11.5
离群值和影响点
离群值是指与其余数据不符的数据观测值。有时也称为极值。在绘制离群值的图表时，离群值看起来与图表的模式不相符。一些离群值是由于出现错误而造成的（例如，将 50 写成 500），而其他离群值则表明有可能发生了异常情况。离群值通常位于垂直方向上远离最小二乘线的位置。它们之间有很大的“误差”，其中，“误差”…
考虑年收入与受教育年限的散点图，并拟合了回归线。
与其他人相比，一名仅接受过几年教育的人却拥有极高的收入。
这种不遵循趋势且在垂直方向上远离回归线的数据点被称为离群值。
从定量角度来看，可以通过残差来识别异常值，残差是指数据点的观测 y 值与根据回归方程预测得到的 y 值之间的差异。
现在，使用其公式计算残差的标准差。
根据经验法则，位于回归线以上或以下至少两个残差标准差的数据点将被标记为潜在的离群值。
此外，数据集可能还包含有影响的点。这些点在水平方向上远离其他大多数数据点。有影响的点的加入或移除会显著改变回归线。
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Q1: What is an outlier in a regression analysis?
An outlier is a data point that does not follow the trend and lies far from the regression line in the vertical direction. For example, a person with few years of schooling but exceptionally high income would be an outlier in an income versus education scatter plot. Outliers appear as extreme values that do not fit the pattern of the graph.
Q2: How do you identify outliers using residuals?
Outliers are identified by calculating residuals, which represent the vertical distance between observed y-values and predicted y-values from the regression equation. Data points located at least two residual standard deviations above or below the regression line are flagged as potential outliers. This quantitative approach helps systematically detect extreme values in your dataset.
Q3: What are influential points and how do they differ from outliers?
Influential points are data points located far from other points in the horizontal direction, unlike outliers which are vertically distant. These points significantly change the regression line's slope when added or removed from the dataset. While outliers have large vertical errors, influential points affect the line's position and angle due to their extreme x-values.
Q4: How can you determine if a point is influential?
To identify an influential point, remove it from the dataset and recalculate the regression line. If the slope changes significantly, the point is influential. Computer software and many calculators automatically identify both outliers and influential points, allowing you to examine their effects on your regression analysis.
Q5: Should outliers always be removed from data analysis?
No. Outliers require careful examination to determine their cause. Some result from data entry errors and should be removed, while others may represent valuable information about the population being studied. The key is investigating what causes a data point to be an outlier before deciding whether to include or exclude it.
Q6: Why do outliers have large residuals?
Outliers have large residuals because the residual represents the vertical distance between the observed data point and the regression line's predicted value. When a point is far from the line vertically, this distance is substantial, resulting in a large error or residual value that signals the point does not fit the overall trend.
Q7: What role do computers play in identifying outliers and influential points?
Computers and calculators automate the identification of outliers and influential points through regression analysis output. This automated detection allows researchers to systematically examine problematic data points without manual calculation. Computer-generated reports help you visualize and evaluate whether these points should be retained or removed from your analysis.