What is of regression coefficient?

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Regression coefficients are the quantities by which the variables in a regression equation are multiplied. The most commonly used type of regression is linear regression. The aim of linear regression is to find the regression coefficients that produce the best-fitted line.

The regression coefficients in linear regression help in predicting the value of an unknown variable using a known variable. In this article, we will learn more about regression coefficients, their formulas as well as see certain associated examples so as to find the best-fitted regression line.

What are Regression Coefficients?

Regression coefficients can be defined as estimates of some unknown parameters to describe the relationship between a predictor variable and the corresponding response. In other words, regression coefficients are used to predict the value of an unknown variable using a known variable. Linear regression is used to quantify how a unit change in an independent variable causes an effect in the dependent variable by determining the equation of the best-fitted straight line. This process is known as regression analysis.

Formula for Regression Coefficients

The goal of linear regression is to find the equation of the straight line that best describes the relationship between two or more variables. For example, suppose a simple regression equation is given by y = 7x - 3, then 7 is the coefficient, x is the predictor and -3 is the constant term. Suppose the equation of the best-fitted line is given by Y = aX + b then, the regression coefficients formula is given as follows:

a = \(\frac{n(\sum xy)-(\sum x)(\sum y)}{n(\sum x^{2})-(\sum x)^{2}}\)

b = \(\frac{(\sum y)(\sum x^{2})-(\sum x)(\sum xy)}{n(\sum x^{2})-(\sum x)^{2}}\)

here, n refers to the number of data points in the given data sets.

How to Find Regression Coefficients?

Before determining the regression coefficients to find the best-fitted line, it is necessary to check whether the variables follow a linear relationship or not. This can be done by using the correlation coefficient and interpreting the corresponding value. Given below are the steps to find the regression coefficients for regression analysis.

• To find the coefficient of X use the formula a = \(\frac{n(\sum xy)-(\sum x)(\sum y)}{n(\sum x^{2})-(\sum x)^{2}}\).
• To find the constant term the formula is b = \(\frac{(\sum y)(\sum x^{2})-(\sum x)(\sum xy)}{n(\sum x^{2})-(\sum x)^{2}}\).
• Now input the regression coefficients in the equation Y = aX + b.
• A scatter plot can also be made so as to visually depict the regression line as shown below.

Regression Coefficients Interpretation

It is necessary to understand the nature of the regression coefficient as this helps to make certain predictions about the unknown variable. It helps to check to what extent a dependent variable will change with a unit change in the independent variable. Given below are the regression coefficients interpretation.

• If the sign of the coefficients is positive it implies that there is a direct relationship between the variables. This means that if the independent variable increases (or decreases) then the dependent variable also increases (or decreases).
• If the sign of the coefficients is negative it means that if the independent variable increases then the dependent variable decreases and vice versa. This means it is an indirect relationship.

Related Articles:

• Probability and Statistics
• Data Handling
• Introduction to Graphing

Important Notes on Regression Coefficients

• Regression coefficients are values that are used in a regression equation to estimate the predictor variable and its response.
• The most commonly used type of regression is linear regression. The equation of the best-fitted line is given by Y = aX + b.
• By using formulas, the values of the regression coefficient can be determined so as to get the regression line for the given variables.

1. Example 1: Find the regression coefficients for the following data:

   Age	Glucose Level
   43	99
   21	65
   25	79
   42	75
   57	87
   59	81

   Solution:

   Age (x)	Glucose Level (y)	xy	x2	y2
   43	99	4257	1849	9801
   21	65	1365	441	4225
   25	79	1975	625	6241
   42	75	3150	1764	5625
   57	87	4959	3249	7569
   59	81	4779	3481	6561
   Total = 247	486	20485	11409	40022

   The formula for finding the regression coefficients are as follows:

   a = \(\frac{n(\sum xy)-(\sum x)(\sum y)}{n(\sum x^{2})-(\sum x)^{2}}\)

   = 0.39

   b = \(\frac{(\sum y)(\sum x^{2})-(\sum x)(\sum xy)}{n(\sum x^{2})-(\sum x)^{2}}\)

   = 65.14

   The regression equation is Y = 0.39X + 65.14

   Answer: a = 0.39 and b = 65.14

2. Example 2: Find the regression line for the following data.

   A	B
   6.25	4.03
   6.5	4.02
   6.5	4.02
   6	4.04
   6.25	4.03
   6.25	4.03

   Solution:

   X	Y	XY	X2	Y2
   6.25	4.03	25.19	39.06	16.24
   6.5	4.02	26.13	42.25	16.16
   6.5	4.02	26.13	42.25	16.16
   6	4.04	24.24	36	16.32
   6.25	4.03	25.19	39.06	16.24
   6.25	4.03	25.19	39.06	16.24
   Total = 37.75	24.17	152.06	237.69	97.37

   The formula for finding the regression coefficients are as follows:

   a = \(\frac{n(\sum xy)-(\sum x)(\sum y)}{n(\sum x^{2})-(\sum x)^{2}}\)

   = -0.04

   b = \(\frac{(\sum y)(\sum x^{2})-(\sum x)(\sum xy)}{n(\sum x^{2})-(\sum x)^{2}}\)

   = 4.28

   The regression equation is Y = -0.04X + 4.28

   Answer: Regression equation is Y = -0.04X + 4.28

3. Example 3: Plot the graph for the following data if the regression coefficients are given as a = -0.07 and b = 68.63

   X	Y
   130	55
   135	56
   140	62
   142	63
   147	63
   156	51

   Solution: The regression coefficients are given as a = -0.07 and b = 68.63

   Thus, the regression line is Y = -0.07X + 68.63

   Thus, the scatter plot can be drawn as follows:

   

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FAQs on Regression Coefficients

In statistics, regression coefficients can be defined as multipliers for variables. They are used in regression equations to estimate the value of the unknown parameters using the known parameters.

What are Regression Coefficients Independent of?

Regression coefficients are independent of the change of scale as well as the origin of the plot.

What is the Formula for Regression Coefficients?

The formula for regression coefficients is given as a = \(\frac{n(\sum xy)-(\sum x)(\sum y)}{n(\sum x^{2})-(\sum x)^{2}}\) and b = \(\frac{(\sum y)(\sum x^{2})-(\sum x)(\sum xy)}{n(\sum x^{2})-(\sum x)^{2}}\).

How are Regression Coefficients used in a Linear Regression Line?

The equation of a linear regression line is given as Y = aX + b, where a and b are the regression coefficients.

How to Interpret Regression Coefficients?

If the value of the regression coefficients is positive then it means that the variables have a direct relationship while negative regression coefficients imply that the variables have an indirect relationship.

How to Calculate Regression Coefficients?

The steps to calculate the regression coefficients are as follows:

• Substitute values to find a (coefficient of X).
• Substitute values for b (constant term).
• Put the values of these regression coefficients in the linear equation Y = aX + b.

What Do Regression Coefficients Tell Us?

Regression Coefficients tell us how much a dependent variable changes with a unit change in the independent variables.