removed from the data swarm.
我们使用两个变量x1和x2的相关系数r，影响函数则变为以下公式给出的双曲线。这里y1和y2是 x1和x2 的平均值，c是一个选定的常数。为了绘制这些双曲线选定c值是具有主观性的（主要是为了包括这些数据点），不过这样就能辨别观察点从数群中的偏离。
Superimposing these hyperbolic contours over the corresponding scatter plot for y1 and y2 allows you to determine which observations are having the greatest effect on the estimate of the correlation coefficient. Points inside the hyperbolae will influence function values greater that +c or less than –c. Points outside the hyperbolae will have influence function values between –c and +c. Figure 3 shows an example of these contours for a case in which c = ± 2.7 and r = 0.81.
叠加图3中y1和y2相关点图周围的双曲线，能够确定哪个观测点对整体相关系数评估起最大的影响作用。双曲线上点所遵循的影响函数值范围大于+c或小于-c，双曲线外的点影响函数值范围在-c和﹢c之间。如图3所示，这一系列双曲线c值为±2.7，r为0.81
Figure 3
图3 相关系数影响函数点图
Detailed procedures exist for interpreting the data points in relation to the contour plots.2,3 Those points located on the side of the data swarm but inside the hyperbolae, such as point A in Figure 3, will decrease the value of the correlation coefficient. Those points located within the hyperbola on the ends of the data swarm, such as points B and C in Figure 3, will increase the correlation coefficient.
下面对数据点和等值线的相对位置关系的细节进行说明：那些位于双曲线内侧，在数据群侧面的离群值如图3中的A，会减小相关系数的值；而那些位于双曲线内侧同时又位于数据群趋势延长线端部的离群点如图3中的点B和点C会增大相关系数。
Influence functions also can be used for detecting outliers in a bivariate sample.4 For example, any point located within the hyperbolae, such as points A, B and C in Figure 3, are subject to removal. In addition, the influence function value in (1) can be computed for any other observation in the sample.
影响函数也能用来定义二元变量的离群值。比如任意一个位于双曲线内侧的数据点如图3中的A、B、C，都是被移除的目标；同时，例子中的任何一个观测点都可以用（1）中的影响函数值来计算。
Influence function example
影响函数举例
Figure 4 contains the scatter plot of 212 observations selected at random from a bivariate normal distribution in which the variables are standardized with a correlation coefficient of 0.812. In this form, the correlation is the same as the covariance between the two variables. These observations are represented in Figure 4 by the points within the ellipse, excluding the one labeled point
图4中的散点图包含212个任意抽取的观测点，这些数据点服从二元正态分布，变量的标准相关系数为0.812，在这个序列中，两个变量的相关关系和协方差相同。图4中这些点在椭圆内进行重新描述，包括一个标记点。
Figure 4
图4 散点图椭圆和双曲线
For illustrative purposes, two additional observations, point 1 with coordinates (-2, 2) and point 2 with coordinates (1, 1), have been added to the plot in Figure 4. Point 1 is outside the data swarm and inside the hyperbolae, indicating it could be an outlier and could possibly have influence on the computation of sample statistics.
为了说明目的，另外在图4中添加2个标记点，点1的坐标为（-2,2），点2的坐标为（1,1）。点1是点群外部双曲线内侧的一个点，说明这是一个会影响整个样本统计结果的点。
This is verified in Table 1 by comparing the sample correlation coefficient values obtained with and without this point (while ignoring point 2). Including point 1 decreases the pairwise correlation between the two variables from 0.812 to 0.776. In addition, Table 1 includes the effect on the sample means and variances of y1 and y2. For y1, the absolute value of the mean increases and the standard deviation slightly increases when point 1 is included. For y2, the absolute value of the mean decreases, but the standard deviation slightly increases when point 1 is included.
表1中给出点1对样本相关系数值的影响结果，分别对比包含点1和不包含点1的相关系数（这里忽略点2的影响），包含点1的话，两两相关相关系数会从0.812减小到0.776。另外，表1还得出了对样本平均值以及变量y1和y2的影响，对于y1而言当包括点1时，平均值的绝对值增加，标准偏差略有增大；对于y2而言当包括点1是，平均值的绝对值减小，标准偏差明显增大。
Table 1
表1 点1 的统计数据
The coordinates of point 1 are (-2, 2). Including this point in the original sample, the value of the correlation coefficient is r = 0.776. These coordinates and the correlation coefficient are needed to compute the value of the influence function for point 1 using the earlier equation. The computed value is -7.1, which is less than the chosen value of c = -2.7. This result independently confirms what we see in Figure 4, namely that point 1 is inside the hyperbolae and has a decreasing effect on the correlation coefficient.
点1的坐标为（-2,2），如果在最初的样本中包含这个点，相关系数的值为0.776，我们需要用这个坐标和相关系数来计算先前用过的等式中点1的影响函数值，这个计算值为-7.1，这个值小于我们当初选定的c的下限-2.7，这个结果就证明了我们在图4中所见的，点1位于双曲线的内侧，对相关系数起减小的作用。
In contrast, point 2, with coordinates (1, 1), is contained within the data swarm (elliptical region) and outside of the hyperbolae in Figure 4. Thus, this point should have minimal effect on the sample estimates. This is confirmed when examining the results in Table 2.
相反的，我们再来看点2，点2的坐标为（1,1），如图4所示该点位于椭圆范围之内，并位于双曲线的外侧，这样这个点对整个样本的评价起到的影响作用不大，从表2中我们就能够看出结论。
Table 2
表2 点2的统计数据
When the summary statistics and the correlations are computed with and without this point (while excluding point 1), small differences are noted in all of the statistics. These results also are confirmed by the small value of the influence function for this point. The computed value using the earlier equation is 0.2, which is between –c = -2.7 and +c = +2.7, and is therefore outside the hyperbolae in Figure 4.
表2中给出的是包括点2和不包括点2的统计结果和相关性计算结果（这个结果不包括点1），我们能看出这一统计结果的差异非常小，这个结果也证明这个点影响函数的值较小。用前期的等式计算这个点的影响函数值为0.2，在c值-2.7和﹢2.7之间，因此这个点就位于图4中双曲线外侧。
As can be seen from these examples, the influence function of a statistic is a key component in robust estimation because it helps you assess the influence that an observation has on the estimation of a statistic. It also is important in detecting outliers in the data.
从例子中能够看出一项统计的影响函数是稳健评估的关键影响因素，因为这能够帮助你评估一个观察点对整个统计结果的影响程度，这也是检测外部点的一个重要方法。
References
参考文献
Robert L. Mason is an institute analyst at Southwest Research Institute in San Antonio. He has a doctorate in statistics from Southern Methodist University in Dallas and is a fellow of ASQ and the American Statistical Association.
Robert L. Mason 是圣安东尼奥西南研究所的一名分析师，他获得达拉斯州南方卫理公会大学统计学博士学位，是美国质量协会和美国统计协会的会议。
John C. Young is a retired statistics professor at McNeese State University in Lake Charles, LA. He received a doctorate in statistics from Southern Methodist University.
John C. Young 是莱克查尔斯麦克尼斯州立大学的一名退休统计学教授，他也获得南方卫理公会大学统计学博士学位
本篇是由xy_persist翻译：
Know how to measure the effect a data point has on a statistic
了解如何测量一个数据点在统计中的影响
by Robert L. Mason and John C. Young
作者：Robert L. Mason，John C. Young
Being able to determine the effect a data point has on summary statistics provides useful insight into the construction of better parameter estimators.
能确定一个数据点的影响已经在总结统计提供了有用的洞察建设更好的参数估计。
Consider, for example, a person with an annual income of more than $1 million in a room with nine others with annual incomes in the $25,000 to $50,000 range. The average income for this group of 10 is greater than $100,000, but this is not a useful summary statistic of the typical income of most members of the group.
举例来说，一个收入超过一百万美元的人和另外九个收入在2.5万~5万美元之间的人，这组人员的十个人平均收入超过10万美元，但这并不是一项有效的统计，它并不能代表这组成员的平均状况，不具有典型性。
From this example, you observe that the inclusion of an observation far removed from the bulk of the observations in a sample can have a great effect on the estimated overall mean. If included, the outlying observation actually pulls the average value of the group toward it.
通过这个例子，你发现如果一个数据与组内的其他数据偏离很大的话，这一数值会对整组的平均值产生很大的影响，如果分析结果包含这一数据，输出的结果将会被拉动到偏向这一值的方向。
In another example, consider the group of bivariate observations contained in the circle in Figure 1. Such a circular region of data points indicates that the correlation between the two variables, x1 and x2, is close to zero; that is, no linear relationship exists between the two variables.
另外一个例子，还是图1中所包含的一组双变量观察数据，如图环形范围内的数据点表明两个变量（x1和x2）的相互关系接近于0，也就是说这两个变量没有线性关系。
Figure 1
Observe the labeled point in the upper right-hand corner of the plot but outside of the circle. As the distance between this point and the mean of the group of points in the circle increases along the drawn 45° line, the correlation between the two variables will increase and approach its maximum value of one. Thus, this single outlying observation can distort the estimated value of the true correlation.
观察图中右上角位于环形之外的数据点，连接该点与环内整组数据点的平均值大约呈45°角，两个变量之间的关系就会沿着这个方向变化并接近各自的最大值，这样一来，这个外部点就会影响真实的相关性结果，造成偏离。
The variance, σ2, of a variable x is defined as the average of the squared deviation of that variable from its population mean, µ. Consequently, the square of the distance that an outlying observation is from its mean, that is, (x – µ)2, can have a great impact on the estimated value of the variance parameter.
一个变量的方差σ2是指变量相对于平均值µ的平均偏差的平方，因此，一个外部点与平均值的距离的平方(x – µ)2能够极大的影响变量参数的平均。
For example, including the outlying point in Figure 1 with the circular group of points in the plot will increase the variances of x1 and x2. This occurs because the outlying point causes the data to be spread wider in both dimensions.
还是以图1中包含一个外部点的环形数据组为例，这个外部点会增加x1和x2之间的方差，这是由于外部点的存在导致数据在两个方向上的散布都增加了。
In two dimensions, scatter plots can be constructed to show how one or more data points can change the estimate of the means, the variances and the correlation coefficient between the variables.
在两个方向上，点图能够解释为什么一个点或多个点能够改变平均值，方差和变量间相关系数的评估。
For example, Figure 2 contains four observations, labeled A, B, C and D, which are removed from the bulk of the data enclosed in the ellipse. The inclusion of points A or C will not affect the correlation coefficient because both support the linear trend in the data.
如图2所示，图中包括四个观测值分别标记为A、B、C、D，这四个点都偏离于图中椭圆形范围内的其它点，图中点A或点C都位于图中点群的线性趋势上，因此并不影响相关系数。
Figure 2
图2 4个会影响平均值，方差或相关性的观测点
Including point A, however, will increase the variances and decrease the means of x1 and x2, while including point C will increase the variances and the means of x1 and x2. The inclusion of points B or D will affect the correlation coefficient between the two variables because both lie in directions opposite of the linear trend of the data. In addition, including point B will decrease the mean but increase the variance of x1, while including point D will increase the mean and variance of x1.
但是，如果包括点A会增加方差，并会减小x1和x2的平均值，而包括点C则会增大方差，也会增加x1和x2的平均值。点B和点D的会影响到两个变量的相关系数，这是因为这两个点位于数据组线性趋势的反方向。另外，如果分析结果时包括点B会使平均值变小，同时会增加x1的方差，包括点D会同时增加x1的平均值和方差。
The influence function
影响函数
Several mathematical procedures exist for determining the effect an observation has on these particular estimates. One popular procedure1 used in developing robust statistical estimators is based on developing an influence curve or influence function for use as a measure of the effect that an observation has on the parameter being estimated.
有一些数学方程能够用来定义这样的一个观测点对整体评估的影响。一个流行的方法就是用稳健统计理论，使用影响曲线或者影响函数作为分析方法来测评观测点对要评估的参数影响度。
When applied to the mean, the influence function is exactly what you would expect: a measure of the difference between an observation and the mean:
x – µ. Likewise, when applied to the variance, you obtain the expected answer that the influence function is the squared distance between an observation and the mean: (x – µ)2 – σ2.
当使用平均值时，影响函数就能给出我们所想看到的结果：观测值与平均值之间的差异x-μ。同样的，当我们运用方差，我们得到影响函数的结果是观测值与平均值之间距离的平方：(x – µ)2 – σ2.
When applied to the sample correlation coefficient, r, between two variables, x1 and x2, the contours of the influence function are a set of hyperbolae given by the formula
,
in which y1 and y2 are the studentized values of x1 and x2, and c is a chosen constant value. The selection of the value of c for drawing these contours is arbitrary (and chosen to include the bulk of the points), but nevertheless serves to identify observations removed from the data swarm.
我们使用两个变量x1和x2的相关系数r，影响函数则变为以下公式给出的双曲线。这里y1和y2是 x1和x2 的平均值，c是一个选定的常数。为了绘制这些双曲线选定c值是具有主观性的（主要是为了包括这些数据点），不过这样就能辨别观察点从数群中的偏离。
Superimposing these hyperbolic contours over the corresponding scatter plot for y1 and y2 allows you to determine which observations are having the greatest effect on the estimate of the correlation coefficient. Points inside the hyperbolae will influence function values greater that +c or less than –c. Points outside the hyperbolae will have influence function values between –c and +c. Figure 3 shows an example of these contours for a case in which c = ± 2.7 and r = 0.81.
叠加图3中y1和y2相关点图周围的双曲线，能够确定哪个观测点对整体相关系数评估起最大的影响作用。双曲线上点所遵循的影响函数值范围大于+c或小于-c，双曲线外的点影响函数值范围在-c和﹢c之间。如图3所示，这一系列双曲线c值为±2.7，r为0.81
Figure 3
图3 相关系数影响函数点图
Detailed procedures exist for interpreting the data points in relation to the contour plots.2,3 Those points located on the side of the data swarm but inside the hyperbolae, such as point A in Figure 3, will decrease the value of the correlation coefficient. Those points located within the hyperbola on the ends of the data swarm, such as points B and C in Figure 3, will increase the correlation coefficient.
下面对数据点和等值线的相对位置关系的细节进行说明：那些位于双曲线内侧，在数据群侧面的离群值如图3中的A，会减小相关系数的值；而那些位于双曲线内侧同时又位于数据群趋势延长线端部的离群点如图3中的点B和点C会增大相关系数。
Influence functions also can be used for detecting outliers in a bivariate sample.4 For example, any point located within the hyperbolae, such as points A, B and C in Figure 3, are subject to removal. In addition, the influence function value in (1) can be computed for any other observation in the sample.
影响函数也能用来定义二元变量的离群值。比如任意一个位于双曲线内侧的数据点如图3中的A、B、C，都是被移除的目标；同时，例子中的任何一个观测点都可以用（1）中的影响函数值来计算。
Influence function example
影响函数举例
Figure 4 contains the scatter plot of 212 observations selected at random from a bivariate normal distribution in which the variables are standardized with a correlation coefficient of 0.812. In this form, the correlation is the same as the covariance between the two variables. These observations are represented in Figure 4 by the points within the ellipse, excluding the one labeled point
图4中的散点图包含212个任意抽取的观测点，这些数据点服从二元正态分布，变量的标准相关系数为0.812，在这个序列中，两个变量的相关关系和协方差相同。图4中这些点在椭圆内进行重新描述，包括一个标记点。
Figure 4
图4 散点图椭圆和双曲线
For illustrative purposes, two additional observations, point 1 with coordinates (-2, 2) and point 2 with coordinates (1, 1), have been added to the plot in Figure 4. Point 1 is outside the data swarm and inside the hyperbolae, indicating it could be an outlier and could possibly have influence on the computation of sample statistics.
为了说明目的，另外在图4中添加2个标记点，点1的坐标为（-2,2），点2的坐标为（1,1）。点1是点群外部双曲线内侧的一个点，说明这是一个会影响整个样本统计结果的点。
This is verified in Table 1 by comparing the sample correlation coefficient values obtained with and without this point (while ignoring point 2). Including point 1 decreases the pairwise correlation between the two variables from 0.812 to 0.776. In addition, Table 1 includes the effect on the sample means and variances of y1 and y2. For y1, the absolute value of the mean increases and the standard deviation slightly increases when point 1 is included. For y2, the absolute value of the mean decreases, but the standard deviation slightly increases when point 1 is included.
表1中给出点1对样本相关系数值的影响结果，分别对比包含点1和不包含点1的相关系数（这里忽略点2的影响），包含点1的话，两两相关相关系数会从0.812减小到0.776。另外，表1还得出了对样本平均值以及变量y1和y2的影响，对于y1而言当包括点1时，平均值的绝对值增加，标准偏差略有增大；对于y2而言当包括点1是，平均值的绝对值减小，标准偏差明显增大。
Table 1
表1 点1 的统计数据
The coordinates of point 1 are (-2, 2). Including this point in the original sample, the value of the correlation coefficient is r = 0.776. These coordinates and the correlation coefficient are needed to compute the value of the influence function for point 1 using the earlier equation. The computed value is -7.1, which is less than the chosen value of c = -2.7. This result independently confirms what we see in Figure 4, namely that point 1 is inside the hyperbolae and has a decreasing effect on the correlation coefficient.
点1的坐标为（-2,2），如果在最初的样本中包含这个点，相关系数的值为0.776，我们需要用这个坐标和相关系数来计算先前用过的等式中点1的影响函数值，这个计算值为-7.1，这个值小于我们当初选定的c的下限-2.7，这个结果就证明了我们在图4中所见的，点1位于双曲线的内侧，对相关系数起减小的作用。
In contrast, point 2, with coordinates (1, 1), is contained within the data swarm (elliptical region) and outside of the hyperbolae in Figure 4. Thus, this point should have minimal effect on the sample estimates. This is confirmed when examining the results in Table 2.
相反的，我们再来看点2，点2的坐标为（1,1），如图4所示该点位于椭圆范围之内，并位于双曲线的外侧，这样这个点对整个样本的评价起到的影响作用不大，从表2中我们就能够看出结论。
Table 2
表2 点2的统计数据
When the summary statistics and the correlations are computed with and without this point (while excluding point 1), small differences are noted in all of the statistics. These results also are confirmed by the small value of the influence function for this point. The computed value using the earlier equation is 0.2, which is between –c = -2.7 and +c = +2.7, and is therefore outside the hyperbolae in Figure 4.
表2中给出的是包括点2和不包括点2的统计结果和相关性计算结果（这个结果不包括点1），我们能看出这一统计结果的差异非常小，这个结果也证明这个点影响函数的值较小。用前期的等式计算这个点的影响函数值为0.2，在c值-2.7和﹢2.7之间，因此这个点就位于图4中双曲线外侧。
As can be seen from these examples, the influence function of a statistic is a key component in robust estimation because it helps you assess the influence that an observation has on the estimation of a statistic. It also is important in detecting outliers in the data.
从例子中能够看出一项统计的影响函数是稳健评估的关键影响因素，因为这能够帮助你评估一个观察点对整个统计结果的影响程度，这也是检测外部点的一个重要方法。
References
参考文献
- Frank R. Hampel, "The Influence Curve and its Role in Robust Estimation," Journal of the American Statistical Association, 1974, pp. 383–393.
- Susan J. Devlin, Ramanathan Gnanadesikan and John R. Kettenring, "Robust Estimation and Outlier Detection With Correlation Coefficients," Biometrika, 1975, pp. 531–545.
- Michael R. Chernick, "The Influence Function and its Application to Data Validation," American Journal of Mathematical and Management Sciences, 1982, pp. 263–288.
- Chernick, "The Influence Function and its Application to Data Validation," American Journal of Mathematical and Management Sciences, see reference 3.
Robert L. Mason is an institute analyst at Southwest Research Institute in San Antonio. He has a doctorate in statistics from Southern Methodist University in Dallas and is a fellow of ASQ and the American Statistical Association.
Robert L. Mason 是圣安东尼奥西南研究所的一名分析师，他获得达拉斯州南方卫理公会大学统计学博士学位，是美国质量协会和美国统计协会的会议。
John C. Young is a retired statistics professor at McNeese State University in Lake Charles, LA. He received a doctorate in statistics from Southern Methodist University.
John C. Young 是莱克查尔斯麦克尼斯州立大学的一名退休统计学教授，他也获得南方卫理公会大学统计学博士学位
2011-07-15 16:26
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