linear correlation coefficient|Correlation and Causation|lurking variables

4.4 Linear Correlation

若由SxxSyySxy定义则为：

所以为了计算方便：

所以，可以明白的是，Sxx和Sx是不一样的！

所以，t r is independent of the choice of units and always lies between −1 and 1

Understanding the Linear Correlation Coefficient

measures the strength of the linear relationship between two variables and that the following properties of r are meaningful only when the data points are scattered about a line.

r reflflects the slope of the scatterplot

如图，若scatterplot为正向分布（平均看来，虽然有2,4可能存在值）即positively linearly correlated（正相关）,，则point必在1,3区域；则Sxy必为正，则r值为正。若scatterplot为负向分布，则point必在2,4区域；则Sxy必为负，则r值为负，即negatively linearly correlated（负相关）。

原因：

则r与b1同号，所以

The magnitude of r indicates the strength of the linear relationship

. A value of r close to −1 or to 1 indicates a strong linear relationship between the variables and that the variable x is a good linear predictor of the variable y

所以，绝对值大证明相关程度高，相关程度高则证明拟合直线的拟合成果好

以下是一些拟合直线和散点图的例子：

使用Linear Correlation Coefficient必须保证数据线性（即分布在一条直线上）

Noted：Correlation does not imply causation!

而对于密切相关的变量之间的关系的成因，可以理解为：Two variables may be strongly correlated because they are both associated with other variables, called lurking variables,For example, a study showed that teachers’ salaries and the dollar amount of liquor sales are positively linearly correlated. A possible explanation for this curious fact might be that both variables are tied to other variables, such as the rate of inflflation, that pull them along together.即有密切关联的两个变量，这两个变量若毫无联系，则可能是因为这两个变量之间的某些中间变量将其联系起来。