主要區別-相關與回歸

相關和回歸是統計學中用來研究變數之間關係的兩種方法。相關性和回歸的主要區別在於相關性衡量兩個變數之間的關聯程度，而回歸是描述兩個變數之間關係的一種方法。回歸還可以更準確地預測因變數對自變數給定值的取值。

什麼是相關性(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:

利用這些值，我們可以計算協方差：

我們還可以計算標準偏差：

現在我們可以計算相關係數：

什麼是回歸(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.

相關性(correlation)和回歸(regression)的區別

描述關係

相關性描述了兩個變數的相關程度。

回歸給出了一種尋找兩個變數之間關係的方法。

做預測

相關性僅僅描述了兩個變數的相關性。分析兩個變數之間的相關性並不能提高對自變數給定值預測因變數值的準確性。

回歸使我們能夠更準確地預測自變數給定值的因變數值。

變數之間的依賴關係

在分析相關性時，哪個變數是獨立的，哪個是獨立的並不重要。

在分析回歸時，有必要區分因變數和自變數。

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***