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Correlation and Regression - CMA

Regression Coefficients

The regression coefficients are a statically measure which is used to measure the average functional relationship between variables. In regression analysis, one variable is dependent and other is independent. Also, it measures the degree of dependence of one variable on the other(s).

The regression coefficient was first used to measure the relationship between the heights of fathers and their sons. Regression coefficients are also known as the slope coefficient. Since it determines the slope of the line which is the change in the independent variable for the unit change in the independent variable.

regression coeffecients

Classification of Regression Coefficient

  • Simple partial and multiple
  • Positive and negative
  • Linear and non-linear

Some of the properties of regression coefficient:

  • It is generally denoted by ‘b’.
  • It is expressed in the form of an original unit of data.
  • If two variables are there say x and y, two values of the regression coefficient are obtained. One will be obtained when x is independent and y is dependent and other when we consider y as independent and x as a dependent. The regression coefficient of y on x is represented by byx and x on y as bxy.
  • Both of the regression coefficients must have the same sign. If byx is positive, bxy will also be positive and it is true for vice versa.
  • If one regression coefficient is greater than unity, then others will be lesser than unity.
  • The geometric mean between the two regression coefficients is equal to the correlation coefficient
    • R=sqrt(byx*bxy)
  • Also, the arithmetic means (am) of both regression coefficients is equal to or greater than the coefficient of correlation.
    • (byx + bxy)/2= equal or greater than r.
  • The regression coefficients are independent of the change of the origin. But, they are not independent of the change of the scale. It means there will be no effect on the regression coefficients if any constant is subtracted from the value of x and y. If x and y are multiplied by any constant, then the regression coefficient will change.

How to interpret regression coefficients?

Linear regression is one of the most popular techniques in statistics. Despite of its popularity, it is sometimes difficult to interpret regression coefficients.

Interpreting the Intercept

Let us assume we have an equation

Y= b0 +b1*x1 + b2*x2+e.

Where y is the response variable x1 is the first predictor variable, x2 is the second predictor variable and e is the residual error.

Here,

B0 is the y-intercept

B1 is the first regression coefficient and

B2 is the second regression coefficient.

B0 can be interpreted as the value you would predict for y if both x1 and x2 are 0.

Interpreting the coefficients of the continuous predictor variable.

Similarly, b2 is interpreted as the difference in the predicted value in y and for each unit difference in x2 if x1 remains constant. Here b2 is the average difference in y between the category for which x2 =0 and the category for which x2=1.

Interpreting the coefficients when the predictor variables are correlated.

Each coefficient is influenced by the other variables in the regression model. Because all the predictor variables are associated with each other.

It means each coefficient will change when other variables are added to or deleted from the model.

Solved Question on Regression Coefficients

Subject Age X Glucose Level Y X*Y X^2 Y^2
1 43 99 4257 1849 9801
2 21 65 1365 441 4225
3 25 79 1975 625 6241
4 42 75 3150 1764 5625
5 57 87 4959 3249 7569
6 59 81 4779 3481 6561
Σ 247 486 20485 11409 40022

You can use the following equations for finding the values of a and b

find a linear regression equation

 

 

 

by putting the values in the formulas you get value

a=65.14

b=.385

so in equation

y= a+bx

y= 65.14 + .385225x

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