## Regression

### Meaning

Regression is the study of the nature of relationship between the variables so that one may be able to predict the unknown value of one variable for a known value of another variable.

### Comparison between Correlation and Regression

1. Correlation study the degree of relationship between variables while the regression study the nature of relationship between variables.
2. Correlation need not imply cause and effect relationship between variables whereas regression implies cause and effect relationship between variables.
3. There may be non-sense correlation between variables whereas in case of regression there is nothing like non-sense regression.
4. The correlation coefficient is independent of change of origin and scale but regression coefficients are independent of only change of origin but not of scale.
5. Correlation coefficient cannot be used for prediction. But regression lines are used for prediction.

### Two lines of regression Two lines of regression
There are two lines of regression for two variables 'x' and 'y'.

#### Regression  equation of line Y on X

When we try to depict the change in 'y' for a given change in 'x', then the regression line of y on x is used.
The regression equation of y on x is given by
y-ybar = byx (x-xbar)
Where  byx = Cov(x,y)/Var(x)
byx is called the regression coefficient of y on x.

#### Regression equation of line X on Y

When we try to depict the change in 'x' for a given change in 'y'. then the regression line of x on y is used
The regression equation of x on y is given by
x-xbar = bxy(y-ybar)
Where bxy= Cov(xy)/Var(y)
bxy is called the regression coefficient of x on y

#### Relation between correlation and regression coefficient

• byx.bxy = r^2

This implies that multiplication of both regression coefficient is equal to the square of correlation coefficient and the value of r^2 lies between 0 and 1
• Correlation coefficient is geometric mean of both regression coefficients i.e.

r = +/- square root( byx. bxy)