主要区别-相关与回归

相关和回归是统计学中用来研究变量之间关系的两种方法。相关性和回归的主要区别在于相关性衡量两个变量之间的关联程度，而回归是描述两个变量之间关系的一种方法。回归还可以更准确地预测因变量对自变量给定值的取值。

什么是相关性(correlation)？

In statistics, we say there is a correlation between two variables if the two variables are related. If the relati***hip between the variables is a linear one, we can express the degree to which they are related using a number called Pearson’s correlation coefficient  .  takes a value between -1 and 1. A value of 0 means that the two variables are uncorrelated. Negative values indicate that the correlation between the variables is negative: i.e. as one variable increases, the other variable decreases. Similarly, a positive value for means that the data is positively correlated (when one variable increases, the other variable increases too).

A value of  that is -1 or 1 gives the strongest possible correlation. When  the variables are said to be completely negatively correlated and when  the values are said to be completely positively correlated. The figure below shows several shapes of scatter plots between two variables and the correlation coefficient for each case:

Pearson’s correlation coefficient for different types of scatter plots

Pearson’s correlation coefficient for two variables and is defined as follows:

Here,  is the covariance between  and :

The terms  and  stand for standard deviati*** of  and  respectively.This is defined as:

and 

Let us see how the correlation coefficient is calculated using an example. We will try to calculate the correlation coefficient for the following set of 20 values for and  :

The values of  are plotted against the values of  on the graph shown below:

Looking at the equati*** needed to calculate the correlation coefficient, we will first calculate values for . These are the mean values of and respectively. We find that:

Next, we will calculate and . We will put these values next to our values of  and  on the table above:

利用这些值，我们可以计算协方差：

我们还可以计算标准偏差：

现在我们可以计算相关系数：

什么是回归(regression)？

Regression is a method for finding the relati***hip between two variables. Specifically, we will look at linear regression, which gives an equation for a “line of best fit” for a given sample of data, where two variables have a linear relati***hip. A straight line can be described with an equation in the form of  where  is the gradient of the line and  axis, and linear regression allows us to calculate the values of  and  . Once we have calculated the correlation coefficient  , we can calculate these values as:

Note that in these cases, is taken to be the dependent variable while is the independent variable. From our previous calculati***, we know that

, and  . Therefore,  .

and  . Therefore,  .

The image below shows the previous scatter plot with the line  :

The data, with the best-fitting straight line obtained from regression ****ysis

As we mentioned before, regression ****ysis aids us to make predicti***. For instance, if the value of the independent variable ( ) was 1.000, then we can predict that  would be close to  . In reality, the value of  may not necessarily be exactly 1.614. Due to uncertainty, the actual value is likely to be different. Note that the accuracy of the prediction is higher for data with a correlation coefficient closer to ±1.

相关性(correlation)和回归(regression)的区别

描述关系

相关性描述了两个变量的相关程度。

回归给出了一种寻找两个变量之间关系的方法。

做预测

相关性仅仅描述了两个变量的相关性。分析两个变量之间的相关性并不能提高对自变量给定值预测因变量值的准确性。

回归使我们能够更准确地预测自变量给定值的因变量值。

变量之间的依赖关系

在分析相关性时，哪个变量是独立的，哪个是独立的并不重要。

在分析回归时，有必要区分因变量和自变量。

Image Courtesy:

“redesign File:Correlation_examples.png using vector graphics (SVG file)” by DenisBoigelot (Own work, original uploader was Imagecreator) [CC0 1.0], via Wikimedia Comm***